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Existence, Data Dependence and Stability of Fixed Points of Multivalued Maps in Incomplete Metric Spaces

Existence, Data Dependence and Stability of Fixed Points of Multivalued Maps in Incomplete Metric... Annales Mathematicae Silesianae 37 (2023), no. 1, 32–48 DOI: 10.2478/amsil-2022-0020 EXISTENCE, DATA DEPENDENCE AND STABILITY OF FIXED POINTS OF MULTIVALUED MAPS IN INCOMPLETE METRIC SPACES Binayak S. Choudhury, Nikhilesh Metiya , Sunirmal Kundu, Debashis Khatua Abstract. In this paper we formulate a setvalued fixed point problem by combining four prevalent trends of fixed point theory. We solve the problem by showing that the set of fixed points is nonempty. Further we have a data dependence result pertaining to the problem and also a stability result for the fixed point sets. The main result is extended to metric spaces with a graph. The results are obtained without the use of metric completeness assumption which is replaced by some other conditions suitable for solving the fixed point problem. There are some consequences of the main result. The main result is illustrated with an example. 1. Introduction and mathematical preliminaries The development of fixed point theory of contractive mappings following the work of Banach has been very extensive and is carried into the recent times even after about hundred years of its initiation. Works like [6, 13, 17, 20, 22, 33, 35] are some instances from this line of research. A very influential form of contraction was proposed by Suzuki ([35]) who generalized the Banach contraction and in the sequel initiated a new trend in fixed point theory. Such Received: 19.10.2021. Accepted: 23.11.2022. Published online: 20.12.2022. (2020) Mathematics Subject Classification: 47H10, 54H10, 54H25. Key words and phrases: metric space, -completeness, -continuity, fixed point, data dependence, stability. c 2022 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution License CC BY (http://creativecommons.org/licenses/by/4.0/). Existence, data dependence and stability of fixed points of multivalued maps ... 33 mappings defined in line with the idea in [35] came to be known as Suzuki type contractions ([2, 16, 29]). Nadler ([28]) extended fixed point theory to the domain of setvalued analysis with the help of the idea of Hausdorff distance. Following the work of Nadler, fixed point studies of setvalued mappings have flourished in a large way. Comprehensive accounts of this development are obtainable in [3, 7, 24]. Again rational contractive inequalities occupy a prominent position in fixed point theory. It was initiated by Dass et al ([14]). The use of rational terms in contractive inequalities has been done in works like [4, 8, 21]. The use of admissibility conditions has come up prominently in fixed point theory. These are certain conditions on the behaviour of the contractive map- ping under consideration and are brought about through a prescribed function. The advantage of using such conditions is that the contraction condition can be restricted to certain suitable pairs of points in which case there is no need to define contraction condition on the whole space. Recently, fixed point re- sults using admissibility conditions have been developed in several works like [12, 19, 33]. The above trends of research have individually contributed very substan- tially to the development of fixed point theory. There are large scopes of putting these ideas together in order to create new results in fixed point the- ory. Accordingly we combine the above four existing trends to formulate a fixed point problem in metric spaces. We do not assume completeness property of the metric space. Rather we use an alternative condition on the metric space which is brought about through a separate function. We establish existence, data dependence and stability results relating to the fixed point problem for- mulated here. We extend our result to the case of a metric space with a graphic structure. Some of our results are illustrated with examples. A data dependence problem is to estimate the distance between the fixed point sets of two operators when the functional value of these mappings at every point differs by a magnitude less than a given positive number. As mul- tivalued mappings often have larger fixed point sets than their singlevalued counterparts, the study of data dependence problem within the domain of setvalued analysis assumes additional importance. It has important applica- tions to differential and integral equations ([9, 32]). Several research papers on data dependence have been published in recent literature of which we mention a few in references [10, 12, 18]. Stability is a concept in dynamical systems related to limiting behaviors. There are various notions of stability both in discrete and continuous dynam- ical systems ([31]). In this article, stability is related limiting behaviour of the fixed point sets associated with a sequence of multivalued mappings to that of the limit function to which the sequence converges. There are several results dealing with the stability of fixed point sets as for instance the works noted in the references [5, 11, 12, 34]. 34 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua In the following we give the technical details required for deduction of our results in the following sections. Let (M; ) be a metric space and CLB(M) be the class of all non-empty closed and bounded subsets of M. Define D(a; B) = inff(a; b) : b 2 Bg; where a 2 M and B 2 CLB(M); D(A; B) = inff(a; b) : a 2 A; b 2 Bg; where A; B 2 CLB(M); H(A; B) = maxfsup D(x; B); sup D(y; A)g; where A; B 2 CLB(M): x2A y2B H is a metric on CLB(M) and is called the Hausdorff–Pompeiu metric on CLB(M). Moreover, if (M; ) is complete then (CLB(M);H) is also com- plete ([28]). Lemma 1.1 ([28]). Let A; B 2 CLB(M) and q > 1. Then for every x 2 A there exists y 2 B satisfying (x; y)  qH(A; B). Definition 1.1 ([28]). A point u 2 M is called a fixed point of a multi- valued mapping T : M ! CLB(M) if u 2 Tu. The fixed point set of T is denoted by F . Definition 1.2 ([11]). A multivalued mapping T : M ! CLB(M) is called continuous at x 2 M if H(Tx ; Tx) ! 0 as n ! 1 for any sequence fx g in M with x ! x as n ! 1. n n The following ideas involve a function : MM ! [0;1). The idea of the -continuity of multivalued mappings has been introduced recently by Kutbi and Sintunavarat ([25]). Definition 1.3 ([25]). A multivalued mapping T : M ! CLB(M) is called -continuous at x 2 M if lim H(Tx ; Tx) = 0, whenever fx g n!1 n n is a sequence in M with x ! x as n ! 1 and (x ; x )  1 for all n > 0. n n n+1 Remark 1.1 ([25]). The continuity of a mapping guarantees its -continuity but the converse may not be true. Recently, the idea of -completeness of a metric space has been introduced by Hussain et al ([19]). Definition 1.4 ([19]). The metric space M is called -complete if every Cauchy sequence fx g in M satisfying (x ; x )  1 for all n > 0 is n n n+1 convergent in M. Existence, data dependence and stability of fixed points of multivalued maps ... 35 Remark 1.2 ([19]). The completeness of a metric space M guarantees its -completeness but the converse is not true. Definition 1.5 ([12]). We say that a metric space M has -regular prop- erty if (x ; x)  1 for all n > 0 whenever fx g is a convergent sequence n n in M having limit x 2 M and satisfying (x ; x )  1 for all n > 0. n n+1 Definition 1.6 ([11]). A multivalued mapping T : M ! CLB(M) is called -admissible if (x; y)  1, for x; y 2 M implies (u; v)  1, where u 2 Tx and v 2 Ty. In the following we define a multivalued contraction of Suzuki-type which unifies and generalizes many Suzuki type contractions in the existing literature [23, 27, 30, 35]. Definition 1.7. A multivalued mapping T : M ! CLB(M) is said to be a Suzuki-type -contraction if for u; v 2 M with (u; v)  1, D(u; Tu)  (u; v) implies H(Tu; Tv)  qQ(u; v); where Q(u; v) = max (u; v);D(u; Tu);D(v; Tv); [D(u; Tv) + D(v; Tu)]; p D(u; Tu)D(v; Tv) p D(u; Tv)D(v; Tu) q ; q p +H(Tu; Tv) r +H(Tu; Tv) and q 2 (0; 1), p; r > 0. 2. Existence of nonempty fixed point set Theorem 2.1. Let (M; ) be a metric space and : M  M ! [0;1) be a mapping such that M is -complete and has -regular property. Let T : M ! CLB(M) be such that (i) T is -admissible, (ii) there exist x 2 M and x 2 Tx such that (x ; x )  1, (iii) T is a Suzuki-type -contraction. Then 1 0 0 1 F is nonempty. T 36 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Proof. By assumption (ii), there exists x 2 M and x 2 Tx such that 0 1 0 (x ; x )  1. Since q 2 (0; 1), we have > 1. As Tx ; Tx 2 CLB(M) and 0 1 0 1 x 2 Tx , by Lemma 1.1, we find x 2 Tx such that 1 0 2 1 (x ; x )  H(Tx ; Tx ): 1 2 0 1 As (x ; x )  1, by assumption (i), we have (x ; x )  1. As Tx ; Tx 2 0 1 1 2 1 2 CLB(M), x 2 Tx and > 1, there exists x 2 Tx such that 2 1 3 2 (x ; x )  H(Tx ; Tx ): 2 3 1 2 As (x ; x )  1, by assumption (i), we have (x ; x )  1. Arguing in this 1 2 2 3 way we construct a sequence fx g in X such that (2.1) x 2 Tx ; for all n  0; n+1 n (2.2) (x ; x )  1; for all n  0; n n+1 (2.3) (x ; x )  H(Tx ; Tx ); for all n  0: n+1 n+2 n n+1 Now, 1 1 (2.4) D(x ; Tx )  (x ; x )  (x ; x ); for all n  0: n n n n+1 n n+1 2 2 Let R = (x ; x ); for all n  0: n n n+1 By (2.2), (2.3) and (2.4), we have (2.5) (x ; x )  H(Tx ; Tx ) n+1 n+2 n n+1 q Q(x ; x ) n n+1 = q Q(x ; x ): n n+1 Existence, data dependence and stability of fixed points of multivalued maps ... 37 Now, Q(x ; x ) = max (x ; x );D(x ; Tx );D(x ; Tx ); n n+1 n n+1 n n n+1 n+1 [D(x ; Tx ) + D(x ; Tx )]; n n+1 n+1 n p D(x ; Tx )D(x ; Tx ) n n n+1 n+1 q ; p +H(Tx ; Tx ) n n+1 p D(x ; Tx )D(x ; Tx ) n n+1 n+1 n r +H(Tx ; Tx ) n n+1 max (x ; x ); (x ; x ); (x ; x ); n n+1 n n+1 n+1 n+2 [(x ; x ) + (x ; x )]; n n+2 n+1 n+1 (x ; x )(x ; x ) (x ; x )(x ; x ) p p n n+1 n+1 n+2 n n+2 n+1 n+1 q p ; q p p + q(x ; x ) r + q(x ; x ) n+1 n+2 n+1 n+2 max (x ; x ); (x ; x ); (x ; x ); n n+1 n n+1 n+1 n+2 1 (x ; x )(x ; x ) n n+1 n+1 n+2 [(x ; x ) + (x ; x )]; ; 0 n n+1 n+1 n+2 2 + (x ; x ) n+1 n+2 max (x ; x ); (x ; x ); (x ; x ); n n+1 n n+1 n+1 n+2 [(x ; x ) + (x ; x )]; (x ; x ); 0 n n+1 n+1 n+2 n n+1 n o = max R ; R ; R ; [R + R ]; R ; 0 n n n+1 n n+1 n h i (2.6) = maxfR ; R g; since [R + R ]  maxfR ; R g : n n+1 n n+1 n n+1 If possible, suppose that R > R  0. From (2.5) and the above inequality, n+1 n we have p p R  q maxfR ; R g = q R < R ; n+1 n n+1 n+1 n+1 which is a contradiction. Therefore, we have (2.7) R  R ; for all n: n+1 n 38 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua From (2.5), (2.6) and (2.7), we get p p R  q maxfR ; R g = q R : n+1 n n+1 n Applying the above inequality repeatedly, we have p p p p 2 3 n+1 R  q R  ( q) R  ( q) R  : : :  ( q) R : n+1 n n1 n2 0 Now, 1 1 1 X X p qR (x ; x ) = R  ( q) R = p < 1: n n+1 n 0 1 q n=1 n=1 n=1 Then fx g is a Cauchy sequence in X with (x ; x )  1; for all n  0. n n n+1 Using the -completeness property of M we have a point x 2 M such that (2.8) lim x = x: n!1 Using (2.2) and -regularity assumption of M, we get (2.9) (x ; x)  1; for all n: If possible, suppose that for some n 2 N, 1 1 D(x ; Tx ) > (x ; x) and D(x ; Tx ) > (x ; x): n n n n+1 n+1 n+1 2 2 Then 1 1 (x ; x ) > (x ; x) and (x ; x ) > (x ; x): n n+1 n n+1 n+2 n+1 2 2 Using (2.7), we have R = (x ; x )  (x ; x) + (x; x ) < [(x ; x ) + (x ; x )] n n n+1 n n+1 n n+1 n+1 n+2 1 1 = [R + R ]  [R + R ] = R ; n n+1 n n n 2 2 which leads to a contradiction. Therefore, for each n 2 N, we have 1 1 either D(x ; Tx )  (x ; x) or D(x ; Tx )  (x ; x): n n n n+1 n+1 n+1 2 2 Existence, data dependence and stability of fixed points of multivalued maps ... 39 Hence, we have a subsequence fx g of fx g for which n(k) n D(x ; Tx )  (x ; x): n(k) n(k) n(k) By (2.8), (2.9) and the above inequality, we have x ! x as k ! 1 and (x ; x)  1 for all k: n(k) n(k) Applying (iii), we get (2.10) D(x ; Tx)  H(Tx ; Tx)  qQ(x ; x): n(k)+1 n(k) n(k) Using (2.1), we have Q(x ; x) = max (x ; x);D(x ; Tx );D(x; Tx); n(k) n(k) n(k) n(k) D(x ; Tx )D(x; Tx) 1 p n(k) n(k) [D(x; Tx ) + D(x ; Tx)]; q ; n(k) n(k) 2 p +H(Tx ; Tx) n(k) D(x ; Tx)D(x; Tx ) n(k) n(k) r +H(Tx ; Tx) n(k) max (x ; x); (x ; x );D(x; Tx); n(k) n(k) n(k)+1 (x ; x )D(x; Tx) 1 p n(k) n(k)+1 [(x; x ) + D(x ; Tx)]; q ; n(k)+1 n(k) 2 p + D(x ; Tx) n(k)+1 D(x ; Tx)(x; x ) n(k) n(k)+1 q : r + D(x ; Tx) n(k)+1 Now, n o D(x; Tx) (2.11) lim sup Q(x ; x)  max 0; 0;D(x; Tx); ; 0; 0 n(k) k!1 = D(x; Tx): Taking lim sup as k ! 1 in (2.10) and applying (2.11), we have D(x; Tx) q D(x; Tx), which implies that D(x; Tx) = 0. Now, D(x; Tx) = 0 implies that x 2 Tx, where Tx is the closure of Tx. Since Tx is closed, we have Tx = Tx. Therefore, x 2 Tx, that is, x 2 F , and so, F is non-empty. T T We have the following observations on Theorem 2.1. 40 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Note 2.1. The conclusion of Theorem 2.1 is still true if one takes the assumption that T is -continuous instead of taking the - regularity as- sumption of the space. Then the portion just after (2.8) of the proof of above theorem is changed in the following way: D(x; Tx) = lim D(x ; Tx)  lim H(Tx ; Tx) = 0: n+1 n n!1 n!1 Therefore, we have D(x; Tx) = 0, which implies that x 2 Tx = Tx, where Tx is the closure of Tx. Hence F is nonempty. Note 2.2. The conclusion of Theorem 2.1 is still true if one considers that T is continuous instead of taking the -regularity assumption of the spaces. Since every continuous mapping is -continuous, the result follows from Note 2.1 and Theorem 2.1. Note 2.3. The conclusion of Theorem 2.1 is still true if one considers that M is complete instead of taking the -completeness assumption of M. Since every complete metric space is -complete, the result follows from The- orem 2.1. Example 2.1. Let M = (10; 10] and (x; y) = jx yj, for x; y 2 M. Let T : M ! CLB(M) be defined as n o ; if 10 < u < 0; h i Tu = 0; ; if 0  u  1; fug; if u > 1: Take q = . Let : M  M ! [0;1) be defined as a+b e ; for a 2 [0; 1] and b 2 0; ; (a; b) = 0; otherwise. Supposefu g is a convergent sequence in M with limit u and (u ; u )  1, n n n+1 for all n. Then u 2 [0; 1] and u 2 0;  [0; 1], for n  2. It follows that 1 n u 2 0; and (u ; u)  1, for all n. Hence M has -regular property. Suppose fu g is a Cauchy sequence in M for which (u ; u )  1, for n n n+1 all n. Then u 2 [0; 1] and u 2 0; , for all n  2. Then there exists 1 n u 2 0; such that u ! u as n ! 1. Hence M is -complete. Take x; y 2 M for which (x; y)  1. Then 0  x  1 and y 2 0; . So, x y 1 we have Tx = 0;  [0; 1] and Ty = 0;  0; . Then (u; v)  1, 16 16 16 whenever u 2 Tx and v 2 Ty. Hence, T is -admissible. Existence, data dependence and stability of fixed points of multivalued maps ... 41 Here 0 2 M, 0 2 T 0 and (0; 0)  1. D(u;Tu) Take u; v 2 M for which (u; v)  1 and  (u; v). Then u 2 [0; 1], (u;v) (u;v) 1 uv 1 1 v 2 0; and H(Tu; Tv) = = =  Q(u; v): Therefore, 16 16 16 4 4 4 all the assumptions of Theorem 2.1 are satisfied and F = f0g[ (1; 10]. Note 2.4. In the above example the metric space M is -complete but not complete. Also, the mapping T is -continuous but not continuous. If (x; y) = 1, for all x; y 2 M, we can obtain various Suzuki-type fixed point theorems from Theorem 2.1. Corollary 2.1. Let (M; ) be a complete metric space and 0 < q < 1. D(u;Tu) Then T has a fixed point if for u; v 2 M,  (u; v) implies one of the following inequalities holds: (i) H(Tu; Tv)  q (u; v); (ii) H(Tu; Tv)  [D(u; Tu) + D(v; Tv)]; (iii) H(Tu; Tv)  [D(u; Tv) + D(v; Tu)]; n o D(u; Tu)+D(v; Tv) D(u; Tv)+D(v; Tu) (iv) H(Tu; Tv)  q max (u; v); ; . 2 2 3. Data dependence result Let T ; T : M ! CLB(M) be such that H(T x; T x)  , for all x 2 M, 1 2 1 2 where  is some positive number. A data dependence problem is to estimate the distance between the fixed point sets of these two mappings. The above is meaningful only if we have an assurance of nonempty fixed point sets of these two operators. There are also some variants of the problem. Our data dependence theorem is the following. Theorem 3.1. Let (M; ) be a metric space and : M  M ! [0;1) be a mapping such that M is -complete and has -regular property. Let T ; T : M ! CLB(M) be two multivalued mappings satisfying H(T x; T x) 1 2 1 2 K, for all x 2 M, where K > 0 is a fixed real number. Suppose that T satisfies the assumptions (i) and (iii) of Theorem 2.1. Assume that F is nonempty and (x; u)  1, for all x 2 F and u 2 T x. Then F 6= ; and T 2 T 1 2 sup D(z; F )  . q(1 q) z2F 1 42 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Proof. Since F 6= ;, we take y 2 F . Since T y is non-empty and T 0 T 2 0 1 1 (y ; u)  1, for all u 2 T y , T satisfies the assumptions (i), (ii) and (iii) of 0 2 0 2 Theorem 2.1 and hence by Theorem 2.1, F is non-empty, that is, F 6= ;. T T 2 2 Since q 2 (0; 1), we have > 1. As T y and T y 2 CLB(M), there exists 1 0 2 0 y 2 T y such that 1 2 0 (3.1) (y ; y )  H(T y ; T y ): 0 1 1 0 2 0 Now, y 2 M and y 2 T y such that (y ; y )  1. Arguing similarly as in 0 1 2 0 0 1 the proof of Theorem 2.1, we construct a sequence fy g in M such that y 2 T y ; (y ; y )  1; n+1 2 n n n+1 > 1 (y ; y )  H(T y ; T y ) n+1 n+2 2 n 2 n+1 (3.2) > and (y ; y )  q(y ; y )  : : : n+1 n+2 n n+1 n+1 ( q) (y ; y ); for all n  0: 0 1 Following the arguments as in the proof of Theorem 2.1, we prove fy g is a Cauchy sequence in M and there exists  2 M such that y !  as n ! 1 and also  is a fixed point of T , that is,  2 T . From (3.1), we have 2 2 1 K (3.3) (y ; y )  H(T y ; T y )  : 0 1 1 0 2 0 q q Using (3.2) and triangular property, we have n n X X (y ; )  (y ; y ) + (y ; )  ( q) (y ; y ) + (y ; ): 0 i i+1 n+1 0 1 n+1 i=0 i=0 Letting n ! 1 in the above inequality and using (3.3), we obtain p (y ; y ) K 0 1 (y ; )  ( q) (y ; y ) = p  p ; 0 0 1 (1 q) q(1 q) i=0 which implies that D(y ; F )  . Since y 2 F is arbitrary, we 0 T 0 T 2 1 q(1 q) obtain sup D(z; F )  . q(1 q) z2F 1 Existence, data dependence and stability of fixed points of multivalued maps ... 43 4. Stability analysis Stability is related limiting behavior of a system which, in this case, is the relation of the fixed point sets associated with a sequence of multival- ued mappings with the limit function to which the sequence converges. Let fT : M ! CLB(M)g be a sequence of multivalued mappings that converges to a mapping T : M ! CLB(M), that is, T = lim T . Suppose that n!1 n fF g is the sequence of fixed point sets of the sequence of mappings fT g T n and F is the fixed point set of T . We say that the fixed point sets F of T T the sequence of multivalued mappings fT : M ! CLB(M)g are stable if H(F ; F ) ! 0 as n ! 1. T T In continuation of the data dependence result of the previous section, by particularly considering a special case in which both the mappings are assumed to satisfy the conditions of the main theorem in Section 2, we establish a stability result for fixed point sets of these mappings. Lemma 4.1. Let (M; ) be a metric space and : M  M ! [0;1). Let fT : M ! CLB(M) : n 2 Ng be a sequence of multivalued mappings con- verging to a mapping T : M ! CLB(M). If each T (n 2 N) is a Suzuki-type -contraction, then T is also a Suzuki-type -contraction. Proof. Take x; y 2 M for which (x; y)  1. Since each T (n 2 N) is a Suzuki-type - contraction, we have D(x; T x)  (x; y) implies H(T x; T y)  q max (x; y);D(x; T x);D(y; T y); [D(x; T y) + D(y; T x)]; n n n n n n p D(x; T x)D(y; T y) p D(x; T y)D(y; T x) n n n n q ; q : p +H(T x; T y) r +H(T x; T y) n n n n Taking limit as n ! 1 in the above inequalities, we get D(x; Tx)  (x; y) implies H(Tx; Ty)  q max (x; y);D(x; Tx);D(y; Ty); [D(x; Ty) + D(y; Tx)]; p D(x; Tx)D(y; Ty) p D(x; Ty)D(y; Tx) q ; q : p +H(Tx; Ty) r +H(Tx; Ty) This shows that T is a Suzuki-type -contraction.  44 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Theorem 4.1. Let (M; ) be a metric space and : M  M ! [0;1) be a mapping such that M is -complete and has -regular property. Let fT : M ! CLB(M) : n 2 Ng be a sequence of multivalued mappings converg- ing to T : M ! CLB(M) uniformly, that is, T ! T uniformly as n ! 1. Suppose that each T (n 2 N) satisfies the assumptions (i), (ii) and (iii) of Theorem 2.1 and also T satisfies the assumptions (i) and (ii) of Theo- rem 2.1. Then F 6= ; for every n and F 6= ;. If (x; u)  1, whenever T T x 2 F (n 2 N) and u 2 Tx or x 2 F and u 2 T x (n 2 N), then the fixed T T n point sets of the sequence fT g are stable. Proof. By Theorem 2.1, F 6= ; for every n 2 N. By Lemma 4.1 and Theorem 2.1, F 6= ;. Let K = sup H(T x; Tx), where n 2 N. Since the T n n x2X sequence fT g is uniformly convergent to T , we have (4.1) lim K = lim sup H(T x; Tx) = 0: n n n!1 n!1 x2X By Theorem 3.1, we obtain K K n n sup D(z; F )  p and sup D(z; F )  p : T T q(1 q) q(1 q) z2F z2F T T Therefore, we have H(F ; F )  ; for all n 2 N: T T q(1 q) Taking limit as n ! 1 in the above inequality and using (4.1), we get lim H(F ; F ) = 0. Therefore, the fixed point sets of mappings of the n!1 T T sequence fT g are stable. 5. Some results on graphic contraction In the present section, we extend our results in metric spaces with an additional structure of graph. Suppose that the metric space (M; ) is endowed with a directed graph G(V; E), that is, G is a directed graph such that its vertex set V (G) coincides with M and the edge set E(G) contains all loops. Assume that G has no parallel edges. Fixed point problem on the structures of metric spaces with a graph is a recent development. Works like [1, 15, 20] are some instances. Existence, data dependence and stability of fixed points of multivalued maps ... 45 Definition 5.1. A multivalued mapping T : M ! CLB(M) is called G- admissible if (x; y) 2 E for x; y 2 M implies (u; v) 2 E, whenever u 2 Tx, v 2 Ty. Definition 5.2. M is called G-regular if (x ; x) 2 E for all n, whenever fx g is a convergent sequence in M with limit x and (x ; x ) 2 E for all n. n n n+1 Definition 5.3. A multivalued mapping T : M ! CLB(M) is called G-continuous at x 2 M if lim H(Tx ; Tx) = 0, whenever fx g is any n!1 n n convergent sequence in M having limit x and (x ; x ) 2 E for all n. n n+1 Definition 5.4. M is called G-complete if every Cauchy sequence fx g in M with (x ; x ) 2 E for all n converges in M. n n+1 Definition 5.5. A multivalued mapping T : M ! CLB(M) is said to be a Suzuki-type graphic contraction if for all u; v 2 M with (u; v) 2 E, D(u; Tu)  (u; v) implies H(Tu; Tv)  q Q(u; v); where Q(u; v), q, p and r are as in Definition 1.7. Theorem 5.1. Let (M; ) be a metric space endowed with a directed graph G(V; E) such that M is G-complete and has G-regular property. Let T : M ! CLB(M) be such that (i) T is G-admissible, (ii) there exist x 2 M and x 2 Tx such that (x ; x ) 2 E, (iii) T is a Suzuki-type G-contraction. Then 1 0 0 1 F is non-empty. 1; if (u; v) 2 E; Proof. Define : M  M ! [0;1) as (u; v) = 0; if (u; v) 2= E: It can be easily verified that all the assumptions of Theorem 2.1 are sat- isfied and hence F is non-empty. Conclusion. In the expanding scenario of research on fixed point theory it is worth seeing how different lines of study coalesce amongst themselves to create new results. In the present paper we have made such an attempt. We think that more of such efforts can enrich the theory of fixed points in a substantial way. The constant q which is taken in the Suzuki-type -contraction considered in Theorem 2.1 may be replaced by a Mizoguchi-Takahashi function ([26]). Here we have not proceeded with it but this can be taken up as an immedi- ate future work. The investigation of possible application of the corresponding theorem to integral and differential inclusion problem is supposed to be of con- siderable interest. The study of different types of stability such as Ulam–Hyers 46 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua stability, asymptotic stability, etc., error estimation and rate of convergence of fixed point sets in the current context or in similar contexts would be an interesting topic for future study. References [1] M.R. Alfuraidan and M.A. Khamsi, Caristi fixed point theorem in metric spaces with a graph, Abstr. Appl. Anal. 2014, Art. ID 303484, 5 pp. [2] I. Altun and A. Erduran, A Suzuki type fixed-point theorem, Int. J. Math. Math. Sci. 2011, Art. ID 736063, 9 pp. [3] Q.H. Ansari, Metric Spaces: Including Fixed Point Theory and Set-valued Maps, Alpha Science International Ltd., Oxford, 2010. [4] G.V.R. Babu and M.V.R. Kameswari, Coupled fixed points of generalized contractive maps with rational expressions in partially ordered metric spaces, J. Adv. Res. Pure Math. 6 (2014), no. 2, 43–57. [5] R.K. Bose and R.N. Mukherjee, Stability of fixed point sets and common fixed points of families of mappings, Indian J. Pure Appl. Math. 11 (1980), no. 9, 1130–1138. [6] D.W. Boyd and J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464. [7] R.F. Brown, M. Furi, L. Górniewicz, and B. Jiang (eds.), Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005. [8] S. Chandok, Some fixed point theorems for ( ; )-admissible Geraghty type contractive mappings and related results, Math. Sci. (Springer) 9 (2015), no. 3, 127–135. [9] C. Chifu and G. Petruşel, Coupled fixed point results for ('; G)-contractions of type (b) in b-metric spaces endowed with a graph, J. Nonlinear Sci. Appl. 10 (2017), no. 2, 671–683. [10] A. Chiş-Novac, R. Precup, and I.A. Rus, Data dependence of fixed points for non-self generalized contractions, Fixed Point Theory 10 (2009), no. 1, 73–87. [11] B.S. Choudhury, N. Metiya, and C. Bandyopadhyay, Fixed points of multivalued - admissible mappings and stability of fixed point sets in metric spaces, Rend. Circ. Mat. Palermo (2) 64 (2015), no. 1, 43–55. [12] B.S. Choudhury, N. Metiya, and S. Kundu, Existence, data-dependence and stability of coupled fixed point sets of some multivalued operators, Chaos Solitons Fractals 133 (2020), 109678, 7 pp. [13] Lj.B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267–273. [14] B.K. Dass and S. Gupta, An extension of Banach contraction principle through rational expression, Indian J. Pure Appl. Math. 6 (1975), no. 12, 1455–1458. [15] M. Dinarvand, Fixed point results for ('; )-contractions in metric spaces endowed with a graph and applications, Mat. Vesnik 69 (2017), no. 1, 23–38. [16] D. Dorić, Z. Kadelburg, and S. Radenović, Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces, Nonlinear Anal. 75 (2012), no. 4, 1927–1932. [17] P.N. Dutta and B.S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl. 2008, Art. ID 406368, 8 pp. [18] R. Espínola and A. Petruşel, Existence and data dependence of fixed points for multi- valued operators on gauge spaces, J. Math. Anal. Appl. 309 (2005), no. 2, 420–432. [19] N. Hussain, M.A. Kutbi, and P. Salimi, Fixed point theory in -complete metric spaces with applications, Abstr. Appl. Anal. 2014, Art. ID 280817, 11 pp. Existence, data dependence and stability of fixed points of multivalued maps ... 47 [20] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1359–1373. [21] D.S. Jaggi and B.K. Dass, An extension of Banach’s fixed point theorem through a ra- tional expression, Bull. Calcutta Math. Soc. 72 (1980), no. 5, 261–262. [22] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76. [23] M. Kikkawa and T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal. 69 (2008), no. 9, 2942–2949. [24] W.A. Kirk and B. Sims (eds.), Handbook of Metric Fixed Point Theory, Springer, Dordrecht, 2001. [25] M.A. Kutbi and W. Sintunavarat, On new fixed point results for ( ; ; )-contractive multi-valued mappings on -complete metric spaces and their consequences, Fixed Point Theory Appl. 2015, 2015:2, 15 pp. [26] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989), no. 1, 177–188. [27] G. Moţ and A. Petruşel, Fixed point theory for a new type of contractive multivalued operators, Nonlinear Anal. 70 (2009), no. 9, 3371–3377. [28] S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488. [29] O. Popescu, Two fixed point theorems for generalized contractions with constants in complete metric space, Cent. Eur. J. Math. 7 (2009), no. 3, 529–538. [30] O. Popescu, A new type of contractive multivalued operators, Bull. Sci. Math. 137 (2013), no. 1, 30–44. [31] C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Second edition, CRC Press, Boca Raton, 1999. [32] I.A. Rus, A. Petruşel, and A. Sîntămărian, Data dependence of the fixed points set of multivalued weakly Picard operators, Studia Univ. Babeş-Bolyai Math. 46 (2001), no. 2, 111–121. [33] B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for - -contractive type map- pings, Nonlinear Anal. 75 (2012), no. 4, 2154–2165. [34] S.L. Singh, S.N. Mishra, and W. Sinkala, A note on fixed point stability for generalized multivalued contractions, Appl. Math. Lett. 25 (2012), no. 11, 1708–1710. [35] T. Suzuki, A generalized Banach contraction principle that characterizes metric com- pleteness, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1861–1869. Binayak S. Choudhury Debashis Khatua Department of Mathematics Indian Institute of Engineering Science and Technology Shibpur Howrah – 711103 West Bengal India e-mail: binayak12@yahoo.co.in, binayak@math.iiests.ac.in e-mail: debashiskhatua@yahoo.com Nikhilesh Metiya Department of Mathematics Sovarani Memorial College Jagatballavpur Howrah – 711408 West Bengal India e-mail: metiya.nikhilesh@gmail.com, nikhileshm@smc.edu.in 48 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Sunirmal Kundu Department of Mathematics Government General Degree College Salboni Paschim Medinipur – 721516 West Bengal India e-mail: sunirmalkundu2009@rediffmail.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Mathematicae Silesianae de Gruyter

Existence, Data Dependence and Stability of Fixed Points of Multivalued Maps in Incomplete Metric Spaces

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Annales Mathematicae Silesianae 37 (2023), no. 1, 32–48 DOI: 10.2478/amsil-2022-0020 EXISTENCE, DATA DEPENDENCE AND STABILITY OF FIXED POINTS OF MULTIVALUED MAPS IN INCOMPLETE METRIC SPACES Binayak S. Choudhury, Nikhilesh Metiya , Sunirmal Kundu, Debashis Khatua Abstract. In this paper we formulate a setvalued fixed point problem by combining four prevalent trends of fixed point theory. We solve the problem by showing that the set of fixed points is nonempty. Further we have a data dependence result pertaining to the problem and also a stability result for the fixed point sets. The main result is extended to metric spaces with a graph. The results are obtained without the use of metric completeness assumption which is replaced by some other conditions suitable for solving the fixed point problem. There are some consequences of the main result. The main result is illustrated with an example. 1. Introduction and mathematical preliminaries The development of fixed point theory of contractive mappings following the work of Banach has been very extensive and is carried into the recent times even after about hundred years of its initiation. Works like [6, 13, 17, 20, 22, 33, 35] are some instances from this line of research. A very influential form of contraction was proposed by Suzuki ([35]) who generalized the Banach contraction and in the sequel initiated a new trend in fixed point theory. Such Received: 19.10.2021. Accepted: 23.11.2022. Published online: 20.12.2022. (2020) Mathematics Subject Classification: 47H10, 54H10, 54H25. Key words and phrases: metric space, -completeness, -continuity, fixed point, data dependence, stability. c 2022 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution License CC BY (http://creativecommons.org/licenses/by/4.0/). Existence, data dependence and stability of fixed points of multivalued maps ... 33 mappings defined in line with the idea in [35] came to be known as Suzuki type contractions ([2, 16, 29]). Nadler ([28]) extended fixed point theory to the domain of setvalued analysis with the help of the idea of Hausdorff distance. Following the work of Nadler, fixed point studies of setvalued mappings have flourished in a large way. Comprehensive accounts of this development are obtainable in [3, 7, 24]. Again rational contractive inequalities occupy a prominent position in fixed point theory. It was initiated by Dass et al ([14]). The use of rational terms in contractive inequalities has been done in works like [4, 8, 21]. The use of admissibility conditions has come up prominently in fixed point theory. These are certain conditions on the behaviour of the contractive map- ping under consideration and are brought about through a prescribed function. The advantage of using such conditions is that the contraction condition can be restricted to certain suitable pairs of points in which case there is no need to define contraction condition on the whole space. Recently, fixed point re- sults using admissibility conditions have been developed in several works like [12, 19, 33]. The above trends of research have individually contributed very substan- tially to the development of fixed point theory. There are large scopes of putting these ideas together in order to create new results in fixed point the- ory. Accordingly we combine the above four existing trends to formulate a fixed point problem in metric spaces. We do not assume completeness property of the metric space. Rather we use an alternative condition on the metric space which is brought about through a separate function. We establish existence, data dependence and stability results relating to the fixed point problem for- mulated here. We extend our result to the case of a metric space with a graphic structure. Some of our results are illustrated with examples. A data dependence problem is to estimate the distance between the fixed point sets of two operators when the functional value of these mappings at every point differs by a magnitude less than a given positive number. As mul- tivalued mappings often have larger fixed point sets than their singlevalued counterparts, the study of data dependence problem within the domain of setvalued analysis assumes additional importance. It has important applica- tions to differential and integral equations ([9, 32]). Several research papers on data dependence have been published in recent literature of which we mention a few in references [10, 12, 18]. Stability is a concept in dynamical systems related to limiting behaviors. There are various notions of stability both in discrete and continuous dynam- ical systems ([31]). In this article, stability is related limiting behaviour of the fixed point sets associated with a sequence of multivalued mappings to that of the limit function to which the sequence converges. There are several results dealing with the stability of fixed point sets as for instance the works noted in the references [5, 11, 12, 34]. 34 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua In the following we give the technical details required for deduction of our results in the following sections. Let (M; ) be a metric space and CLB(M) be the class of all non-empty closed and bounded subsets of M. Define D(a; B) = inff(a; b) : b 2 Bg; where a 2 M and B 2 CLB(M); D(A; B) = inff(a; b) : a 2 A; b 2 Bg; where A; B 2 CLB(M); H(A; B) = maxfsup D(x; B); sup D(y; A)g; where A; B 2 CLB(M): x2A y2B H is a metric on CLB(M) and is called the Hausdorff–Pompeiu metric on CLB(M). Moreover, if (M; ) is complete then (CLB(M);H) is also com- plete ([28]). Lemma 1.1 ([28]). Let A; B 2 CLB(M) and q > 1. Then for every x 2 A there exists y 2 B satisfying (x; y)  qH(A; B). Definition 1.1 ([28]). A point u 2 M is called a fixed point of a multi- valued mapping T : M ! CLB(M) if u 2 Tu. The fixed point set of T is denoted by F . Definition 1.2 ([11]). A multivalued mapping T : M ! CLB(M) is called continuous at x 2 M if H(Tx ; Tx) ! 0 as n ! 1 for any sequence fx g in M with x ! x as n ! 1. n n The following ideas involve a function : MM ! [0;1). The idea of the -continuity of multivalued mappings has been introduced recently by Kutbi and Sintunavarat ([25]). Definition 1.3 ([25]). A multivalued mapping T : M ! CLB(M) is called -continuous at x 2 M if lim H(Tx ; Tx) = 0, whenever fx g n!1 n n is a sequence in M with x ! x as n ! 1 and (x ; x )  1 for all n > 0. n n n+1 Remark 1.1 ([25]). The continuity of a mapping guarantees its -continuity but the converse may not be true. Recently, the idea of -completeness of a metric space has been introduced by Hussain et al ([19]). Definition 1.4 ([19]). The metric space M is called -complete if every Cauchy sequence fx g in M satisfying (x ; x )  1 for all n > 0 is n n n+1 convergent in M. Existence, data dependence and stability of fixed points of multivalued maps ... 35 Remark 1.2 ([19]). The completeness of a metric space M guarantees its -completeness but the converse is not true. Definition 1.5 ([12]). We say that a metric space M has -regular prop- erty if (x ; x)  1 for all n > 0 whenever fx g is a convergent sequence n n in M having limit x 2 M and satisfying (x ; x )  1 for all n > 0. n n+1 Definition 1.6 ([11]). A multivalued mapping T : M ! CLB(M) is called -admissible if (x; y)  1, for x; y 2 M implies (u; v)  1, where u 2 Tx and v 2 Ty. In the following we define a multivalued contraction of Suzuki-type which unifies and generalizes many Suzuki type contractions in the existing literature [23, 27, 30, 35]. Definition 1.7. A multivalued mapping T : M ! CLB(M) is said to be a Suzuki-type -contraction if for u; v 2 M with (u; v)  1, D(u; Tu)  (u; v) implies H(Tu; Tv)  qQ(u; v); where Q(u; v) = max (u; v);D(u; Tu);D(v; Tv); [D(u; Tv) + D(v; Tu)]; p D(u; Tu)D(v; Tv) p D(u; Tv)D(v; Tu) q ; q p +H(Tu; Tv) r +H(Tu; Tv) and q 2 (0; 1), p; r > 0. 2. Existence of nonempty fixed point set Theorem 2.1. Let (M; ) be a metric space and : M  M ! [0;1) be a mapping such that M is -complete and has -regular property. Let T : M ! CLB(M) be such that (i) T is -admissible, (ii) there exist x 2 M and x 2 Tx such that (x ; x )  1, (iii) T is a Suzuki-type -contraction. Then 1 0 0 1 F is nonempty. T 36 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Proof. By assumption (ii), there exists x 2 M and x 2 Tx such that 0 1 0 (x ; x )  1. Since q 2 (0; 1), we have > 1. As Tx ; Tx 2 CLB(M) and 0 1 0 1 x 2 Tx , by Lemma 1.1, we find x 2 Tx such that 1 0 2 1 (x ; x )  H(Tx ; Tx ): 1 2 0 1 As (x ; x )  1, by assumption (i), we have (x ; x )  1. As Tx ; Tx 2 0 1 1 2 1 2 CLB(M), x 2 Tx and > 1, there exists x 2 Tx such that 2 1 3 2 (x ; x )  H(Tx ; Tx ): 2 3 1 2 As (x ; x )  1, by assumption (i), we have (x ; x )  1. Arguing in this 1 2 2 3 way we construct a sequence fx g in X such that (2.1) x 2 Tx ; for all n  0; n+1 n (2.2) (x ; x )  1; for all n  0; n n+1 (2.3) (x ; x )  H(Tx ; Tx ); for all n  0: n+1 n+2 n n+1 Now, 1 1 (2.4) D(x ; Tx )  (x ; x )  (x ; x ); for all n  0: n n n n+1 n n+1 2 2 Let R = (x ; x ); for all n  0: n n n+1 By (2.2), (2.3) and (2.4), we have (2.5) (x ; x )  H(Tx ; Tx ) n+1 n+2 n n+1 q Q(x ; x ) n n+1 = q Q(x ; x ): n n+1 Existence, data dependence and stability of fixed points of multivalued maps ... 37 Now, Q(x ; x ) = max (x ; x );D(x ; Tx );D(x ; Tx ); n n+1 n n+1 n n n+1 n+1 [D(x ; Tx ) + D(x ; Tx )]; n n+1 n+1 n p D(x ; Tx )D(x ; Tx ) n n n+1 n+1 q ; p +H(Tx ; Tx ) n n+1 p D(x ; Tx )D(x ; Tx ) n n+1 n+1 n r +H(Tx ; Tx ) n n+1 max (x ; x ); (x ; x ); (x ; x ); n n+1 n n+1 n+1 n+2 [(x ; x ) + (x ; x )]; n n+2 n+1 n+1 (x ; x )(x ; x ) (x ; x )(x ; x ) p p n n+1 n+1 n+2 n n+2 n+1 n+1 q p ; q p p + q(x ; x ) r + q(x ; x ) n+1 n+2 n+1 n+2 max (x ; x ); (x ; x ); (x ; x ); n n+1 n n+1 n+1 n+2 1 (x ; x )(x ; x ) n n+1 n+1 n+2 [(x ; x ) + (x ; x )]; ; 0 n n+1 n+1 n+2 2 + (x ; x ) n+1 n+2 max (x ; x ); (x ; x ); (x ; x ); n n+1 n n+1 n+1 n+2 [(x ; x ) + (x ; x )]; (x ; x ); 0 n n+1 n+1 n+2 n n+1 n o = max R ; R ; R ; [R + R ]; R ; 0 n n n+1 n n+1 n h i (2.6) = maxfR ; R g; since [R + R ]  maxfR ; R g : n n+1 n n+1 n n+1 If possible, suppose that R > R  0. From (2.5) and the above inequality, n+1 n we have p p R  q maxfR ; R g = q R < R ; n+1 n n+1 n+1 n+1 which is a contradiction. Therefore, we have (2.7) R  R ; for all n: n+1 n 38 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua From (2.5), (2.6) and (2.7), we get p p R  q maxfR ; R g = q R : n+1 n n+1 n Applying the above inequality repeatedly, we have p p p p 2 3 n+1 R  q R  ( q) R  ( q) R  : : :  ( q) R : n+1 n n1 n2 0 Now, 1 1 1 X X p qR (x ; x ) = R  ( q) R = p < 1: n n+1 n 0 1 q n=1 n=1 n=1 Then fx g is a Cauchy sequence in X with (x ; x )  1; for all n  0. n n n+1 Using the -completeness property of M we have a point x 2 M such that (2.8) lim x = x: n!1 Using (2.2) and -regularity assumption of M, we get (2.9) (x ; x)  1; for all n: If possible, suppose that for some n 2 N, 1 1 D(x ; Tx ) > (x ; x) and D(x ; Tx ) > (x ; x): n n n n+1 n+1 n+1 2 2 Then 1 1 (x ; x ) > (x ; x) and (x ; x ) > (x ; x): n n+1 n n+1 n+2 n+1 2 2 Using (2.7), we have R = (x ; x )  (x ; x) + (x; x ) < [(x ; x ) + (x ; x )] n n n+1 n n+1 n n+1 n+1 n+2 1 1 = [R + R ]  [R + R ] = R ; n n+1 n n n 2 2 which leads to a contradiction. Therefore, for each n 2 N, we have 1 1 either D(x ; Tx )  (x ; x) or D(x ; Tx )  (x ; x): n n n n+1 n+1 n+1 2 2 Existence, data dependence and stability of fixed points of multivalued maps ... 39 Hence, we have a subsequence fx g of fx g for which n(k) n D(x ; Tx )  (x ; x): n(k) n(k) n(k) By (2.8), (2.9) and the above inequality, we have x ! x as k ! 1 and (x ; x)  1 for all k: n(k) n(k) Applying (iii), we get (2.10) D(x ; Tx)  H(Tx ; Tx)  qQ(x ; x): n(k)+1 n(k) n(k) Using (2.1), we have Q(x ; x) = max (x ; x);D(x ; Tx );D(x; Tx); n(k) n(k) n(k) n(k) D(x ; Tx )D(x; Tx) 1 p n(k) n(k) [D(x; Tx ) + D(x ; Tx)]; q ; n(k) n(k) 2 p +H(Tx ; Tx) n(k) D(x ; Tx)D(x; Tx ) n(k) n(k) r +H(Tx ; Tx) n(k) max (x ; x); (x ; x );D(x; Tx); n(k) n(k) n(k)+1 (x ; x )D(x; Tx) 1 p n(k) n(k)+1 [(x; x ) + D(x ; Tx)]; q ; n(k)+1 n(k) 2 p + D(x ; Tx) n(k)+1 D(x ; Tx)(x; x ) n(k) n(k)+1 q : r + D(x ; Tx) n(k)+1 Now, n o D(x; Tx) (2.11) lim sup Q(x ; x)  max 0; 0;D(x; Tx); ; 0; 0 n(k) k!1 = D(x; Tx): Taking lim sup as k ! 1 in (2.10) and applying (2.11), we have D(x; Tx) q D(x; Tx), which implies that D(x; Tx) = 0. Now, D(x; Tx) = 0 implies that x 2 Tx, where Tx is the closure of Tx. Since Tx is closed, we have Tx = Tx. Therefore, x 2 Tx, that is, x 2 F , and so, F is non-empty. T T We have the following observations on Theorem 2.1. 40 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Note 2.1. The conclusion of Theorem 2.1 is still true if one takes the assumption that T is -continuous instead of taking the - regularity as- sumption of the space. Then the portion just after (2.8) of the proof of above theorem is changed in the following way: D(x; Tx) = lim D(x ; Tx)  lim H(Tx ; Tx) = 0: n+1 n n!1 n!1 Therefore, we have D(x; Tx) = 0, which implies that x 2 Tx = Tx, where Tx is the closure of Tx. Hence F is nonempty. Note 2.2. The conclusion of Theorem 2.1 is still true if one considers that T is continuous instead of taking the -regularity assumption of the spaces. Since every continuous mapping is -continuous, the result follows from Note 2.1 and Theorem 2.1. Note 2.3. The conclusion of Theorem 2.1 is still true if one considers that M is complete instead of taking the -completeness assumption of M. Since every complete metric space is -complete, the result follows from The- orem 2.1. Example 2.1. Let M = (10; 10] and (x; y) = jx yj, for x; y 2 M. Let T : M ! CLB(M) be defined as n o ; if 10 < u < 0; h i Tu = 0; ; if 0  u  1; fug; if u > 1: Take q = . Let : M  M ! [0;1) be defined as a+b e ; for a 2 [0; 1] and b 2 0; ; (a; b) = 0; otherwise. Supposefu g is a convergent sequence in M with limit u and (u ; u )  1, n n n+1 for all n. Then u 2 [0; 1] and u 2 0;  [0; 1], for n  2. It follows that 1 n u 2 0; and (u ; u)  1, for all n. Hence M has -regular property. Suppose fu g is a Cauchy sequence in M for which (u ; u )  1, for n n n+1 all n. Then u 2 [0; 1] and u 2 0; , for all n  2. Then there exists 1 n u 2 0; such that u ! u as n ! 1. Hence M is -complete. Take x; y 2 M for which (x; y)  1. Then 0  x  1 and y 2 0; . So, x y 1 we have Tx = 0;  [0; 1] and Ty = 0;  0; . Then (u; v)  1, 16 16 16 whenever u 2 Tx and v 2 Ty. Hence, T is -admissible. Existence, data dependence and stability of fixed points of multivalued maps ... 41 Here 0 2 M, 0 2 T 0 and (0; 0)  1. D(u;Tu) Take u; v 2 M for which (u; v)  1 and  (u; v). Then u 2 [0; 1], (u;v) (u;v) 1 uv 1 1 v 2 0; and H(Tu; Tv) = = =  Q(u; v): Therefore, 16 16 16 4 4 4 all the assumptions of Theorem 2.1 are satisfied and F = f0g[ (1; 10]. Note 2.4. In the above example the metric space M is -complete but not complete. Also, the mapping T is -continuous but not continuous. If (x; y) = 1, for all x; y 2 M, we can obtain various Suzuki-type fixed point theorems from Theorem 2.1. Corollary 2.1. Let (M; ) be a complete metric space and 0 < q < 1. D(u;Tu) Then T has a fixed point if for u; v 2 M,  (u; v) implies one of the following inequalities holds: (i) H(Tu; Tv)  q (u; v); (ii) H(Tu; Tv)  [D(u; Tu) + D(v; Tv)]; (iii) H(Tu; Tv)  [D(u; Tv) + D(v; Tu)]; n o D(u; Tu)+D(v; Tv) D(u; Tv)+D(v; Tu) (iv) H(Tu; Tv)  q max (u; v); ; . 2 2 3. Data dependence result Let T ; T : M ! CLB(M) be such that H(T x; T x)  , for all x 2 M, 1 2 1 2 where  is some positive number. A data dependence problem is to estimate the distance between the fixed point sets of these two mappings. The above is meaningful only if we have an assurance of nonempty fixed point sets of these two operators. There are also some variants of the problem. Our data dependence theorem is the following. Theorem 3.1. Let (M; ) be a metric space and : M  M ! [0;1) be a mapping such that M is -complete and has -regular property. Let T ; T : M ! CLB(M) be two multivalued mappings satisfying H(T x; T x) 1 2 1 2 K, for all x 2 M, where K > 0 is a fixed real number. Suppose that T satisfies the assumptions (i) and (iii) of Theorem 2.1. Assume that F is nonempty and (x; u)  1, for all x 2 F and u 2 T x. Then F 6= ; and T 2 T 1 2 sup D(z; F )  . q(1 q) z2F 1 42 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Proof. Since F 6= ;, we take y 2 F . Since T y is non-empty and T 0 T 2 0 1 1 (y ; u)  1, for all u 2 T y , T satisfies the assumptions (i), (ii) and (iii) of 0 2 0 2 Theorem 2.1 and hence by Theorem 2.1, F is non-empty, that is, F 6= ;. T T 2 2 Since q 2 (0; 1), we have > 1. As T y and T y 2 CLB(M), there exists 1 0 2 0 y 2 T y such that 1 2 0 (3.1) (y ; y )  H(T y ; T y ): 0 1 1 0 2 0 Now, y 2 M and y 2 T y such that (y ; y )  1. Arguing similarly as in 0 1 2 0 0 1 the proof of Theorem 2.1, we construct a sequence fy g in M such that y 2 T y ; (y ; y )  1; n+1 2 n n n+1 > 1 (y ; y )  H(T y ; T y ) n+1 n+2 2 n 2 n+1 (3.2) > and (y ; y )  q(y ; y )  : : : n+1 n+2 n n+1 n+1 ( q) (y ; y ); for all n  0: 0 1 Following the arguments as in the proof of Theorem 2.1, we prove fy g is a Cauchy sequence in M and there exists  2 M such that y !  as n ! 1 and also  is a fixed point of T , that is,  2 T . From (3.1), we have 2 2 1 K (3.3) (y ; y )  H(T y ; T y )  : 0 1 1 0 2 0 q q Using (3.2) and triangular property, we have n n X X (y ; )  (y ; y ) + (y ; )  ( q) (y ; y ) + (y ; ): 0 i i+1 n+1 0 1 n+1 i=0 i=0 Letting n ! 1 in the above inequality and using (3.3), we obtain p (y ; y ) K 0 1 (y ; )  ( q) (y ; y ) = p  p ; 0 0 1 (1 q) q(1 q) i=0 which implies that D(y ; F )  . Since y 2 F is arbitrary, we 0 T 0 T 2 1 q(1 q) obtain sup D(z; F )  . q(1 q) z2F 1 Existence, data dependence and stability of fixed points of multivalued maps ... 43 4. Stability analysis Stability is related limiting behavior of a system which, in this case, is the relation of the fixed point sets associated with a sequence of multival- ued mappings with the limit function to which the sequence converges. Let fT : M ! CLB(M)g be a sequence of multivalued mappings that converges to a mapping T : M ! CLB(M), that is, T = lim T . Suppose that n!1 n fF g is the sequence of fixed point sets of the sequence of mappings fT g T n and F is the fixed point set of T . We say that the fixed point sets F of T T the sequence of multivalued mappings fT : M ! CLB(M)g are stable if H(F ; F ) ! 0 as n ! 1. T T In continuation of the data dependence result of the previous section, by particularly considering a special case in which both the mappings are assumed to satisfy the conditions of the main theorem in Section 2, we establish a stability result for fixed point sets of these mappings. Lemma 4.1. Let (M; ) be a metric space and : M  M ! [0;1). Let fT : M ! CLB(M) : n 2 Ng be a sequence of multivalued mappings con- verging to a mapping T : M ! CLB(M). If each T (n 2 N) is a Suzuki-type -contraction, then T is also a Suzuki-type -contraction. Proof. Take x; y 2 M for which (x; y)  1. Since each T (n 2 N) is a Suzuki-type - contraction, we have D(x; T x)  (x; y) implies H(T x; T y)  q max (x; y);D(x; T x);D(y; T y); [D(x; T y) + D(y; T x)]; n n n n n n p D(x; T x)D(y; T y) p D(x; T y)D(y; T x) n n n n q ; q : p +H(T x; T y) r +H(T x; T y) n n n n Taking limit as n ! 1 in the above inequalities, we get D(x; Tx)  (x; y) implies H(Tx; Ty)  q max (x; y);D(x; Tx);D(y; Ty); [D(x; Ty) + D(y; Tx)]; p D(x; Tx)D(y; Ty) p D(x; Ty)D(y; Tx) q ; q : p +H(Tx; Ty) r +H(Tx; Ty) This shows that T is a Suzuki-type -contraction.  44 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Theorem 4.1. Let (M; ) be a metric space and : M  M ! [0;1) be a mapping such that M is -complete and has -regular property. Let fT : M ! CLB(M) : n 2 Ng be a sequence of multivalued mappings converg- ing to T : M ! CLB(M) uniformly, that is, T ! T uniformly as n ! 1. Suppose that each T (n 2 N) satisfies the assumptions (i), (ii) and (iii) of Theorem 2.1 and also T satisfies the assumptions (i) and (ii) of Theo- rem 2.1. Then F 6= ; for every n and F 6= ;. If (x; u)  1, whenever T T x 2 F (n 2 N) and u 2 Tx or x 2 F and u 2 T x (n 2 N), then the fixed T T n point sets of the sequence fT g are stable. Proof. By Theorem 2.1, F 6= ; for every n 2 N. By Lemma 4.1 and Theorem 2.1, F 6= ;. Let K = sup H(T x; Tx), where n 2 N. Since the T n n x2X sequence fT g is uniformly convergent to T , we have (4.1) lim K = lim sup H(T x; Tx) = 0: n n n!1 n!1 x2X By Theorem 3.1, we obtain K K n n sup D(z; F )  p and sup D(z; F )  p : T T q(1 q) q(1 q) z2F z2F T T Therefore, we have H(F ; F )  ; for all n 2 N: T T q(1 q) Taking limit as n ! 1 in the above inequality and using (4.1), we get lim H(F ; F ) = 0. Therefore, the fixed point sets of mappings of the n!1 T T sequence fT g are stable. 5. Some results on graphic contraction In the present section, we extend our results in metric spaces with an additional structure of graph. Suppose that the metric space (M; ) is endowed with a directed graph G(V; E), that is, G is a directed graph such that its vertex set V (G) coincides with M and the edge set E(G) contains all loops. Assume that G has no parallel edges. Fixed point problem on the structures of metric spaces with a graph is a recent development. Works like [1, 15, 20] are some instances. Existence, data dependence and stability of fixed points of multivalued maps ... 45 Definition 5.1. A multivalued mapping T : M ! CLB(M) is called G- admissible if (x; y) 2 E for x; y 2 M implies (u; v) 2 E, whenever u 2 Tx, v 2 Ty. Definition 5.2. M is called G-regular if (x ; x) 2 E for all n, whenever fx g is a convergent sequence in M with limit x and (x ; x ) 2 E for all n. n n n+1 Definition 5.3. A multivalued mapping T : M ! CLB(M) is called G-continuous at x 2 M if lim H(Tx ; Tx) = 0, whenever fx g is any n!1 n n convergent sequence in M having limit x and (x ; x ) 2 E for all n. n n+1 Definition 5.4. M is called G-complete if every Cauchy sequence fx g in M with (x ; x ) 2 E for all n converges in M. n n+1 Definition 5.5. A multivalued mapping T : M ! CLB(M) is said to be a Suzuki-type graphic contraction if for all u; v 2 M with (u; v) 2 E, D(u; Tu)  (u; v) implies H(Tu; Tv)  q Q(u; v); where Q(u; v), q, p and r are as in Definition 1.7. Theorem 5.1. Let (M; ) be a metric space endowed with a directed graph G(V; E) such that M is G-complete and has G-regular property. Let T : M ! CLB(M) be such that (i) T is G-admissible, (ii) there exist x 2 M and x 2 Tx such that (x ; x ) 2 E, (iii) T is a Suzuki-type G-contraction. Then 1 0 0 1 F is non-empty. 1; if (u; v) 2 E; Proof. Define : M  M ! [0;1) as (u; v) = 0; if (u; v) 2= E: It can be easily verified that all the assumptions of Theorem 2.1 are sat- isfied and hence F is non-empty. Conclusion. In the expanding scenario of research on fixed point theory it is worth seeing how different lines of study coalesce amongst themselves to create new results. In the present paper we have made such an attempt. We think that more of such efforts can enrich the theory of fixed points in a substantial way. The constant q which is taken in the Suzuki-type -contraction considered in Theorem 2.1 may be replaced by a Mizoguchi-Takahashi function ([26]). Here we have not proceeded with it but this can be taken up as an immedi- ate future work. The investigation of possible application of the corresponding theorem to integral and differential inclusion problem is supposed to be of con- siderable interest. The study of different types of stability such as Ulam–Hyers 46 Binayak S. 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Choudhury Debashis Khatua Department of Mathematics Indian Institute of Engineering Science and Technology Shibpur Howrah – 711103 West Bengal India e-mail: binayak12@yahoo.co.in, binayak@math.iiests.ac.in e-mail: debashiskhatua@yahoo.com Nikhilesh Metiya Department of Mathematics Sovarani Memorial College Jagatballavpur Howrah – 711408 West Bengal India e-mail: metiya.nikhilesh@gmail.com, nikhileshm@smc.edu.in 48 Binayak S. Choudhury, Nikhilesh Metiya, Sunirmal Kundu, Debashis Khatua Sunirmal Kundu Department of Mathematics Government General Degree College Salboni Paschim Medinipur – 721516 West Bengal India e-mail: sunirmalkundu2009@rediffmail.com

Journal

Annales Mathematicae Silesianae – de Gruyter

Published: Mar 1, 2023

Keywords: metric space; α-completeness; α-continuity; fixed point; data dependence; stability; 47H10; 54H10; 54H25

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