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Convergence theorems for monotone vector field inclusions and minimization problems in Hadamard spaces

Convergence theorems for monotone vector field inclusions and minimization problems in Hadamard... 1IntroductionOne of the most applicable optimization problems in a Hilbert space HHis to find an element (1.1)w∈D(A)≔{z∈H:Az≠∅}such that0∈Aw,w\in {\mathbb{D}}\left(A):= \left\{z\in H:Az\ne \varnothing \right\}\hspace{1em}\hspace{0.1em}\text{such that}\hspace{0.1em}\hspace{0.33em}0\in Aw,where AAis an operator from HHto 2H{2}^{H}. This problem is called monotone inclusion problem (MIP), provided the operator AAis monotone in the sense that ⟨u−w,x−y⟩≥0,∀u,w∈D(A),x∈Au,y∈Aw.\langle u-w,x-y\rangle \ge 0,\hspace{1em}\forall u,w\in {\mathbb{D}}\left(A),\hspace{0.33em}x\in Au,y\in Aw.Applications of MIP can be found in several fields of science and engineering such as in inverse problems, image recovery, signal processing, fuzzy theory, game theory, robotic control, etc. As a result of these applications, several schemes for approximating a solution of (1.1) have been developed. The classical and prominent scheme for approximating MIP in linear spaces is proximal point algorithm (PPA), introduced in [29] and [37]. Rockafellar [37] proved under some conditions that any sequence generated by the PPA converges weakly to a solution of (1.1). Thereafter, several researchers have developed and studied different modifications of PPA with convergence analysis (see e.g. [18,21,33,47] and references therein).It is known that many applicable optimization problems come as constrained problems, where the constrains are non-convex, non-smooth, and nonlinear, among other things. However, it is recently observed that Hadamard spaces view some non-smooth and non-convex constrained optimization problems as smooth and convex unconstrained problems. For this reason, several notions related to optimization including that of monotone operator are being extended from linear spaces and Hadamard manifolds to Hadamard spaces (see e.g. [6,7,11,27] and references therein).Let (W,d)\left(W,d)be a Hadamard space. In 2008, Berg and Nikolaev [5] denoted (u,w)∈W×W\left(u,w)\in W\times Wby uw→\overrightarrow{uw}and defined a quasilinearization map ⟨⋅,⋅⟩:(W×W)×(W×W)→R\langle \cdot ,\cdot \rangle :\left(W\times W)\times \left(W\times W)\to {\mathbb{R}}by ⟨uw→,vy→⟩=12(d2(u,y)+d2(w,v)−d2(u,v)−d2(w,y)),(u,v,w,y∈W).\langle \overrightarrow{uw},\overrightarrow{vy}\rangle =\frac{1}{2}({d}^{2}\left(u,y)+{d}^{2}\left(w,v)-{d}^{2}\left(u,v)-{d}^{2}\left(w,y)),\hspace{1.0em}\left(u,v,w,y\in W).Using this notion of quasilinearization, Ahmadi Kakavandi [1] introduced the dual space of a Hadamard space (W,d)\left(W,d)as follows. Let ϕ:W→R\phi :W\to {\mathbb{R}}be a function, L(ϕ)≔supϕ(w)−ϕ(v)d(w,v):w,v∈W,w≠vL\left(\phi ):= \sup \left\{\frac{\phi \left(w)-\phi \left(v)}{d\left(w,v)}:w,v\in W,\hspace{0.33em}w\ne v\right\}and consider Θ:R×(W×W)→C(X,R)\Theta :{\mathbb{R}}\times \left(W\times W)\to C\left(X,{\mathbb{R}})defined by Θ(t,u,w)(x)=t⟨uw→,ux→⟩\Theta \left(t,u,w)\left(x)=t\langle \overrightarrow{uw},\overrightarrow{ux}\rangle for all t∈R,u,w,x∈Wt\in {\mathbb{R}},\hspace{0.33em}u,w,x\in W, where C(W,R)C\left(W,R)denotes the space of continuous real-valued functions on WW. The map DDon R×W×W{\mathbb{R}}\times W\times Wdefined by D((t,u,w),(s,x,y))=L(Θ(t,u,w)−Θ(s,x,y))D(\left(t,u,w),\left(s,x,y))=L(\Theta \left(t,u,w)-\Theta \left(s,x,y))is a pseudometric on R×W×W{\mathbb{R}}\times W\times W. Moreover, DDforms equivalence relation on R×W×W{\mathbb{R}}\times W\times Wwith the equivalence class of (t,u,w)\left(t,u,w)as [tuw→]≔{sxy→:D((t,u,w),(s,x,y))=0}\left[t\overrightarrow{uw}]:= \{s\overrightarrow{xy}:D(\left(t,u,w),\left(s,x,y))=0\}. The dual space of (W,d)\left(W,d)is (W∗,D)\left({W}^{\ast },D), where W∗≔{tuw→:(t,u,w)∈R×W×W}{W}^{\ast }:= \{t\overrightarrow{uw}:\left(t,u,w)\in {\mathbb{R}}\times W\times W\}, which acts on W×WW\times Wby ⟨x∗,uw→⟩=t⟨xy→,uw→⟩,\langle {x}^{\ast },\overrightarrow{uw}\rangle =t\langle \overrightarrow{xy},\overrightarrow{uw}\rangle ,for x∗=[txy→]∈W∗{x}^{\ast }=\left[t\overrightarrow{xy}]\in {W}^{\ast }, u,w∈Wu,w\in W. Moreover, the author observed that if WWis a Hilbert space, then [txy→]=t(y−x)\left[t\overrightarrow{xy}]=t(y-x).In 2017, Khatibzadeh and Ranjbar [24] introduced the concept of monotonicity in a Hadamard space WWwith dual space W∗{W}^{\ast }through the following definitions. A mapping A:W→2W∗A:W\to {2}^{{W}^{\ast }}is called monotone if ⟨u∗−w∗,wu→⟩≥0,∀u,w∈{z∈W:Az≠∅},u∗∈Au,w∗∈Aw.\langle {u}^{\ast }-{w}^{\ast },\overrightarrow{wu}\rangle \ge 0,\hspace{0.33em}\forall u,w\in \{z\in W:Az\ne \varnothing \},{u}^{\ast }\in Au,{w}^{\ast }\in Aw.Also, for λ>0\lambda \gt 0the authors considered λ\lambda -resolvent of AAas (1.2)JλAz≔w∈W:1λwz→∈Aw{J}_{\lambda }^{A}z:= \left\{\phantom{\rule[-1.25em]{}{0ex}},w\in W:\left[\frac{1}{\lambda }\overrightarrow{wz}\right]\in Aw\right\}and proved some Δ\Delta -convergence theorems. In the same year, Ranjbar and Khatibsadeh [35] proposed two schemes for approximating a solution of MIP. One is Mann-type PPA as follows: (1.3)wn+1=σnwn⊕(1−σn)JλnAwn,w1∈W,{w}_{n+1}={\sigma }_{n}{w}_{n}\oplus \left(1-{\sigma }_{n}){J}_{{\lambda }_{n}}^{A}{w}_{n},\hspace{0.33em}{w}_{1}\in W,and the other is Halpern-type PPA as follows: (1.4)wn+1=σnw⊕(1−σn)JλnAwn,w1,w∈W,{w}_{n+1}={\sigma }_{n}w\oplus \left(1-{\sigma }_{n}){J}_{{\lambda }_{n}}^{A}{w}_{n},\hspace{0.33em}{w}_{1},w\in W,where {λn}⊂(0,∞)\left\{{\lambda }_{n}\right\}\subset \left(0,\infty )and {σn}⊂[0,1]\left\{{\sigma }_{n}\right\}\subset \left[0,1]. The authors obtained a Δ\Delta -convergent result for the Mann-type PPA and a strong convergence result for the Halpern-type PPA.In 2018, Okeke and Izuchukwu [34] proposed the following Halpern-type PPA for finding a common solution of MIP, minimization problem (MP), and fixed-point problem in Hadamard spaces: (1.5)w,w1∈Wzn=JλA∘argminw∈Wh(w)+12μd2(w,wn)wn+1=σnw⊕(1−σn)Tzn,n≥1,\left\{\begin{array}{l}w,{w}_{1}\in W\hspace{1.0em}\\ {z}_{n}={J}_{\lambda }^{A}\circ \mathop{{\rm{argmin}}}\limits_{w\in W}\left\{\phantom{\rule[-1.25em]{}{0ex}},h\left(w)+\frac{1}{2\mu }{d}^{2}\left(w,{w}_{n})\right\}\hspace{1.0em}\\ {w}_{n+1}={\sigma }_{n}w\displaystyle \oplus \left(1-{\sigma }_{n})T{z}_{n},\hspace{0.33em}n\ge 1,\hspace{1.0em}\end{array}\right.where μ,λ∈(0,∞)\mu ,\lambda \in \left(0,\infty ), {σn}⊂(0,1)\left\{{\sigma }_{n}\right\}\subset \left(0,1), hhis a proper convex and lower semi-continuous function, and TTis a singlevalued nonexpansive mapping on WW. The authors proved a strong convergence result using the assumptions that limn→∞σn=0{\mathrm{lim}}_{n\to \infty }{\sigma }_{n}=0, ∑n=1∞σn=∞{\sum }_{n=1}^{\infty }{\sigma }_{n}=\infty , and ∑n=1∞∣σn−σn−1∣<∞{\sum }_{n=1}^{\infty }| {\sigma }_{n}-{\sigma }_{n-1}| \lt \infty . For other related development see e.g. [10,22,23,31,38–40,42,46] and references therein.In 2021, Chaipunya et al. [9] observed that although Hadamard spaces extend Hilbert spaces and Hadamard manifolds, the prior notion of monotonicity barely has a relationship with the Hadamard manifolds. For that reason, the authors introduced a new notion of monotonicity called monotone vector field using tangent spaces. They analysed that this notion coincides with the notion of monotonicity found in both Hilbert spaces and Hadamard manifolds better than that of Khatibzadeh and Ranjbar [24].Inspired by the work of Chaipunya et al. [9] and motivated by the work of Okeke and Izuchukwu [34] and research in this direction, we propose and analyse two schemes. Both for approximating a common solution of finite family of monotone vector field inclusion problems (MVFIP) that is also a common fixed point of multivalued demicontractive mappings at the same time a solution of MP in the framework of Hadamard spaces. One scheme is Mann-type PPA and the other is viscosity-type PPA (motivated by the fact that for appropriate contraction mapping, a viscosity-type scheme converges at a rate faster than Halpern-type, see [19,43]). We establish some convergence results for the proposed schemes and then apply our results to find mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We give a numerical example in a nonlinear space to show the applicability of the proposed schemes. Our results extend and complement the results of Suparatulatorn et al. [46], Khatibzadeh and Ranjbar [24], Ranjbar and Khatibzadeh [35], Okeke and Izuchukwu [34], and some equivalent results in Hilbert spaces.2PreliminariesRecall that for a metric space (W,d)\left(W,d)with nonempty subset YY, a Hausdorff metric is the map H:Cℬ(W)×Cℬ(W)→[0,+∞)H:{\mathcal{C {\mathcal B} }}\left(W)\times {\mathcal{C {\mathcal B} }}\left(W)\to \left[0,+\infty )defined by H(A,B)≔max{supa∈Adist(a,B),supb∈Bdist(b,A)},(A,B∈Cℬ(W)),H\left(A,B):= \max \{\mathop{\sup }\limits_{a\in A}{\rm{dist}}\left(a,B),\mathop{\sup }\limits_{b\in B}{\rm{dist}}\left(b,A)\},\hspace{1.0em}(A,B\in {\mathcal{C {\mathcal B} }}\left(W)),where dist(z,Y)≔inf{d(z,y):y∈Y}{\rm{dist}}\left(z,Y):= \inf \left\{d\left(z,y):y\in Y\right\}for z∈Wz\in Wand Cℬ(W){\mathcal{C {\mathcal B} }}\left(W)denotes the family of nonempty closed bounded subsets of YY. A point w∈Yw\in Yis called a fixed point of multi-valued map T:Y→2YT:Y\to {2}^{Y}if w∈Tww\in Tw. In the sequel, we denote the fixed points set of the map TTby F(T)F\left(T), that is, F(T)={w∈Y:w∈Tw}F\left(T)=\left\{w\in Y:w\in Tw\right\}. The map TTis said to be (i)nonexpansive if H(Tv,Tw)≤d(u,w)for allv,w∈Y.H\left(Tv,Tw)\le d\left(u,w)\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}v,w\in Y.(ii)quasi-nonexpansive if F(T)≠∅F\left(T)\ne \varnothing and H(Tv,Tp)≤d(v,p)for allv∈Yandp∈F(T).H\left(Tv,Tp)\le d\left(v,p)\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}v\in Y\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}p\in F\left(T).(iii)κ\kappa -demicontractive (demicontractive for short) if F(T)≠∅F\left(T)\ne \varnothing and there exists κ∈[0,1)\kappa \in \left[0,1)such that H(Tw,Tp)≤d(w,p)+κdist(w,Tw)for allw∈Yandp∈F(T).H\left(Tw,Tp)\le d\left(w,p)+\kappa \hspace{0.1em}\text{dist}\hspace{0.1em}\left(w,Tw)\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}w\in Y\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}p\in F\left(T).It follows that every nonexpansive mapping with fixed points is quasi-nonexpansive and every quasi-nonexpansive mapping is κ\kappa -demicontractive but not the converse.Definition 2.1Let (W,d)\left(W,d)be a metric space and let {wn}\left\{{w}_{n}\right\}be a bounded sequence in WW. Then the asymptotic centre A({wn})A\left(\left\{{w}_{n}\right\})of {wn}\left\{{w}_{n}\right\}is defined by A({wn})≔{u∈Y:limsupn→∞d(u,wn)=infz∈Ylimsupn→∞d(z,wn)}.A\left(\left\{{w}_{n}\right\}):= \left\{\phantom{\rule[-1.25em]{}{0ex}}u\in Y:\mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left(u,{w}_{n})=\mathop{\inf }\limits_{z\in Y}\mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left(z,{w}_{n})\right\}.Remark 2.2It is shown in [14, Proposition 7] that in a Hadamard space A({yn})A\left(\{{y}_{n}\})has only one element.Definition 2.3A bounded sequence {yn}\{{y}_{n}\}in a metric space (W,d)Δ\left(W,d)\hspace{0.33em}\Delta -converges to a point wwin YYif {w}\left\{w\right\}is the unique asymptotic centre for every subsequence {ynk}\{{y}_{{n}_{k}}\}of {yn}\{{y}_{n}\}and strongly converges to wwif limn→∞d(yn,w)=0{\mathrm{lim}}_{n\to \infty }d({y}_{n},w)=0.Let (W,d)\left(W,d)be a metric space and uu, wwbe two points in WW. A map τuw:[0,ℓ]⊂R→W{\tau }_{u}^{w}:\left[0,\ell ]\subset {\mathbb{R}}\to Wis called a geodesic path from uuto wwif τuw(0)=u,τuw(ℓ)=w{\tau }_{u}^{w}\left(0)=u,\hspace{0.33em}{\tau }_{u}^{w}\left(\ell )=wand d(τuw(α1),τuw(α2))=∣α1−α2∣d({\tau }_{u}^{w}\left({\alpha }_{1}),{\tau }_{u}^{w}\left({\alpha }_{2}))=| {\alpha }_{1}-{\alpha }_{2}| for every α1{\alpha }_{1}, α2∈[0,ℓ]{\alpha }_{2}\in \left[0,\ell ]. The image of τuw{\tau }_{u}^{w}is called a geodesic segment joining uuand ww. Where there is no ambiguity, we shall denote the image by [u,w]\left[u,w]. A metric space (W,d)\left(W,d)is a geodesic space if every two elements uu, wwin WWare joined by a geodesic segment and is said to be uniquely geodesic space if every two points uu, wware joined by a unique geodesic segment [u,w]\left[u,w]in WW. A subset YYof WWis convex if all geodesic segments connecting any two points of YYare in YY. A geodesic space that satisfies the following CN-inequality of Bruhat and Tits [8] is called a CAT(0) space. Let w,v∈Ww,v\in Wand zzbe a midpoint of a geodesic segment connecting wwand vv, then (2.1)d2(z,y)≤12d2(w,y)+12d2(v,y)−14d2(w,v),{d}^{2}\left(z,y)\le \frac{1}{2}{d}^{2}\left(w,y)+\frac{1}{2}{d}^{2}\left(v,y)-\frac{1}{4}{d}^{2}\left(w,v),for every y∈Wy\in W. CAT(0) spaces include pre-Hilbert spaces, Hilbert balls, Euclidean buildings, R{\mathbb{R}}-trees, and Hadamard manifolds. A complete CAT(0) space is known as Hadamard space. For details on CAT(0) spaces, see [6,7,17,25,36].Let ∡¯y(u,w){\bar{\measuredangle }}_{y}\left(u,w)denote the comparison angle between uuand wwat yy, i.e.∡¯y(y,y)≔0,∡¯y(y,w)=∡¯y(w,y)≔π2,cos∡¯y(u,w)≔⟨u¯−y¯,w¯−y¯⟩‖u¯−y¯‖2‖w¯−y¯‖2,{\bar{\measuredangle }}_{y}(y,y):= 0,\hspace{0.33em}{\bar{\measuredangle }}_{y}(y,w)={\bar{\measuredangle }}_{y}\left(w,y):= \frac{\pi }{2},\hspace{0.33em}\cos {\bar{\measuredangle }}_{y}\left(u,w):= \frac{\langle \bar{u}-\bar{y},\bar{w}-\bar{y}\rangle }{\Vert \bar{u}-\bar{y}{\Vert }_{2}\Vert \bar{w}-\bar{y}{\Vert }_{2}},where u,w∈Y\{y}u,w\left\in Y\backslash \{y\}and △¯(u¯,w¯,y¯)\bar{\bigtriangleup }\left(\bar{u},\bar{w},\bar{y})is the comparison triangle of △(u,w,y)\bigtriangleup \left(u,w,y). Then the Alexandrov angle between two geodesic issuing from a common point y∈Yy\in Yis defined by αy(τ1,τ2)=lims,t→0+∡¯y(τ1(t),τ2(s)).{\alpha }_{y}({\tau }_{1},{\tau }_{2})=\mathop{\mathrm{lim}}\limits_{s,t\to {0}^{+}}{\bar{\measuredangle }}_{y}({\tau }_{1}\left(t),{\tau }_{2}\left(s)).The Alexandrov angle αy{\alpha }_{y}defines a pseudometric on the set of all geodesics issuing from yy. We denote the metric identification of the pseudometric space by (Sy,∡y)\left({S}_{y},{\measuredangle }_{y}). In this work, the element of Sy{S}_{y}is denoted by τ≡[τ]\tau \equiv \left[\tau ]. Moreover, as in [9], ∼\sim forms an equivalence relation on [0,∞)×Sy\left[0,\infty )\times {S}_{y}in the sense that (t,τ1)∼(s,τ2)\left(t,{\tau }_{1})\hspace{0.33em} \sim \hspace{0.33em}\left(s,{\tau }_{2})if and only if tη(τ1)=sη(τ2)=0ortη(τ1)=sη(τ2)>0withτ1=τ2,t\eta \left({\tau }_{1})=s\eta \left({\tau }_{2})=0\hspace{1em}\hspace{0.1em}\text{or}\hspace{0.1em}\hspace{1em}t\eta \left({\tau }_{1})=s\eta \left({\tau }_{2})\gt 0\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{\tau }_{1}={\tau }_{2},where η(τ)≔0\eta \left(\tau ):= 0if τ\tau is a geodesic connecting only one point and η(τ)≔1\eta \left(\tau ):= 1otherwise. Then TyY≔([0,∞)×Sy)/∼{T}_{y}Y:= (\left[0,\infty )\times {S}_{y})\hspace{0.1em}\text{/}\hspace{0.1em} \sim together with the metric dy{d}_{y}defined by dy(tτ1,sτ2)≔t2η(τ1)+s2η(τ2)−2stη(τ1)η(τ2)cos∡y(τ1,τ2){d}_{y}\left(t{\tau }_{1},s{\tau }_{2}):= \sqrt{{t}^{2}\eta \left({\tau }_{1})+{s}^{2}\eta \left({\tau }_{2})-2st\eta \left({\tau }_{1})\eta \left({\tau }_{2})\cos {\measuredangle }_{y}\left({\tau }_{1},{\tau }_{2})}form a metric space (TyY,dy)\left({T}_{y}Y,{d}_{y})known as the tangent space of YY. For more details see [32].In the sequel, we denote a complete CAT(0) space by (W,d)\left(W,d)and a nonempty convex closed subset of WWby YY, the tangent space of YYat yyby (TyY,dy)\left({T}_{y}Y,{d}_{y}). We shall denote the tangent bundle of YY, ⋃u∈YTyY{\bigcup }_{u\in Y}{T}_{y}Yby TYTY, and adopt the notation 0≔{0y:y∈Y}{\bf{0}}:= \left\{{0}_{y}:y\in Y\right\}, where 0y≔0τ=sτyy{0}_{y}:= 0\tau =s{\tau }_{y}^{y}for which s>0s\gt 0and τ∈Sy\tau \in {S}_{y}. We shall say that a vector field A:Y→TYA:Y\to TYsatisfies condition (S)\left(S)if for any s>0s\gt 0and y∈Yy\in Y, there exists u∈Yu\in Ysuch that sd(u,y)τuy∈Ausd\left(u,y){\tau }_{u}^{y}\in Au.Definition 2.4[9] A vector field A:Y→TYA:Y\to TYis said to be monotone if Gy(ξ,τuw)+Gy(ϕ,τwu)≤0,{G}_{y}(\xi ,{\tau }_{u}^{w})+{G}_{y}(\phi ,{\tau }_{w}^{u})\le 0,for every (u,ξ),(w,ϕ)∈{(y,u)∈Y×TY:u∈Ay},\left(u,\xi ),\left(w,\phi )\in \left\{(y,u)\in Y\times TY:u\in Ay\right\},where Gy(tτ1,sτ2)=stη(τ1)η(τ2)cos∡y(τ1,τ2).{G}_{y}\left(t{\tau }_{1},s{\tau }_{2})=st\eta \left({\tau }_{1})\eta \left({\tau }_{2})\cos {\measuredangle }_{y}\left({\tau }_{1},{\tau }_{2}).In what follows, A−1(0){A}^{-1}\left({\bf{0}})denotes the solution set of MVFIP and Jμ{J}_{\mu }denotes the μ\mu -resolvent of AAdefined by Jμ(z)≔w∈X:1μd(w,z)τwz∈Aw,∀z∈X.{J}_{\mu }\left(z):= \left\{\phantom{\rule[-1.25em]{}{0ex}},w\in X:\frac{1}{\mu }d\left(w,z){\tau }_{w}^{z}\in Aw\right\},\hspace{1em}\forall z\in X.Lemma 2.5[9, p. 15] Let A:Y→TYA:Y\to TYbe a monotone vector field satisfying condition (S)\left(S)and let Jμ{J}_{\mu }be the μ\mu -resolvent of A. Then(i)Jμ{J}_{\mu }is well defined and singlevalued on YY,(ii)d(Jμ(x),Jμ(y))≤d(x,y)d({J}_{\mu }\left(x),{J}_{\mu }(y))\le d\left(x,y)for every xx, yyin YY,(iii)A−1(0)={x∈X:x=Jμ(x)}{A}^{-1}\left({\bf{0}})=\left\{x\in X:x={J}_{\mu }\left(x)\right\},(iv)Jλ(y)=Jμ1−μλJλ(y)⊕μλy{J}_{\lambda }(y)={J}_{\mu }\left(\left(1-\frac{\mu }{\lambda }\right){J}_{\lambda }(y)\oplus \frac{\mu }{\lambda }y\right), ∀y∈Y\forall \hspace{-0.3em}y\in Y, μ,λ∈R\mu ,\lambda \in {\mathbb{R}}such that λ≥μ>0\lambda \ge \mu \gt 0.Lemma 2.6[26, Proposition 3.7] Let TTbe a singlevalued nonexpansive mapping on YYand {wn}\left\{{w}_{n}\right\}be a sequence in YY. If {wn}Δ\left\{{w}_{n}\right\}\hspace{0.25em}\Delta -converges to wwand d(wn,Twn)→0,d\left({w}_{n},T{w}_{n})\to 0,then w=Tww=Tw.Remark 2.7Similar result of Lemma 2.6 holds for multivalued nonexpansive mappings.Lemma 2.8[13] The asymptotic centre of any bounded sequence in YYis in YY.Lemma 2.9[26, Proposition 3.6] Every bounded sequence {yn}\{{y}_{n}\}in YYhas a Δ\Delta -convergent subsequence {ynk}\{{y}_{{n}_{k}}\}.Lemma 2.10[15, Lemma 2.10] Let {yn}\{{y}_{n}\}be a sequence in YYwith A({yn})={v}A\left(\{{y}_{n}\})=\left\{v\right\}. Suppose that {ynk}\{{y}_{{n}_{k}}\}is a subsequence of {yn}\{{y}_{n}\}with A({ynk})={w}A\left(\{{y}_{{n}_{k}}\})=\left\{w\right\}and the sequence {d(yn,w)}\left\{d({y}_{n},w)\right\}converges, then v=wv=w.Lemma 2.11[15, Lemma 2.1 (iv)] Let u,y∈Yu,y\in Y. Then for each t∈[0,1]t\in \left[0,1], there exists a unique point w∈[u,y]w\in \left[u,y]such thatd(u,w)=td(u,y)andd(y,w)=(1−t)d(u,y).d\left(u,w)=td\left(u,y)\hspace{1.0em}{and}\hspace{1.0em}d(y,w)=\left(1-t)d\left(u,y).In this article, such a point wwis denoted by (1−t)u⊕ty\left(1-t)u\oplus ty. Moreover, for finite elements {yj}1m⊂Y{\{{y}_{j}\}}_{1}^{m}\subset Yand {tj}1m⊂(0,1){\left\{{t}_{j}\right\}}_{1}^{m}\subset \left(0,1), the notation ⊕j=1mtjyj{\oplus }_{j=1}^{m}{t}_{j}{y}_{j}is adopted from Dhompongsa et al. [12, p. 460], which is defined orderly as follows: ⊕j=1mtjyj≔(1−tm)t11−tmy1⊕t21−tmy2⊕⋯⊕tm−11−tm⊕tmym.\underset{j=1}{\overset{m}{\oplus }}{t}_{j}{y}_{j}:= \left(1-{t}_{m})\left(\frac{{t}_{1}}{1-{t}_{m}}{y}_{1}\oplus \frac{{t}_{2}}{1-{t}_{m}}{y}_{2}\oplus \cdots \oplus \frac{{t}_{m-1}}{1-{t}_{m}}\right)\oplus {t}_{m}{y}_{m}.Lemma 2.12[15, Lemma 2.5] Let y1,y2{y}_{1},{y}_{2}be points in YYand t∈[0,1]t\in \left[0,1]. Thend((1−t)y1⊕ty2,y3)≤(1−t)d(y1,y3)+td(y2,y3),d\left(\left(1-t){y}_{1}\oplus t{y}_{2},{y}_{3})\le \left(1-t)d({y}_{1},{y}_{3})+td({y}_{2},{y}_{3}),for every y3∈Y{y}_{3}\in Y.Lemma 2.13[15, Lemma 2.6] Let y1,y2,y3{y}_{1},{y}_{2},{y}_{3}be points in YYand t∈[0,1]t\in \left[0,1]. Thend2((1−t)y1⊕ty2,y3)≤(1−t)d2(y1,y3)+td2(y2,y3)−t(1−t)d2(y1,y2),{d}^{2}\left(\left(1-t){y}_{1}\oplus t{y}_{2},{y}_{3})\le \left(1-t){d}^{2}({y}_{1},{y}_{3})+t{d}^{2}({y}_{2},{y}_{3})-t\left(1-t){d}^{2}({y}_{1},{y}_{2}),for every y3∈Y{y}_{3}\in Y.As immediate consequence of Lemma 2.15, we have the following lemma.Lemma 2.14Let y1,y2{y}_{1},{y}_{2}be points in YYand t∈[0,1]t\in \left[0,1]. Thend2((1−t)y1⊕ty2,y3)≤(1−t)2d2(y1,y3)+t2d2(y2,y3)+2t(1−t)⟨y1y3→,y2y3→⟩,{d}^{2}\left(\left(1-t){y}_{1}\oplus t{y}_{2},{y}_{3})\le {\left(1-t)}^{2}{d}^{2}({y}_{1},{y}_{3})+{t}^{2}{d}^{2}({y}_{2},{y}_{3})+2t\left(1-t)\langle \overrightarrow{{y}_{1}{y}_{3}},\overrightarrow{{y}_{2}{y}_{3}}\rangle ,for every y3∈Y{y}_{3}\in Y.Lemma 2.15[1, Theorem 2.6] A bounded sequence {wn}\left\{{w}_{n}\right\}Δ\Delta -converge to a point wwin WWif and only if limsupn→∞⟨wnw→,zw→⟩≤0{\mathrm{limsup}}_{n\to \infty }\langle \overrightarrow{{w}_{n}w},\overrightarrow{zw}\rangle \le 0for all zzin WW.A function h:Y→R∪{+∞}h:Y\to {\mathbb{R}}\cup \left\{+\infty \right\}is called convex if for every a∈(0,1)a\in \left(0,1)and u,v∈Yu,v\in Y, h(au⊕(1−a)v)≤ah(u)+(1−a)h(v).h\left(au\oplus \left(1-a)v)\le ah\left(u)+\left(1-a)h\left(v).If the set D(h)≔{u∈Y:h(u)<+∞}≠∅D\left(h):= \left\{u\in Y:h\left(u)\lt +\infty \right\}\ne \varnothing , then hhis said to be proper. The function hhis said to be lower semi-continuous at a point w∈D(h)w\in D\left(h)if h(w)≤liminfn→∞h(wn)h\left(w)\le {\mathrm{liminf}}_{n\to \infty }h\left({w}_{n})for any convergent sequence {wn}\left\{{w}_{n}\right\}in D(h)D\left(h)with limit ww. If hhis lower semi-continuous at every point in D(h)D\left(h), then it is lower semi-continuous on D(h)D\left(h). For example of a proper convex lower semi-continuous function in a Hadamard space, see e.g. [11].Lemma 2.16[30, Lemma 1.10] Let h:Y→(−∞,+∞]h:Y\to \left(-\infty ,+\infty ]be a convex proper lower semi-continuous function. ThenJμhw=Jλhμ−λμJμhw⊕λμw,foreveryw∈Yandμ>λ>0,{J}_{\mu }^{h}w={J}_{\lambda }^{h}\left(\frac{\mu -\lambda }{\mu }{J}_{\mu }^{h}w\oplus \frac{\lambda }{\mu }w\right),\hspace{1em}{for}\hspace{0.33em}{every}\hspace{0.33em}w\in Y\hspace{1em}\hspace{0.1em}{\text{and}}\hspace{0.1em}\hspace{1em}\mu \gt \lambda \gt 0,whereJμhv≔argminw∈Yh(w)+12μd2(w,v).{J}_{\mu }^{h}v:= \mathop{{\rm{argmin}}}\limits_{w\in Y}\left\{\phantom{\rule[-1.25em]{}{0ex}},h\left(w)+\frac{1}{2\mu }{d}^{2}\left(w,v)\right\}.Lemma 2.17[2, p. 11] Let h:Y→(−∞,+∞]h:Y\to \left(-\infty ,+\infty ]be a convex proper lower semi-continuous function. Then for every u,v∈Yu,v\in Yand μ>0\mu \gt 0, the following hold:d2(w,Jμhw)≤d2(w,z)−d2(z,Jμhw)+2μ(h(z)−h(Jμhw)).{d}^{2}\left(w,{J}_{\mu }^{h}w)\le {d}^{2}\left(w,z)-{d}^{2}\left(z,{J}_{\mu }^{h}w)+2\mu (h\left(z)-h\left({J}_{\mu }^{h}w)).Lemma 2.18[3, Proposition 6.5] Let h:Y→(−∞,+∞]h:Y\to \left(-\infty ,+\infty ]be a convex proper lower semi-continuous function and Jλh{J}_{\lambda }^{h}be the λ\lambda -resolvent operator of hh. Then the fixed point set of Jλh{J}_{\lambda }^{h}coincides with the solution set of minimizers of hh.Lemma 2.19[20, Lemma 4] The λ\lambda -resolvent operator Jλh{J}_{\lambda }^{h}of a convex proper lower semi-continuous function h:Y→(−∞,+∞]h:Y\to \left(-\infty ,+\infty ]is nonexpansive and single-valued map.Lemma 2.20[28, Lemma 3.1] Let {θn}\left\{{\theta }_{n}\right\}be a sequence in R{\mathbb{R}}such that there exists a subsequence {nj}\left\{{n}_{j}\right\}of {n}\left\{n\right\}with θnj<θnj+1{\theta }_{{n}_{j}}\lt {\theta }_{{n}_{j}+1}for every j∈Nj\in {\mathbb{N}}. Then there exists a nondecreasing sequence {mk}⊂N\left\{{m}_{k}\right\}\subset {\mathbb{N}}such that mk→∞{m}_{k}\to \infty and for sufficiently large numbers k∈Nk\in {\mathbb{N}}, θmk≤θmk+1andθk≤θmk+1.{\theta }_{{m}_{k}}\le {\theta }_{{m}_{k}+1}\hspace{1.0em}{and}\hspace{1.0em}{\theta }_{k}\le {\theta }_{{m}_{k}+1}.In fact, mk=max{i≤k:θi<θi+1}.{m}_{k}=\max \left\{i\le k:{\theta }_{i}\lt {\theta }_{i+1}\right\}.Lemma 2.21[48, Lemma 2.5] Let {θn}\left\{{\theta }_{n}\right\}be a sequence in [0,+∞)⊂R\left[0,+\infty )\subset {\mathbb{R}}with(2.2)θn+1≤(1−σn)θn+σnϕn+γn,n≥1,{\theta }_{n+1}\le \left(1-{\sigma }_{n}){\theta }_{n}+{\sigma }_{n}{\phi }_{n}+{\gamma }_{n},\hspace{1.0em}n\ge 1,where {σn}\left\{{\sigma }_{n}\right\}, {ϕn}\left\{{\phi }_{n}\right\}, and {γn}\left\{{\gamma }_{n}\right\}satisfy the following conditions:(i){σn}⊂[0,1]\left\{{\sigma }_{n}\right\}\subset \left[0,1], ∑n=1∞σn=∞{\sum }_{n=1}^{\infty }{\sigma }_{n}=\infty ,(ii)limsupn→∞ϕn≤0{\mathrm{limsup}}_{n\to \infty }{\phi }_{n}\le 0, and(iii){γn}⊂[0,∞)\left\{{\gamma }_{n}\right\}\subset \left[0,\infty ), ∑n=1∞γn<∞{\sum }_{n=1}^{\infty }{\gamma }_{n}\lt \infty .Then limn→∞θn=0{\mathrm{lim}}_{n\to \infty }{\theta }_{n}=0.3Main resultsLet (W,d)\left(W,d)be a complete CAT(0) space and YYa nonempty closed convex subset of WW. Suppose that TYTYdenotes the tangent bundle of YY, h:Y→Rh:Y\to {\mathbb{R}}is a convex proper lower semicontinuous function, Aj:Y→TY{A}_{j}:Y\to TY, j=1,2,…,M1j=1,2,\ldots ,{M}_{1}are monotone vector fields with corresponding μ\mu -resolvent Jμnj{J}_{{\mu }_{n}}^{j}and Tp:Y→Cℬ(Y){T}_{p}:Y\to {\mathcal{C {\mathcal B} }}\left(Y), p=1,2,…,M2p=1,2,\ldots ,{M}_{2}are multivalued κp{\kappa }_{p}-demicontractive mappings. We shall assume that Γ≔argminy∈Yh(y)∩⋂j=1M1Aj−1(0)∩⋂p=1M2F(Tp)≠∅\Gamma := {{\rm{argmin}}}_{y\in Y}h(y)\cap {\bigcap }_{j=1}^{{M}_{1}}{A}_{j}^{-1}\left({\bf{0}})\cap {\bigcap }_{p=1}^{{M}_{2}}F\left({T}_{p})\ne \varnothing and each Tp(w)={w}{T}_{p}\left(w)=\left\{w\right\}for w∈Γw\in \Gamma . Moreover, for validation of the proposed algorithms, we let Jμn0{J}_{{\mu }_{n}}^{0}to be the identity map on YY, M≔max{M1,M2}M:= \max \left\{{M}_{1},{M}_{2}\right\}and if M1<M{M}_{1}\lt Mwe take Tj(x)≔{x}{T}_{j}\left(x):= \left\{x\right\}on YYfor j∈(M1,M]∩Nj\in \left({M}_{1},M]\cap {\mathbb{N}}and if M2<M{M}_{2}\lt Mwe set Jμnj=Jμn0{J}_{{\mu }_{n}}^{j}={J}_{{\mu }_{n}}^{0}for j∈(M2,M]∩Nj\in \left({M}_{2},M]\cap {\mathbb{N}}. In the convergence analysis, we shall need the assumption that each map Tp{T}_{p}satisfies demiclosedness-type property, that is if {wn}Δ\left\{{w}_{n}\right\}\hspace{0.33em}\Delta -converges to wwand dist(wn,Tpwn)→0,\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{n},{T}_{p}{w}_{n})\to 0,then w∈Tpww\in {T}_{p}w.Algorithm 1: Mann-type PPAInitialization: Choose {μn}⊂(μ,+∞),{λn}⊂(μ,+∞)\left\{{\mu }_{n}\right\}\subset \left(\mu ,+\infty ),\left\{{\lambda }_{n}\right\}\subset \left(\mu ,+\infty )for some μ>0\mu \gt 0, {βnj}⊂(0,1)\left\{{\beta }_{n}^{j}\right\}\subset \left(0,1), j=0,1,…,Mj=0,1,\ldots ,Msuch that ∑j=0Mβnj=1{\sum }_{j=0}^{M}{\beta }_{n}^{j}=1and {αnj}⊂[κj,1]\left\{{\alpha }_{n}^{j}\right\}\subset \left[{\kappa }_{j},1], j=1,…,Mj=1,\ldots ,M. Let w1∈Y{w}_{1}\in Y.Step 1: Set n=1n=1and compute yn0=Jλnh(wn).{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}\left({w}_{n}).Step 2: For each j∈{1,2,…,M},j\in \left\{1,2,\ldots ,M\right\},choose wnj∈Tjwn{w}_{n}^{j}\in {T}_{j}{w}_{n}and compute ynj=αnjwn⊕(1−αnj)wnj.{y}_{n}^{j}={\alpha }_{n}^{j}{w}_{n}\oplus \left(1-{\alpha }_{n}^{j}){w}_{n}^{j}.Step 3: Compute wn+1=⊕j=0MβnjJμnj(ynj).{w}_{n+1}=\underset{j=0}{\overset{M}{\oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}).Set n≔n+1n:= n+1and repeat all the steps.Lemma 3.1Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 1. Then for every w∈Γw\in \Gamma , the sequence {d(wn,w)}\left\{d\left({w}_{n},w)\right\}converges in R{\mathbb{R}}. Moreover, if each {βnk}⊂[b,1)\left\{{\beta }_{n}^{k}\right\}\subset \left[b,1)for some b∈(0,1)b\in \left(0,1), then {d(ynk,w)}\left\{d({y}_{n}^{k},w)\right\}converges for every w∈Γw\in \Gamma andlimn→∞d(ynk,w)=limn→∞d(wn,w)foreveryk∈{0,1,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({y}_{n}^{k},w)=\mathop{\mathrm{lim}}\limits_{n\to \infty }d\left({w}_{n},w)\hspace{0.33em}{for}\hspace{0.33em}{every}\hspace{0.33em}k\in \left\{0,1,\ldots ,M\right\}.ProofLet w∈Γw\in \Gamma and let n∈Nn\in {\mathbb{N}}. Then by Step 1 of Algorithm 1 and Lemma 2.19, we have (3.1)d(yn0,w)=d(Jλnh(wn),Jλnh(w))≤d(wn,w).d({y}_{n}^{0},w)=d({J}_{{\lambda }_{n}}^{h}\left({w}_{n}),{J}_{{\lambda }_{n}}^{h}\left(w))\le d\left({w}_{n},w).For each j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, we obtain from Step 2 of Algorithm 1, Lemma 2.13, and the assumption that Tj{T}_{j}is κj{\kappa }_{j}-demicontractive that (3.2)d2(ynj,w)≤αnjd2(wn,w)+(1−αnj)d2(wnj,w)−αnj(1−αnj)d2(wn,wnj)=αnjd2(wn,w)+(1−αnj)dist2(wnj,Tjw)−αnj(1−αnj)d2(wn,wnj)≤αnjd2(wn,w)+(1−αnj)H2(Tjwn,Tjw)−αnj(1−αnj)d2(wn,wnj)≤αnjd2(wn,w)+(1−αnj)[d2(wn,w)+κjdist2(wn,Tjwn)]−αnj(1−αnj)d2(wn,wnj)=d2(wn,w)+(1−αnj)(κj−αnj)d2(wn,wnj)=d2(wn,w)−(1−αnj)(αnj−κj)d2(wn,wnj)\begin{array}{rcl}{d}^{2}({y}_{n}^{j},w)& \le & {\alpha }_{n}^{j}{d}^{2}({w}_{n},w)+\left(1-{\alpha }_{n}^{j}){d}^{2}({w}_{n}^{j},w)-{\alpha }_{n}^{j}\left(1-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & =& {\alpha }_{n}^{j}{d}^{2}\left({w}_{n},w)+\left(1-{\alpha }_{n}^{j}){\text{dist}}^{2}\left({w}_{n}^{j},{T}_{j}w)-{\alpha }_{n}^{j}\left(1-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & \le & {\alpha }_{n}^{j}{d}^{2}\left({w}_{n},w)+\left(1-{\alpha }_{n}^{j}){H}^{2}\left({T}_{j}{w}_{n},{T}_{j}w)-{\alpha }_{n}^{j}\left(1-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & \le & {\alpha }_{n}^{j}{d}^{2}\left({w}_{n},w)+\left(1-{\alpha }_{n}^{j}){[}{d}^{2}\left({w}_{n},w)+{\kappa }_{j}{\text{dist}}^{2}({w}_{n},{T}_{j}{w}_{n})]-{\alpha }_{n}^{j}\left(1-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & =& {d}^{2}\left({w}_{n},w)+\left(1-{\alpha }_{n}^{j})\left({\kappa }_{j}-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & =& {d}^{2}\left({w}_{n},w)-\left(1-{\alpha }_{n}^{j})\left({\alpha }_{n}^{j}-{\kappa }_{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\end{array}(3.3)≤d2(wn,w).\le \hspace{0.33em}{d}^{2}\left({w}_{n},w).\hspace{23.6em}By Step 3 of Algorithm 1 and Lemma 2.12, we obtain (3.4)d(wn+1,w)=d⊕j=0MβnjJμnj(ynj),w≤∑j=0Mβnjd(Jμnj(ynj),w)=∑j=0Mβnjd(Jμnj(ynj),Jμnj(w))≤∑j=0Mβnjd(ynj,w).\begin{array}{rcl}d({w}_{n+1},w)& =& d\hspace{0.08em}\left(\underset{j=0}{\overset{M}{\displaystyle \oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),w\right)\\ & \le & \mathop{\displaystyle \sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),w)\\ & =& \mathop{\displaystyle \sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),{J}_{{\mu }_{n}}^{j}\left(w))\\ & \le & \mathop{\displaystyle \sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w).\end{array}This, (3.1) and (3.3) imply (3.5)d(wn+1,w)≤∑j=0Mβnjd(ynj,w)≤∑j=0Mβnjd(wn,w)≤d(wn,w).d({w}_{n+1},w)\le \mathop{\sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w)\le \mathop{\sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({w}_{n},w)\le d({w}_{n},w).This implies that {d(wn,w)}\left\{d\left({w}_{n},w)\right\}converges in R{\mathbb{R}}. Let k∈{0,1,…,M}k\in \left\{0,1,\ldots ,M\right\}, then from (3.4), (3.1), and (3.3), we have (3.6)d(wn+1,w)≤βnkd(ynk,w)+∑j=1,j≠kMβnjd(ynj,w)≤βnkd(ynk,w)+∑j=1,j≠kMβnjd(wn,w)=βnkd(ynk,w)+(1−βnk)d(wn,w).\begin{array}{rcl}d({w}_{n+1},w)& \le & {\beta }_{n}^{k}d({y}_{n}^{k},w)+\mathop{\displaystyle \sum }\limits_{j=1,j\ne k}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w)\\ & \le & {\beta }_{n}^{k}d({y}_{n}^{k},w)+\mathop{\displaystyle \sum }\limits_{j=1,j\ne k}^{M}{\beta }_{n}^{j}d\left({w}_{n},w)\\ & =& {\beta }_{n}^{k}d({y}_{n}^{k},w)+\left(1-{\beta }_{n}^{k})d\left({w}_{n},w).\end{array}It follows from (3.6), (3.3), and (3.1) that (3.7)1βnk[d(wn+1,w)−d(wn,w)]+d(wn,w)≤d(ynk,w)≤d(wn,w).\frac{1}{{\beta }_{n}^{k}}{[}d\left({w}_{n+1},w)-d\left({w}_{n},w)]+d\left({w}_{n},w)\le d({y}_{n}^{k},w)\le d\left({w}_{n},w).Using the fact that {d(wn,w)}\left\{d\left({w}_{n},w)\right\}converges and letting n→∞n\to \infty in (3.7), we have the complete proof.□Lemma 3.2Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 1. If {βnk}⊂[b,1)\left\{{\beta }_{n}^{k}\right\}\subset \left[b,1)for some b∈(0,1)b\in \left(0,1)and limn→∞αnj∈(κj,1){\mathrm{lim}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({\kappa }_{j},1), thenlimn→∞d(wn,Jμh(wn))=limn→∞d(wn,Jμj(wn))=limn→∞dist(wn,Tjwn)=0,\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{J}_{\mu }^{h}\left({w}_{n}))=\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{J}_{\mu }^{j}\left({w}_{n}))=\mathop{\mathrm{lim}}\limits_{n\to \infty }\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{n},{T}_{j}{w}_{n})=0,for every j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}.ProofLet w∈Γw\in \Gamma . By Lemma 2.17 and the fact that h(w)≤h(wn)h\left(w)\le h\left({w}_{n}), we obtain (3.8)d2(yn0,wn)≤d2(yn0,w)−d2(wn,w)+2μ(h(w)−h(wn))≤d2(yn0,w)−d2(wn,w).{d}^{2}({y}_{n}^{0},{w}_{n})\le {d}^{2}({y}_{n}^{0},w)-{d}^{2}\left({w}_{n},w)+2\mu (h\left(w)-h\left({w}_{n}))\le {d}^{2}({y}_{n}^{0},w)-{d}^{2}\left({w}_{n},w).This and Lemma 3.1 imply that (3.9)limn→∞d(wn,yn0)=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{y}_{n}^{0})=0.Consequently, it follows from Lemmas 2.16, 2.19, and 2.12 that (3.10)d(wn,Jμhwn)≤d(wn,yn0)+d(yn0,Jμhwn)=d(wn,yn0)+dJμh(wn),Jμhλn−μλnJλnh(wn)⊕μμnwn≤d(wn,yn0)+dwn,λn−μλnyn0⊕μλnwn≤d(wn,yn0)+1−μλnd(wn,yn0)≤2−μλnd(wn,yn0)→0,asn→∞.\begin{array}{rcl}d\left({w}_{n},{J}_{\mu }^{h}{w}_{n})& \le & d({w}_{n},{y}_{n}^{0})+d({y}_{n}^{0},{J}_{\mu }^{h}{w}_{n})\\ & =& d({w}_{n},{y}_{n}^{0})+d\hspace{0.08em}\left(\phantom{\rule[-.2em]{}{0ex}},{J}_{\mu }^{h}\left({w}_{n}),{J}_{\mu }^{h}\left(\frac{{\lambda }_{n}-\mu }{{\lambda }_{n}}{J}_{{\lambda }_{n}}^{h}\left({w}_{n})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{w}_{n}\right)\right)\\ & \le & d({w}_{n},{y}_{n}^{0})+d\hspace{0.08em}\left(\phantom{\rule[-.2em]{}{0ex}},{w}_{n},\left(\frac{{\lambda }_{n}-\mu }{{\lambda }_{n}}{y}_{n}^{0}\displaystyle \oplus \frac{\mu }{{\lambda }_{n}}{w}_{n}\right)\right)\\ & \le & d({w}_{n},{y}_{n}^{0})+\left(1-\frac{\mu }{{\lambda }_{n}}\right)\hspace{0.08em}d\hspace{0.08em}\left({w}_{n},{y}_{n}^{0})\\ & \le & \left(2-\frac{\mu }{{\lambda }_{n}}\right)\hspace{0.08em}d\left({w}_{n},{y}_{n}^{0})\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .\end{array}Let n∈Nn\in {\mathbb{N}}and set ϕk(n)=⊕j=0kβnjγkJμnj(ynj){\phi }_{k}^{\left(n)}={\oplus }_{j=0}^{k}\frac{{\beta }_{n}^{j}}{{\gamma }_{k}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}), where γk≔∑j=0kβnj{\gamma }_{k}:= {\sum }_{j=0}^{k}{\beta }_{n}^{j}, k∈{0,1,2,…,M}k\in \left\{0,1,2,\ldots ,M\right\}. Then γk∈[b,1){\gamma }_{k}\in \left[b,1), γ0=βn0{\gamma }_{0}={\beta }_{n}^{0}, γM=1{\gamma }_{M}=1, and ϕ0(n)=Jμn0(yn0)=yn0{\phi }_{0}^{\left(n)}={J}_{{\mu }_{n}}^{0}({y}_{n}^{0})={y}_{n}^{0}. Moreover, (3.11)γk−1γk≥βn0and\frac{{\gamma }_{k-1}}{{\gamma }_{k}}\ge {\beta }_{n}^{0}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}(3.12)ϕk(n)=γk−1γkϕk−1(n)⊕βnkγkJμk(ynk),{\phi }_{k}^{\left(n)}=\frac{{\gamma }_{k-1}}{{\gamma }_{k}}{\phi }_{k-1}^{\left(n)}\oplus \frac{{\beta }_{n}^{k}}{{\gamma }_{k}}{J}_{\mu }^{k}({y}_{n}^{k}),for every k∈{1,2,…,M}k\in \left\{1,2,\ldots ,M\right\}. Let w∈Γw\in \Gamma . Then from (3.12), Lemma 2.13, and (3.11), we have the following inequality for every k∈{1,2,…,M}k\in \left\{1,2,\ldots ,M\right\}: (3.13)d2(ϕk(n),w)=d2γk−1γkϕk−1(n)⊕βnkγkJμnk(ynk),w≤1γkγk−1d2(ϕk−1(n),w)+βnkd2(Jμnk(ynk),w)−γk−1βnkγkd2(ϕk−1(n),Jμnk(ynk))≤1γk[γk−1d2(ϕk−1(n),w)+βnkd2(Jμnk(ynk),w)−βn0βnkd2(ϕk−1(n),Jμnk(ynk))].\begin{array}{rcl}{d}^{2}({\phi }_{k}^{\left(n)},w)& =& {d}^{2}\left(\frac{{\gamma }_{k-1}}{{\gamma }_{k}}{\phi }_{k-1}^{\left(n)}\displaystyle \oplus \frac{{\beta }_{n}^{k}}{{\gamma }_{k}}{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w\right)\\ & \le & \frac{1}{{\gamma }_{k}}\left[{\gamma }_{k-1}{d}^{2}({\phi }_{k-1}^{\left(n)},w)+{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-\frac{{\gamma }_{k-1}{\beta }_{n}^{k}}{{\gamma }_{k}}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\right]\\ & \le & \frac{1}{{\gamma }_{k}}{[}{\gamma }_{k-1}{d}^{2}({\phi }_{k-1}^{\left(n)},w)+{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))].\end{array}Moreover, from Algorithm 1 and (3.13), we have d2(wn+1,w)=d2(ϕM(n),w)≤1γM[γM−1d2(ϕM−1(n),w)+βnMd2(JμnM(ynM),w)−βn0βnMd2(ϕM−1(n),JμnM(ynM))]≤γM−1d2(ϕM−1(n),w)+βnMd2(JμnM(ynM),w)−βn0βnMd2(ϕM−1(n),JμnM(ynM))≤[γM−2d2(ϕM−2(n),w)+βnM−1d2(JμnM−1(ynM−1),w)−βn0βnM−1d2(ϕM−2(n),JμnM−1(ynM−1))]+βnMd2(JμnM(ynM),w)−βn0βnMd2(ϕM−1(n),JμnM(ynj))=γM−2d2(ϕM−2(n),w)+∑k=M−1Mβnkd2(Jμnk(wnk),w)−βn0∑k=M−1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& =& {d}^{2}({\phi }_{M}^{\left(n)},w)\\ & \le & \frac{1}{{\gamma }_{M}}{[}{\gamma }_{M-1}{d}^{2}({\phi }_{M-1}^{\left(n)},w)+{\beta }_{n}^{M}{d}^{2}({J}_{{\mu }_{n}}^{M}({y}_{n}^{M}),w)-{\beta }_{n}^{0}{\beta }_{n}^{M}{d}^{2}({\phi }_{M-1}^{\left(n)},{J}_{{\mu }_{n}}^{M}({y}_{n}^{M}))]\\ & \le & {\gamma }_{M-1}{d}^{2}({\phi }_{M-1}^{\left(n)},w)+{\beta }_{n}^{M}{d}^{2}({J}_{{\mu }_{n}}^{M}({y}_{n}^{M}),w)-{\beta }_{n}^{0}{\beta }_{n}^{M}{d}^{2}({\phi }_{M-1}^{\left(n)},{J}_{{\mu }_{n}}^{M}({y}_{n}^{M}))\\ & \le & {[}{\gamma }_{M-2}{d}^{2}({\phi }_{M-2}^{\left(n)},w)+{\beta }_{n}^{M-1}{d}^{2}({J}_{{\mu }_{n}}^{M-1}({y}_{n}^{M-1}),w)-{\beta }_{n}^{0}{\beta }_{n}^{M-1}{d}^{2}({\phi }_{M-2}^{\left(n)},{J}_{{\mu }_{n}}^{M-1}({y}_{n}^{M-1}))]\\ & & +{\beta }_{n}^{M}{d}^{2}({J}_{{\mu }_{n}}^{M}({y}_{n}^{M}),w)-{\beta }_{n}^{0}{\beta }_{n}^{M}{d}^{2}({\phi }_{M-1}^{\left(n)},{J}_{{\mu }_{n}}^{M}({y}_{n}^{j}))\\ & =& {\gamma }_{M-2}{d}^{2}({\phi }_{M-2}^{\left(n)},w)+\mathop{\displaystyle \sum }\limits_{k=M-1}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}\left({w}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=M-1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}Continuing in this pattern, we obtain that d2(wn+1,w)≤γM−3d2(ϕM−3(n),w)+∑k=M−2Mβnkd2(Jμnk(ynk),w)−βn0∑k=M−2Mβnkd2(ϕk−1(n),Jμnk(ynk))⋮≤γ1d2(ϕ1(n),w)+∑k=2Mβnkd2(Jμnk(ynk),w)−βn0∑k=2Mβnkd2(ϕk−1(n),Jμnk(ynk))\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & {\gamma }_{M-3}{d}^{2}({\phi }_{M-3}^{\left(n)},w)+\mathop{\displaystyle \sum }\limits_{k=M-2}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=M-2}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \vdots & \\ & \le & {\gamma }_{1}{d}^{2}({\phi }_{1}^{\left(n)},w)+\mathop{\displaystyle \sum }\limits_{k=2}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=2}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\end{array}≤γ0d2(ϕ0(n),w)+∑k=1Mβnkd2(Jμnk(ynk),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))=∑k=0Mβnkd2(Jμnk(ynk),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}& \le & {\gamma }_{0}{d}^{2}({\phi }_{0}^{\left(n)},w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & =& \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}This and Lemma 2.5(iv) imply that (3.14)d2(wn+1,w)≤∑k=0Mβnkd2(Jμnk(ynk),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))=∑k=0Mβnkd2(Jμnk(ynk),Jμnk(w))−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤∑k=0Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & =& \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),{J}_{{\mu }_{n}}^{k}\left(w))-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}Consequently, by (3.3), we have d2(wn+1,w)≤∑k=0Mβnkd2(wn,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))=d2(wn,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({w}_{n},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & =& {d}^{2}({w}_{n},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}This implies that for every j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, βn0βnjd2(ϕj−1(n),Jμnj(ynj))≤βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynj))≤d2(wn,w)−d2(wn+1,w).{\beta }_{n}^{0}{\beta }_{n}^{j}{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\le {\beta }_{n}^{0}\mathop{\sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{j}))\le {d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w).Therefore, (3.15)d2(ϕj−1(n),Jμnj(ynj))≤1βn0βnj[d2(wn,w)−d2(wn+1,w)]≤1b2[d2(wn,w)−d2(wn+1,w)].{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\le \frac{1}{{\beta }_{n}^{0}{\beta }_{n}^{j}}{[}{d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)]\le \frac{1}{{b}^{2}}{[}{d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)].It follows from Lemma 3.1 and (3.15) that limn→∞d2(ϕj−1(n),Jμnj(ynj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.Consequently, (3.16)limn→∞d(ϕj−1(n),Jμnj(ynj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.From (3.12), Lemma 2.11, and (3.16), we obtain that for every j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}, (3.17)d(ϕj−1(n),ϕj(n))≤d(ϕj−1(n),Jμnj(ynj))→0,asn→∞.d({\phi }_{j-1}^{\left(n)},{\phi }_{j}^{\left(n)})\le d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .Furthermore, for any j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}, we have (3.18)d(wn,Jμnj(ynj))=d(ϕ0(n),Jμnj(ynj))≤d(ϕ0(n),ϕ1(n))+d(ϕ1(n),ϕ2(n))+⋯+d(ϕj−2(n),ϕj−1(n))+d(ϕj−1(n),Jμnj(ynj))=∑k=1j−1d(ϕk−1(n),ϕk(n))+d(ϕj−1(n),Jμnj(ynj))\begin{array}{rcl}d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))& =& d({\phi }_{0}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\\ & \le & d({\phi }_{0}^{\left(n)},{\phi }_{1}^{\left(n)})+d({\phi }_{1}^{\left(n)},{\phi }_{2}^{\left(n)})+\cdots +d({\phi }_{j-2}^{\left(n)},{\phi }_{j-1}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\\ & =& \mathop{\displaystyle \sum }\limits_{k=1}^{j-1}d({\phi }_{k-1}^{\left(n)},{\phi }_{k}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\end{array}(3.19)≤∑k=1Md(ϕk−1(n),ϕk(n))+d(ϕj−1(n),Jμnj(ynj)).\le \hspace{0.33em}\mathop{\sum }\limits_{k=1}^{M}d({\phi }_{k-1}^{\left(n)},{\phi }_{k}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})).\hspace{7.3em}It follows from (3.19), (3.16), and (3.17) that (3.20)limn→∞d(wn,Jμnj(ynj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.Also, for each j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}and for w∈Γw\in \Gamma , we have from (3.2) that (3.21)(1−αnj)(αnj−κj)d2(wn,wnj)≤d2(wn,w)−d2(wn+1,w).\left(1-{\alpha }_{n}^{j})\left({\alpha }_{n}^{j}-{\kappa }_{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\le {d}^{2}\left({w}_{n},w)-{d}^{2}\left({w}_{n+1},w).This, Lemma 3.1, and the assumption that liminfn→∞αnj∈(kj,1){\mathrm{liminf}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({k}_{j},1)imply limn→∞d2(wn,wnj)=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }{d}^{2}\left({w}_{n},{w}_{n}^{j})=0.Consequently, (3.22)limn→∞d(wn,wnj)=0,for everyj∈{1,2,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }d\left({w}_{n},{w}_{n}^{j})=0,\hspace{1em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}j\in \left\{1,2,\ldots ,M\right\}.Hence, for each j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, (3.23)dist(wn,Tjwn)≤d(wn,wnj)→0,asn→∞.\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{n},{T}_{j}{w}_{n})\le d\left({w}_{n},{w}_{n}^{j})\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .Furthermore, it follows from Lemma 2.5, Lemma 2.12, (3.20), and (3.22) that (3.24)d(wn,Jμj(wn))≤d(wn,Jμnj(ynj))+d(Jμnj(ynj),Jμj(wn))=d(wn,Jμnj(ynj))+dJμj(wn),Jμjμn−μμnJμnj(ynj)⊕μμnynj≤d(wn,Jμnj(ynj))+dwn,μn−μμnJμnj(ynj)⊕μμnynj≤2−μμnd(wn,Jμnj(ynj))+μμnd(wn,ynj)≤2d(wn,Jμnj(ynj))+d(wn,ynj)≤2d(wn,Jμnj(ynj))+d(wn,wnj)→0,asn→∞.\begin{array}{rcl}d({w}_{n},{J}_{\mu }^{j}\left({w}_{n}))& \le & d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d({J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),{J}_{\mu }^{j}\left({w}_{n}))\\ & =& d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\hspace{0.08em}\left({J}_{\mu }^{j}\left({w}_{n}),{J}_{\mu }^{j}\left(\frac{{\mu }_{n}-\mu }{{\mu }_{n}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{y}_{n}^{j}\right)\right)\\ & \le & d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\hspace{0.08em}\left({w}_{n},\left(\frac{{\mu }_{n}-\mu }{{\mu }_{n}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{y}_{n}^{j}\right)\right)\\ & \le & \left(2-\frac{\mu }{{\mu }_{n}}\right)\hspace{0.08em}d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+\frac{\mu }{{\mu }_{n}}d\left({w}_{n},{y}_{n}^{j})\\ & \le & 2d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\left({w}_{n},{y}_{n}^{j})\\ & \le & 2d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\left({w}_{n},{w}_{n}^{j})\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .\end{array}By (3.35), (3.24), and (3.23), we have the complete proof.□Theorem 3.3Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 1 and let each {βnk}⊂[b,1)\left\{{\beta }_{n}^{k}\right\}\subset \left[b,1)for some b∈(0,1)b\in \left(0,1)and limn→∞αnj∈(κj,1){\mathrm{lim}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({\kappa }_{j},1). Suppose that for each p∈{1,2,…,M2}p\in \left\{1,2,\ldots ,{M}_{2}\right\}, Tp{T}_{p}has demiclosedness-type property. Then {wn}Δ\left\{{w}_{n}\right\}\hspace{0.33em}\Delta -converges to an element of Γ\Gamma . Moreover, if YYis compact, then the convergence is strong.ProofFrom Lemma 3.1, we have that {wn}\left\{{w}_{n}\right\}is bounded. Let ωA(wn){\omega }_{A}\left({w}_{n})denote the union of asymptotic centres of all subsequences of {wn}\left\{{w}_{n}\right\}. Suppose that z∈ωA(wn)z\in {\omega }_{A}\left({w}_{n})with A({wnk})={z}A\left(\left\{{w}_{{n}_{k}}\right\})=\left\{z\right\}. By Lemma 2.9, there exists a subsequence {wnkp}\{{w}_{{n}_{{k}_{p}}}\}of {wnk}\left\{{w}_{{n}_{k}}\right\}that Δ\Delta -converges to vv, and Lemma 2.8 implies that v∈Yv\in Y. Now by Lemmas 2.19, 2.6, and 3.2, we have that v∈argminy∈Yh(y)v\in {{\rm{argmin}}}_{y\in Y}h(y). Similarly, by Lemmas 2.5(ii), 2.7, and 3.2, we have that v∈⋂j=1M1Aj−1(0)v\in {\bigcap }_{j=1}^{{M}_{1}}{A}_{j}^{-1}\left({\bf{0}}). Also, from Lemma 3.2 and the assumption that each Tp{T}_{p}has demiclosedness-type property, we have v∈⋂p=1M2F(Tp)v\in {\bigcap }_{p=1}^{{M}_{2}}F\left({T}_{p}). Therefore, v∈Γv\in \Gamma . So, by Lemma 3.1, {d(wn,v)}\left\{d\left({w}_{n},v)\right\}converges and by Lemma 2.10, z=vz=v. Thus, ωA(wn)⊂Γ{\omega }_{A}\left({w}_{n})\subset \Gamma . To complete the proof, it suffices to show that ωA(wn){\omega }_{A}\left({w}_{n})consists of only one element. Let A({wn})={w}A\left(\left\{{w}_{n}\right\})=\left\{w\right\}and suppose there exists y∈ωA(wn)y\in {\omega }_{A}\left({w}_{n})with y≠wy\ne w. Now, let {wnk}\left\{{w}_{{n}_{k}}\right\}be the subsequence of {wn}\left\{{w}_{n}\right\}with A({wnk})={y}A\left(\left\{{w}_{{n}_{k}}\right\})=\{y\}. Then y∈Γy\in \Gamma , since ωA(wn)⊂Γ{\omega }_{A}\left({w}_{n})\subset \Gamma . By Lemma 3.1 and the definition of asymptotic centre, we have limsupn→∞d(wn,w)<limsupn→∞d(wn,y)=limn→∞d(wn,y)=limsupk→∞d(wnk,y)\begin{array}{rcl}\mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left({w}_{n},w)& \lt & \mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left({w}_{n},y)\\ & =& \mathop{\mathrm{lim}}\limits_{n\to \infty }d\left({w}_{n},y)\\ & =& \mathop{\mathrm{limsup}}\limits_{k\to \infty }d\left({w}_{{n}_{k}},y)\end{array}<limsupk→∞d(wnk,w)≤limsupn→∞d(wn,w),\begin{array}{rcl}& \lt & \mathop{\mathrm{limsup}}\limits_{k\to \infty }d\left({w}_{{n}_{k}},w)\\ & \le & \mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left({w}_{n},w),\end{array}which contradicts y≠wy\ne w. Therefore, ωA(wn){\omega }_{A}\left({w}_{n})consists of exactly one element.Suppose that YYis compact, then there exists a subsequence {wnk}\left\{{w}_{{n}_{k}}\right\}of {wn}\left\{{w}_{n}\right\}that converges strongly to some point yyin YY. Thus, {wnk}Δ\left\{{w}_{{n}_{k}}\right\}\hspace{0.33em}\Delta -converges to y∈Yy\in Y. By first part of the proof, we have that y∈ωA(wn)⊂Γy\in {\omega }_{A}\left({w}_{n})\subset \Gamma . Consequently, by Lemma 3.1, we have limn→∞d(wn,y)=limk→∞d(wnk,y)=0,\mathop{\mathrm{lim}}\limits_{n\to \infty }d\left({w}_{n},y)=\mathop{\mathrm{lim}}\limits_{k\to \infty }d\left({w}_{{n}_{k}},y)=0,which completes the proof.□The following corollary is obtained from the fact that every quasi-nonexpansive mapping is 0-demicontractive.Corollary 3.4Let Tp,p=1,2,…,M2{T}_{p},\hspace{0.33em}p=1,2,\ldots ,{M}_{2}be quasi-nonexpansive mappings with demiclosedness-type property. Suppose that {αnj},{βnj},{wn}\left\{{\alpha }_{n}^{j}\right\},\hspace{0.33em}\left\{{\beta }_{n}^{j}\right\},\hspace{0.33em}\left\{{w}_{n}\right\}, and Γ\Gamma are as in Theorem 3.3. Then {wn}Δ\left\{{w}_{n}\right\}\hspace{0.33em}\Delta -converges to an element of Γ\Gamma . Moreover, if YYis compact, then the convergence is strong.Since every nonexpansive mapping with fixed points is quasi-nonexpansive and by Remark 2 has demiclosedness-type property, we have the following corollary.Corollary 3.5Let Tp,p=1,2,…,M2{T}_{p},\hspace{0.33em}p=1,2,\ldots ,{M}_{2}be nonexpansive mappings and suppose that {αnj},{βnj},{wn}\left\{{\alpha }_{n}^{j}\right\},\hspace{0.33em}\left\{{\beta }_{n}^{j}\right\},\hspace{0.33em}\left\{{w}_{n}\right\}, and Γ\Gamma are as in Theorem 3.3. Then {wn}Δ\left\{{w}_{n}\right\}\hspace{0.25em}\Delta -converges to an element of Γ\Gamma . Moreover, if YYis compact, then the convergence is strong.We are now ready to analyse viscosity-type PPA for strong convergence resultAlgorithm 2: Viscosity-type PPAInitialization: Choose {φn}⊂[0,1]\left\{{\varphi }_{n}\right\}\subset \left[0,1], {μn}⊂(μ,+∞),{λn}⊂(μ,+∞)\left\{{\mu }_{n}\right\}\subset \left(\mu ,+\infty ),\hspace{0.33em}\left\{{\lambda }_{n}\right\}\subset \left(\mu ,+\infty )for some μ>0\mu \gt 0, {βnj}⊂(0,1)\left\{{\beta }_{n}^{j}\right\}\subset \left(0,1), j=0,1,…,Mj=0,1,\ldots ,Msuch that ∑j=0Mβnj=1{\sum }_{j=0}^{M}{\beta }_{n}^{j}=1, and {αnj}⊂[κj,1]\left\{{\alpha }_{n}^{j}\right\}\subset \left[{\kappa }_{j},1], j=1,…,Mj=1,\ldots ,M. Let ffbe a η\eta -contraction map on YYand w1∈Y{w}_{1}\in Y.Step 1: Set n=1n=1and compute yn0=Jλnh((1−φn)wn⊕φnf(wn)).{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\oplus {\varphi }_{n}f\left({w}_{n})).Step 2: For each j∈{1,2,…,M},j\in \left\{1,2,\ldots ,M\right\},choose wnj∈Tjwn{w}_{n}^{j}\in {T}_{j}{w}_{n}and compute ynj=αnjwn⊕(1−αnj)wnj.{y}_{n}^{j}={\alpha }_{n}^{j}{w}_{n}\oplus \left(1-{\alpha }_{n}^{j}){w}_{n}^{j}.Step 3: Compute wn+1=⊕j=0MβnjJμnj(ynj).{w}_{n+1}=\underset{j=0}{\overset{M}{\oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}).Set n≔n+1n:= n+1and repeat all the steps.Lemma 3.6Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 2. Then {wn}\left\{{w}_{n}\right\}is bounded.ProofLet w∈Γw\in \Gamma . Then by Lemmas 2.19, 2.12, and the assumption that ffis contractive, we have (3.25)d(yn0,w)=d(Jλnh((1−φn)wn⊕φnf(wn)),Jλnh(w))≤d((1−φn)wn⊕φnf(wn),w)≤(1−φn)d(wn,w)+φnd(f(wn),w)≤(1−φn)d(wn,w)+φnd(f(wn),f(w))+φnd(f(w),w)=(1−(1−η)φn)d(wn,w)+φnd(f(w),w).\begin{array}{rcl}d({y}_{n}^{0},w)& =& d({J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),{J}_{{\lambda }_{n}}^{h}\left(w))\\ & \le & d(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n}),w)\\ & \le & \left(1-{\varphi }_{n})d\left({w}_{n},w)+{\varphi }_{n}d(f\left({w}_{n}),w)\\ & \le & \left(1-{\varphi }_{n})d\left({w}_{n},w)+{\varphi }_{n}d(f\left({w}_{n}),f\left(w))+{\varphi }_{n}d(f\left(w),w)\\ & =& \left(1-\left(1-\eta ){\varphi }_{n})d\left({w}_{n},w)+{\varphi }_{n}d(f\left(w),w).\end{array}Moreover, following similar line as in (3.3), we have (3.26)d(ynj,w)≤d(wn,w),∀j∈{1,2,…,M}.d({y}_{n}^{j},w)\le d\left({w}_{n},w),\hspace{0.33em}\forall j\in \left\{1,2,\ldots ,M\right\}.Also, as in (3.4), we have (3.27)d(wn+1,w)≤∑j=0Mβnjd(ynj,w).d({w}_{n+1},w)\le \mathop{\sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w).It follows from (3.27), (3.26), and (3.25) that (3.28)d(wn+1,w)≤βn0d(yn0,w)+∑j=1Mβnjd(ynj,w)≤βn0[(1−(1−η)φn)d(wn,w)+φnd(f(w),w)]+∑j=1Mβnjd(wn,w)=(1−(1−η)βn0φn)d(wn,w)+βn0φnd(f(w),w).\begin{array}{rcl}d({w}_{n+1},w)& \le & {\beta }_{n}^{0}d({y}_{n}^{0},w)+\mathop{\displaystyle \sum }\limits_{j=1}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w)\\ & \le & {\beta }_{n}^{0}{[}\left(1-\left(1-\eta ){\varphi }_{n})d\left({w}_{n},w)+{\varphi }_{n}d(f\left(w),w)]+\mathop{\displaystyle \sum }\limits_{j=1}^{M}{\beta }_{n}^{j}d({w}_{n},w)\\ & =& (1-\left(1-\eta ){\beta }_{n}^{0}{\varphi }_{n})d({w}_{n},w)+{\beta }_{n}^{0}{\varphi }_{n}d(f\left(w),w).\end{array}This implies that (3.29)d(wn+1,w)≤(1−(1−η)βn0φn)d(wn,w)+βn0φnd(f(w),w)≤maxd(wn,w),1(1−η)d(f(w),w)⋮≤maxd(w1,w),1(1−η)d(f(w),w).\begin{array}{rcl}d({w}_{n+1},w)& \le & (1-\left(1-\eta ){\beta }_{n}^{0}{\varphi }_{n})d({w}_{n},w)+{\beta }_{n}^{0}{\varphi }_{n}d(f\left(w),w)\\ & \le & \max \left\{\phantom{\rule[-1.25em]{}{0ex}},d\left({w}_{n},w),\frac{1}{\left(1-\eta )}d(f\left(w),w)\right\}\\ & \vdots & \\ & \le & \max \left\{\phantom{\rule[-1.25em]{}{0ex}},d\left({w}_{1},w),\frac{1}{\left(1-\eta )}d(f\left(w),w)\right\}.\end{array}Therefore, {wn}\left\{{w}_{n}\right\}is bounded.□Lemma 3.7Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 2 and w∈Gw\in G. If limn→∞φn=0{\mathrm{lim}}_{n\to \infty }{\varphi }_{n}=0, then there exists a subsequence {mk}⊂N\left\{{m}_{k}\right\}\subset {\mathbb{N}}such that the sequence {d2(wmk+1,w)−d2(wmk,w)}\{{d}^{2}({w}_{{m}_{k}+1},w)-{d}^{2}({w}_{{m}_{k}},w)\}converges to 0.ProofLet w∈Γw\in \Gamma . Then the following similar arguments as in (3.14), we have d2(wn+1,w)≤∑k=0Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤∑k=0Mβnkd2(ynk,w).{d}^{2}({w}_{n+1},w)\le \mathop{\sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\le \mathop{\sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w).This, (3.26), and Lemma 2.13 imply (3.30)d2(wn+1,w)≤βn0d2(yn0,w)+∑j=1Mβnjd2(ynj,w)≤βn0d2(yn0,w)+(1−βn0)d2(wn,w)=βn0d2(Jλnh((1−φn)wn⊕φnf(wn)),Jλnh(w))+(1−βn0)d2(wn,w)\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},w)+\mathop{\displaystyle \sum }\limits_{j=1}^{M}{\beta }_{n}^{j}{d}^{2}({y}_{n}^{j},w)\\ & \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},w)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\\ & =& {\beta }_{n}^{0}{d}^{2}({J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),{J}_{{\lambda }_{n}}^{h}\left(w))+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\end{array}≤βn0d2((1−φn)wn⊕φnf(wn),w)+(1−βn0)d2(wn,w)≤(1−φn)βn0d2(wn,w)+φnβn0d2(f(wn),w)+(1−βn0)d2(wn,w)≤βn0d2(wn,w)+φnd2(f(wn),w)+(1−βn0)d2(wn,w)\begin{array}{rcl}& \le & {\beta }_{n}^{0}{d}^{2}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n}),w)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\\ & \le & \left(1-{\varphi }_{n}){\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+{\varphi }_{n}{\beta }_{n}^{0}{d}^{2}(f\left({w}_{n}),w)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\\ & \le & {\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\end{array}(3.31)≤d2(wn,w)+φnd2(f(wn),w).\le \hspace{0.25em}{d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w).\hspace{13.2em}Now, by Lemma 3.6, if d2(wn+1,w)≤d2(wn,w){d}^{2}\left({w}_{n+1},w)\le {d}^{2}\left({w}_{n},w)for n∈Nn\in {\mathbb{N}}, then {d2(wn,w)}\left\{{d}^{2}\left({w}_{n},w)\right\}converges and consequently the proof is complete. Otherwise, there exists a subsequence {nk}\left\{{n}_{k}\right\}of {n}\left\{n\right\}such that d2(wnk,w)<d2(wnk+1,w){d}^{2}\left({w}_{{n}_{k}},w)\lt {d}^{2}\left({w}_{{n}_{k}+1},w)for every k∈Nk\in {\mathbb{N}}. Thus, by Lemma 2.20, there exists a subsequence {mk}⊂N\left\{{m}_{k}\right\}\subset {\mathbb{N}}such that mk→∞{m}_{k}\to \infty , d2(wmk,w)≤d2(wmk+1,w)andd2(wk,w)≤d2(wmk+1,w).{d}^{2}({w}_{{m}_{k}},w)\le {d}^{2}({w}_{{m}_{k}+1},w)\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{d}^{2}({w}_{k},w)\le {d}^{2}({w}_{{m}_{k}+1},w).Consequently, we obtain from (3.31) that □0≤limsupmk→∞(d2(wmk+1,w)−d2(wmk,w))≤limsupn→∞(d2(wn+1,w)−d2(wn,w))≤limsupn→∞(d2(wn,w)+φnd2(f(w),w)−d2(wn,w))≤limsupn→∞(φnd2(f(w),w))=0.\begin{array}{rcl}0& \le & \mathop{\mathrm{limsup}}\limits_{{m}_{k}\to \infty }({d}^{2}\left({w}_{{m}_{k}+1},w)-{d}^{2}\left({w}_{{m}_{k}},w))\\ & \le & \mathop{\mathrm{limsup}}\limits_{n\to \infty }({d}^{2}\left({w}_{n+1},w)-{d}^{2}\left({w}_{n},w))\\ & \le & \mathop{\mathrm{limsup}}\limits_{n\to \infty }({d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left(w),w)-{d}^{2}\left({w}_{n},w))\\ & \le & \mathop{\mathrm{limsup}}\limits_{n\to \infty }({\varphi }_{n}{d}^{2}(f\left(w),w))\\ & =& 0.\end{array}Lemma 3.8Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 2 and let {βnk}⊂[b,1)\left\{{\beta }_{n}^{k}\right\}\subset \left[b,1)for some b∈(0,1)b\in \left(0,1), limn→∞φn=0{\mathrm{lim}}_{n\to \infty }{\varphi }_{n}=0, and limn→∞αnj∈(κj,1){\mathrm{lim}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({\kappa }_{j},1). Suppose that {mk}\left\{{m}_{k}\right\}is the subsequence in Lemma 3.7. Then for each j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, the sequences {d(wn,Jμh(wn))}\{d({w}_{n},{J}_{\mu }^{h}\left({w}_{n}))\}, {d(wmk,Jμj(wmk))}\{d({w}_{{m}_{k}},{J}_{\mu }^{j}\left({w}_{{m}_{k}}))\}, and {dist(wmk,Tjwmk)}\{\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{{m}_{k}},{T}_{j}{w}_{{m}_{k}})\}converge to 0.ProofLet w∈Γw\in \Gamma . It follows from Lemma 2.17 and the fact that h(w)≤h(wn)h\left(w)\le h\left({w}_{n})that (3.32)d2(yn0,wn)≤d2(yn0,w)−d2(wn,w)+2μ(h(w)−h(wn))≤d2(yn0,w)−d2(wn,w)=d2(Jλnh((1−φn)wn⊕φnf(wn)),Jλnh(w))−d2(wn,w)≤d2((1−φn)wn⊕φnf(wn),w)−d2(wn,w)≤(1−φn)d2(wn,w)+φnd2(f(wn),w)−d2(wn,w)≤d2(wn,w)+φnd2(f(wn),w)−d2(wn,w)\begin{array}{rcl}{d}^{2}({y}_{n}^{0},{w}_{n})& \le & {d}^{2}({y}_{n}^{0},w)-{d}^{2}\left({w}_{n},w)+2\mu (h\left(w)-h\left({w}_{n}))\\ & \le & {d}^{2}({y}_{n}^{0},w)-{d}^{2}\left({w}_{n},w)\\ & =& {d}^{2}({J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),{J}_{{\lambda }_{n}}^{h}\left(w))-{d}^{2}\left({w}_{n},w)\\ & \le & {d}^{2}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n}),w)-{d}^{2}\left({w}_{n},w)\\ & \le & \left(1-{\varphi }_{n}){d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{d}^{2}\left({w}_{n},w)\\ & \le & {d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{d}^{2}\left({w}_{n},w)\end{array}(3.33)≤φnd2(f(wn),w).\le \hspace{0.33em}{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w).\hspace{9.7em}This and the assumptions on {φn}\left\{{\varphi }_{n}\right\}imply that (3.34)limn→∞d(wn,yn0)=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{y}_{n}^{0})=0.Consequently, it follows from Lemmas 2.16, 2.19, and 2.12 that (3.35)d(wn,Jμhwn)≤d(wn,yn0)+d(yn0,Jμhwn)=d(wn,yn0)+dJμh(wn),Jμhλn−μλnyn0⊕μμnwn≤d(wn,yn0)+dwn,λn−μλnyn0⊕μλnwn≤d(wn,yn0)+1−μλnd(wn,yn0)≤2−μλnd(wn,yn0)→0,asn→∞.\begin{array}{rcl}d\left({w}_{n},{J}_{\mu }^{h}{w}_{n})& \le & d({w}_{n},{y}_{n}^{0})+d({y}_{n}^{0},{J}_{\mu }^{h}{w}_{n})\\ & =& d({w}_{n},{y}_{n}^{0})+d\hspace{0.08em}\left({J}_{\mu }^{h}\left({w}_{n}),{J}_{\mu }^{h}\left(\frac{{\lambda }_{n}-\mu }{{\lambda }_{n}}{y}_{n}^{0}\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{w}_{n}\right)\right)\\ & \le & d({w}_{n},{y}_{n}^{0})+d\hspace{0.08em}\left({w}_{n},\left(\frac{{\lambda }_{n}-\mu }{{\lambda }_{n}}{y}_{n}^{0}\displaystyle \oplus \frac{\mu }{{\lambda }_{n}}{w}_{n}\right)\right)\\ & \le & d({w}_{n},{y}_{n}^{0})+\left(1-\frac{\mu }{{\lambda }_{n}}\right)\hspace{0.08em}d\left({w}_{n},{y}_{n}^{0})\\ & \le & \left(2-\frac{\mu }{{\lambda }_{n}}\right)\hspace{0.08em}d\left({w}_{n},{y}_{n}^{0})\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .\end{array}Moreover, as in similar arguments to (3.14), we obtain (3.36)d2(wn+1,w)≤∑k=0Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤βn0d2(yn0,w)+∑k=1Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}This and (3.32) imply that (3.37)d2(wn+1,w)≤βn0d2(yn0,w)+∑k=1Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤βn0d2(wn,w)+βn0φnd2(f(wn),w)−βn0d2(wn,w)+∑k=1Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤βn0d2(wn,w)+φnd2(f(wn),w)+∑k=1Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & {\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+{\beta }_{n}^{0}{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & {\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}It follows from (3.26) and (3.37) that d2(wn+1,w)≤∑k=0Mβnkd2(wn,w)+φnd2(f(wn),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤d2(wn,w)+φnd2(f(wn),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & {d}^{2}({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}This implies that for every j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, βn0βnjd2(ϕj−1(n),Jμnj(ynj))≤βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynj))≤d2(wn,w)−d2(wn+1,w)+φnd2(f(wn),w).\begin{array}{rcl}{\beta }_{n}^{0}{\beta }_{n}^{j}{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))& \le & {\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{j}))\\ & \le & {d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w).\end{array}Therefore, (3.38)d2(ϕj−1(n),Jμnj(ynj))≤1βn0βnj[d2(wn,w)−d2(wn+1,w)+φnd2(f(wn),w)]≤1b2[d2(wn,w)−d2(wn+1,w)+φnd2(f(wn),w)].\begin{array}{rcl}{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))& \le & \frac{1}{{\beta }_{n}^{0}{\beta }_{n}^{j}}{[}{d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)]\\ & \le & \frac{1}{{b}^{2}}{[}{d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)].\end{array}It follows from Lemma 3.6, (3.38), and the assumption that limn→∞φn=0{\mathrm{lim}}_{n\to \infty }{\varphi }_{n}=0that limmk→∞d2(ϕj−1(mk),Jμmkj(ymkj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }{d}^{2}({\phi }_{j-1}^{\left({m}_{k})},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.Consequently, (3.39)limmk→∞d(ϕj−1(mk),Jμmkj(ymkj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }d({\phi }_{j-1}^{\left({m}_{k})},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.As in (3.17), we obtain that for every j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}, (3.40)d(ϕj−1(mk),ϕj(mk))≤d(ϕj−1(mk),Jμmkj(ymkj))→0,asmk→∞.d({\phi }_{j-1}^{\left({m}_{k})},{\phi }_{j}^{\left({m}_{k})})\le d({\phi }_{j-1}^{\left({m}_{k})},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}{m}_{k}\to \infty .Furthermore, for any j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}, we have (3.41)d(wn,Jμnj(ynj))=d(ϕ0(n),Jμnj(ynj))≤d(ϕ0(n),ϕ1(n))+d(ϕ1(n),ϕ2(n))+⋯+d(ϕj−2(n),ϕj−1(n))+d(ϕj−1(n),Jμnj(ynj))=∑k=1j−1d(ϕk−1(n),ϕk(n))+d(ϕj−1(n),Jμnj(ynj))\begin{array}{rcl}d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))& =& d({\phi }_{0}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\\ & \le & d({\phi }_{0}^{\left(n)},{\phi }_{1}^{\left(n)})+d({\phi }_{1}^{\left(n)},{\phi }_{2}^{\left(n)})+\cdots +d({\phi }_{j-2}^{\left(n)},{\phi }_{j-1}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\\ & =& \mathop{\displaystyle \sum }\limits_{k=1}^{j-1}d({\phi }_{k-1}^{\left(n)},{\phi }_{k}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\end{array}(3.42)≤∑k=1Md(ϕk−1(n),ϕk(n))+d(ϕj−1(n),Jμnj(ynj)).\le \hspace{0.33em}\mathop{\sum }\limits_{k=1}^{M}d({\phi }_{k-1}^{\left(n)},{\phi }_{k}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})).\hspace{7em}It follows from (3.42), (3.39), and (3.40) that (3.43)limmk→∞d(wmk,Jμmkj(ymkj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }d({w}_{{m}_{k}},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.Also, as in (3.2), we have for each j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}and for w∈Γw\in \Gamma that (3.44)(1−αmkj)(αmkj−κj)d2(wmk,wmkj)≤d2(wmk,w)−d2(wmk+1,w).\left(1-{\alpha }_{{m}_{k}}^{j})\left({\alpha }_{{m}_{k}}^{j}-{\kappa }_{j}){d}^{2}\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})\le {d}^{2}\left({w}_{{m}_{k}},w)-{d}^{2}\left({w}_{{m}_{k}+1},w).This, Lemma 3.7, and the assumption that liminfn→∞αnj∈(κj,1){\mathrm{liminf}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({\kappa }_{j},1)imply limmk→∞d2(wmk,wmkj)=0.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }{d}^{2}\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})=0.Consequently, (3.45)limmk→∞d(wmk,wmkj)=0,for everyj∈{1,2,…,M}.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }d\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})=0,\hspace{1em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}j\in \left\{1,2,\ldots ,M\right\}.Hence, for each j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, (3.46)dist(wmk,Tjwmk)≤d(wmk,wmkj)→0,asmk→∞.\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{{m}_{k}},{T}_{j}{w}_{{m}_{k}})\le d\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}{m}_{k}\to \infty .Moreover, it follows from Lemmas 2.5 and 2.12 that (3.47)d(wn,Jμj(wn))≤d(wn,Jμnj(ynj))+d(Jμnj(ynj),Jμj(wn))=d(wn,Jμnj(ynj))+dJμj(wn),Jμjμn−μμnJμnj(ynj)⊕μμnynj≤d(wn,Jμnj(ynj))+dwn,μn−μμnJμnj(ynj)⊕μμnynj≤2−μμnd(wn,Jμnj(ynj))+μμnd(wn,ynj)≤2d(wn,Jμnj(ynj))+d(wn,ynj)≤2d(wn,Jμnj(ynj))+d(wn,wnj).\begin{array}{rcl}d({w}_{n},{J}_{\mu }^{j}\left({w}_{n}))& \le & d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d({J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),{J}_{\mu }^{j}\left({w}_{n}))\\ & =& d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\hspace{0.08em}\left({J}_{\mu }^{j}\left({w}_{n}),{J}_{\mu }^{j}\left(\frac{{\mu }_{n}-\mu }{{\mu }_{n}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{y}_{n}^{j}\right)\right)\\ & \le & d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\hspace{0.08em}\left({w}_{n},\left(\frac{{\mu }_{n}-\mu }{{\mu }_{n}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{y}_{n}^{j}\right)\right)\\ & \le & \left(2-\frac{\mu }{{\mu }_{n}}\right)d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+\frac{\mu }{{\mu }_{n}}d\left({w}_{n},{y}_{n}^{j})\\ & \le & 2d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\left({w}_{n},{y}_{n}^{j})\\ & \le & 2d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\left({w}_{n},{w}_{n}^{j}).\end{array}This, (3.43) and (3.45) imply that (3.48)d(wmk,Jμj(wmk))≤2d(wmk,Jμmkj(ymkj))+d(wmk,wmkj)→0,d({w}_{{m}_{k}},{J}_{\mu }^{j}\left({w}_{{m}_{k}}))\le 2d({w}_{{m}_{k}},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))+d\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})\to 0,for every j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}. Therefore, by (3.35), (3.48), and (3.46), we have the complete proof.□Theorem 3.9Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 2 and let {βnk}\left\{{\beta }_{n}^{k}\right\}, {φn}\left\{{\varphi }_{n}\right\}, and {αn}\left\{{\alpha }_{n}\right\}be as in Lemma (3.8). Suppose that ∑n=1∞βn0φn=+∞{\sum }_{n=1}^{\infty }{\beta }_{n}^{0}{\varphi }_{n}=+\infty . Then the sequence {wn}\left\{{w}_{n}\right\}converges strongly to an element of Γ\Gamma .ProofFrom Lemma 3.6, we have that {wn}\left\{{w}_{n}\right\}is bounded and consequently {wmk}\left\{{w}_{{m}_{k}}\right\}is bounded. Then by Lemmas 2.9 and 2.8, there exists a subsequence {zmk}\left\{{z}_{{m}_{k}}\right\}of {wmk}\left\{{w}_{{m}_{k}}\right\}that Δ\Delta -converges to zzfor some z∈Yz\in Y. By Lemma (3.7) and the assumption that each Tp{T}_{p}satisfies demiclosedness type property, we have (3.49)z∈Γ.z\in \Gamma .Without loss of generality, we may assume that {xmk}\left\{{x}_{{m}_{k}}\right\}is a subsequence of {wmk}\left\{{w}_{{m}_{k}}\right\}that Δ\Delta -converges to zzin YYand (3.50)limsupmk→∞⟨wmkz→,f(z)z→⟩=limmk→∞⟨xmkz→,f(z)z→⟩.\mathop{\mathrm{limsup}}\limits_{{m}_{k}\to \infty }\langle \overrightarrow{{w}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle =\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }\langle \overrightarrow{{x}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle .This and Lemma 2.15 imply that (3.51)limsupmk→∞⟨wmkz→,f(z)z→⟩=limmk→∞⟨xmkz→,f(z)z→⟩≤0.\mathop{\mathrm{limsup}}\limits_{{m}_{k}\to \infty }\langle \overrightarrow{{w}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle =\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }\langle \overrightarrow{{x}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle \le 0.It follows from (3.30), Lemma 2.12, and (3.49) that (3.52)d2(wn+1,z)≤βn0d2(yn0,z)+(1−βn0)d2(wn,z)=βn0d2(Jλnh((1−φn)wn⊕φnf(wn)),Jλnh(z))+(1−βn0)d2(wn,z)≤βn0d2((1−φn)wn⊕φnf(wn),z)+(1−βn0)d2(wn,z).\begin{array}{rcl}{d}^{2}({w}_{n+1},z)& \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},z)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},z)\\ & =& {\beta }_{n}^{0}{d}^{2}({J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),{J}_{{\lambda }_{n}}^{h}\left(z))+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},z)\\ & \le & {\beta }_{n}^{0}{d}^{2}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n}),z)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},z).\end{array}This and Lemma 2.14 imply that d2(wn+1,z)≤βn0(1−φn)2d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φn(1−φn)⟨wnz→,f(wn)z→⟩+(1−βn0)d2(wn,z)=(1−2βn0φn+βn0φn2)d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φn(1−φn)[⟨wnz→,f(wn)f(z)→⟩+⟨wnz→,f(z)z→⟩].\begin{array}{rcl}{d}^{2}({w}_{n+1},z)& \le & {\beta }_{n}^{0}{\left(1-{\varphi }_{n})}^{2}{d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n})\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left({w}_{n})z}\rangle +\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},z)\\ & =& (1-2{\beta }_{n}^{0}{\varphi }_{n}+{\beta }_{n}^{0}{\varphi }_{n}^{2}){d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n}){[}\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left({w}_{n})f\left(z)}\rangle +\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle ].\end{array}This, Cauchy-Schwartz inequality, and the assumption that ffis contraction mapping imply d2(wn+1,z)≤(1−2βn0φn+βn0φn2)d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φn(1−φn)[d(wn,z)⋅d(f(wn),f(z))+⟨wnz→,f(z)z→⟩]≤(1−2βn0φn+βn0φn2)d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φn(1−φn)ηd2(wn,z)+2βn0φn(1−φn)⟨wnz→,f(z)z→⟩≤(1−2βn0φn+βn0φn2)d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φnηd2(wn,z)+2βn0φn(1−φn)⟨wnz→,f(z)z→⟩=(1−2βn0φn(1−η))d2(wn,z)+2βn0φn(1−η)φn2(1−η)d2(wn,z)+φn2(1−η)d2(f(wn),z)+(1−φn)(1−η)⟨wnz→,f(z)z→⟩.\begin{array}{rcl}{d}^{2}({w}_{n+1},z)& \le & (1-2{\beta }_{n}^{0}{\varphi }_{n}+{\beta }_{n}^{0}{\varphi }_{n}^{2}){d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n}){[}d\left({w}_{n},z)\cdot d(f\left({w}_{n}),f\left(z))+\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle ]\\ & \le & (1-2{\beta }_{n}^{0}{\varphi }_{n}+{\beta }_{n}^{0}{\varphi }_{n}^{2}){d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n})\eta {d}^{2}\left({w}_{n},z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n})\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle \\ & \le & (1-2{\beta }_{n}^{0}{\varphi }_{n}+{\beta }_{n}^{0}{\varphi }_{n}^{2}){d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\eta {d}^{2}\left({w}_{n},z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n})\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle \\ & =& (1-2{\beta }_{n}^{0}{\varphi }_{n}\left(1-\eta )){d}^{2}({w}_{n},z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-\eta )\left[\frac{{\varphi }_{n}}{2\left(1-\eta )}{d}^{2}({w}_{n},z)+\frac{{\varphi }_{n}}{2\left(1-\eta )}{d}^{2}(f\left({w}_{n}),z)+\frac{\left(1-{\varphi }_{n})}{\left(1-\eta )}\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle \right].\end{array}This implies that (3.53)d2(wmk+1,z)≤(1−σmk)d2(wmk,w)+σmkϕmk,{d}^{2}\left({w}_{{m}_{k}+1},z)\le \left(1-{\sigma }_{{m}_{k}}){d}^{2}\left({w}_{{m}_{k}},w)+{\sigma }_{{m}_{k}}{\phi }_{{m}_{k}},where σmk=2βmk0φmk(1−η){\sigma }_{{m}_{k}}=2{\beta }_{{m}_{k}}^{0}{\varphi }_{{m}_{k}}\left(1-\eta )and ϕmk=φmk2(1−η)d2(wmk,z)+φmk2(1−η)d2(f(wmk),z)+(1−φmk)(1−η)⟨wmkz→,f(z)z→⟩.{\phi }_{{m}_{k}}=\frac{{\varphi }_{{m}_{k}}}{2\left(1-\eta )}{d}^{2}({w}_{{m}_{k}},z)+\frac{{\varphi }_{{m}_{k}}}{2\left(1-\eta )}{d}^{2}(f\left({w}_{{m}_{k}}),z)+\frac{\left(1-{\varphi }_{{m}_{k}})}{\left(1-\eta )}\langle \overrightarrow{{w}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle .Therefore, from (3.53), (3.51), and the assumptions on {φn}\left\{{\varphi }_{n}\right\}, we conclude by Lemma 2.21 that {wmk}\left\{{w}_{{m}_{k}}\right\}converges strongly to zz. Moreover, since (3.54)d2(wk,z)≤d2(wmk+1,z),{d}^{2}({w}_{k},z)\le {d}^{2}({w}_{{m}_{k}+1},z),we have that {wn}\left\{{w}_{n}\right\}converges strongly to zz.□The following corollary is obtained from the fact that every quasi-nonexpansive mapping is 0-demicontractive.Corollary 3.10Let Tp,p=1,2,…,M2{T}_{p},\hspace{0.33em}p=1,2,\ldots ,{M}_{2}be quasi-nonexpansive mappings with demiclosedness-type property. Suppose that {αnj},{βnj},{wn}\left\{{\alpha }_{n}^{j}\right\},\left\{{\beta }_{n}^{j}\right\},\left\{{w}_{n}\right\}, and Γ\Gamma are as in Theorem 3.9. Then {wn}\left\{{w}_{n}\right\}converges strongly to a member of Γ\Gamma .Corollary 3.11Let Tp,p=1,2,…,M2{T}_{p},\hspace{0.33em}p=1,2,\ldots ,{M}_{2}be nonexpansive mappings and suppose {αnj},{βnj},{wn}\left\{{\alpha }_{n}^{j}\right\},\hspace{0.33em}\left\{{\beta }_{n}^{j}\right\},\hspace{0.33em}\left\{{w}_{n}\right\}, and Γ\Gamma are as in Theorem 3.9. Then {wn}\left\{{w}_{n}\right\}converges strongly to a member of Γ\Gamma .4Applications and numerical exampleIn this section, we apply our results to find mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We also give a numerical example to support the proposed methods. We shall maintain the notation (W,d)\left(W,d)for a Hadamard space and YYits nonempty convex closed subset.Let M=1M=1, TTbe a 0-demicontractive map, {μn}⊂(μ,+∞),{λn}⊂(μ,+∞)\left\{{\mu }_{n}\right\}\subset \left(\mu ,+\infty ),\hspace{0.33em}\left\{{\lambda }_{n}\right\}\subset \left(\mu ,+\infty )for some μ>0\mu \gt 0, {βn0}⊂(0,1)\left\{{\beta }_{n}^{0}\right\}\subset \left(0,1), {φn}\left\{{\varphi }_{n}\right\}, {αn0}⊂[0,1]\left\{{\alpha }_{n}^{0}\right\}\subset \left[0,1], and ffas η\eta -contraction map on YY. Then Algorithms 1 and 2 reduce to the following: (4.1)yn0=Jλnh(wn),yn1=αn0wn⊕(1−αn0)wn∗,wn∗∈Twn,wn+1=βn0yn0⊕(1−βn0)Jμn(yn1),\left\{\begin{array}{l}{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}({w}_{n}),\hspace{1.0em}\\ {y}_{n}^{1}={\alpha }_{n}^{0}{w}_{n}\displaystyle \oplus \left(1-{\alpha }_{n}^{0}){w}_{n}^{\ast },\hspace{0.33em}{w}_{n}^{\ast }\in T{w}_{n},\hspace{1.0em}\\ {w}_{n+1}={\beta }_{n}^{0}{y}_{n}^{0}\displaystyle \oplus \left(1-{\beta }_{n}^{0}){J}_{{\mu }_{n}}({y}_{n}^{1}),\hspace{1.0em}\end{array}\right.and (4.2)yn0=Jλnh((1−φn)wn⊕φnf(wn)),yn1=αn0wn⊕(1−αn0)wn∗,wn∗∈Twn,wn+1=βn0yn0⊕(1−βn0)Jμn(yn1),\left\{\begin{array}{l}{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),\hspace{1.0em}\\ {y}_{n}^{1}={\alpha }_{n}^{0}{w}_{n}\displaystyle \oplus \left(1-{\alpha }_{n}^{0}){w}_{n}^{\ast },\hspace{0.33em}{w}_{n}^{\ast }\in T{w}_{n},\hspace{1.0em}\\ {w}_{n+1}={\beta }_{n}^{0}{y}_{n}^{0}\displaystyle \oplus \left(1-{\beta }_{n}^{0}){J}_{{\mu }_{n}}({y}_{n}^{1}),\hspace{1.0em}\end{array}\right.respectively.4.1Mean and median of probabilitiesGiven a probability measure μ∈P2(W)\mu \in {{\mathcal{P}}}^{2}\left(W). Let ffand ggbe two real-valued functions on YYdefined by f(w)=∫d(w,z)dμ(z),g(w)=∫d2(w,z)dμ(z),for everyw∈Y.f\left(w)=\int d\left(w,z)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(z),\hspace{0.33em}g\left(w)=\int {d}^{2}\left(w,z)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(z),\hspace{0.1em}\text{for every}\hspace{0.1em}w\in Y.Then the minimizers of ffand ggare called median and mean of probability, respectively (see e.g. [4]). Moreover, by properties of metric dd, ff, and ggare convex proper and lower semi-continuous functions on YY.Take A:Y→TYA:Y\to TYsuch that Aw={0}Aw=\left\{{\bf{0}}\right\}and TTbe such that Tz={z}Tz=\left\{z\right\}. Then AAis MVF that satisfies condition (S)\left(S)and TTis 0-demicontractive mapping with demiclosedness-type property and Γ≠∅\Gamma \ne \varnothing . For h=fh=f, the sequences in (4.1) and (4.2) approximate a median, and for h=gh=g, the sequences in (4.1) and (4.2) approximate the mean.4.2Optimal energy for measurable mappingsLet (Y,χ,μ)\left(Y,\chi ,\mu )be a measure space with measure μ\mu and let h:Y→Wh:Y\to Wbe a measurable mapping. Consider the following set of measurable mappings: (4.3)L2(Y,W,h)≔{g:Y→Wmeasurable:d(g(⋅),h(⋅))∈L2(Y)}{L}^{2}\left(Y,W,h):= \{g:Y\to W\hspace{0.33em}\hspace{0.1em}\text{measurable:}\hspace{0.1em}\hspace{0.33em}d(g\left(\cdot ),h\left(\cdot ))\in {L}^{2}\left(Y)\}equipped with the L2{L}^{2}-metric (4.4)d2(g,f)≔∫Yd(g(w),f(w))2dμ(w),g,f∈L2(Y,W,h).{d}_{2}(g,f):= \mathop{\int }\limits_{Y}d(g\left(w),f\left(w){)}^{2}\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(w),\hspace{0.33em}g,f\in {L}^{2}\left(Y,W,h).Then, as in [44], L2(Y,W,h){L}^{2}\left(Y,W,h)together with metric d2{d}_{2}forms a Hadamard space. Furthermore, the energy of a measurable mapping g:Y→Wg:Y\to Wis given by (4.5)E(g)≔12∫Y∫Yd(g(u),g(w))2ρ(u,dw)dμ(u),E\left(g):= \frac{1}{2}\mathop{\int }\limits_{Y}\mathop{\int }\limits_{Y}d(g\left(u),g\left(w){)}^{2}\rho \left(u,\hspace{0.1em}\text{d}\hspace{0.1em}w)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(u),where ρ(u,dw)\rho \left(u,\hspace{0.1em}\text{d}\hspace{0.1em}w)is a Markov kernel that is symmetric with respect to μ\mu in the sense that ρ(u,dw)dμ(u)=ρ(w,du)dμ(w)\rho \left(u,\hspace{0.1em}\text{d}\hspace{0.1em}w)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(u)=\rho \left(w,\hspace{0.1em}\text{d}\hspace{0.1em}u)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(w).It follows from [44] that the set of optimal energy coincides with the fixed-point set of Markov operator PP. Moreover, the Markov operator is a singlevalued nonexpansive mapping on L2(Y,W,h){L}^{2}\left(Y,W,h). Now, let A:Y→TYA:Y\to TYbe defined by Aw={0}Aw=\left\{{\bf{0}}\right\}, TTbe defined by Tz={Pz}Tz=\left\{Pz\right\}, and hhbe a zero functional. Then, it is easy to see that the assumptions on AA, TT, and hhare all satisfied. Thus, the sequences in (4.1) and (4.2), with AA, TT, hhimmediate, approximate an optimal energy.4.3Two-arm robotic motion controlLet k∈{1,2,…,m}k\in \left\{1,2,\ldots ,m\right\}, hk{h}_{k}be defined by hk(w)=d2(w,ξk){h}_{k}\left(w)={d}^{2}\left(w,{\xi }_{k}), for every w∈Yw\in Yand some ξk∈Y{\xi }_{k}\in Y. Then by properties of metric dd, each hk{h}_{k}is convex proper and lower semi-continuous functions. The problem of finding the minimizers of hk{h}_{k}at each kksolves large optimization problems. We now consider a special case of Y=R2Y={{\mathbb{R}}}^{2}equipped with the Euclidean distance ddand analyse a discrete-time kinematics problem of two-arm robotic manipulator. That is the problem of solving (4.6)minhk(δk),\min {h}_{k}\left({\delta }_{k}),at each time tk{t}_{k}, where δk=g(θk){\delta }_{k}=g\left({\theta }_{k})is the end effector and ggis the kinematic mapping as given in [45]. For three-arm robotic motion see [49]. In this experiment, we shall use Algorithm 2 (which reduces to (4.2)) to track the following curve: ξk=1.5+0.2sin(tk)0.5+0.2sin3tk+π6.{\xi }_{k}=\left[\begin{array}{c}1.5+0.2\sin \left({t}_{k})\\ 0.5+0.2\sin \left(3{t}_{k}+\frac{\pi }{6}\right)\end{array}\right].For full numerical display, we set Tw={w}Tw=\left\{w\right\}, f(w)=w5f\left(w)=\frac{w}{5}, Aw={0}Aw=\left\{{\bf{0}}\right\}for every w∈Yw\in Yand take Jλnhk{J}_{{\lambda }_{n}}^{{h}_{k}}be the resolvent of hk{h}_{k}at each k=1,2,…,mk=1,2,\ldots ,m. We take equal arms’ length as 1, set the time range of 10 s into 200 parts, that is m=200m=200, and take the starting point w1=0,π4T{w}_{1}={\left(0,\frac{\pi }{4}\right)}^{T}. Other control parameters are as follows: λn=μn=20{\lambda }_{n}={\mu }_{n}=20, βn0=12{\beta }_{n}^{0}=\frac{1}{2}, φn=1n{\varphi }_{n}=\frac{1}{n}, and αn0=n2n+3{\alpha }_{n}^{0}=\frac{n}{2n+3}. Then, at each k∈{1,2,…,200}k\in \left\{1,2,\ldots ,200\right\}, (4.2) reduces to the following: (4.7)wn+1=12J20hk5n−45nwn+12wn,{w}_{n+1}=\frac{1}{2}{J}_{20}^{{h}_{k}}\left(\frac{5n-4}{5n}{w}_{n}\right)+\frac{1}{2}{w}_{n},which yield the result in Figure 1.Figure 1Numerical display for two-arm robotic motion control. (a) Synthesized trajectory. (b) End effector trajectory and desired path. (c) Tracking error on horizontal axis. (d) Tracking error on vertical axis.Remark 4.1Figure 1(a) and (b) shows that the process is completed successfully. Furthermore, Figure 1(c) and (d) signifies that the residual error is optimal.4.4Numerical exampleIn this subsection, we present a numerical example in a non-Hilbert Hadamard space to show the applicability of our results. All codes are written and executed in Matlab R2021b and run on Acer laptop (Swift SF514-55TA, 11th Gen Intel(R) Core(TM) i5-1135G7).Example 4.2Let Y=W=R2Y=W={{\mathbb{R}}}^{2}with the metric dY:R2×R2→R{d}_{Y}:{{\mathbb{R}}}^{2}\times {{\mathbb{R}}}^{2}\to {\mathbb{R}}defined by dY((w1,w2),(z1,z2))=(w1−z1)2+(w12−w2−z12+z2)2.{d}_{Y}(\left({w}_{1},{w}_{2}),\left({z}_{1},{z}_{2}))=\sqrt{{\left({w}_{1}-{z}_{1})}^{2}+{\left({w}_{1}^{2}-{w}_{2}-{z}_{1}^{2}+{z}_{2})}^{2}}.Then (W,dY)(W,{d}_{Y})is a non-Hilbert Hadamard space (see e.g. [16, Example 5.2]) and the geodesic operation, (1−t)(w1,w2)⊕t(v1,v2)\left(1-t)\left({w}_{1},{w}_{2})\oplus t\left({v}_{1},{v}_{2}), is given by ((1−t)w1+tv1,((1−t)w1+tv1)2−((1−t)(w12−w2)+t(v12−v2))).(\left(1-t){w}_{1}+t{v}_{1},(\left(1-t){w}_{1}+t{v}_{1}{)}^{2}-(\left(1-t)\left({w}_{1}^{2}-{w}_{2})+t\left({v}_{1}^{2}-{v}_{2}))).Let h,g:Y→Rh,g:Y\to {\mathbb{R}}and Tj:Y→Cℬ(Y),j=1,2,3,…,10,{T}_{j}:Y\to {\mathcal{C {\mathcal B} }}\left(Y),\hspace{0.33em}j=1,2,3,\ldots ,10,be defined by h(w1,w2)=200((w2+1)−(w1+1)2)2+w12,∀(w1,w2)∈Y,g(w1,w2)=50w12∀(w1,w2)∈Y,Tj(w1,w2)=0,j10w1×j10w2,0,ifw2<0<w1,0,j10∣w1∣×{0},otherwise.\begin{array}{rcl}h({w}_{1},{w}_{2})& =& 200(\left({w}_{2}+1)-{({w}_{1}+1)}^{2}{)}^{2}+{w}_{1}^{2},\hspace{0.33em}\forall \left({w}_{1},{w}_{2})\in Y,\\ g({w}_{1},{w}_{2})& =& 50{w}_{1}^{2}\hspace{0.33em}\forall \left({w}_{1},{w}_{2})\in Y,\\ {T}_{j}({w}_{1},{w}_{2})& =& \left\{\begin{array}{ll}\left[0,\sqrt{\frac{j}{10}}{w}_{1}\right]\times \left[\frac{j}{10}{w}_{2},0\right],\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{w}_{2}\lt 0\lt {w}_{1},\\ \left[0,\sqrt{\frac{j}{10}}| {w}_{1}| \right]\times \{0\},\hspace{1.0em}& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.\end{array}It follows from [41, Example 1] that each Tj{T}_{j}is multivalued 0-demicontractive mapping. Also, hhand ggare convex proper lower semicontinuous functions. Now, setting A1{A}_{1}to be the subdifferential of gg, then A1{A}_{1}and hhsatisfy the required conditions and Γ=(0,0)\Gamma =\left(0,0). Moreover, Algorithms 1 and 2, respectively, reduce to the following: (4.8)yn0=Jλnh(wn),ynj=αnjwn⊕(1−αnj)wnj,wnj∈Tjwn,j=1,2,3,…,10,wn+1=⊕j=010βnjJμnj(ynj),\left\{\begin{array}{l}{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}({w}_{n}),\hspace{1.0em}\\ {y}_{n}^{j}={\alpha }_{n}^{j}{w}_{n}\displaystyle \oplus \left(1-{\alpha }_{n}^{j}){w}_{n}^{j},\hspace{0.33em}{w}_{n}^{j}\in {T}_{j}{w}_{n},\hspace{0.33em}j=1,2,3,\ldots ,10,\hspace{1.0em}\\ {w}_{n+1}=\underset{j=0}{\overset{10}{\displaystyle \oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),\hspace{1.0em}\end{array}\right.and (4.9)yn0=Jλnh((1−φn)wn⊕φnf(wn)),ynj=αnjwn⊕(1−αnj)wnj,wnj∈Tjwn,j=1,2,3,…,10,wn+1=⊕j=010βnjJμnj(ynj),\left\{\begin{array}{l}{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),\hspace{1.0em}\\ {y}_{n}^{j}={\alpha }_{n}^{j}{w}_{n}\displaystyle \oplus \left(1-{\alpha }_{n}^{j}){w}_{n}^{j},\hspace{0.33em}{w}_{n}^{j}\in {T}_{j}{w}_{n},\hspace{0.33em}j=1,2,3,\ldots ,10,\hspace{1.0em}\\ {w}_{n+1}=\underset{j=0}{\overset{10}{\displaystyle \oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),\hspace{1.0em}\end{array}\right.where Jλnhv=argminw∈Yh(w)+12λndY2(w,v),Jμn1=argminw∈Yg(w)+12μndY2(w,v){J}_{{\lambda }_{n}}^{h}v={{\rm{argmin}}}_{w\in Y}\left\{h\left(w)+\frac{1}{2{\lambda }_{n}}{d}_{Y}^{2}\left(w,v)\right\},\hspace{0.33em}{J}_{{\mu }_{n}}^{1}={{\rm{argmin}}}_{w\in Y}\left\{g\left(w)+\frac{1}{2{\mu }_{n}}{d}_{Y}^{2}\left(w,v)\right\}, and Jμnj:x↦x{J}_{{\mu }_{n}}^{j}:x\mapsto xfor j=0,2,3,4,…,10.j=0,2,3,4,\ldots ,10.In the experiment, we take f:w↦w5f:w\mapsto \frac{w}{5}, λn=μn=50{\lambda }_{n}={\mu }_{n}=50, φn=13n{\varphi }_{n}=\frac{1}{3n}, αnj=1j+1{\alpha }_{n}^{j}=\frac{1}{j+1}, βnj=10242047(2j){\beta }_{n}^{j}=\frac{1024}{2047\left({2}^{j})}, j=0,1,2,…,10j=0,1,2,\ldots ,10, and we choose wnj=j10wn1,j10wn2{w}_{n}^{j}=\left(\sqrt{\frac{j}{10}}{w}_{{n}_{1}},\frac{j}{10}{w}_{{n}_{2}}\right)if wn2<0<wn1{w}_{{n}_{2}}\lt 0\lt {w}_{{n}_{1}}and wnj=j10∣wn1∣,0{w}_{n}^{j}=\left(\sqrt{\frac{j}{10}}| {w}_{{n}_{1}}| ,0\right)otherwise. We use Matlab function “fminsearch” for evaluation of argmin with initial search term as the input value and in reporting the results, we denote Algorithm 1 (resp. Algorithm 2) by Alg1 (resp. Alg2). We test the proposed algorithm using the starting points w1=(−23,7){w}_{1}=\left(-23,7), w1=(50,20){w}_{1}=\left(50,20), w1=(0.9,−16){w}_{1}=\left(0.9,-16)for both algorithms. The obtained results are shown in Tables 1 and 2.Table 1Few values {wn}\left\{{w}_{n}\right\}by Alg1 and Alg2 from Example 4.2wn{w}_{n}nnAlg1Alg2Alg1Alg2Alg1Alg21(−23,7-23,7)(−23,7-23,7)(50,2050,20)(50,2050,20)(0.9,−160.9,-16)(0.9,−160.9,-16)2(−8.9256,−67.7637)\left(-8.9256,-67.7637)(−7.6596,−86.2257)\left(-7.6596,-86.2257)(−4.7251,−655.1734)\left(-4.7251,-655.1734)(0.45729,−666.7701)\left(0.45729,-666.7701)(−0.0055069,−4.7165)\left(-0.0055069,-4.7165)(0.033795,−4.6367)\left(0.033795,-4.6367)3(−2.9384,−28.2297)\left(-2.9384,-28.2297)(−2.5211,−28.1613)\left(-2.5211,-28.1613)(−7.1975,−93.4627)\left(-7.1975,-93.4627)(−4.353,−164.6805)\left(-4.353,-164.6805)(−0.006031,−0.92149)\left(-0.006031,-0.92149)(−0.032863,−1.2912)\left(-0.032863,-1.2912)4(−0.85612,−7.7147)\left(-0.85612,-7.7147)(−0.73399,−7.1938)\left(-0.73399,-7.1938)(−2.6208,−27.602)\left(-2.6208,-27.602)(−2.3506,−34.5071)\left(-2.3506,-34.5071)(−0.0067264,−0.18944)\left(-0.0067264,-0.18944)(−0.016925,−0.27236)\left(-0.016925,-0.27236)5(−0.22759,−1.8001)\left(-0.22759,-1.8001)(−0.19503,−1.638)\left(-0.19503,-1.638)(−0.77876,−7.2087)\left(-0.77876,-7.2087)(−0.76101,−8.273)\left(-0.76101,-8.273)(−0.0029538,−0.040273)\left(-0.0029538,-0.040273)(−0.0053266,−0.057669)\left(-0.0053266,-0.057669)6(−0.056753,−0.39563)\left(-0.056753,-0.39563)(−0.048658,−0.35625)\left(-0.048658,-0.35625)(−0.20827,−1.6648)\left(-0.20827,-1.6648)(−0.21077,−1.8732)\left(-0.21077,-1.8732)(−0.002957,−0.012734)\left(-0.002957,-0.012734)(−0.0049952,−0.019333)\left(-0.0049952,-0.019333)⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots 85(−1.9078×10−9,(-1.9078\times 1{0}^{-9},−7.8638×10−14)-7.8638\times 1{0}^{-14})(−5.0291×10−10,(-5.0291\times 1{0}^{-10},−1.889×10−14)-1.889\times 1{0}^{-14})(−2.0876×10−9,(-2.0876\times 1{0}^{-9},−9.9453×10−14)-9.9453\times 1{0}^{-14})(−7.14×10−10,(-7.14\times 1{0}^{-10},−2.6563×10−14)-2.6563\times 1{0}^{-14})(−1.1067×1−9,(-1.1067\times {1}^{-9},−3.2876×10−14)-3.2876\times 1{0}^{-14})(−1.0945×1−9,(-1.0945\times {1}^{-9},−3.0742×10−14)-3.0742\times 1{0}^{-14})86(−1.5871×10−9,(-1.5871\times 1{0}^{-9},−5.6499×10−14)-5.6499\times 1{0}^{-14})(−4.1787×10−10,(-4.1787\times 1{0}^{-10},−1.3543×10−14)-1.3543\times 1{0}^{-14})(−1.7367×10−9,(-1.7367\times 1{0}^{-9},−7.1453×10−14)-7.1453\times 1{0}^{-14})(−5.9326×10−10,(-5.9326\times 1{0}^{-10},−1.9044×10−14)-1.9044\times 1{0}^{-14})(−9.2069×10−10,(-9.2069\times 1{0}^{-10},−2.362×10−14)-2.362\times 1{0}^{-14})(−9.0946×10−10,(-9.0946\times 1{0}^{-10},−2.2039×10−14)-2.2039\times 1{0}^{-14})87(−1.3203×10−9,(-1.3203\times 1{0}^{-9},−4.0592×10−14)-4.0592\times 1{0}^{-14})(−3.4721×10−10,(-3.4721\times 1{0}^{-10},−9.7092×10−15)-9.7092\times 1{0}^{-15})(−1.4447×10−9,(-1.4447\times 1{0}^{-9},−5.1336×10−14)-5.1336\times 1{0}^{-14})(−4.9295×10−10,(-4.9295\times 1{0}^{-10},−1.3653×10−14)-1.3653\times 1{0}^{-14})(−7.659×10−10,(-7.659\times 1{0}^{-10},−1.697×10−14)-1.697\times 1{0}^{-14})(−7.5568×10−10,(-7.5568\times 1{0}^{-10},−1.5801×10−14)-1.5801\times 1{0}^{-14})88(−1.0983×10−9,(-1.0983\times 1{0}^{-9},−2.9164×10−14)-2.9164\times 1{0}^{-14})(−2.8851×10−10,(-2.8851\times 1{0}^{-10},−6.961×10−15)-6.961\times 1{0}^{-15})(−1.2018×10−9,(-1.2018\times 1{0}^{-9},−3.6883×10−14)-3.6883\times 1{0}^{-14})(−4.096×10−10,(-4.096\times 1{0}^{-10},−9.7886×10−15)-9.7886\times 1{0}^{-15})(−6.3714×10−10,(-6.3714\times 1{0}^{-10},−1.2192×10−14)-1.2192\times 1{0}^{-14})(−6.2791×10−10,(-6.2791\times 1{0}^{-10},−1.1329×10−14)-1.1329\times 1{0}^{-14})89(−9.1366×10−10,(-9.1366\times 1{0}^{-10},−2.0953×10−14)-2.0953\times 1{0}^{-14})(−2.3973×10−10,(-2.3973\times 1{0}^{-10},−4.9908×10−15)-4.9908\times 1{0}^{-15})(−9.9977×10−10,(-9.9977\times 1{0}^{-10},−2.6499×10−14)-2.6499\times 1{0}^{-14})(−3.4035×10−10,(-3.4035\times 1{0}^{-10},−7.0181×10−15)-7.0181\times 1{0}^{-15})(−5.3002×10−10,(-5.3002\times 1{0}^{-10},−8.7598×10−15)-8.7598\times 1{0}^{-15})(−5.2175×10−10,(-5.2175\times 1{0}^{-10},−8.1222×10−15)-8.1222\times 1{0}^{-15})90(−7.6006×10−10,(-7.6006\times 1{0}^{-10},−1.5054×10−14)-1.5054\times 1{0}^{-14})(−1.992×10−10,(-1.992\times 1{0}^{-10},−3.5784×10−15)-3.5784\times 1{0}^{-15})(−8.3169×10−10,(-8.3169\times 1{0}^{-10},−1.9039×10−14)-1.9039\times 1{0}^{-14})(−2.8282×10−10,(-2.8282\times 1{0}^{-10},−5.0319×10−15)-5.0319\times 1{0}^{-15})(−4.4092×10−10,(-4.4092\times 1{0}^{-10},−6.2936×10−15)-6.2936\times 1{0}^{-15})(−4.3355×10−10,(-4.3355\times 1{0}^{-10},−5.8235×10−15)-5.8235\times 1{0}^{-15})Table 2Few values {dY(wn,(0,0))}\left\{{d}_{Y}\left({w}_{n},\left(0,0))\right\}by Alg1 and Alg2 from Example 4.2dY(wn,(0,0)){d}_{Y}\left({w}_{n},\left(0,0))nnAlg1Alg2Alg1Alg2Alg1Alg21522.5065522.50652480.5042480.50416.834116.83412147.7003145.0973677.5161666.97944.71654.6379336.980634.609145.4444183.68110.921551.292748.49097.767334.570240.10150.18960.2731751.86581.68737.85398.88480.040390.05794360.402870.36191.72081.92920.0130820.019992⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots 851.9078×10−91.9078\times 1{0}^{-9}5.0291×10−105.0291\times 1{0}^{-10}2.0876×10−92.0876\times 1{0}^{-9}7.14×10−107.14\times 1{0}^{-10}1.1067×10−91.1067\times 1{0}^{-9}1.0945×10−91.0945\times 1{0}^{-9}861.5871×10−91.5871\times 1{0}^{-9}4.1787×10−104.1787\times 1{0}^{-10}1.7367×10−91.7367\times 1{0}^{-9}5.9326×10−105.9326\times 1{0}^{-10}9.2069×10−109.2069\times 1{0}^{-10}9.0946×10−109.0946\times 1{0}^{-10}871.3203×10−91.3203\times 1{0}^{-9}3.4721×10−103.4721\times 1{0}^{-10}1.4447×10−91.4447\times 1{0}^{-9}4.9295×10−104.9295\times 1{0}^{-10}7.659×10−107.659\times 1{0}^{-10}7.5568×10−107.5568\times 1{0}^{-10}881.0983×10−91.0983\times 1{0}^{-9}2.8851×10−102.8851\times 1{0}^{-10}1.2018×10−91.2018\times 1{0}^{-9}4.096×10−104.096\times 1{0}^{-10}6.3714×10−106.3714\times 1{0}^{-10}6.2791×10−106.2791\times 1{0}^{-10}899.1366×10−109.1366\times 1{0}^{-10}2.3973×10−102.3973\times 1{0}^{-10}9.9977×10−109.9977\times 1{0}^{-10}3.4035×10−103.4035\times 1{0}^{-10}5.3002×10−105.3002\times 1{0}^{-10}5.2175×10−105.2175\times 1{0}^{-10}907.6006×10−107.6006\times 1{0}^{-10}1.992×10−101.992\times 1{0}^{-10}8.3169×10−108.3169\times 1{0}^{-10}2.8282×10−102.8282\times 1{0}^{-10}4.4092×10−104.4092\times 1{0}^{-10}4.3355×10−104.3355\times 1{0}^{-10}5Concluding remarksThis work analysed Mann-type and viscosity-type PPAs under which we approximated a common solution of MVFIPs, an MP, and a common fixed point of multivalued demicontractive mappings in Hadamard spaces. Using the results herein, we computed mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We also gave a numerical example in a nonlinear Hadamard space to support the findings. Our results are based on the newly introduced concept of monotonicity. Moreover, the results complement and extend several results in the literature. In particular, Theorem 3.9 generalises the result of Okeke and Izuchukwu [34] from a singlevalued nonexpansive mapping to a family of more general mappings (multivalued demicontractive), from one monotone mapping to finite family of monotone mappings yet considering a newly generalised notion of monotonicity as introduced in [9]. Also, we used viscosity-type PPA which is known to be more general and faster than the Halpern-type PPA used in [34].Our results generalise the results of Ranjbar and Khatibzadeh [35], Suparatulatorn et al. [46], and Khatibzadeh and Ranjbar [24] to more general problems and to finite family of monotone mappings.We extend the results of Takahashi and Shimoji [47] and Rockafellar [37] from linear spaces to CAT(0) spaces with a more general problem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Geometry in Metric Spaces de Gruyter

Convergence theorems for monotone vector field inclusions and minimization problems in Hadamard spaces

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de Gruyter
Copyright
© 2023 the author(s), published by De Gruyter
ISSN
2299-3274
eISSN
2299-3274
DOI
10.1515/agms-2022-0150
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Abstract

1IntroductionOne of the most applicable optimization problems in a Hilbert space HHis to find an element (1.1)w∈D(A)≔{z∈H:Az≠∅}such that0∈Aw,w\in {\mathbb{D}}\left(A):= \left\{z\in H:Az\ne \varnothing \right\}\hspace{1em}\hspace{0.1em}\text{such that}\hspace{0.1em}\hspace{0.33em}0\in Aw,where AAis an operator from HHto 2H{2}^{H}. This problem is called monotone inclusion problem (MIP), provided the operator AAis monotone in the sense that ⟨u−w,x−y⟩≥0,∀u,w∈D(A),x∈Au,y∈Aw.\langle u-w,x-y\rangle \ge 0,\hspace{1em}\forall u,w\in {\mathbb{D}}\left(A),\hspace{0.33em}x\in Au,y\in Aw.Applications of MIP can be found in several fields of science and engineering such as in inverse problems, image recovery, signal processing, fuzzy theory, game theory, robotic control, etc. As a result of these applications, several schemes for approximating a solution of (1.1) have been developed. The classical and prominent scheme for approximating MIP in linear spaces is proximal point algorithm (PPA), introduced in [29] and [37]. Rockafellar [37] proved under some conditions that any sequence generated by the PPA converges weakly to a solution of (1.1). Thereafter, several researchers have developed and studied different modifications of PPA with convergence analysis (see e.g. [18,21,33,47] and references therein).It is known that many applicable optimization problems come as constrained problems, where the constrains are non-convex, non-smooth, and nonlinear, among other things. However, it is recently observed that Hadamard spaces view some non-smooth and non-convex constrained optimization problems as smooth and convex unconstrained problems. For this reason, several notions related to optimization including that of monotone operator are being extended from linear spaces and Hadamard manifolds to Hadamard spaces (see e.g. [6,7,11,27] and references therein).Let (W,d)\left(W,d)be a Hadamard space. In 2008, Berg and Nikolaev [5] denoted (u,w)∈W×W\left(u,w)\in W\times Wby uw→\overrightarrow{uw}and defined a quasilinearization map ⟨⋅,⋅⟩:(W×W)×(W×W)→R\langle \cdot ,\cdot \rangle :\left(W\times W)\times \left(W\times W)\to {\mathbb{R}}by ⟨uw→,vy→⟩=12(d2(u,y)+d2(w,v)−d2(u,v)−d2(w,y)),(u,v,w,y∈W).\langle \overrightarrow{uw},\overrightarrow{vy}\rangle =\frac{1}{2}({d}^{2}\left(u,y)+{d}^{2}\left(w,v)-{d}^{2}\left(u,v)-{d}^{2}\left(w,y)),\hspace{1.0em}\left(u,v,w,y\in W).Using this notion of quasilinearization, Ahmadi Kakavandi [1] introduced the dual space of a Hadamard space (W,d)\left(W,d)as follows. Let ϕ:W→R\phi :W\to {\mathbb{R}}be a function, L(ϕ)≔supϕ(w)−ϕ(v)d(w,v):w,v∈W,w≠vL\left(\phi ):= \sup \left\{\frac{\phi \left(w)-\phi \left(v)}{d\left(w,v)}:w,v\in W,\hspace{0.33em}w\ne v\right\}and consider Θ:R×(W×W)→C(X,R)\Theta :{\mathbb{R}}\times \left(W\times W)\to C\left(X,{\mathbb{R}})defined by Θ(t,u,w)(x)=t⟨uw→,ux→⟩\Theta \left(t,u,w)\left(x)=t\langle \overrightarrow{uw},\overrightarrow{ux}\rangle for all t∈R,u,w,x∈Wt\in {\mathbb{R}},\hspace{0.33em}u,w,x\in W, where C(W,R)C\left(W,R)denotes the space of continuous real-valued functions on WW. The map DDon R×W×W{\mathbb{R}}\times W\times Wdefined by D((t,u,w),(s,x,y))=L(Θ(t,u,w)−Θ(s,x,y))D(\left(t,u,w),\left(s,x,y))=L(\Theta \left(t,u,w)-\Theta \left(s,x,y))is a pseudometric on R×W×W{\mathbb{R}}\times W\times W. Moreover, DDforms equivalence relation on R×W×W{\mathbb{R}}\times W\times Wwith the equivalence class of (t,u,w)\left(t,u,w)as [tuw→]≔{sxy→:D((t,u,w),(s,x,y))=0}\left[t\overrightarrow{uw}]:= \{s\overrightarrow{xy}:D(\left(t,u,w),\left(s,x,y))=0\}. The dual space of (W,d)\left(W,d)is (W∗,D)\left({W}^{\ast },D), where W∗≔{tuw→:(t,u,w)∈R×W×W}{W}^{\ast }:= \{t\overrightarrow{uw}:\left(t,u,w)\in {\mathbb{R}}\times W\times W\}, which acts on W×WW\times Wby ⟨x∗,uw→⟩=t⟨xy→,uw→⟩,\langle {x}^{\ast },\overrightarrow{uw}\rangle =t\langle \overrightarrow{xy},\overrightarrow{uw}\rangle ,for x∗=[txy→]∈W∗{x}^{\ast }=\left[t\overrightarrow{xy}]\in {W}^{\ast }, u,w∈Wu,w\in W. Moreover, the author observed that if WWis a Hilbert space, then [txy→]=t(y−x)\left[t\overrightarrow{xy}]=t(y-x).In 2017, Khatibzadeh and Ranjbar [24] introduced the concept of monotonicity in a Hadamard space WWwith dual space W∗{W}^{\ast }through the following definitions. A mapping A:W→2W∗A:W\to {2}^{{W}^{\ast }}is called monotone if ⟨u∗−w∗,wu→⟩≥0,∀u,w∈{z∈W:Az≠∅},u∗∈Au,w∗∈Aw.\langle {u}^{\ast }-{w}^{\ast },\overrightarrow{wu}\rangle \ge 0,\hspace{0.33em}\forall u,w\in \{z\in W:Az\ne \varnothing \},{u}^{\ast }\in Au,{w}^{\ast }\in Aw.Also, for λ>0\lambda \gt 0the authors considered λ\lambda -resolvent of AAas (1.2)JλAz≔w∈W:1λwz→∈Aw{J}_{\lambda }^{A}z:= \left\{\phantom{\rule[-1.25em]{}{0ex}},w\in W:\left[\frac{1}{\lambda }\overrightarrow{wz}\right]\in Aw\right\}and proved some Δ\Delta -convergence theorems. In the same year, Ranjbar and Khatibsadeh [35] proposed two schemes for approximating a solution of MIP. One is Mann-type PPA as follows: (1.3)wn+1=σnwn⊕(1−σn)JλnAwn,w1∈W,{w}_{n+1}={\sigma }_{n}{w}_{n}\oplus \left(1-{\sigma }_{n}){J}_{{\lambda }_{n}}^{A}{w}_{n},\hspace{0.33em}{w}_{1}\in W,and the other is Halpern-type PPA as follows: (1.4)wn+1=σnw⊕(1−σn)JλnAwn,w1,w∈W,{w}_{n+1}={\sigma }_{n}w\oplus \left(1-{\sigma }_{n}){J}_{{\lambda }_{n}}^{A}{w}_{n},\hspace{0.33em}{w}_{1},w\in W,where {λn}⊂(0,∞)\left\{{\lambda }_{n}\right\}\subset \left(0,\infty )and {σn}⊂[0,1]\left\{{\sigma }_{n}\right\}\subset \left[0,1]. The authors obtained a Δ\Delta -convergent result for the Mann-type PPA and a strong convergence result for the Halpern-type PPA.In 2018, Okeke and Izuchukwu [34] proposed the following Halpern-type PPA for finding a common solution of MIP, minimization problem (MP), and fixed-point problem in Hadamard spaces: (1.5)w,w1∈Wzn=JλA∘argminw∈Wh(w)+12μd2(w,wn)wn+1=σnw⊕(1−σn)Tzn,n≥1,\left\{\begin{array}{l}w,{w}_{1}\in W\hspace{1.0em}\\ {z}_{n}={J}_{\lambda }^{A}\circ \mathop{{\rm{argmin}}}\limits_{w\in W}\left\{\phantom{\rule[-1.25em]{}{0ex}},h\left(w)+\frac{1}{2\mu }{d}^{2}\left(w,{w}_{n})\right\}\hspace{1.0em}\\ {w}_{n+1}={\sigma }_{n}w\displaystyle \oplus \left(1-{\sigma }_{n})T{z}_{n},\hspace{0.33em}n\ge 1,\hspace{1.0em}\end{array}\right.where μ,λ∈(0,∞)\mu ,\lambda \in \left(0,\infty ), {σn}⊂(0,1)\left\{{\sigma }_{n}\right\}\subset \left(0,1), hhis a proper convex and lower semi-continuous function, and TTis a singlevalued nonexpansive mapping on WW. The authors proved a strong convergence result using the assumptions that limn→∞σn=0{\mathrm{lim}}_{n\to \infty }{\sigma }_{n}=0, ∑n=1∞σn=∞{\sum }_{n=1}^{\infty }{\sigma }_{n}=\infty , and ∑n=1∞∣σn−σn−1∣<∞{\sum }_{n=1}^{\infty }| {\sigma }_{n}-{\sigma }_{n-1}| \lt \infty . For other related development see e.g. [10,22,23,31,38–40,42,46] and references therein.In 2021, Chaipunya et al. [9] observed that although Hadamard spaces extend Hilbert spaces and Hadamard manifolds, the prior notion of monotonicity barely has a relationship with the Hadamard manifolds. For that reason, the authors introduced a new notion of monotonicity called monotone vector field using tangent spaces. They analysed that this notion coincides with the notion of monotonicity found in both Hilbert spaces and Hadamard manifolds better than that of Khatibzadeh and Ranjbar [24].Inspired by the work of Chaipunya et al. [9] and motivated by the work of Okeke and Izuchukwu [34] and research in this direction, we propose and analyse two schemes. Both for approximating a common solution of finite family of monotone vector field inclusion problems (MVFIP) that is also a common fixed point of multivalued demicontractive mappings at the same time a solution of MP in the framework of Hadamard spaces. One scheme is Mann-type PPA and the other is viscosity-type PPA (motivated by the fact that for appropriate contraction mapping, a viscosity-type scheme converges at a rate faster than Halpern-type, see [19,43]). We establish some convergence results for the proposed schemes and then apply our results to find mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We give a numerical example in a nonlinear space to show the applicability of the proposed schemes. Our results extend and complement the results of Suparatulatorn et al. [46], Khatibzadeh and Ranjbar [24], Ranjbar and Khatibzadeh [35], Okeke and Izuchukwu [34], and some equivalent results in Hilbert spaces.2PreliminariesRecall that for a metric space (W,d)\left(W,d)with nonempty subset YY, a Hausdorff metric is the map H:Cℬ(W)×Cℬ(W)→[0,+∞)H:{\mathcal{C {\mathcal B} }}\left(W)\times {\mathcal{C {\mathcal B} }}\left(W)\to \left[0,+\infty )defined by H(A,B)≔max{supa∈Adist(a,B),supb∈Bdist(b,A)},(A,B∈Cℬ(W)),H\left(A,B):= \max \{\mathop{\sup }\limits_{a\in A}{\rm{dist}}\left(a,B),\mathop{\sup }\limits_{b\in B}{\rm{dist}}\left(b,A)\},\hspace{1.0em}(A,B\in {\mathcal{C {\mathcal B} }}\left(W)),where dist(z,Y)≔inf{d(z,y):y∈Y}{\rm{dist}}\left(z,Y):= \inf \left\{d\left(z,y):y\in Y\right\}for z∈Wz\in Wand Cℬ(W){\mathcal{C {\mathcal B} }}\left(W)denotes the family of nonempty closed bounded subsets of YY. A point w∈Yw\in Yis called a fixed point of multi-valued map T:Y→2YT:Y\to {2}^{Y}if w∈Tww\in Tw. In the sequel, we denote the fixed points set of the map TTby F(T)F\left(T), that is, F(T)={w∈Y:w∈Tw}F\left(T)=\left\{w\in Y:w\in Tw\right\}. The map TTis said to be (i)nonexpansive if H(Tv,Tw)≤d(u,w)for allv,w∈Y.H\left(Tv,Tw)\le d\left(u,w)\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}v,w\in Y.(ii)quasi-nonexpansive if F(T)≠∅F\left(T)\ne \varnothing and H(Tv,Tp)≤d(v,p)for allv∈Yandp∈F(T).H\left(Tv,Tp)\le d\left(v,p)\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}v\in Y\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}p\in F\left(T).(iii)κ\kappa -demicontractive (demicontractive for short) if F(T)≠∅F\left(T)\ne \varnothing and there exists κ∈[0,1)\kappa \in \left[0,1)such that H(Tw,Tp)≤d(w,p)+κdist(w,Tw)for allw∈Yandp∈F(T).H\left(Tw,Tp)\le d\left(w,p)+\kappa \hspace{0.1em}\text{dist}\hspace{0.1em}\left(w,Tw)\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}w\in Y\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}p\in F\left(T).It follows that every nonexpansive mapping with fixed points is quasi-nonexpansive and every quasi-nonexpansive mapping is κ\kappa -demicontractive but not the converse.Definition 2.1Let (W,d)\left(W,d)be a metric space and let {wn}\left\{{w}_{n}\right\}be a bounded sequence in WW. Then the asymptotic centre A({wn})A\left(\left\{{w}_{n}\right\})of {wn}\left\{{w}_{n}\right\}is defined by A({wn})≔{u∈Y:limsupn→∞d(u,wn)=infz∈Ylimsupn→∞d(z,wn)}.A\left(\left\{{w}_{n}\right\}):= \left\{\phantom{\rule[-1.25em]{}{0ex}}u\in Y:\mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left(u,{w}_{n})=\mathop{\inf }\limits_{z\in Y}\mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left(z,{w}_{n})\right\}.Remark 2.2It is shown in [14, Proposition 7] that in a Hadamard space A({yn})A\left(\{{y}_{n}\})has only one element.Definition 2.3A bounded sequence {yn}\{{y}_{n}\}in a metric space (W,d)Δ\left(W,d)\hspace{0.33em}\Delta -converges to a point wwin YYif {w}\left\{w\right\}is the unique asymptotic centre for every subsequence {ynk}\{{y}_{{n}_{k}}\}of {yn}\{{y}_{n}\}and strongly converges to wwif limn→∞d(yn,w)=0{\mathrm{lim}}_{n\to \infty }d({y}_{n},w)=0.Let (W,d)\left(W,d)be a metric space and uu, wwbe two points in WW. A map τuw:[0,ℓ]⊂R→W{\tau }_{u}^{w}:\left[0,\ell ]\subset {\mathbb{R}}\to Wis called a geodesic path from uuto wwif τuw(0)=u,τuw(ℓ)=w{\tau }_{u}^{w}\left(0)=u,\hspace{0.33em}{\tau }_{u}^{w}\left(\ell )=wand d(τuw(α1),τuw(α2))=∣α1−α2∣d({\tau }_{u}^{w}\left({\alpha }_{1}),{\tau }_{u}^{w}\left({\alpha }_{2}))=| {\alpha }_{1}-{\alpha }_{2}| for every α1{\alpha }_{1}, α2∈[0,ℓ]{\alpha }_{2}\in \left[0,\ell ]. The image of τuw{\tau }_{u}^{w}is called a geodesic segment joining uuand ww. Where there is no ambiguity, we shall denote the image by [u,w]\left[u,w]. A metric space (W,d)\left(W,d)is a geodesic space if every two elements uu, wwin WWare joined by a geodesic segment and is said to be uniquely geodesic space if every two points uu, wware joined by a unique geodesic segment [u,w]\left[u,w]in WW. A subset YYof WWis convex if all geodesic segments connecting any two points of YYare in YY. A geodesic space that satisfies the following CN-inequality of Bruhat and Tits [8] is called a CAT(0) space. Let w,v∈Ww,v\in Wand zzbe a midpoint of a geodesic segment connecting wwand vv, then (2.1)d2(z,y)≤12d2(w,y)+12d2(v,y)−14d2(w,v),{d}^{2}\left(z,y)\le \frac{1}{2}{d}^{2}\left(w,y)+\frac{1}{2}{d}^{2}\left(v,y)-\frac{1}{4}{d}^{2}\left(w,v),for every y∈Wy\in W. CAT(0) spaces include pre-Hilbert spaces, Hilbert balls, Euclidean buildings, R{\mathbb{R}}-trees, and Hadamard manifolds. A complete CAT(0) space is known as Hadamard space. For details on CAT(0) spaces, see [6,7,17,25,36].Let ∡¯y(u,w){\bar{\measuredangle }}_{y}\left(u,w)denote the comparison angle between uuand wwat yy, i.e.∡¯y(y,y)≔0,∡¯y(y,w)=∡¯y(w,y)≔π2,cos∡¯y(u,w)≔⟨u¯−y¯,w¯−y¯⟩‖u¯−y¯‖2‖w¯−y¯‖2,{\bar{\measuredangle }}_{y}(y,y):= 0,\hspace{0.33em}{\bar{\measuredangle }}_{y}(y,w)={\bar{\measuredangle }}_{y}\left(w,y):= \frac{\pi }{2},\hspace{0.33em}\cos {\bar{\measuredangle }}_{y}\left(u,w):= \frac{\langle \bar{u}-\bar{y},\bar{w}-\bar{y}\rangle }{\Vert \bar{u}-\bar{y}{\Vert }_{2}\Vert \bar{w}-\bar{y}{\Vert }_{2}},where u,w∈Y\{y}u,w\left\in Y\backslash \{y\}and △¯(u¯,w¯,y¯)\bar{\bigtriangleup }\left(\bar{u},\bar{w},\bar{y})is the comparison triangle of △(u,w,y)\bigtriangleup \left(u,w,y). Then the Alexandrov angle between two geodesic issuing from a common point y∈Yy\in Yis defined by αy(τ1,τ2)=lims,t→0+∡¯y(τ1(t),τ2(s)).{\alpha }_{y}({\tau }_{1},{\tau }_{2})=\mathop{\mathrm{lim}}\limits_{s,t\to {0}^{+}}{\bar{\measuredangle }}_{y}({\tau }_{1}\left(t),{\tau }_{2}\left(s)).The Alexandrov angle αy{\alpha }_{y}defines a pseudometric on the set of all geodesics issuing from yy. We denote the metric identification of the pseudometric space by (Sy,∡y)\left({S}_{y},{\measuredangle }_{y}). In this work, the element of Sy{S}_{y}is denoted by τ≡[τ]\tau \equiv \left[\tau ]. Moreover, as in [9], ∼\sim forms an equivalence relation on [0,∞)×Sy\left[0,\infty )\times {S}_{y}in the sense that (t,τ1)∼(s,τ2)\left(t,{\tau }_{1})\hspace{0.33em} \sim \hspace{0.33em}\left(s,{\tau }_{2})if and only if tη(τ1)=sη(τ2)=0ortη(τ1)=sη(τ2)>0withτ1=τ2,t\eta \left({\tau }_{1})=s\eta \left({\tau }_{2})=0\hspace{1em}\hspace{0.1em}\text{or}\hspace{0.1em}\hspace{1em}t\eta \left({\tau }_{1})=s\eta \left({\tau }_{2})\gt 0\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{\tau }_{1}={\tau }_{2},where η(τ)≔0\eta \left(\tau ):= 0if τ\tau is a geodesic connecting only one point and η(τ)≔1\eta \left(\tau ):= 1otherwise. Then TyY≔([0,∞)×Sy)/∼{T}_{y}Y:= (\left[0,\infty )\times {S}_{y})\hspace{0.1em}\text{/}\hspace{0.1em} \sim together with the metric dy{d}_{y}defined by dy(tτ1,sτ2)≔t2η(τ1)+s2η(τ2)−2stη(τ1)η(τ2)cos∡y(τ1,τ2){d}_{y}\left(t{\tau }_{1},s{\tau }_{2}):= \sqrt{{t}^{2}\eta \left({\tau }_{1})+{s}^{2}\eta \left({\tau }_{2})-2st\eta \left({\tau }_{1})\eta \left({\tau }_{2})\cos {\measuredangle }_{y}\left({\tau }_{1},{\tau }_{2})}form a metric space (TyY,dy)\left({T}_{y}Y,{d}_{y})known as the tangent space of YY. For more details see [32].In the sequel, we denote a complete CAT(0) space by (W,d)\left(W,d)and a nonempty convex closed subset of WWby YY, the tangent space of YYat yyby (TyY,dy)\left({T}_{y}Y,{d}_{y}). We shall denote the tangent bundle of YY, ⋃u∈YTyY{\bigcup }_{u\in Y}{T}_{y}Yby TYTY, and adopt the notation 0≔{0y:y∈Y}{\bf{0}}:= \left\{{0}_{y}:y\in Y\right\}, where 0y≔0τ=sτyy{0}_{y}:= 0\tau =s{\tau }_{y}^{y}for which s>0s\gt 0and τ∈Sy\tau \in {S}_{y}. We shall say that a vector field A:Y→TYA:Y\to TYsatisfies condition (S)\left(S)if for any s>0s\gt 0and y∈Yy\in Y, there exists u∈Yu\in Ysuch that sd(u,y)τuy∈Ausd\left(u,y){\tau }_{u}^{y}\in Au.Definition 2.4[9] A vector field A:Y→TYA:Y\to TYis said to be monotone if Gy(ξ,τuw)+Gy(ϕ,τwu)≤0,{G}_{y}(\xi ,{\tau }_{u}^{w})+{G}_{y}(\phi ,{\tau }_{w}^{u})\le 0,for every (u,ξ),(w,ϕ)∈{(y,u)∈Y×TY:u∈Ay},\left(u,\xi ),\left(w,\phi )\in \left\{(y,u)\in Y\times TY:u\in Ay\right\},where Gy(tτ1,sτ2)=stη(τ1)η(τ2)cos∡y(τ1,τ2).{G}_{y}\left(t{\tau }_{1},s{\tau }_{2})=st\eta \left({\tau }_{1})\eta \left({\tau }_{2})\cos {\measuredangle }_{y}\left({\tau }_{1},{\tau }_{2}).In what follows, A−1(0){A}^{-1}\left({\bf{0}})denotes the solution set of MVFIP and Jμ{J}_{\mu }denotes the μ\mu -resolvent of AAdefined by Jμ(z)≔w∈X:1μd(w,z)τwz∈Aw,∀z∈X.{J}_{\mu }\left(z):= \left\{\phantom{\rule[-1.25em]{}{0ex}},w\in X:\frac{1}{\mu }d\left(w,z){\tau }_{w}^{z}\in Aw\right\},\hspace{1em}\forall z\in X.Lemma 2.5[9, p. 15] Let A:Y→TYA:Y\to TYbe a monotone vector field satisfying condition (S)\left(S)and let Jμ{J}_{\mu }be the μ\mu -resolvent of A. Then(i)Jμ{J}_{\mu }is well defined and singlevalued on YY,(ii)d(Jμ(x),Jμ(y))≤d(x,y)d({J}_{\mu }\left(x),{J}_{\mu }(y))\le d\left(x,y)for every xx, yyin YY,(iii)A−1(0)={x∈X:x=Jμ(x)}{A}^{-1}\left({\bf{0}})=\left\{x\in X:x={J}_{\mu }\left(x)\right\},(iv)Jλ(y)=Jμ1−μλJλ(y)⊕μλy{J}_{\lambda }(y)={J}_{\mu }\left(\left(1-\frac{\mu }{\lambda }\right){J}_{\lambda }(y)\oplus \frac{\mu }{\lambda }y\right), ∀y∈Y\forall \hspace{-0.3em}y\in Y, μ,λ∈R\mu ,\lambda \in {\mathbb{R}}such that λ≥μ>0\lambda \ge \mu \gt 0.Lemma 2.6[26, Proposition 3.7] Let TTbe a singlevalued nonexpansive mapping on YYand {wn}\left\{{w}_{n}\right\}be a sequence in YY. If {wn}Δ\left\{{w}_{n}\right\}\hspace{0.25em}\Delta -converges to wwand d(wn,Twn)→0,d\left({w}_{n},T{w}_{n})\to 0,then w=Tww=Tw.Remark 2.7Similar result of Lemma 2.6 holds for multivalued nonexpansive mappings.Lemma 2.8[13] The asymptotic centre of any bounded sequence in YYis in YY.Lemma 2.9[26, Proposition 3.6] Every bounded sequence {yn}\{{y}_{n}\}in YYhas a Δ\Delta -convergent subsequence {ynk}\{{y}_{{n}_{k}}\}.Lemma 2.10[15, Lemma 2.10] Let {yn}\{{y}_{n}\}be a sequence in YYwith A({yn})={v}A\left(\{{y}_{n}\})=\left\{v\right\}. Suppose that {ynk}\{{y}_{{n}_{k}}\}is a subsequence of {yn}\{{y}_{n}\}with A({ynk})={w}A\left(\{{y}_{{n}_{k}}\})=\left\{w\right\}and the sequence {d(yn,w)}\left\{d({y}_{n},w)\right\}converges, then v=wv=w.Lemma 2.11[15, Lemma 2.1 (iv)] Let u,y∈Yu,y\in Y. Then for each t∈[0,1]t\in \left[0,1], there exists a unique point w∈[u,y]w\in \left[u,y]such thatd(u,w)=td(u,y)andd(y,w)=(1−t)d(u,y).d\left(u,w)=td\left(u,y)\hspace{1.0em}{and}\hspace{1.0em}d(y,w)=\left(1-t)d\left(u,y).In this article, such a point wwis denoted by (1−t)u⊕ty\left(1-t)u\oplus ty. Moreover, for finite elements {yj}1m⊂Y{\{{y}_{j}\}}_{1}^{m}\subset Yand {tj}1m⊂(0,1){\left\{{t}_{j}\right\}}_{1}^{m}\subset \left(0,1), the notation ⊕j=1mtjyj{\oplus }_{j=1}^{m}{t}_{j}{y}_{j}is adopted from Dhompongsa et al. [12, p. 460], which is defined orderly as follows: ⊕j=1mtjyj≔(1−tm)t11−tmy1⊕t21−tmy2⊕⋯⊕tm−11−tm⊕tmym.\underset{j=1}{\overset{m}{\oplus }}{t}_{j}{y}_{j}:= \left(1-{t}_{m})\left(\frac{{t}_{1}}{1-{t}_{m}}{y}_{1}\oplus \frac{{t}_{2}}{1-{t}_{m}}{y}_{2}\oplus \cdots \oplus \frac{{t}_{m-1}}{1-{t}_{m}}\right)\oplus {t}_{m}{y}_{m}.Lemma 2.12[15, Lemma 2.5] Let y1,y2{y}_{1},{y}_{2}be points in YYand t∈[0,1]t\in \left[0,1]. Thend((1−t)y1⊕ty2,y3)≤(1−t)d(y1,y3)+td(y2,y3),d\left(\left(1-t){y}_{1}\oplus t{y}_{2},{y}_{3})\le \left(1-t)d({y}_{1},{y}_{3})+td({y}_{2},{y}_{3}),for every y3∈Y{y}_{3}\in Y.Lemma 2.13[15, Lemma 2.6] Let y1,y2,y3{y}_{1},{y}_{2},{y}_{3}be points in YYand t∈[0,1]t\in \left[0,1]. Thend2((1−t)y1⊕ty2,y3)≤(1−t)d2(y1,y3)+td2(y2,y3)−t(1−t)d2(y1,y2),{d}^{2}\left(\left(1-t){y}_{1}\oplus t{y}_{2},{y}_{3})\le \left(1-t){d}^{2}({y}_{1},{y}_{3})+t{d}^{2}({y}_{2},{y}_{3})-t\left(1-t){d}^{2}({y}_{1},{y}_{2}),for every y3∈Y{y}_{3}\in Y.As immediate consequence of Lemma 2.15, we have the following lemma.Lemma 2.14Let y1,y2{y}_{1},{y}_{2}be points in YYand t∈[0,1]t\in \left[0,1]. Thend2((1−t)y1⊕ty2,y3)≤(1−t)2d2(y1,y3)+t2d2(y2,y3)+2t(1−t)⟨y1y3→,y2y3→⟩,{d}^{2}\left(\left(1-t){y}_{1}\oplus t{y}_{2},{y}_{3})\le {\left(1-t)}^{2}{d}^{2}({y}_{1},{y}_{3})+{t}^{2}{d}^{2}({y}_{2},{y}_{3})+2t\left(1-t)\langle \overrightarrow{{y}_{1}{y}_{3}},\overrightarrow{{y}_{2}{y}_{3}}\rangle ,for every y3∈Y{y}_{3}\in Y.Lemma 2.15[1, Theorem 2.6] A bounded sequence {wn}\left\{{w}_{n}\right\}Δ\Delta -converge to a point wwin WWif and only if limsupn→∞⟨wnw→,zw→⟩≤0{\mathrm{limsup}}_{n\to \infty }\langle \overrightarrow{{w}_{n}w},\overrightarrow{zw}\rangle \le 0for all zzin WW.A function h:Y→R∪{+∞}h:Y\to {\mathbb{R}}\cup \left\{+\infty \right\}is called convex if for every a∈(0,1)a\in \left(0,1)and u,v∈Yu,v\in Y, h(au⊕(1−a)v)≤ah(u)+(1−a)h(v).h\left(au\oplus \left(1-a)v)\le ah\left(u)+\left(1-a)h\left(v).If the set D(h)≔{u∈Y:h(u)<+∞}≠∅D\left(h):= \left\{u\in Y:h\left(u)\lt +\infty \right\}\ne \varnothing , then hhis said to be proper. The function hhis said to be lower semi-continuous at a point w∈D(h)w\in D\left(h)if h(w)≤liminfn→∞h(wn)h\left(w)\le {\mathrm{liminf}}_{n\to \infty }h\left({w}_{n})for any convergent sequence {wn}\left\{{w}_{n}\right\}in D(h)D\left(h)with limit ww. If hhis lower semi-continuous at every point in D(h)D\left(h), then it is lower semi-continuous on D(h)D\left(h). For example of a proper convex lower semi-continuous function in a Hadamard space, see e.g. [11].Lemma 2.16[30, Lemma 1.10] Let h:Y→(−∞,+∞]h:Y\to \left(-\infty ,+\infty ]be a convex proper lower semi-continuous function. ThenJμhw=Jλhμ−λμJμhw⊕λμw,foreveryw∈Yandμ>λ>0,{J}_{\mu }^{h}w={J}_{\lambda }^{h}\left(\frac{\mu -\lambda }{\mu }{J}_{\mu }^{h}w\oplus \frac{\lambda }{\mu }w\right),\hspace{1em}{for}\hspace{0.33em}{every}\hspace{0.33em}w\in Y\hspace{1em}\hspace{0.1em}{\text{and}}\hspace{0.1em}\hspace{1em}\mu \gt \lambda \gt 0,whereJμhv≔argminw∈Yh(w)+12μd2(w,v).{J}_{\mu }^{h}v:= \mathop{{\rm{argmin}}}\limits_{w\in Y}\left\{\phantom{\rule[-1.25em]{}{0ex}},h\left(w)+\frac{1}{2\mu }{d}^{2}\left(w,v)\right\}.Lemma 2.17[2, p. 11] Let h:Y→(−∞,+∞]h:Y\to \left(-\infty ,+\infty ]be a convex proper lower semi-continuous function. Then for every u,v∈Yu,v\in Yand μ>0\mu \gt 0, the following hold:d2(w,Jμhw)≤d2(w,z)−d2(z,Jμhw)+2μ(h(z)−h(Jμhw)).{d}^{2}\left(w,{J}_{\mu }^{h}w)\le {d}^{2}\left(w,z)-{d}^{2}\left(z,{J}_{\mu }^{h}w)+2\mu (h\left(z)-h\left({J}_{\mu }^{h}w)).Lemma 2.18[3, Proposition 6.5] Let h:Y→(−∞,+∞]h:Y\to \left(-\infty ,+\infty ]be a convex proper lower semi-continuous function and Jλh{J}_{\lambda }^{h}be the λ\lambda -resolvent operator of hh. Then the fixed point set of Jλh{J}_{\lambda }^{h}coincides with the solution set of minimizers of hh.Lemma 2.19[20, Lemma 4] The λ\lambda -resolvent operator Jλh{J}_{\lambda }^{h}of a convex proper lower semi-continuous function h:Y→(−∞,+∞]h:Y\to \left(-\infty ,+\infty ]is nonexpansive and single-valued map.Lemma 2.20[28, Lemma 3.1] Let {θn}\left\{{\theta }_{n}\right\}be a sequence in R{\mathbb{R}}such that there exists a subsequence {nj}\left\{{n}_{j}\right\}of {n}\left\{n\right\}with θnj<θnj+1{\theta }_{{n}_{j}}\lt {\theta }_{{n}_{j}+1}for every j∈Nj\in {\mathbb{N}}. Then there exists a nondecreasing sequence {mk}⊂N\left\{{m}_{k}\right\}\subset {\mathbb{N}}such that mk→∞{m}_{k}\to \infty and for sufficiently large numbers k∈Nk\in {\mathbb{N}}, θmk≤θmk+1andθk≤θmk+1.{\theta }_{{m}_{k}}\le {\theta }_{{m}_{k}+1}\hspace{1.0em}{and}\hspace{1.0em}{\theta }_{k}\le {\theta }_{{m}_{k}+1}.In fact, mk=max{i≤k:θi<θi+1}.{m}_{k}=\max \left\{i\le k:{\theta }_{i}\lt {\theta }_{i+1}\right\}.Lemma 2.21[48, Lemma 2.5] Let {θn}\left\{{\theta }_{n}\right\}be a sequence in [0,+∞)⊂R\left[0,+\infty )\subset {\mathbb{R}}with(2.2)θn+1≤(1−σn)θn+σnϕn+γn,n≥1,{\theta }_{n+1}\le \left(1-{\sigma }_{n}){\theta }_{n}+{\sigma }_{n}{\phi }_{n}+{\gamma }_{n},\hspace{1.0em}n\ge 1,where {σn}\left\{{\sigma }_{n}\right\}, {ϕn}\left\{{\phi }_{n}\right\}, and {γn}\left\{{\gamma }_{n}\right\}satisfy the following conditions:(i){σn}⊂[0,1]\left\{{\sigma }_{n}\right\}\subset \left[0,1], ∑n=1∞σn=∞{\sum }_{n=1}^{\infty }{\sigma }_{n}=\infty ,(ii)limsupn→∞ϕn≤0{\mathrm{limsup}}_{n\to \infty }{\phi }_{n}\le 0, and(iii){γn}⊂[0,∞)\left\{{\gamma }_{n}\right\}\subset \left[0,\infty ), ∑n=1∞γn<∞{\sum }_{n=1}^{\infty }{\gamma }_{n}\lt \infty .Then limn→∞θn=0{\mathrm{lim}}_{n\to \infty }{\theta }_{n}=0.3Main resultsLet (W,d)\left(W,d)be a complete CAT(0) space and YYa nonempty closed convex subset of WW. Suppose that TYTYdenotes the tangent bundle of YY, h:Y→Rh:Y\to {\mathbb{R}}is a convex proper lower semicontinuous function, Aj:Y→TY{A}_{j}:Y\to TY, j=1,2,…,M1j=1,2,\ldots ,{M}_{1}are monotone vector fields with corresponding μ\mu -resolvent Jμnj{J}_{{\mu }_{n}}^{j}and Tp:Y→Cℬ(Y){T}_{p}:Y\to {\mathcal{C {\mathcal B} }}\left(Y), p=1,2,…,M2p=1,2,\ldots ,{M}_{2}are multivalued κp{\kappa }_{p}-demicontractive mappings. We shall assume that Γ≔argminy∈Yh(y)∩⋂j=1M1Aj−1(0)∩⋂p=1M2F(Tp)≠∅\Gamma := {{\rm{argmin}}}_{y\in Y}h(y)\cap {\bigcap }_{j=1}^{{M}_{1}}{A}_{j}^{-1}\left({\bf{0}})\cap {\bigcap }_{p=1}^{{M}_{2}}F\left({T}_{p})\ne \varnothing and each Tp(w)={w}{T}_{p}\left(w)=\left\{w\right\}for w∈Γw\in \Gamma . Moreover, for validation of the proposed algorithms, we let Jμn0{J}_{{\mu }_{n}}^{0}to be the identity map on YY, M≔max{M1,M2}M:= \max \left\{{M}_{1},{M}_{2}\right\}and if M1<M{M}_{1}\lt Mwe take Tj(x)≔{x}{T}_{j}\left(x):= \left\{x\right\}on YYfor j∈(M1,M]∩Nj\in \left({M}_{1},M]\cap {\mathbb{N}}and if M2<M{M}_{2}\lt Mwe set Jμnj=Jμn0{J}_{{\mu }_{n}}^{j}={J}_{{\mu }_{n}}^{0}for j∈(M2,M]∩Nj\in \left({M}_{2},M]\cap {\mathbb{N}}. In the convergence analysis, we shall need the assumption that each map Tp{T}_{p}satisfies demiclosedness-type property, that is if {wn}Δ\left\{{w}_{n}\right\}\hspace{0.33em}\Delta -converges to wwand dist(wn,Tpwn)→0,\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{n},{T}_{p}{w}_{n})\to 0,then w∈Tpww\in {T}_{p}w.Algorithm 1: Mann-type PPAInitialization: Choose {μn}⊂(μ,+∞),{λn}⊂(μ,+∞)\left\{{\mu }_{n}\right\}\subset \left(\mu ,+\infty ),\left\{{\lambda }_{n}\right\}\subset \left(\mu ,+\infty )for some μ>0\mu \gt 0, {βnj}⊂(0,1)\left\{{\beta }_{n}^{j}\right\}\subset \left(0,1), j=0,1,…,Mj=0,1,\ldots ,Msuch that ∑j=0Mβnj=1{\sum }_{j=0}^{M}{\beta }_{n}^{j}=1and {αnj}⊂[κj,1]\left\{{\alpha }_{n}^{j}\right\}\subset \left[{\kappa }_{j},1], j=1,…,Mj=1,\ldots ,M. Let w1∈Y{w}_{1}\in Y.Step 1: Set n=1n=1and compute yn0=Jλnh(wn).{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}\left({w}_{n}).Step 2: For each j∈{1,2,…,M},j\in \left\{1,2,\ldots ,M\right\},choose wnj∈Tjwn{w}_{n}^{j}\in {T}_{j}{w}_{n}and compute ynj=αnjwn⊕(1−αnj)wnj.{y}_{n}^{j}={\alpha }_{n}^{j}{w}_{n}\oplus \left(1-{\alpha }_{n}^{j}){w}_{n}^{j}.Step 3: Compute wn+1=⊕j=0MβnjJμnj(ynj).{w}_{n+1}=\underset{j=0}{\overset{M}{\oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}).Set n≔n+1n:= n+1and repeat all the steps.Lemma 3.1Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 1. Then for every w∈Γw\in \Gamma , the sequence {d(wn,w)}\left\{d\left({w}_{n},w)\right\}converges in R{\mathbb{R}}. Moreover, if each {βnk}⊂[b,1)\left\{{\beta }_{n}^{k}\right\}\subset \left[b,1)for some b∈(0,1)b\in \left(0,1), then {d(ynk,w)}\left\{d({y}_{n}^{k},w)\right\}converges for every w∈Γw\in \Gamma andlimn→∞d(ynk,w)=limn→∞d(wn,w)foreveryk∈{0,1,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({y}_{n}^{k},w)=\mathop{\mathrm{lim}}\limits_{n\to \infty }d\left({w}_{n},w)\hspace{0.33em}{for}\hspace{0.33em}{every}\hspace{0.33em}k\in \left\{0,1,\ldots ,M\right\}.ProofLet w∈Γw\in \Gamma and let n∈Nn\in {\mathbb{N}}. Then by Step 1 of Algorithm 1 and Lemma 2.19, we have (3.1)d(yn0,w)=d(Jλnh(wn),Jλnh(w))≤d(wn,w).d({y}_{n}^{0},w)=d({J}_{{\lambda }_{n}}^{h}\left({w}_{n}),{J}_{{\lambda }_{n}}^{h}\left(w))\le d\left({w}_{n},w).For each j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, we obtain from Step 2 of Algorithm 1, Lemma 2.13, and the assumption that Tj{T}_{j}is κj{\kappa }_{j}-demicontractive that (3.2)d2(ynj,w)≤αnjd2(wn,w)+(1−αnj)d2(wnj,w)−αnj(1−αnj)d2(wn,wnj)=αnjd2(wn,w)+(1−αnj)dist2(wnj,Tjw)−αnj(1−αnj)d2(wn,wnj)≤αnjd2(wn,w)+(1−αnj)H2(Tjwn,Tjw)−αnj(1−αnj)d2(wn,wnj)≤αnjd2(wn,w)+(1−αnj)[d2(wn,w)+κjdist2(wn,Tjwn)]−αnj(1−αnj)d2(wn,wnj)=d2(wn,w)+(1−αnj)(κj−αnj)d2(wn,wnj)=d2(wn,w)−(1−αnj)(αnj−κj)d2(wn,wnj)\begin{array}{rcl}{d}^{2}({y}_{n}^{j},w)& \le & {\alpha }_{n}^{j}{d}^{2}({w}_{n},w)+\left(1-{\alpha }_{n}^{j}){d}^{2}({w}_{n}^{j},w)-{\alpha }_{n}^{j}\left(1-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & =& {\alpha }_{n}^{j}{d}^{2}\left({w}_{n},w)+\left(1-{\alpha }_{n}^{j}){\text{dist}}^{2}\left({w}_{n}^{j},{T}_{j}w)-{\alpha }_{n}^{j}\left(1-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & \le & {\alpha }_{n}^{j}{d}^{2}\left({w}_{n},w)+\left(1-{\alpha }_{n}^{j}){H}^{2}\left({T}_{j}{w}_{n},{T}_{j}w)-{\alpha }_{n}^{j}\left(1-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & \le & {\alpha }_{n}^{j}{d}^{2}\left({w}_{n},w)+\left(1-{\alpha }_{n}^{j}){[}{d}^{2}\left({w}_{n},w)+{\kappa }_{j}{\text{dist}}^{2}({w}_{n},{T}_{j}{w}_{n})]-{\alpha }_{n}^{j}\left(1-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & =& {d}^{2}\left({w}_{n},w)+\left(1-{\alpha }_{n}^{j})\left({\kappa }_{j}-{\alpha }_{n}^{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\\ & =& {d}^{2}\left({w}_{n},w)-\left(1-{\alpha }_{n}^{j})\left({\alpha }_{n}^{j}-{\kappa }_{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\end{array}(3.3)≤d2(wn,w).\le \hspace{0.33em}{d}^{2}\left({w}_{n},w).\hspace{23.6em}By Step 3 of Algorithm 1 and Lemma 2.12, we obtain (3.4)d(wn+1,w)=d⊕j=0MβnjJμnj(ynj),w≤∑j=0Mβnjd(Jμnj(ynj),w)=∑j=0Mβnjd(Jμnj(ynj),Jμnj(w))≤∑j=0Mβnjd(ynj,w).\begin{array}{rcl}d({w}_{n+1},w)& =& d\hspace{0.08em}\left(\underset{j=0}{\overset{M}{\displaystyle \oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),w\right)\\ & \le & \mathop{\displaystyle \sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),w)\\ & =& \mathop{\displaystyle \sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),{J}_{{\mu }_{n}}^{j}\left(w))\\ & \le & \mathop{\displaystyle \sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w).\end{array}This, (3.1) and (3.3) imply (3.5)d(wn+1,w)≤∑j=0Mβnjd(ynj,w)≤∑j=0Mβnjd(wn,w)≤d(wn,w).d({w}_{n+1},w)\le \mathop{\sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w)\le \mathop{\sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({w}_{n},w)\le d({w}_{n},w).This implies that {d(wn,w)}\left\{d\left({w}_{n},w)\right\}converges in R{\mathbb{R}}. Let k∈{0,1,…,M}k\in \left\{0,1,\ldots ,M\right\}, then from (3.4), (3.1), and (3.3), we have (3.6)d(wn+1,w)≤βnkd(ynk,w)+∑j=1,j≠kMβnjd(ynj,w)≤βnkd(ynk,w)+∑j=1,j≠kMβnjd(wn,w)=βnkd(ynk,w)+(1−βnk)d(wn,w).\begin{array}{rcl}d({w}_{n+1},w)& \le & {\beta }_{n}^{k}d({y}_{n}^{k},w)+\mathop{\displaystyle \sum }\limits_{j=1,j\ne k}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w)\\ & \le & {\beta }_{n}^{k}d({y}_{n}^{k},w)+\mathop{\displaystyle \sum }\limits_{j=1,j\ne k}^{M}{\beta }_{n}^{j}d\left({w}_{n},w)\\ & =& {\beta }_{n}^{k}d({y}_{n}^{k},w)+\left(1-{\beta }_{n}^{k})d\left({w}_{n},w).\end{array}It follows from (3.6), (3.3), and (3.1) that (3.7)1βnk[d(wn+1,w)−d(wn,w)]+d(wn,w)≤d(ynk,w)≤d(wn,w).\frac{1}{{\beta }_{n}^{k}}{[}d\left({w}_{n+1},w)-d\left({w}_{n},w)]+d\left({w}_{n},w)\le d({y}_{n}^{k},w)\le d\left({w}_{n},w).Using the fact that {d(wn,w)}\left\{d\left({w}_{n},w)\right\}converges and letting n→∞n\to \infty in (3.7), we have the complete proof.□Lemma 3.2Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 1. If {βnk}⊂[b,1)\left\{{\beta }_{n}^{k}\right\}\subset \left[b,1)for some b∈(0,1)b\in \left(0,1)and limn→∞αnj∈(κj,1){\mathrm{lim}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({\kappa }_{j},1), thenlimn→∞d(wn,Jμh(wn))=limn→∞d(wn,Jμj(wn))=limn→∞dist(wn,Tjwn)=0,\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{J}_{\mu }^{h}\left({w}_{n}))=\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{J}_{\mu }^{j}\left({w}_{n}))=\mathop{\mathrm{lim}}\limits_{n\to \infty }\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{n},{T}_{j}{w}_{n})=0,for every j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}.ProofLet w∈Γw\in \Gamma . By Lemma 2.17 and the fact that h(w)≤h(wn)h\left(w)\le h\left({w}_{n}), we obtain (3.8)d2(yn0,wn)≤d2(yn0,w)−d2(wn,w)+2μ(h(w)−h(wn))≤d2(yn0,w)−d2(wn,w).{d}^{2}({y}_{n}^{0},{w}_{n})\le {d}^{2}({y}_{n}^{0},w)-{d}^{2}\left({w}_{n},w)+2\mu (h\left(w)-h\left({w}_{n}))\le {d}^{2}({y}_{n}^{0},w)-{d}^{2}\left({w}_{n},w).This and Lemma 3.1 imply that (3.9)limn→∞d(wn,yn0)=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{y}_{n}^{0})=0.Consequently, it follows from Lemmas 2.16, 2.19, and 2.12 that (3.10)d(wn,Jμhwn)≤d(wn,yn0)+d(yn0,Jμhwn)=d(wn,yn0)+dJμh(wn),Jμhλn−μλnJλnh(wn)⊕μμnwn≤d(wn,yn0)+dwn,λn−μλnyn0⊕μλnwn≤d(wn,yn0)+1−μλnd(wn,yn0)≤2−μλnd(wn,yn0)→0,asn→∞.\begin{array}{rcl}d\left({w}_{n},{J}_{\mu }^{h}{w}_{n})& \le & d({w}_{n},{y}_{n}^{0})+d({y}_{n}^{0},{J}_{\mu }^{h}{w}_{n})\\ & =& d({w}_{n},{y}_{n}^{0})+d\hspace{0.08em}\left(\phantom{\rule[-.2em]{}{0ex}},{J}_{\mu }^{h}\left({w}_{n}),{J}_{\mu }^{h}\left(\frac{{\lambda }_{n}-\mu }{{\lambda }_{n}}{J}_{{\lambda }_{n}}^{h}\left({w}_{n})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{w}_{n}\right)\right)\\ & \le & d({w}_{n},{y}_{n}^{0})+d\hspace{0.08em}\left(\phantom{\rule[-.2em]{}{0ex}},{w}_{n},\left(\frac{{\lambda }_{n}-\mu }{{\lambda }_{n}}{y}_{n}^{0}\displaystyle \oplus \frac{\mu }{{\lambda }_{n}}{w}_{n}\right)\right)\\ & \le & d({w}_{n},{y}_{n}^{0})+\left(1-\frac{\mu }{{\lambda }_{n}}\right)\hspace{0.08em}d\hspace{0.08em}\left({w}_{n},{y}_{n}^{0})\\ & \le & \left(2-\frac{\mu }{{\lambda }_{n}}\right)\hspace{0.08em}d\left({w}_{n},{y}_{n}^{0})\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .\end{array}Let n∈Nn\in {\mathbb{N}}and set ϕk(n)=⊕j=0kβnjγkJμnj(ynj){\phi }_{k}^{\left(n)}={\oplus }_{j=0}^{k}\frac{{\beta }_{n}^{j}}{{\gamma }_{k}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}), where γk≔∑j=0kβnj{\gamma }_{k}:= {\sum }_{j=0}^{k}{\beta }_{n}^{j}, k∈{0,1,2,…,M}k\in \left\{0,1,2,\ldots ,M\right\}. Then γk∈[b,1){\gamma }_{k}\in \left[b,1), γ0=βn0{\gamma }_{0}={\beta }_{n}^{0}, γM=1{\gamma }_{M}=1, and ϕ0(n)=Jμn0(yn0)=yn0{\phi }_{0}^{\left(n)}={J}_{{\mu }_{n}}^{0}({y}_{n}^{0})={y}_{n}^{0}. Moreover, (3.11)γk−1γk≥βn0and\frac{{\gamma }_{k-1}}{{\gamma }_{k}}\ge {\beta }_{n}^{0}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}(3.12)ϕk(n)=γk−1γkϕk−1(n)⊕βnkγkJμk(ynk),{\phi }_{k}^{\left(n)}=\frac{{\gamma }_{k-1}}{{\gamma }_{k}}{\phi }_{k-1}^{\left(n)}\oplus \frac{{\beta }_{n}^{k}}{{\gamma }_{k}}{J}_{\mu }^{k}({y}_{n}^{k}),for every k∈{1,2,…,M}k\in \left\{1,2,\ldots ,M\right\}. Let w∈Γw\in \Gamma . Then from (3.12), Lemma 2.13, and (3.11), we have the following inequality for every k∈{1,2,…,M}k\in \left\{1,2,\ldots ,M\right\}: (3.13)d2(ϕk(n),w)=d2γk−1γkϕk−1(n)⊕βnkγkJμnk(ynk),w≤1γkγk−1d2(ϕk−1(n),w)+βnkd2(Jμnk(ynk),w)−γk−1βnkγkd2(ϕk−1(n),Jμnk(ynk))≤1γk[γk−1d2(ϕk−1(n),w)+βnkd2(Jμnk(ynk),w)−βn0βnkd2(ϕk−1(n),Jμnk(ynk))].\begin{array}{rcl}{d}^{2}({\phi }_{k}^{\left(n)},w)& =& {d}^{2}\left(\frac{{\gamma }_{k-1}}{{\gamma }_{k}}{\phi }_{k-1}^{\left(n)}\displaystyle \oplus \frac{{\beta }_{n}^{k}}{{\gamma }_{k}}{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w\right)\\ & \le & \frac{1}{{\gamma }_{k}}\left[{\gamma }_{k-1}{d}^{2}({\phi }_{k-1}^{\left(n)},w)+{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-\frac{{\gamma }_{k-1}{\beta }_{n}^{k}}{{\gamma }_{k}}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\right]\\ & \le & \frac{1}{{\gamma }_{k}}{[}{\gamma }_{k-1}{d}^{2}({\phi }_{k-1}^{\left(n)},w)+{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))].\end{array}Moreover, from Algorithm 1 and (3.13), we have d2(wn+1,w)=d2(ϕM(n),w)≤1γM[γM−1d2(ϕM−1(n),w)+βnMd2(JμnM(ynM),w)−βn0βnMd2(ϕM−1(n),JμnM(ynM))]≤γM−1d2(ϕM−1(n),w)+βnMd2(JμnM(ynM),w)−βn0βnMd2(ϕM−1(n),JμnM(ynM))≤[γM−2d2(ϕM−2(n),w)+βnM−1d2(JμnM−1(ynM−1),w)−βn0βnM−1d2(ϕM−2(n),JμnM−1(ynM−1))]+βnMd2(JμnM(ynM),w)−βn0βnMd2(ϕM−1(n),JμnM(ynj))=γM−2d2(ϕM−2(n),w)+∑k=M−1Mβnkd2(Jμnk(wnk),w)−βn0∑k=M−1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& =& {d}^{2}({\phi }_{M}^{\left(n)},w)\\ & \le & \frac{1}{{\gamma }_{M}}{[}{\gamma }_{M-1}{d}^{2}({\phi }_{M-1}^{\left(n)},w)+{\beta }_{n}^{M}{d}^{2}({J}_{{\mu }_{n}}^{M}({y}_{n}^{M}),w)-{\beta }_{n}^{0}{\beta }_{n}^{M}{d}^{2}({\phi }_{M-1}^{\left(n)},{J}_{{\mu }_{n}}^{M}({y}_{n}^{M}))]\\ & \le & {\gamma }_{M-1}{d}^{2}({\phi }_{M-1}^{\left(n)},w)+{\beta }_{n}^{M}{d}^{2}({J}_{{\mu }_{n}}^{M}({y}_{n}^{M}),w)-{\beta }_{n}^{0}{\beta }_{n}^{M}{d}^{2}({\phi }_{M-1}^{\left(n)},{J}_{{\mu }_{n}}^{M}({y}_{n}^{M}))\\ & \le & {[}{\gamma }_{M-2}{d}^{2}({\phi }_{M-2}^{\left(n)},w)+{\beta }_{n}^{M-1}{d}^{2}({J}_{{\mu }_{n}}^{M-1}({y}_{n}^{M-1}),w)-{\beta }_{n}^{0}{\beta }_{n}^{M-1}{d}^{2}({\phi }_{M-2}^{\left(n)},{J}_{{\mu }_{n}}^{M-1}({y}_{n}^{M-1}))]\\ & & +{\beta }_{n}^{M}{d}^{2}({J}_{{\mu }_{n}}^{M}({y}_{n}^{M}),w)-{\beta }_{n}^{0}{\beta }_{n}^{M}{d}^{2}({\phi }_{M-1}^{\left(n)},{J}_{{\mu }_{n}}^{M}({y}_{n}^{j}))\\ & =& {\gamma }_{M-2}{d}^{2}({\phi }_{M-2}^{\left(n)},w)+\mathop{\displaystyle \sum }\limits_{k=M-1}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}\left({w}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=M-1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}Continuing in this pattern, we obtain that d2(wn+1,w)≤γM−3d2(ϕM−3(n),w)+∑k=M−2Mβnkd2(Jμnk(ynk),w)−βn0∑k=M−2Mβnkd2(ϕk−1(n),Jμnk(ynk))⋮≤γ1d2(ϕ1(n),w)+∑k=2Mβnkd2(Jμnk(ynk),w)−βn0∑k=2Mβnkd2(ϕk−1(n),Jμnk(ynk))\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & {\gamma }_{M-3}{d}^{2}({\phi }_{M-3}^{\left(n)},w)+\mathop{\displaystyle \sum }\limits_{k=M-2}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=M-2}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \vdots & \\ & \le & {\gamma }_{1}{d}^{2}({\phi }_{1}^{\left(n)},w)+\mathop{\displaystyle \sum }\limits_{k=2}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=2}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\end{array}≤γ0d2(ϕ0(n),w)+∑k=1Mβnkd2(Jμnk(ynk),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))=∑k=0Mβnkd2(Jμnk(ynk),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}& \le & {\gamma }_{0}{d}^{2}({\phi }_{0}^{\left(n)},w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & =& \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}This and Lemma 2.5(iv) imply that (3.14)d2(wn+1,w)≤∑k=0Mβnkd2(Jμnk(ynk),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))=∑k=0Mβnkd2(Jμnk(ynk),Jμnk(w))−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤∑k=0Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & =& \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({J}_{{\mu }_{n}}^{k}({y}_{n}^{k}),{J}_{{\mu }_{n}}^{k}\left(w))-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}Consequently, by (3.3), we have d2(wn+1,w)≤∑k=0Mβnkd2(wn,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))=d2(wn,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({w}_{n},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & =& {d}^{2}({w}_{n},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}This implies that for every j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, βn0βnjd2(ϕj−1(n),Jμnj(ynj))≤βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynj))≤d2(wn,w)−d2(wn+1,w).{\beta }_{n}^{0}{\beta }_{n}^{j}{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\le {\beta }_{n}^{0}\mathop{\sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{j}))\le {d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w).Therefore, (3.15)d2(ϕj−1(n),Jμnj(ynj))≤1βn0βnj[d2(wn,w)−d2(wn+1,w)]≤1b2[d2(wn,w)−d2(wn+1,w)].{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\le \frac{1}{{\beta }_{n}^{0}{\beta }_{n}^{j}}{[}{d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)]\le \frac{1}{{b}^{2}}{[}{d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)].It follows from Lemma 3.1 and (3.15) that limn→∞d2(ϕj−1(n),Jμnj(ynj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.Consequently, (3.16)limn→∞d(ϕj−1(n),Jμnj(ynj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.From (3.12), Lemma 2.11, and (3.16), we obtain that for every j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}, (3.17)d(ϕj−1(n),ϕj(n))≤d(ϕj−1(n),Jμnj(ynj))→0,asn→∞.d({\phi }_{j-1}^{\left(n)},{\phi }_{j}^{\left(n)})\le d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .Furthermore, for any j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}, we have (3.18)d(wn,Jμnj(ynj))=d(ϕ0(n),Jμnj(ynj))≤d(ϕ0(n),ϕ1(n))+d(ϕ1(n),ϕ2(n))+⋯+d(ϕj−2(n),ϕj−1(n))+d(ϕj−1(n),Jμnj(ynj))=∑k=1j−1d(ϕk−1(n),ϕk(n))+d(ϕj−1(n),Jμnj(ynj))\begin{array}{rcl}d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))& =& d({\phi }_{0}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\\ & \le & d({\phi }_{0}^{\left(n)},{\phi }_{1}^{\left(n)})+d({\phi }_{1}^{\left(n)},{\phi }_{2}^{\left(n)})+\cdots +d({\phi }_{j-2}^{\left(n)},{\phi }_{j-1}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\\ & =& \mathop{\displaystyle \sum }\limits_{k=1}^{j-1}d({\phi }_{k-1}^{\left(n)},{\phi }_{k}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\end{array}(3.19)≤∑k=1Md(ϕk−1(n),ϕk(n))+d(ϕj−1(n),Jμnj(ynj)).\le \hspace{0.33em}\mathop{\sum }\limits_{k=1}^{M}d({\phi }_{k-1}^{\left(n)},{\phi }_{k}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})).\hspace{7.3em}It follows from (3.19), (3.16), and (3.17) that (3.20)limn→∞d(wn,Jμnj(ynj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.Also, for each j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}and for w∈Γw\in \Gamma , we have from (3.2) that (3.21)(1−αnj)(αnj−κj)d2(wn,wnj)≤d2(wn,w)−d2(wn+1,w).\left(1-{\alpha }_{n}^{j})\left({\alpha }_{n}^{j}-{\kappa }_{j}){d}^{2}\left({w}_{n},{w}_{n}^{j})\le {d}^{2}\left({w}_{n},w)-{d}^{2}\left({w}_{n+1},w).This, Lemma 3.1, and the assumption that liminfn→∞αnj∈(kj,1){\mathrm{liminf}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({k}_{j},1)imply limn→∞d2(wn,wnj)=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }{d}^{2}\left({w}_{n},{w}_{n}^{j})=0.Consequently, (3.22)limn→∞d(wn,wnj)=0,for everyj∈{1,2,…,M}.\mathop{\mathrm{lim}}\limits_{n\to \infty }d\left({w}_{n},{w}_{n}^{j})=0,\hspace{1em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}j\in \left\{1,2,\ldots ,M\right\}.Hence, for each j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, (3.23)dist(wn,Tjwn)≤d(wn,wnj)→0,asn→∞.\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{n},{T}_{j}{w}_{n})\le d\left({w}_{n},{w}_{n}^{j})\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .Furthermore, it follows from Lemma 2.5, Lemma 2.12, (3.20), and (3.22) that (3.24)d(wn,Jμj(wn))≤d(wn,Jμnj(ynj))+d(Jμnj(ynj),Jμj(wn))=d(wn,Jμnj(ynj))+dJμj(wn),Jμjμn−μμnJμnj(ynj)⊕μμnynj≤d(wn,Jμnj(ynj))+dwn,μn−μμnJμnj(ynj)⊕μμnynj≤2−μμnd(wn,Jμnj(ynj))+μμnd(wn,ynj)≤2d(wn,Jμnj(ynj))+d(wn,ynj)≤2d(wn,Jμnj(ynj))+d(wn,wnj)→0,asn→∞.\begin{array}{rcl}d({w}_{n},{J}_{\mu }^{j}\left({w}_{n}))& \le & d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d({J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),{J}_{\mu }^{j}\left({w}_{n}))\\ & =& d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\hspace{0.08em}\left({J}_{\mu }^{j}\left({w}_{n}),{J}_{\mu }^{j}\left(\frac{{\mu }_{n}-\mu }{{\mu }_{n}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{y}_{n}^{j}\right)\right)\\ & \le & d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\hspace{0.08em}\left({w}_{n},\left(\frac{{\mu }_{n}-\mu }{{\mu }_{n}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{y}_{n}^{j}\right)\right)\\ & \le & \left(2-\frac{\mu }{{\mu }_{n}}\right)\hspace{0.08em}d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+\frac{\mu }{{\mu }_{n}}d\left({w}_{n},{y}_{n}^{j})\\ & \le & 2d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\left({w}_{n},{y}_{n}^{j})\\ & \le & 2d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\left({w}_{n},{w}_{n}^{j})\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .\end{array}By (3.35), (3.24), and (3.23), we have the complete proof.□Theorem 3.3Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 1 and let each {βnk}⊂[b,1)\left\{{\beta }_{n}^{k}\right\}\subset \left[b,1)for some b∈(0,1)b\in \left(0,1)and limn→∞αnj∈(κj,1){\mathrm{lim}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({\kappa }_{j},1). Suppose that for each p∈{1,2,…,M2}p\in \left\{1,2,\ldots ,{M}_{2}\right\}, Tp{T}_{p}has demiclosedness-type property. Then {wn}Δ\left\{{w}_{n}\right\}\hspace{0.33em}\Delta -converges to an element of Γ\Gamma . Moreover, if YYis compact, then the convergence is strong.ProofFrom Lemma 3.1, we have that {wn}\left\{{w}_{n}\right\}is bounded. Let ωA(wn){\omega }_{A}\left({w}_{n})denote the union of asymptotic centres of all subsequences of {wn}\left\{{w}_{n}\right\}. Suppose that z∈ωA(wn)z\in {\omega }_{A}\left({w}_{n})with A({wnk})={z}A\left(\left\{{w}_{{n}_{k}}\right\})=\left\{z\right\}. By Lemma 2.9, there exists a subsequence {wnkp}\{{w}_{{n}_{{k}_{p}}}\}of {wnk}\left\{{w}_{{n}_{k}}\right\}that Δ\Delta -converges to vv, and Lemma 2.8 implies that v∈Yv\in Y. Now by Lemmas 2.19, 2.6, and 3.2, we have that v∈argminy∈Yh(y)v\in {{\rm{argmin}}}_{y\in Y}h(y). Similarly, by Lemmas 2.5(ii), 2.7, and 3.2, we have that v∈⋂j=1M1Aj−1(0)v\in {\bigcap }_{j=1}^{{M}_{1}}{A}_{j}^{-1}\left({\bf{0}}). Also, from Lemma 3.2 and the assumption that each Tp{T}_{p}has demiclosedness-type property, we have v∈⋂p=1M2F(Tp)v\in {\bigcap }_{p=1}^{{M}_{2}}F\left({T}_{p}). Therefore, v∈Γv\in \Gamma . So, by Lemma 3.1, {d(wn,v)}\left\{d\left({w}_{n},v)\right\}converges and by Lemma 2.10, z=vz=v. Thus, ωA(wn)⊂Γ{\omega }_{A}\left({w}_{n})\subset \Gamma . To complete the proof, it suffices to show that ωA(wn){\omega }_{A}\left({w}_{n})consists of only one element. Let A({wn})={w}A\left(\left\{{w}_{n}\right\})=\left\{w\right\}and suppose there exists y∈ωA(wn)y\in {\omega }_{A}\left({w}_{n})with y≠wy\ne w. Now, let {wnk}\left\{{w}_{{n}_{k}}\right\}be the subsequence of {wn}\left\{{w}_{n}\right\}with A({wnk})={y}A\left(\left\{{w}_{{n}_{k}}\right\})=\{y\}. Then y∈Γy\in \Gamma , since ωA(wn)⊂Γ{\omega }_{A}\left({w}_{n})\subset \Gamma . By Lemma 3.1 and the definition of asymptotic centre, we have limsupn→∞d(wn,w)<limsupn→∞d(wn,y)=limn→∞d(wn,y)=limsupk→∞d(wnk,y)\begin{array}{rcl}\mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left({w}_{n},w)& \lt & \mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left({w}_{n},y)\\ & =& \mathop{\mathrm{lim}}\limits_{n\to \infty }d\left({w}_{n},y)\\ & =& \mathop{\mathrm{limsup}}\limits_{k\to \infty }d\left({w}_{{n}_{k}},y)\end{array}<limsupk→∞d(wnk,w)≤limsupn→∞d(wn,w),\begin{array}{rcl}& \lt & \mathop{\mathrm{limsup}}\limits_{k\to \infty }d\left({w}_{{n}_{k}},w)\\ & \le & \mathop{\mathrm{limsup}}\limits_{n\to \infty }d\left({w}_{n},w),\end{array}which contradicts y≠wy\ne w. Therefore, ωA(wn){\omega }_{A}\left({w}_{n})consists of exactly one element.Suppose that YYis compact, then there exists a subsequence {wnk}\left\{{w}_{{n}_{k}}\right\}of {wn}\left\{{w}_{n}\right\}that converges strongly to some point yyin YY. Thus, {wnk}Δ\left\{{w}_{{n}_{k}}\right\}\hspace{0.33em}\Delta -converges to y∈Yy\in Y. By first part of the proof, we have that y∈ωA(wn)⊂Γy\in {\omega }_{A}\left({w}_{n})\subset \Gamma . Consequently, by Lemma 3.1, we have limn→∞d(wn,y)=limk→∞d(wnk,y)=0,\mathop{\mathrm{lim}}\limits_{n\to \infty }d\left({w}_{n},y)=\mathop{\mathrm{lim}}\limits_{k\to \infty }d\left({w}_{{n}_{k}},y)=0,which completes the proof.□The following corollary is obtained from the fact that every quasi-nonexpansive mapping is 0-demicontractive.Corollary 3.4Let Tp,p=1,2,…,M2{T}_{p},\hspace{0.33em}p=1,2,\ldots ,{M}_{2}be quasi-nonexpansive mappings with demiclosedness-type property. Suppose that {αnj},{βnj},{wn}\left\{{\alpha }_{n}^{j}\right\},\hspace{0.33em}\left\{{\beta }_{n}^{j}\right\},\hspace{0.33em}\left\{{w}_{n}\right\}, and Γ\Gamma are as in Theorem 3.3. Then {wn}Δ\left\{{w}_{n}\right\}\hspace{0.33em}\Delta -converges to an element of Γ\Gamma . Moreover, if YYis compact, then the convergence is strong.Since every nonexpansive mapping with fixed points is quasi-nonexpansive and by Remark 2 has demiclosedness-type property, we have the following corollary.Corollary 3.5Let Tp,p=1,2,…,M2{T}_{p},\hspace{0.33em}p=1,2,\ldots ,{M}_{2}be nonexpansive mappings and suppose that {αnj},{βnj},{wn}\left\{{\alpha }_{n}^{j}\right\},\hspace{0.33em}\left\{{\beta }_{n}^{j}\right\},\hspace{0.33em}\left\{{w}_{n}\right\}, and Γ\Gamma are as in Theorem 3.3. Then {wn}Δ\left\{{w}_{n}\right\}\hspace{0.25em}\Delta -converges to an element of Γ\Gamma . Moreover, if YYis compact, then the convergence is strong.We are now ready to analyse viscosity-type PPA for strong convergence resultAlgorithm 2: Viscosity-type PPAInitialization: Choose {φn}⊂[0,1]\left\{{\varphi }_{n}\right\}\subset \left[0,1], {μn}⊂(μ,+∞),{λn}⊂(μ,+∞)\left\{{\mu }_{n}\right\}\subset \left(\mu ,+\infty ),\hspace{0.33em}\left\{{\lambda }_{n}\right\}\subset \left(\mu ,+\infty )for some μ>0\mu \gt 0, {βnj}⊂(0,1)\left\{{\beta }_{n}^{j}\right\}\subset \left(0,1), j=0,1,…,Mj=0,1,\ldots ,Msuch that ∑j=0Mβnj=1{\sum }_{j=0}^{M}{\beta }_{n}^{j}=1, and {αnj}⊂[κj,1]\left\{{\alpha }_{n}^{j}\right\}\subset \left[{\kappa }_{j},1], j=1,…,Mj=1,\ldots ,M. Let ffbe a η\eta -contraction map on YYand w1∈Y{w}_{1}\in Y.Step 1: Set n=1n=1and compute yn0=Jλnh((1−φn)wn⊕φnf(wn)).{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\oplus {\varphi }_{n}f\left({w}_{n})).Step 2: For each j∈{1,2,…,M},j\in \left\{1,2,\ldots ,M\right\},choose wnj∈Tjwn{w}_{n}^{j}\in {T}_{j}{w}_{n}and compute ynj=αnjwn⊕(1−αnj)wnj.{y}_{n}^{j}={\alpha }_{n}^{j}{w}_{n}\oplus \left(1-{\alpha }_{n}^{j}){w}_{n}^{j}.Step 3: Compute wn+1=⊕j=0MβnjJμnj(ynj).{w}_{n+1}=\underset{j=0}{\overset{M}{\oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}).Set n≔n+1n:= n+1and repeat all the steps.Lemma 3.6Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 2. Then {wn}\left\{{w}_{n}\right\}is bounded.ProofLet w∈Γw\in \Gamma . Then by Lemmas 2.19, 2.12, and the assumption that ffis contractive, we have (3.25)d(yn0,w)=d(Jλnh((1−φn)wn⊕φnf(wn)),Jλnh(w))≤d((1−φn)wn⊕φnf(wn),w)≤(1−φn)d(wn,w)+φnd(f(wn),w)≤(1−φn)d(wn,w)+φnd(f(wn),f(w))+φnd(f(w),w)=(1−(1−η)φn)d(wn,w)+φnd(f(w),w).\begin{array}{rcl}d({y}_{n}^{0},w)& =& d({J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),{J}_{{\lambda }_{n}}^{h}\left(w))\\ & \le & d(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n}),w)\\ & \le & \left(1-{\varphi }_{n})d\left({w}_{n},w)+{\varphi }_{n}d(f\left({w}_{n}),w)\\ & \le & \left(1-{\varphi }_{n})d\left({w}_{n},w)+{\varphi }_{n}d(f\left({w}_{n}),f\left(w))+{\varphi }_{n}d(f\left(w),w)\\ & =& \left(1-\left(1-\eta ){\varphi }_{n})d\left({w}_{n},w)+{\varphi }_{n}d(f\left(w),w).\end{array}Moreover, following similar line as in (3.3), we have (3.26)d(ynj,w)≤d(wn,w),∀j∈{1,2,…,M}.d({y}_{n}^{j},w)\le d\left({w}_{n},w),\hspace{0.33em}\forall j\in \left\{1,2,\ldots ,M\right\}.Also, as in (3.4), we have (3.27)d(wn+1,w)≤∑j=0Mβnjd(ynj,w).d({w}_{n+1},w)\le \mathop{\sum }\limits_{j=0}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w).It follows from (3.27), (3.26), and (3.25) that (3.28)d(wn+1,w)≤βn0d(yn0,w)+∑j=1Mβnjd(ynj,w)≤βn0[(1−(1−η)φn)d(wn,w)+φnd(f(w),w)]+∑j=1Mβnjd(wn,w)=(1−(1−η)βn0φn)d(wn,w)+βn0φnd(f(w),w).\begin{array}{rcl}d({w}_{n+1},w)& \le & {\beta }_{n}^{0}d({y}_{n}^{0},w)+\mathop{\displaystyle \sum }\limits_{j=1}^{M}{\beta }_{n}^{j}d({y}_{n}^{j},w)\\ & \le & {\beta }_{n}^{0}{[}\left(1-\left(1-\eta ){\varphi }_{n})d\left({w}_{n},w)+{\varphi }_{n}d(f\left(w),w)]+\mathop{\displaystyle \sum }\limits_{j=1}^{M}{\beta }_{n}^{j}d({w}_{n},w)\\ & =& (1-\left(1-\eta ){\beta }_{n}^{0}{\varphi }_{n})d({w}_{n},w)+{\beta }_{n}^{0}{\varphi }_{n}d(f\left(w),w).\end{array}This implies that (3.29)d(wn+1,w)≤(1−(1−η)βn0φn)d(wn,w)+βn0φnd(f(w),w)≤maxd(wn,w),1(1−η)d(f(w),w)⋮≤maxd(w1,w),1(1−η)d(f(w),w).\begin{array}{rcl}d({w}_{n+1},w)& \le & (1-\left(1-\eta ){\beta }_{n}^{0}{\varphi }_{n})d({w}_{n},w)+{\beta }_{n}^{0}{\varphi }_{n}d(f\left(w),w)\\ & \le & \max \left\{\phantom{\rule[-1.25em]{}{0ex}},d\left({w}_{n},w),\frac{1}{\left(1-\eta )}d(f\left(w),w)\right\}\\ & \vdots & \\ & \le & \max \left\{\phantom{\rule[-1.25em]{}{0ex}},d\left({w}_{1},w),\frac{1}{\left(1-\eta )}d(f\left(w),w)\right\}.\end{array}Therefore, {wn}\left\{{w}_{n}\right\}is bounded.□Lemma 3.7Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 2 and w∈Gw\in G. If limn→∞φn=0{\mathrm{lim}}_{n\to \infty }{\varphi }_{n}=0, then there exists a subsequence {mk}⊂N\left\{{m}_{k}\right\}\subset {\mathbb{N}}such that the sequence {d2(wmk+1,w)−d2(wmk,w)}\{{d}^{2}({w}_{{m}_{k}+1},w)-{d}^{2}({w}_{{m}_{k}},w)\}converges to 0.ProofLet w∈Γw\in \Gamma . Then the following similar arguments as in (3.14), we have d2(wn+1,w)≤∑k=0Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤∑k=0Mβnkd2(ynk,w).{d}^{2}({w}_{n+1},w)\le \mathop{\sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\le \mathop{\sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w).This, (3.26), and Lemma 2.13 imply (3.30)d2(wn+1,w)≤βn0d2(yn0,w)+∑j=1Mβnjd2(ynj,w)≤βn0d2(yn0,w)+(1−βn0)d2(wn,w)=βn0d2(Jλnh((1−φn)wn⊕φnf(wn)),Jλnh(w))+(1−βn0)d2(wn,w)\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},w)+\mathop{\displaystyle \sum }\limits_{j=1}^{M}{\beta }_{n}^{j}{d}^{2}({y}_{n}^{j},w)\\ & \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},w)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\\ & =& {\beta }_{n}^{0}{d}^{2}({J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),{J}_{{\lambda }_{n}}^{h}\left(w))+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\end{array}≤βn0d2((1−φn)wn⊕φnf(wn),w)+(1−βn0)d2(wn,w)≤(1−φn)βn0d2(wn,w)+φnβn0d2(f(wn),w)+(1−βn0)d2(wn,w)≤βn0d2(wn,w)+φnd2(f(wn),w)+(1−βn0)d2(wn,w)\begin{array}{rcl}& \le & {\beta }_{n}^{0}{d}^{2}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n}),w)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\\ & \le & \left(1-{\varphi }_{n}){\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+{\varphi }_{n}{\beta }_{n}^{0}{d}^{2}(f\left({w}_{n}),w)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\\ & \le & {\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},w)\end{array}(3.31)≤d2(wn,w)+φnd2(f(wn),w).\le \hspace{0.25em}{d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w).\hspace{13.2em}Now, by Lemma 3.6, if d2(wn+1,w)≤d2(wn,w){d}^{2}\left({w}_{n+1},w)\le {d}^{2}\left({w}_{n},w)for n∈Nn\in {\mathbb{N}}, then {d2(wn,w)}\left\{{d}^{2}\left({w}_{n},w)\right\}converges and consequently the proof is complete. Otherwise, there exists a subsequence {nk}\left\{{n}_{k}\right\}of {n}\left\{n\right\}such that d2(wnk,w)<d2(wnk+1,w){d}^{2}\left({w}_{{n}_{k}},w)\lt {d}^{2}\left({w}_{{n}_{k}+1},w)for every k∈Nk\in {\mathbb{N}}. Thus, by Lemma 2.20, there exists a subsequence {mk}⊂N\left\{{m}_{k}\right\}\subset {\mathbb{N}}such that mk→∞{m}_{k}\to \infty , d2(wmk,w)≤d2(wmk+1,w)andd2(wk,w)≤d2(wmk+1,w).{d}^{2}({w}_{{m}_{k}},w)\le {d}^{2}({w}_{{m}_{k}+1},w)\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{d}^{2}({w}_{k},w)\le {d}^{2}({w}_{{m}_{k}+1},w).Consequently, we obtain from (3.31) that □0≤limsupmk→∞(d2(wmk+1,w)−d2(wmk,w))≤limsupn→∞(d2(wn+1,w)−d2(wn,w))≤limsupn→∞(d2(wn,w)+φnd2(f(w),w)−d2(wn,w))≤limsupn→∞(φnd2(f(w),w))=0.\begin{array}{rcl}0& \le & \mathop{\mathrm{limsup}}\limits_{{m}_{k}\to \infty }({d}^{2}\left({w}_{{m}_{k}+1},w)-{d}^{2}\left({w}_{{m}_{k}},w))\\ & \le & \mathop{\mathrm{limsup}}\limits_{n\to \infty }({d}^{2}\left({w}_{n+1},w)-{d}^{2}\left({w}_{n},w))\\ & \le & \mathop{\mathrm{limsup}}\limits_{n\to \infty }({d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left(w),w)-{d}^{2}\left({w}_{n},w))\\ & \le & \mathop{\mathrm{limsup}}\limits_{n\to \infty }({\varphi }_{n}{d}^{2}(f\left(w),w))\\ & =& 0.\end{array}Lemma 3.8Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 2 and let {βnk}⊂[b,1)\left\{{\beta }_{n}^{k}\right\}\subset \left[b,1)for some b∈(0,1)b\in \left(0,1), limn→∞φn=0{\mathrm{lim}}_{n\to \infty }{\varphi }_{n}=0, and limn→∞αnj∈(κj,1){\mathrm{lim}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({\kappa }_{j},1). Suppose that {mk}\left\{{m}_{k}\right\}is the subsequence in Lemma 3.7. Then for each j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, the sequences {d(wn,Jμh(wn))}\{d({w}_{n},{J}_{\mu }^{h}\left({w}_{n}))\}, {d(wmk,Jμj(wmk))}\{d({w}_{{m}_{k}},{J}_{\mu }^{j}\left({w}_{{m}_{k}}))\}, and {dist(wmk,Tjwmk)}\{\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{{m}_{k}},{T}_{j}{w}_{{m}_{k}})\}converge to 0.ProofLet w∈Γw\in \Gamma . It follows from Lemma 2.17 and the fact that h(w)≤h(wn)h\left(w)\le h\left({w}_{n})that (3.32)d2(yn0,wn)≤d2(yn0,w)−d2(wn,w)+2μ(h(w)−h(wn))≤d2(yn0,w)−d2(wn,w)=d2(Jλnh((1−φn)wn⊕φnf(wn)),Jλnh(w))−d2(wn,w)≤d2((1−φn)wn⊕φnf(wn),w)−d2(wn,w)≤(1−φn)d2(wn,w)+φnd2(f(wn),w)−d2(wn,w)≤d2(wn,w)+φnd2(f(wn),w)−d2(wn,w)\begin{array}{rcl}{d}^{2}({y}_{n}^{0},{w}_{n})& \le & {d}^{2}({y}_{n}^{0},w)-{d}^{2}\left({w}_{n},w)+2\mu (h\left(w)-h\left({w}_{n}))\\ & \le & {d}^{2}({y}_{n}^{0},w)-{d}^{2}\left({w}_{n},w)\\ & =& {d}^{2}({J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),{J}_{{\lambda }_{n}}^{h}\left(w))-{d}^{2}\left({w}_{n},w)\\ & \le & {d}^{2}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n}),w)-{d}^{2}\left({w}_{n},w)\\ & \le & \left(1-{\varphi }_{n}){d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{d}^{2}\left({w}_{n},w)\\ & \le & {d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{d}^{2}\left({w}_{n},w)\end{array}(3.33)≤φnd2(f(wn),w).\le \hspace{0.33em}{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w).\hspace{9.7em}This and the assumptions on {φn}\left\{{\varphi }_{n}\right\}imply that (3.34)limn→∞d(wn,yn0)=0.\mathop{\mathrm{lim}}\limits_{n\to \infty }d({w}_{n},{y}_{n}^{0})=0.Consequently, it follows from Lemmas 2.16, 2.19, and 2.12 that (3.35)d(wn,Jμhwn)≤d(wn,yn0)+d(yn0,Jμhwn)=d(wn,yn0)+dJμh(wn),Jμhλn−μλnyn0⊕μμnwn≤d(wn,yn0)+dwn,λn−μλnyn0⊕μλnwn≤d(wn,yn0)+1−μλnd(wn,yn0)≤2−μλnd(wn,yn0)→0,asn→∞.\begin{array}{rcl}d\left({w}_{n},{J}_{\mu }^{h}{w}_{n})& \le & d({w}_{n},{y}_{n}^{0})+d({y}_{n}^{0},{J}_{\mu }^{h}{w}_{n})\\ & =& d({w}_{n},{y}_{n}^{0})+d\hspace{0.08em}\left({J}_{\mu }^{h}\left({w}_{n}),{J}_{\mu }^{h}\left(\frac{{\lambda }_{n}-\mu }{{\lambda }_{n}}{y}_{n}^{0}\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{w}_{n}\right)\right)\\ & \le & d({w}_{n},{y}_{n}^{0})+d\hspace{0.08em}\left({w}_{n},\left(\frac{{\lambda }_{n}-\mu }{{\lambda }_{n}}{y}_{n}^{0}\displaystyle \oplus \frac{\mu }{{\lambda }_{n}}{w}_{n}\right)\right)\\ & \le & d({w}_{n},{y}_{n}^{0})+\left(1-\frac{\mu }{{\lambda }_{n}}\right)\hspace{0.08em}d\left({w}_{n},{y}_{n}^{0})\\ & \le & \left(2-\frac{\mu }{{\lambda }_{n}}\right)\hspace{0.08em}d\left({w}_{n},{y}_{n}^{0})\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty .\end{array}Moreover, as in similar arguments to (3.14), we obtain (3.36)d2(wn+1,w)≤∑k=0Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤βn0d2(yn0,w)+∑k=1Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}This and (3.32) imply that (3.37)d2(wn+1,w)≤βn0d2(yn0,w)+∑k=1Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤βn0d2(wn,w)+βn0φnd2(f(wn),w)−βn0d2(wn,w)+∑k=1Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤βn0d2(wn,w)+φnd2(f(wn),w)+∑k=1Mβnkd2(ynk,w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & {\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+{\beta }_{n}^{0}{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & {\beta }_{n}^{0}{d}^{2}\left({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)+\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({y}_{n}^{k},w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}It follows from (3.26) and (3.37) that d2(wn+1,w)≤∑k=0Mβnkd2(wn,w)+φnd2(f(wn),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk))≤d2(wn,w)+φnd2(f(wn),w)−βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynk)).\begin{array}{rcl}{d}^{2}({w}_{n+1},w)& \le & \mathop{\displaystyle \sum }\limits_{k=0}^{M}{\beta }_{n}^{k}{d}^{2}({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k}))\\ & \le & {d}^{2}({w}_{n},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)-{\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{k})).\end{array}This implies that for every j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, βn0βnjd2(ϕj−1(n),Jμnj(ynj))≤βn0∑k=1Mβnkd2(ϕk−1(n),Jμnk(ynj))≤d2(wn,w)−d2(wn+1,w)+φnd2(f(wn),w).\begin{array}{rcl}{\beta }_{n}^{0}{\beta }_{n}^{j}{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))& \le & {\beta }_{n}^{0}\mathop{\displaystyle \sum }\limits_{k=1}^{M}{\beta }_{n}^{k}{d}^{2}({\phi }_{k-1}^{\left(n)},{J}_{{\mu }_{n}}^{k}({y}_{n}^{j}))\\ & \le & {d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w).\end{array}Therefore, (3.38)d2(ϕj−1(n),Jμnj(ynj))≤1βn0βnj[d2(wn,w)−d2(wn+1,w)+φnd2(f(wn),w)]≤1b2[d2(wn,w)−d2(wn+1,w)+φnd2(f(wn),w)].\begin{array}{rcl}{d}^{2}({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))& \le & \frac{1}{{\beta }_{n}^{0}{\beta }_{n}^{j}}{[}{d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)]\\ & \le & \frac{1}{{b}^{2}}{[}{d}^{2}({w}_{n},w)-{d}^{2}({w}_{n+1},w)+{\varphi }_{n}{d}^{2}(f\left({w}_{n}),w)].\end{array}It follows from Lemma 3.6, (3.38), and the assumption that limn→∞φn=0{\mathrm{lim}}_{n\to \infty }{\varphi }_{n}=0that limmk→∞d2(ϕj−1(mk),Jμmkj(ymkj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }{d}^{2}({\phi }_{j-1}^{\left({m}_{k})},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.Consequently, (3.39)limmk→∞d(ϕj−1(mk),Jμmkj(ymkj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }d({\phi }_{j-1}^{\left({m}_{k})},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.As in (3.17), we obtain that for every j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}, (3.40)d(ϕj−1(mk),ϕj(mk))≤d(ϕj−1(mk),Jμmkj(ymkj))→0,asmk→∞.d({\phi }_{j-1}^{\left({m}_{k})},{\phi }_{j}^{\left({m}_{k})})\le d({\phi }_{j-1}^{\left({m}_{k})},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}{m}_{k}\to \infty .Furthermore, for any j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}, we have (3.41)d(wn,Jμnj(ynj))=d(ϕ0(n),Jμnj(ynj))≤d(ϕ0(n),ϕ1(n))+d(ϕ1(n),ϕ2(n))+⋯+d(ϕj−2(n),ϕj−1(n))+d(ϕj−1(n),Jμnj(ynj))=∑k=1j−1d(ϕk−1(n),ϕk(n))+d(ϕj−1(n),Jμnj(ynj))\begin{array}{rcl}d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))& =& d({\phi }_{0}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\\ & \le & d({\phi }_{0}^{\left(n)},{\phi }_{1}^{\left(n)})+d({\phi }_{1}^{\left(n)},{\phi }_{2}^{\left(n)})+\cdots +d({\phi }_{j-2}^{\left(n)},{\phi }_{j-1}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\\ & =& \mathop{\displaystyle \sum }\limits_{k=1}^{j-1}d({\phi }_{k-1}^{\left(n)},{\phi }_{k}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))\end{array}(3.42)≤∑k=1Md(ϕk−1(n),ϕk(n))+d(ϕj−1(n),Jμnj(ynj)).\le \hspace{0.33em}\mathop{\sum }\limits_{k=1}^{M}d({\phi }_{k-1}^{\left(n)},{\phi }_{k}^{\left(n)})+d({\phi }_{j-1}^{\left(n)},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})).\hspace{7em}It follows from (3.42), (3.39), and (3.40) that (3.43)limmk→∞d(wmk,Jμmkj(ymkj))=0,∀j∈{1,…,M}.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }d({w}_{{m}_{k}},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))=0,\hspace{0.33em}\forall j\in \left\{1,\ldots ,M\right\}.Also, as in (3.2), we have for each j∈{1,…,M}j\in \left\{1,\ldots ,M\right\}and for w∈Γw\in \Gamma that (3.44)(1−αmkj)(αmkj−κj)d2(wmk,wmkj)≤d2(wmk,w)−d2(wmk+1,w).\left(1-{\alpha }_{{m}_{k}}^{j})\left({\alpha }_{{m}_{k}}^{j}-{\kappa }_{j}){d}^{2}\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})\le {d}^{2}\left({w}_{{m}_{k}},w)-{d}^{2}\left({w}_{{m}_{k}+1},w).This, Lemma 3.7, and the assumption that liminfn→∞αnj∈(κj,1){\mathrm{liminf}}_{n\to \infty }{\alpha }_{n}^{j}\in \left({\kappa }_{j},1)imply limmk→∞d2(wmk,wmkj)=0.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }{d}^{2}\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})=0.Consequently, (3.45)limmk→∞d(wmk,wmkj)=0,for everyj∈{1,2,…,M}.\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }d\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})=0,\hspace{1em}\hspace{0.1em}\text{for every}\hspace{0.1em}\hspace{0.33em}j\in \left\{1,2,\ldots ,M\right\}.Hence, for each j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}, (3.46)dist(wmk,Tjwmk)≤d(wmk,wmkj)→0,asmk→∞.\hspace{0.1em}\text{dist}\hspace{0.1em}\left({w}_{{m}_{k}},{T}_{j}{w}_{{m}_{k}})\le d\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})\to 0,\hspace{0.33em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}{m}_{k}\to \infty .Moreover, it follows from Lemmas 2.5 and 2.12 that (3.47)d(wn,Jμj(wn))≤d(wn,Jμnj(ynj))+d(Jμnj(ynj),Jμj(wn))=d(wn,Jμnj(ynj))+dJμj(wn),Jμjμn−μμnJμnj(ynj)⊕μμnynj≤d(wn,Jμnj(ynj))+dwn,μn−μμnJμnj(ynj)⊕μμnynj≤2−μμnd(wn,Jμnj(ynj))+μμnd(wn,ynj)≤2d(wn,Jμnj(ynj))+d(wn,ynj)≤2d(wn,Jμnj(ynj))+d(wn,wnj).\begin{array}{rcl}d({w}_{n},{J}_{\mu }^{j}\left({w}_{n}))& \le & d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d({J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),{J}_{\mu }^{j}\left({w}_{n}))\\ & =& d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\hspace{0.08em}\left({J}_{\mu }^{j}\left({w}_{n}),{J}_{\mu }^{j}\left(\frac{{\mu }_{n}-\mu }{{\mu }_{n}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{y}_{n}^{j}\right)\right)\\ & \le & d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\hspace{0.08em}\left({w}_{n},\left(\frac{{\mu }_{n}-\mu }{{\mu }_{n}}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j})\displaystyle \oplus \frac{\mu }{{\mu }_{n}}{y}_{n}^{j}\right)\right)\\ & \le & \left(2-\frac{\mu }{{\mu }_{n}}\right)d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+\frac{\mu }{{\mu }_{n}}d\left({w}_{n},{y}_{n}^{j})\\ & \le & 2d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\left({w}_{n},{y}_{n}^{j})\\ & \le & 2d({w}_{n},{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}))+d\left({w}_{n},{w}_{n}^{j}).\end{array}This, (3.43) and (3.45) imply that (3.48)d(wmk,Jμj(wmk))≤2d(wmk,Jμmkj(ymkj))+d(wmk,wmkj)→0,d({w}_{{m}_{k}},{J}_{\mu }^{j}\left({w}_{{m}_{k}}))\le 2d({w}_{{m}_{k}},{J}_{{\mu }_{{m}_{k}}}^{j}({y}_{{m}_{k}}^{j}))+d\left({w}_{{m}_{k}},{w}_{{m}_{k}}^{j})\to 0,for every j∈{1,2,…,M}j\in \left\{1,2,\ldots ,M\right\}. Therefore, by (3.35), (3.48), and (3.46), we have the complete proof.□Theorem 3.9Let {wn}\left\{{w}_{n}\right\}be a sequence generated by Algorithm 2 and let {βnk}\left\{{\beta }_{n}^{k}\right\}, {φn}\left\{{\varphi }_{n}\right\}, and {αn}\left\{{\alpha }_{n}\right\}be as in Lemma (3.8). Suppose that ∑n=1∞βn0φn=+∞{\sum }_{n=1}^{\infty }{\beta }_{n}^{0}{\varphi }_{n}=+\infty . Then the sequence {wn}\left\{{w}_{n}\right\}converges strongly to an element of Γ\Gamma .ProofFrom Lemma 3.6, we have that {wn}\left\{{w}_{n}\right\}is bounded and consequently {wmk}\left\{{w}_{{m}_{k}}\right\}is bounded. Then by Lemmas 2.9 and 2.8, there exists a subsequence {zmk}\left\{{z}_{{m}_{k}}\right\}of {wmk}\left\{{w}_{{m}_{k}}\right\}that Δ\Delta -converges to zzfor some z∈Yz\in Y. By Lemma (3.7) and the assumption that each Tp{T}_{p}satisfies demiclosedness type property, we have (3.49)z∈Γ.z\in \Gamma .Without loss of generality, we may assume that {xmk}\left\{{x}_{{m}_{k}}\right\}is a subsequence of {wmk}\left\{{w}_{{m}_{k}}\right\}that Δ\Delta -converges to zzin YYand (3.50)limsupmk→∞⟨wmkz→,f(z)z→⟩=limmk→∞⟨xmkz→,f(z)z→⟩.\mathop{\mathrm{limsup}}\limits_{{m}_{k}\to \infty }\langle \overrightarrow{{w}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle =\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }\langle \overrightarrow{{x}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle .This and Lemma 2.15 imply that (3.51)limsupmk→∞⟨wmkz→,f(z)z→⟩=limmk→∞⟨xmkz→,f(z)z→⟩≤0.\mathop{\mathrm{limsup}}\limits_{{m}_{k}\to \infty }\langle \overrightarrow{{w}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle =\mathop{\mathrm{lim}}\limits_{{m}_{k}\to \infty }\langle \overrightarrow{{x}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle \le 0.It follows from (3.30), Lemma 2.12, and (3.49) that (3.52)d2(wn+1,z)≤βn0d2(yn0,z)+(1−βn0)d2(wn,z)=βn0d2(Jλnh((1−φn)wn⊕φnf(wn)),Jλnh(z))+(1−βn0)d2(wn,z)≤βn0d2((1−φn)wn⊕φnf(wn),z)+(1−βn0)d2(wn,z).\begin{array}{rcl}{d}^{2}({w}_{n+1},z)& \le & {\beta }_{n}^{0}{d}^{2}({y}_{n}^{0},z)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},z)\\ & =& {\beta }_{n}^{0}{d}^{2}({J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),{J}_{{\lambda }_{n}}^{h}\left(z))+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},z)\\ & \le & {\beta }_{n}^{0}{d}^{2}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n}),z)+\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},z).\end{array}This and Lemma 2.14 imply that d2(wn+1,z)≤βn0(1−φn)2d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φn(1−φn)⟨wnz→,f(wn)z→⟩+(1−βn0)d2(wn,z)=(1−2βn0φn+βn0φn2)d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φn(1−φn)[⟨wnz→,f(wn)f(z)→⟩+⟨wnz→,f(z)z→⟩].\begin{array}{rcl}{d}^{2}({w}_{n+1},z)& \le & {\beta }_{n}^{0}{\left(1-{\varphi }_{n})}^{2}{d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n})\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left({w}_{n})z}\rangle +\left(1-{\beta }_{n}^{0}){d}^{2}({w}_{n},z)\\ & =& (1-2{\beta }_{n}^{0}{\varphi }_{n}+{\beta }_{n}^{0}{\varphi }_{n}^{2}){d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n}){[}\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left({w}_{n})f\left(z)}\rangle +\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle ].\end{array}This, Cauchy-Schwartz inequality, and the assumption that ffis contraction mapping imply d2(wn+1,z)≤(1−2βn0φn+βn0φn2)d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φn(1−φn)[d(wn,z)⋅d(f(wn),f(z))+⟨wnz→,f(z)z→⟩]≤(1−2βn0φn+βn0φn2)d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φn(1−φn)ηd2(wn,z)+2βn0φn(1−φn)⟨wnz→,f(z)z→⟩≤(1−2βn0φn+βn0φn2)d2(wn,z)+βn0φn2d2(f(wn),z)+2βn0φnηd2(wn,z)+2βn0φn(1−φn)⟨wnz→,f(z)z→⟩=(1−2βn0φn(1−η))d2(wn,z)+2βn0φn(1−η)φn2(1−η)d2(wn,z)+φn2(1−η)d2(f(wn),z)+(1−φn)(1−η)⟨wnz→,f(z)z→⟩.\begin{array}{rcl}{d}^{2}({w}_{n+1},z)& \le & (1-2{\beta }_{n}^{0}{\varphi }_{n}+{\beta }_{n}^{0}{\varphi }_{n}^{2}){d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n}){[}d\left({w}_{n},z)\cdot d(f\left({w}_{n}),f\left(z))+\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle ]\\ & \le & (1-2{\beta }_{n}^{0}{\varphi }_{n}+{\beta }_{n}^{0}{\varphi }_{n}^{2}){d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n})\eta {d}^{2}\left({w}_{n},z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n})\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle \\ & \le & (1-2{\beta }_{n}^{0}{\varphi }_{n}+{\beta }_{n}^{0}{\varphi }_{n}^{2}){d}^{2}({w}_{n},z)+{\beta }_{n}^{0}{\varphi }_{n}^{2}{d}^{2}(f\left({w}_{n}),z)+2{\beta }_{n}^{0}{\varphi }_{n}\eta {d}^{2}\left({w}_{n},z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-{\varphi }_{n})\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle \\ & =& (1-2{\beta }_{n}^{0}{\varphi }_{n}\left(1-\eta )){d}^{2}({w}_{n},z)+2{\beta }_{n}^{0}{\varphi }_{n}\left(1-\eta )\left[\frac{{\varphi }_{n}}{2\left(1-\eta )}{d}^{2}({w}_{n},z)+\frac{{\varphi }_{n}}{2\left(1-\eta )}{d}^{2}(f\left({w}_{n}),z)+\frac{\left(1-{\varphi }_{n})}{\left(1-\eta )}\langle \overrightarrow{{w}_{n}z},\overrightarrow{f\left(z)z}\rangle \right].\end{array}This implies that (3.53)d2(wmk+1,z)≤(1−σmk)d2(wmk,w)+σmkϕmk,{d}^{2}\left({w}_{{m}_{k}+1},z)\le \left(1-{\sigma }_{{m}_{k}}){d}^{2}\left({w}_{{m}_{k}},w)+{\sigma }_{{m}_{k}}{\phi }_{{m}_{k}},where σmk=2βmk0φmk(1−η){\sigma }_{{m}_{k}}=2{\beta }_{{m}_{k}}^{0}{\varphi }_{{m}_{k}}\left(1-\eta )and ϕmk=φmk2(1−η)d2(wmk,z)+φmk2(1−η)d2(f(wmk),z)+(1−φmk)(1−η)⟨wmkz→,f(z)z→⟩.{\phi }_{{m}_{k}}=\frac{{\varphi }_{{m}_{k}}}{2\left(1-\eta )}{d}^{2}({w}_{{m}_{k}},z)+\frac{{\varphi }_{{m}_{k}}}{2\left(1-\eta )}{d}^{2}(f\left({w}_{{m}_{k}}),z)+\frac{\left(1-{\varphi }_{{m}_{k}})}{\left(1-\eta )}\langle \overrightarrow{{w}_{{m}_{k}}z},\overrightarrow{f\left(z)z}\rangle .Therefore, from (3.53), (3.51), and the assumptions on {φn}\left\{{\varphi }_{n}\right\}, we conclude by Lemma 2.21 that {wmk}\left\{{w}_{{m}_{k}}\right\}converges strongly to zz. Moreover, since (3.54)d2(wk,z)≤d2(wmk+1,z),{d}^{2}({w}_{k},z)\le {d}^{2}({w}_{{m}_{k}+1},z),we have that {wn}\left\{{w}_{n}\right\}converges strongly to zz.□The following corollary is obtained from the fact that every quasi-nonexpansive mapping is 0-demicontractive.Corollary 3.10Let Tp,p=1,2,…,M2{T}_{p},\hspace{0.33em}p=1,2,\ldots ,{M}_{2}be quasi-nonexpansive mappings with demiclosedness-type property. Suppose that {αnj},{βnj},{wn}\left\{{\alpha }_{n}^{j}\right\},\left\{{\beta }_{n}^{j}\right\},\left\{{w}_{n}\right\}, and Γ\Gamma are as in Theorem 3.9. Then {wn}\left\{{w}_{n}\right\}converges strongly to a member of Γ\Gamma .Corollary 3.11Let Tp,p=1,2,…,M2{T}_{p},\hspace{0.33em}p=1,2,\ldots ,{M}_{2}be nonexpansive mappings and suppose {αnj},{βnj},{wn}\left\{{\alpha }_{n}^{j}\right\},\hspace{0.33em}\left\{{\beta }_{n}^{j}\right\},\hspace{0.33em}\left\{{w}_{n}\right\}, and Γ\Gamma are as in Theorem 3.9. Then {wn}\left\{{w}_{n}\right\}converges strongly to a member of Γ\Gamma .4Applications and numerical exampleIn this section, we apply our results to find mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We also give a numerical example to support the proposed methods. We shall maintain the notation (W,d)\left(W,d)for a Hadamard space and YYits nonempty convex closed subset.Let M=1M=1, TTbe a 0-demicontractive map, {μn}⊂(μ,+∞),{λn}⊂(μ,+∞)\left\{{\mu }_{n}\right\}\subset \left(\mu ,+\infty ),\hspace{0.33em}\left\{{\lambda }_{n}\right\}\subset \left(\mu ,+\infty )for some μ>0\mu \gt 0, {βn0}⊂(0,1)\left\{{\beta }_{n}^{0}\right\}\subset \left(0,1), {φn}\left\{{\varphi }_{n}\right\}, {αn0}⊂[0,1]\left\{{\alpha }_{n}^{0}\right\}\subset \left[0,1], and ffas η\eta -contraction map on YY. Then Algorithms 1 and 2 reduce to the following: (4.1)yn0=Jλnh(wn),yn1=αn0wn⊕(1−αn0)wn∗,wn∗∈Twn,wn+1=βn0yn0⊕(1−βn0)Jμn(yn1),\left\{\begin{array}{l}{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}({w}_{n}),\hspace{1.0em}\\ {y}_{n}^{1}={\alpha }_{n}^{0}{w}_{n}\displaystyle \oplus \left(1-{\alpha }_{n}^{0}){w}_{n}^{\ast },\hspace{0.33em}{w}_{n}^{\ast }\in T{w}_{n},\hspace{1.0em}\\ {w}_{n+1}={\beta }_{n}^{0}{y}_{n}^{0}\displaystyle \oplus \left(1-{\beta }_{n}^{0}){J}_{{\mu }_{n}}({y}_{n}^{1}),\hspace{1.0em}\end{array}\right.and (4.2)yn0=Jλnh((1−φn)wn⊕φnf(wn)),yn1=αn0wn⊕(1−αn0)wn∗,wn∗∈Twn,wn+1=βn0yn0⊕(1−βn0)Jμn(yn1),\left\{\begin{array}{l}{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),\hspace{1.0em}\\ {y}_{n}^{1}={\alpha }_{n}^{0}{w}_{n}\displaystyle \oplus \left(1-{\alpha }_{n}^{0}){w}_{n}^{\ast },\hspace{0.33em}{w}_{n}^{\ast }\in T{w}_{n},\hspace{1.0em}\\ {w}_{n+1}={\beta }_{n}^{0}{y}_{n}^{0}\displaystyle \oplus \left(1-{\beta }_{n}^{0}){J}_{{\mu }_{n}}({y}_{n}^{1}),\hspace{1.0em}\end{array}\right.respectively.4.1Mean and median of probabilitiesGiven a probability measure μ∈P2(W)\mu \in {{\mathcal{P}}}^{2}\left(W). Let ffand ggbe two real-valued functions on YYdefined by f(w)=∫d(w,z)dμ(z),g(w)=∫d2(w,z)dμ(z),for everyw∈Y.f\left(w)=\int d\left(w,z)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(z),\hspace{0.33em}g\left(w)=\int {d}^{2}\left(w,z)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(z),\hspace{0.1em}\text{for every}\hspace{0.1em}w\in Y.Then the minimizers of ffand ggare called median and mean of probability, respectively (see e.g. [4]). Moreover, by properties of metric dd, ff, and ggare convex proper and lower semi-continuous functions on YY.Take A:Y→TYA:Y\to TYsuch that Aw={0}Aw=\left\{{\bf{0}}\right\}and TTbe such that Tz={z}Tz=\left\{z\right\}. Then AAis MVF that satisfies condition (S)\left(S)and TTis 0-demicontractive mapping with demiclosedness-type property and Γ≠∅\Gamma \ne \varnothing . For h=fh=f, the sequences in (4.1) and (4.2) approximate a median, and for h=gh=g, the sequences in (4.1) and (4.2) approximate the mean.4.2Optimal energy for measurable mappingsLet (Y,χ,μ)\left(Y,\chi ,\mu )be a measure space with measure μ\mu and let h:Y→Wh:Y\to Wbe a measurable mapping. Consider the following set of measurable mappings: (4.3)L2(Y,W,h)≔{g:Y→Wmeasurable:d(g(⋅),h(⋅))∈L2(Y)}{L}^{2}\left(Y,W,h):= \{g:Y\to W\hspace{0.33em}\hspace{0.1em}\text{measurable:}\hspace{0.1em}\hspace{0.33em}d(g\left(\cdot ),h\left(\cdot ))\in {L}^{2}\left(Y)\}equipped with the L2{L}^{2}-metric (4.4)d2(g,f)≔∫Yd(g(w),f(w))2dμ(w),g,f∈L2(Y,W,h).{d}_{2}(g,f):= \mathop{\int }\limits_{Y}d(g\left(w),f\left(w){)}^{2}\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(w),\hspace{0.33em}g,f\in {L}^{2}\left(Y,W,h).Then, as in [44], L2(Y,W,h){L}^{2}\left(Y,W,h)together with metric d2{d}_{2}forms a Hadamard space. Furthermore, the energy of a measurable mapping g:Y→Wg:Y\to Wis given by (4.5)E(g)≔12∫Y∫Yd(g(u),g(w))2ρ(u,dw)dμ(u),E\left(g):= \frac{1}{2}\mathop{\int }\limits_{Y}\mathop{\int }\limits_{Y}d(g\left(u),g\left(w){)}^{2}\rho \left(u,\hspace{0.1em}\text{d}\hspace{0.1em}w)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(u),where ρ(u,dw)\rho \left(u,\hspace{0.1em}\text{d}\hspace{0.1em}w)is a Markov kernel that is symmetric with respect to μ\mu in the sense that ρ(u,dw)dμ(u)=ρ(w,du)dμ(w)\rho \left(u,\hspace{0.1em}\text{d}\hspace{0.1em}w)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(u)=\rho \left(w,\hspace{0.1em}\text{d}\hspace{0.1em}u)\hspace{0.1em}\text{d}\hspace{0.1em}\mu \left(w).It follows from [44] that the set of optimal energy coincides with the fixed-point set of Markov operator PP. Moreover, the Markov operator is a singlevalued nonexpansive mapping on L2(Y,W,h){L}^{2}\left(Y,W,h). Now, let A:Y→TYA:Y\to TYbe defined by Aw={0}Aw=\left\{{\bf{0}}\right\}, TTbe defined by Tz={Pz}Tz=\left\{Pz\right\}, and hhbe a zero functional. Then, it is easy to see that the assumptions on AA, TT, and hhare all satisfied. Thus, the sequences in (4.1) and (4.2), with AA, TT, hhimmediate, approximate an optimal energy.4.3Two-arm robotic motion controlLet k∈{1,2,…,m}k\in \left\{1,2,\ldots ,m\right\}, hk{h}_{k}be defined by hk(w)=d2(w,ξk){h}_{k}\left(w)={d}^{2}\left(w,{\xi }_{k}), for every w∈Yw\in Yand some ξk∈Y{\xi }_{k}\in Y. Then by properties of metric dd, each hk{h}_{k}is convex proper and lower semi-continuous functions. The problem of finding the minimizers of hk{h}_{k}at each kksolves large optimization problems. We now consider a special case of Y=R2Y={{\mathbb{R}}}^{2}equipped with the Euclidean distance ddand analyse a discrete-time kinematics problem of two-arm robotic manipulator. That is the problem of solving (4.6)minhk(δk),\min {h}_{k}\left({\delta }_{k}),at each time tk{t}_{k}, where δk=g(θk){\delta }_{k}=g\left({\theta }_{k})is the end effector and ggis the kinematic mapping as given in [45]. For three-arm robotic motion see [49]. In this experiment, we shall use Algorithm 2 (which reduces to (4.2)) to track the following curve: ξk=1.5+0.2sin(tk)0.5+0.2sin3tk+π6.{\xi }_{k}=\left[\begin{array}{c}1.5+0.2\sin \left({t}_{k})\\ 0.5+0.2\sin \left(3{t}_{k}+\frac{\pi }{6}\right)\end{array}\right].For full numerical display, we set Tw={w}Tw=\left\{w\right\}, f(w)=w5f\left(w)=\frac{w}{5}, Aw={0}Aw=\left\{{\bf{0}}\right\}for every w∈Yw\in Yand take Jλnhk{J}_{{\lambda }_{n}}^{{h}_{k}}be the resolvent of hk{h}_{k}at each k=1,2,…,mk=1,2,\ldots ,m. We take equal arms’ length as 1, set the time range of 10 s into 200 parts, that is m=200m=200, and take the starting point w1=0,π4T{w}_{1}={\left(0,\frac{\pi }{4}\right)}^{T}. Other control parameters are as follows: λn=μn=20{\lambda }_{n}={\mu }_{n}=20, βn0=12{\beta }_{n}^{0}=\frac{1}{2}, φn=1n{\varphi }_{n}=\frac{1}{n}, and αn0=n2n+3{\alpha }_{n}^{0}=\frac{n}{2n+3}. Then, at each k∈{1,2,…,200}k\in \left\{1,2,\ldots ,200\right\}, (4.2) reduces to the following: (4.7)wn+1=12J20hk5n−45nwn+12wn,{w}_{n+1}=\frac{1}{2}{J}_{20}^{{h}_{k}}\left(\frac{5n-4}{5n}{w}_{n}\right)+\frac{1}{2}{w}_{n},which yield the result in Figure 1.Figure 1Numerical display for two-arm robotic motion control. (a) Synthesized trajectory. (b) End effector trajectory and desired path. (c) Tracking error on horizontal axis. (d) Tracking error on vertical axis.Remark 4.1Figure 1(a) and (b) shows that the process is completed successfully. Furthermore, Figure 1(c) and (d) signifies that the residual error is optimal.4.4Numerical exampleIn this subsection, we present a numerical example in a non-Hilbert Hadamard space to show the applicability of our results. All codes are written and executed in Matlab R2021b and run on Acer laptop (Swift SF514-55TA, 11th Gen Intel(R) Core(TM) i5-1135G7).Example 4.2Let Y=W=R2Y=W={{\mathbb{R}}}^{2}with the metric dY:R2×R2→R{d}_{Y}:{{\mathbb{R}}}^{2}\times {{\mathbb{R}}}^{2}\to {\mathbb{R}}defined by dY((w1,w2),(z1,z2))=(w1−z1)2+(w12−w2−z12+z2)2.{d}_{Y}(\left({w}_{1},{w}_{2}),\left({z}_{1},{z}_{2}))=\sqrt{{\left({w}_{1}-{z}_{1})}^{2}+{\left({w}_{1}^{2}-{w}_{2}-{z}_{1}^{2}+{z}_{2})}^{2}}.Then (W,dY)(W,{d}_{Y})is a non-Hilbert Hadamard space (see e.g. [16, Example 5.2]) and the geodesic operation, (1−t)(w1,w2)⊕t(v1,v2)\left(1-t)\left({w}_{1},{w}_{2})\oplus t\left({v}_{1},{v}_{2}), is given by ((1−t)w1+tv1,((1−t)w1+tv1)2−((1−t)(w12−w2)+t(v12−v2))).(\left(1-t){w}_{1}+t{v}_{1},(\left(1-t){w}_{1}+t{v}_{1}{)}^{2}-(\left(1-t)\left({w}_{1}^{2}-{w}_{2})+t\left({v}_{1}^{2}-{v}_{2}))).Let h,g:Y→Rh,g:Y\to {\mathbb{R}}and Tj:Y→Cℬ(Y),j=1,2,3,…,10,{T}_{j}:Y\to {\mathcal{C {\mathcal B} }}\left(Y),\hspace{0.33em}j=1,2,3,\ldots ,10,be defined by h(w1,w2)=200((w2+1)−(w1+1)2)2+w12,∀(w1,w2)∈Y,g(w1,w2)=50w12∀(w1,w2)∈Y,Tj(w1,w2)=0,j10w1×j10w2,0,ifw2<0<w1,0,j10∣w1∣×{0},otherwise.\begin{array}{rcl}h({w}_{1},{w}_{2})& =& 200(\left({w}_{2}+1)-{({w}_{1}+1)}^{2}{)}^{2}+{w}_{1}^{2},\hspace{0.33em}\forall \left({w}_{1},{w}_{2})\in Y,\\ g({w}_{1},{w}_{2})& =& 50{w}_{1}^{2}\hspace{0.33em}\forall \left({w}_{1},{w}_{2})\in Y,\\ {T}_{j}({w}_{1},{w}_{2})& =& \left\{\begin{array}{ll}\left[0,\sqrt{\frac{j}{10}}{w}_{1}\right]\times \left[\frac{j}{10}{w}_{2},0\right],\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{w}_{2}\lt 0\lt {w}_{1},\\ \left[0,\sqrt{\frac{j}{10}}| {w}_{1}| \right]\times \{0\},\hspace{1.0em}& \hspace{0.1em}\text{otherwise}\hspace{0.1em}.\end{array}\right.\end{array}It follows from [41, Example 1] that each Tj{T}_{j}is multivalued 0-demicontractive mapping. Also, hhand ggare convex proper lower semicontinuous functions. Now, setting A1{A}_{1}to be the subdifferential of gg, then A1{A}_{1}and hhsatisfy the required conditions and Γ=(0,0)\Gamma =\left(0,0). Moreover, Algorithms 1 and 2, respectively, reduce to the following: (4.8)yn0=Jλnh(wn),ynj=αnjwn⊕(1−αnj)wnj,wnj∈Tjwn,j=1,2,3,…,10,wn+1=⊕j=010βnjJμnj(ynj),\left\{\begin{array}{l}{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}({w}_{n}),\hspace{1.0em}\\ {y}_{n}^{j}={\alpha }_{n}^{j}{w}_{n}\displaystyle \oplus \left(1-{\alpha }_{n}^{j}){w}_{n}^{j},\hspace{0.33em}{w}_{n}^{j}\in {T}_{j}{w}_{n},\hspace{0.33em}j=1,2,3,\ldots ,10,\hspace{1.0em}\\ {w}_{n+1}=\underset{j=0}{\overset{10}{\displaystyle \oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),\hspace{1.0em}\end{array}\right.and (4.9)yn0=Jλnh((1−φn)wn⊕φnf(wn)),ynj=αnjwn⊕(1−αnj)wnj,wnj∈Tjwn,j=1,2,3,…,10,wn+1=⊕j=010βnjJμnj(ynj),\left\{\begin{array}{l}{y}_{n}^{0}={J}_{{\lambda }_{n}}^{h}(\left(1-{\varphi }_{n}){w}_{n}\displaystyle \oplus {\varphi }_{n}f\left({w}_{n})),\hspace{1.0em}\\ {y}_{n}^{j}={\alpha }_{n}^{j}{w}_{n}\displaystyle \oplus \left(1-{\alpha }_{n}^{j}){w}_{n}^{j},\hspace{0.33em}{w}_{n}^{j}\in {T}_{j}{w}_{n},\hspace{0.33em}j=1,2,3,\ldots ,10,\hspace{1.0em}\\ {w}_{n+1}=\underset{j=0}{\overset{10}{\displaystyle \oplus }}{\beta }_{n}^{j}{J}_{{\mu }_{n}}^{j}({y}_{n}^{j}),\hspace{1.0em}\end{array}\right.where Jλnhv=argminw∈Yh(w)+12λndY2(w,v),Jμn1=argminw∈Yg(w)+12μndY2(w,v){J}_{{\lambda }_{n}}^{h}v={{\rm{argmin}}}_{w\in Y}\left\{h\left(w)+\frac{1}{2{\lambda }_{n}}{d}_{Y}^{2}\left(w,v)\right\},\hspace{0.33em}{J}_{{\mu }_{n}}^{1}={{\rm{argmin}}}_{w\in Y}\left\{g\left(w)+\frac{1}{2{\mu }_{n}}{d}_{Y}^{2}\left(w,v)\right\}, and Jμnj:x↦x{J}_{{\mu }_{n}}^{j}:x\mapsto xfor j=0,2,3,4,…,10.j=0,2,3,4,\ldots ,10.In the experiment, we take f:w↦w5f:w\mapsto \frac{w}{5}, λn=μn=50{\lambda }_{n}={\mu }_{n}=50, φn=13n{\varphi }_{n}=\frac{1}{3n}, αnj=1j+1{\alpha }_{n}^{j}=\frac{1}{j+1}, βnj=10242047(2j){\beta }_{n}^{j}=\frac{1024}{2047\left({2}^{j})}, j=0,1,2,…,10j=0,1,2,\ldots ,10, and we choose wnj=j10wn1,j10wn2{w}_{n}^{j}=\left(\sqrt{\frac{j}{10}}{w}_{{n}_{1}},\frac{j}{10}{w}_{{n}_{2}}\right)if wn2<0<wn1{w}_{{n}_{2}}\lt 0\lt {w}_{{n}_{1}}and wnj=j10∣wn1∣,0{w}_{n}^{j}=\left(\sqrt{\frac{j}{10}}| {w}_{{n}_{1}}| ,0\right)otherwise. We use Matlab function “fminsearch” for evaluation of argmin with initial search term as the input value and in reporting the results, we denote Algorithm 1 (resp. Algorithm 2) by Alg1 (resp. Alg2). We test the proposed algorithm using the starting points w1=(−23,7){w}_{1}=\left(-23,7), w1=(50,20){w}_{1}=\left(50,20), w1=(0.9,−16){w}_{1}=\left(0.9,-16)for both algorithms. The obtained results are shown in Tables 1 and 2.Table 1Few values {wn}\left\{{w}_{n}\right\}by Alg1 and Alg2 from Example 4.2wn{w}_{n}nnAlg1Alg2Alg1Alg2Alg1Alg21(−23,7-23,7)(−23,7-23,7)(50,2050,20)(50,2050,20)(0.9,−160.9,-16)(0.9,−160.9,-16)2(−8.9256,−67.7637)\left(-8.9256,-67.7637)(−7.6596,−86.2257)\left(-7.6596,-86.2257)(−4.7251,−655.1734)\left(-4.7251,-655.1734)(0.45729,−666.7701)\left(0.45729,-666.7701)(−0.0055069,−4.7165)\left(-0.0055069,-4.7165)(0.033795,−4.6367)\left(0.033795,-4.6367)3(−2.9384,−28.2297)\left(-2.9384,-28.2297)(−2.5211,−28.1613)\left(-2.5211,-28.1613)(−7.1975,−93.4627)\left(-7.1975,-93.4627)(−4.353,−164.6805)\left(-4.353,-164.6805)(−0.006031,−0.92149)\left(-0.006031,-0.92149)(−0.032863,−1.2912)\left(-0.032863,-1.2912)4(−0.85612,−7.7147)\left(-0.85612,-7.7147)(−0.73399,−7.1938)\left(-0.73399,-7.1938)(−2.6208,−27.602)\left(-2.6208,-27.602)(−2.3506,−34.5071)\left(-2.3506,-34.5071)(−0.0067264,−0.18944)\left(-0.0067264,-0.18944)(−0.016925,−0.27236)\left(-0.016925,-0.27236)5(−0.22759,−1.8001)\left(-0.22759,-1.8001)(−0.19503,−1.638)\left(-0.19503,-1.638)(−0.77876,−7.2087)\left(-0.77876,-7.2087)(−0.76101,−8.273)\left(-0.76101,-8.273)(−0.0029538,−0.040273)\left(-0.0029538,-0.040273)(−0.0053266,−0.057669)\left(-0.0053266,-0.057669)6(−0.056753,−0.39563)\left(-0.056753,-0.39563)(−0.048658,−0.35625)\left(-0.048658,-0.35625)(−0.20827,−1.6648)\left(-0.20827,-1.6648)(−0.21077,−1.8732)\left(-0.21077,-1.8732)(−0.002957,−0.012734)\left(-0.002957,-0.012734)(−0.0049952,−0.019333)\left(-0.0049952,-0.019333)⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots 85(−1.9078×10−9,(-1.9078\times 1{0}^{-9},−7.8638×10−14)-7.8638\times 1{0}^{-14})(−5.0291×10−10,(-5.0291\times 1{0}^{-10},−1.889×10−14)-1.889\times 1{0}^{-14})(−2.0876×10−9,(-2.0876\times 1{0}^{-9},−9.9453×10−14)-9.9453\times 1{0}^{-14})(−7.14×10−10,(-7.14\times 1{0}^{-10},−2.6563×10−14)-2.6563\times 1{0}^{-14})(−1.1067×1−9,(-1.1067\times {1}^{-9},−3.2876×10−14)-3.2876\times 1{0}^{-14})(−1.0945×1−9,(-1.0945\times {1}^{-9},−3.0742×10−14)-3.0742\times 1{0}^{-14})86(−1.5871×10−9,(-1.5871\times 1{0}^{-9},−5.6499×10−14)-5.6499\times 1{0}^{-14})(−4.1787×10−10,(-4.1787\times 1{0}^{-10},−1.3543×10−14)-1.3543\times 1{0}^{-14})(−1.7367×10−9,(-1.7367\times 1{0}^{-9},−7.1453×10−14)-7.1453\times 1{0}^{-14})(−5.9326×10−10,(-5.9326\times 1{0}^{-10},−1.9044×10−14)-1.9044\times 1{0}^{-14})(−9.2069×10−10,(-9.2069\times 1{0}^{-10},−2.362×10−14)-2.362\times 1{0}^{-14})(−9.0946×10−10,(-9.0946\times 1{0}^{-10},−2.2039×10−14)-2.2039\times 1{0}^{-14})87(−1.3203×10−9,(-1.3203\times 1{0}^{-9},−4.0592×10−14)-4.0592\times 1{0}^{-14})(−3.4721×10−10,(-3.4721\times 1{0}^{-10},−9.7092×10−15)-9.7092\times 1{0}^{-15})(−1.4447×10−9,(-1.4447\times 1{0}^{-9},−5.1336×10−14)-5.1336\times 1{0}^{-14})(−4.9295×10−10,(-4.9295\times 1{0}^{-10},−1.3653×10−14)-1.3653\times 1{0}^{-14})(−7.659×10−10,(-7.659\times 1{0}^{-10},−1.697×10−14)-1.697\times 1{0}^{-14})(−7.5568×10−10,(-7.5568\times 1{0}^{-10},−1.5801×10−14)-1.5801\times 1{0}^{-14})88(−1.0983×10−9,(-1.0983\times 1{0}^{-9},−2.9164×10−14)-2.9164\times 1{0}^{-14})(−2.8851×10−10,(-2.8851\times 1{0}^{-10},−6.961×10−15)-6.961\times 1{0}^{-15})(−1.2018×10−9,(-1.2018\times 1{0}^{-9},−3.6883×10−14)-3.6883\times 1{0}^{-14})(−4.096×10−10,(-4.096\times 1{0}^{-10},−9.7886×10−15)-9.7886\times 1{0}^{-15})(−6.3714×10−10,(-6.3714\times 1{0}^{-10},−1.2192×10−14)-1.2192\times 1{0}^{-14})(−6.2791×10−10,(-6.2791\times 1{0}^{-10},−1.1329×10−14)-1.1329\times 1{0}^{-14})89(−9.1366×10−10,(-9.1366\times 1{0}^{-10},−2.0953×10−14)-2.0953\times 1{0}^{-14})(−2.3973×10−10,(-2.3973\times 1{0}^{-10},−4.9908×10−15)-4.9908\times 1{0}^{-15})(−9.9977×10−10,(-9.9977\times 1{0}^{-10},−2.6499×10−14)-2.6499\times 1{0}^{-14})(−3.4035×10−10,(-3.4035\times 1{0}^{-10},−7.0181×10−15)-7.0181\times 1{0}^{-15})(−5.3002×10−10,(-5.3002\times 1{0}^{-10},−8.7598×10−15)-8.7598\times 1{0}^{-15})(−5.2175×10−10,(-5.2175\times 1{0}^{-10},−8.1222×10−15)-8.1222\times 1{0}^{-15})90(−7.6006×10−10,(-7.6006\times 1{0}^{-10},−1.5054×10−14)-1.5054\times 1{0}^{-14})(−1.992×10−10,(-1.992\times 1{0}^{-10},−3.5784×10−15)-3.5784\times 1{0}^{-15})(−8.3169×10−10,(-8.3169\times 1{0}^{-10},−1.9039×10−14)-1.9039\times 1{0}^{-14})(−2.8282×10−10,(-2.8282\times 1{0}^{-10},−5.0319×10−15)-5.0319\times 1{0}^{-15})(−4.4092×10−10,(-4.4092\times 1{0}^{-10},−6.2936×10−15)-6.2936\times 1{0}^{-15})(−4.3355×10−10,(-4.3355\times 1{0}^{-10},−5.8235×10−15)-5.8235\times 1{0}^{-15})Table 2Few values {dY(wn,(0,0))}\left\{{d}_{Y}\left({w}_{n},\left(0,0))\right\}by Alg1 and Alg2 from Example 4.2dY(wn,(0,0)){d}_{Y}\left({w}_{n},\left(0,0))nnAlg1Alg2Alg1Alg2Alg1Alg21522.5065522.50652480.5042480.50416.834116.83412147.7003145.0973677.5161666.97944.71654.6379336.980634.609145.4444183.68110.921551.292748.49097.767334.570240.10150.18960.2731751.86581.68737.85398.88480.040390.05794360.402870.36191.72081.92920.0130820.019992⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots ⋮\vdots 851.9078×10−91.9078\times 1{0}^{-9}5.0291×10−105.0291\times 1{0}^{-10}2.0876×10−92.0876\times 1{0}^{-9}7.14×10−107.14\times 1{0}^{-10}1.1067×10−91.1067\times 1{0}^{-9}1.0945×10−91.0945\times 1{0}^{-9}861.5871×10−91.5871\times 1{0}^{-9}4.1787×10−104.1787\times 1{0}^{-10}1.7367×10−91.7367\times 1{0}^{-9}5.9326×10−105.9326\times 1{0}^{-10}9.2069×10−109.2069\times 1{0}^{-10}9.0946×10−109.0946\times 1{0}^{-10}871.3203×10−91.3203\times 1{0}^{-9}3.4721×10−103.4721\times 1{0}^{-10}1.4447×10−91.4447\times 1{0}^{-9}4.9295×10−104.9295\times 1{0}^{-10}7.659×10−107.659\times 1{0}^{-10}7.5568×10−107.5568\times 1{0}^{-10}881.0983×10−91.0983\times 1{0}^{-9}2.8851×10−102.8851\times 1{0}^{-10}1.2018×10−91.2018\times 1{0}^{-9}4.096×10−104.096\times 1{0}^{-10}6.3714×10−106.3714\times 1{0}^{-10}6.2791×10−106.2791\times 1{0}^{-10}899.1366×10−109.1366\times 1{0}^{-10}2.3973×10−102.3973\times 1{0}^{-10}9.9977×10−109.9977\times 1{0}^{-10}3.4035×10−103.4035\times 1{0}^{-10}5.3002×10−105.3002\times 1{0}^{-10}5.2175×10−105.2175\times 1{0}^{-10}907.6006×10−107.6006\times 1{0}^{-10}1.992×10−101.992\times 1{0}^{-10}8.3169×10−108.3169\times 1{0}^{-10}2.8282×10−102.8282\times 1{0}^{-10}4.4092×10−104.4092\times 1{0}^{-10}4.3355×10−104.3355\times 1{0}^{-10}5Concluding remarksThis work analysed Mann-type and viscosity-type PPAs under which we approximated a common solution of MVFIPs, an MP, and a common fixed point of multivalued demicontractive mappings in Hadamard spaces. Using the results herein, we computed mean and median values of probabilities, minimize energy of measurable mappings, and solve a kinematic problem in robotic motion control. We also gave a numerical example in a nonlinear Hadamard space to support the findings. Our results are based on the newly introduced concept of monotonicity. Moreover, the results complement and extend several results in the literature. In particular, Theorem 3.9 generalises the result of Okeke and Izuchukwu [34] from a singlevalued nonexpansive mapping to a family of more general mappings (multivalued demicontractive), from one monotone mapping to finite family of monotone mappings yet considering a newly generalised notion of monotonicity as introduced in [9]. Also, we used viscosity-type PPA which is known to be more general and faster than the Halpern-type PPA used in [34].Our results generalise the results of Ranjbar and Khatibzadeh [35], Suparatulatorn et al. [46], and Khatibzadeh and Ranjbar [24] to more general problems and to finite family of monotone mappings.We extend the results of Takahashi and Shimoji [47] and Rockafellar [37] from linear spaces to CAT(0) spaces with a more general problem.

Journal

Analysis and Geometry in Metric Spacesde Gruyter

Published: Jan 1, 2023

Keywords: fixed point; monotone vector field; proximal point algorithm; resolvent operator; tangent space; 47H05; 47J25; 49J40; 65K10; 65K15

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