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Strong m-Convexity of Set-Valued Functions

Strong m-Convexity of Set-Valued Functions Annales Mathematicae Silesianae 37 (2023), no. 1, 82–94 DOI: 10.2478/amsil-2023-0003 Teodoro Lara , Nelson Merentes, Roy Quintero, Edgar Rosales Abstract. In this research we introduce the concept of strong m-convexity for set-valued functions defined on m-convex subsets of real linear normed spaces, a variety of properties and examples of these functions are shown, an inclusion of Jensen type is also exhibited. 1. Introduction In this research we introduce the notion of a strongly m-convex set-valued function, which represents a generalization of the usual concept of m-convexity for the real case that can be found in [3] and references therein. The idea of this new approach involves the concepts of strong convexity and m-convexity of set-valued functions. This is the main reason for which we start off by recalling both definitions. Along this paper X; Y will denote any real normed linear spaces, D an m-convex subset of X ([1]), B the closed unit ball in Y and n(Y ) the family of all nonempty subsets of Y . Received: 11.10.2022. Accepted: 24.01.2023. Published online: 07.02.2023. (2020) Mathematics Subject Classification: 26A51, 52A30. Key words and phrases: m-convex set, strongly m-convex set-valued function, Jensen type inclusion, normed space. 2023 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/). Strong m-convexity of set-valued functions 83 Definition 1.1 ([4]). Let c > 0. A set-valued function F : D ! n(Y ) is called strongly convex with modulus c if it satisfies the inclusion tF (x) + (1 t)F (y) + ct(1 t)kx yk B  F (tx + (1 t)y); for all x; y 2 D and t 2 [0; 1]. Definition 1.2 ([3]). Let m 2 [0; 1]. A set-valued function F : D ! n(Y ) is called m-convex if the inclusion tF (x) + m(1 t)F (y)  F (tx + m(1 t)y); holds for all x; y 2 D and t 2 [0; 1]. Our first definition runs as follows: Definition 1.3. Let c > 0 and m 2 [0; 1]. A set-valued function F : D ! n(Y ) is called strongly m-convex with modulus c if (1.1) tF (x) + m(1 t)F (y) + cmt(1 t)kx yk B  F (tx + m(1 t)y); for any x; y 2 D, t 2 [0; 1]. Remark 1.4. Notice that (1.1) is equivalent to mtF (x) + (1 t)F (y) + cmt(1 t)kx yk B  F (mtx + (1 t)y); with x; y; t as before. Remark 1.5. If a set-valued function F is strongly m-convex with mod- ulus c, then it is also m-convex. It follows immediately from the fact that 0 2 B. The converse in the foregoing remark is not true. Namely, we have the following. Example 1.1. The set-valued function F : [0; 1]  R ! n(R); given by F (x) = [0; x]; is m-convex ([3, Example 2.17]). But for all x; y; t 2 [0; 1] tF (x)+m(1 t)F (y) + cmt(1 t)kx yk B 2 2 = cmt(1 t)kx yk ; tx + m(1 t)y + cmt(1 t)kx yk ; 84 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales while that F (tx + m(1 t)y) = [0; tx + m(1 t)y]; so F can not be a strongly m-convex function. Example 1.2. If b > 0 and f; g : [0; b] ! R are two real functions, f and g being strongly m-convex with the same modulus ([2]) and f  g on [0; b], it is not difficult to verify (by reasoning as in Example 2.2 from [3]) that the set-valued functions F ; F ; F : [0; b]  R ! n(R) given by 1 2 3 F (x) = [f(x); g(x)]; F (x) = [f(x); +1); F (x) = (1; g(x)] 1 2 3 are strongly m-convex (with the same modulus). So, for example, functions f ; g : [0; 1] ! R defined as f (x) = 0 and g (x) = 1 are clearly m- 1 1 1 1 1 1 2 2 convex ([5, 6]), while functions f(x) = x , g(x) = 1 x are such that 2 2 f and g are strongly m-convex with modulus c = ; moreover f  g on [0; 1]. Consequently the set-valued function F : [0; 1] ! n(R) defined by 1 2 1 2 1 F (x) = x ; 1 x is strongly m-convex with modulus , and so is G(x) = 2 2 2 1 1 2 2 x 1; x . The graphs of F and G are shown in Figures 1 and 2, re- 2 2 spectively. 0:5 0:5 Figure 1. Graph of F Figure 2. Graph of G 2. Results In this section we present some set-properties of the unit ball B. At the same time, a characterization of the family of all the strongly m-convex functions is given and illustrate with an interesting example. We begin with Strong m-convexity of set-valued functions 85 a lemma related to two well-known properties of convexity whose proofs are omitted. Lemma 2.1. (1) If 0   ; then B  B: 1 2 1 2 (2) If  0; then ( + )B = B + B: 1 2 1 2 1 2 Proposition 2.2. A set-valued function F : D ! n(Y ) is strongly m- convex with modulus c if and only if (2.1) tF (A )+m(1t)F (A )+cmt(1t)kA A k B  F (A +m(1t)A ) 1 2 1 2 1 2 for all A ; A  D and t 2 [0; 1]; where F (A ) = fF (x) : x 2 A g (i = 1; 2) 1 2 i i and kA A k = inffkx yk : x 2 A ; y 2 A g: 1 2 1 2 Proof. ()) Let A ; A be two fixed but arbitrary subsets of D and z 2 1 2 tF (A ) + m(1 t)F (A ) + cmt(1 t)kA A k B: Then 1 2 1 2 (2.2) z 2 tF (a) + m(1 t)F (b) + cmt(1 t)kA A k B 1 2 for some a 2 A and b 2 A . Since 0  kA A k  ka bk, 0  cmt(1 1 2 1 2 2 2 t)kA A k  cmt(1 t)ka bk and from Lemma 2.1(1), the inclusion 1 2 2 2 cmt(1 t)kA A k B  cmt(1 t)ka bk B takes place. Hence, 1 2 (2.3) tF (a) + m(1 t)F (b) + cmt(1 t)kA A k B 1 2 tF (a) + m(1 t)F (b) + cmt(1 t)ka bk B: Furthermore, since ta + m(1 t)b 2 tA + m(1 t)A ; it is clear that 1 2 (2.4) F (ta + m(1 t)b)  F (tA + m(1 t)A ): 1 2 So, (2.1) follows from (2.2), (2.3), the strong m-convexity of F and (2.4). (() Let x; y 2 D and t 2 [0; 1]: The strong m-convexity with modu- lus c of F is obtained by considering in (2.1) the singletons A = fxg and A = fyg. Proposition 2.3. Let b 2 Rnf0g and D = [minf0; bg; maxf0; bg]  R: If F : D ! n(Y ) is strongly m-convex with modulus c, and 0 < n  m < 1; then F is strongly n-convex with modulus c. 86 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales Proof. If b < 0, then D = [b; 0]: Let t 2 [0; 1] and x; y 2 D with x  y. n 2 n So, x y  x y  0 and therefore, kx yk  x y : Since F is m m strongly m-convex with modulus c; F is m-convex (Remark 1.5). Thus, from [3, Proposition 2.11], Lemma 2.1(1), and the strong m-convexity of F , tF (x)+n(1 t)F (y) + cnt(1 t)kx yk B n n = tF (x) + m(1 t) F (y) + cmt(1 t) kx yk B m m n n tF (x) + m(1 t)F y + cmt(1 t) x y B m m F (tx + n(1 t)y): And for y < x, kx yk  x y , hence ntF (x)+(1 t)F (y) + cnt(1 t)kx yk B n n = mt F (x) + (1 t)F (y) + cmt(1 t) kx yk B m m n n mtF x + (1 t)F (y) + cmt(1 t) x y B m m F (ntx + (1 t)y); where the last inclusion arises from the strong m-convexity of F and Re- mark 1.4. If b > 0, D = [0; b] and the proof runs in a similar way, this time for x  y, 2 n we obtainkxyk  x y , and the result follows from Remark 1.4; while 2 n for y < x, kx yk  x y and the conclusion follows from (1.1). For the next proposition, X is a real inner product space, cc(Y ) denotes the subfamily of n(Y ) of all convex closed sets. We also recall the cancellation law of Rådström ([4]): Lemma 2.4. Let A; B; C be subsets of X such that A + C  B + C: If B is convex closed and C is nonempty bounded, then A  B: Proposition 2.5. If F : D  X ! n(Y ) is m-convex, c > 0; and there exists a function G: D ! cc(Y ) such F (x) = G(x) + ckxk B for all x 2 D; then G is strongly m-convex with modulus c. Strong m-convexity of set-valued functions 87 Proof. Let x; y 2 D and t 2 [0; 1]: By the m-convexity of F; 2 2 t[G(x) + ckxk B] + m(1 t)[G(y) + ckyk B] G(tx + m(1 t)y) + cktx + m(1 t)yk B; which in turn implies, multiplying by t+m(1t) and applying Lemma 2.1(1), (2.5) [t + m(1 t)](tG(x) + m(1 t)G(y)) 2 2 + [t + m(1 t)](ctkxk B + cm(1 t)kyk B) [t + m(1 t)]G(tx + m(1 t)y) + cktx + m(1 t)yk B; or 2 2 [t + m(1 t)](tkxk + m(1 t)kyk ) 2 2 = mt(1 t)kx yk +ktx + m(1 t)yk : So, by this equality, (2.5), and Lemma 2.1(2), we obtain 2 2 [t+m(1t)](tG(x)+m(1t)G(y))+cmt(1t)kxyk B+cktx+m(1t)yk B [t + m(1 t)]G(tx + m(1 t)y) + cktx + m(1 t)yk B: On the other hand, Lemma 2.1(1) implies 2 2 (2.6) [t + m(1 t)]cmt(1 t)kx yk B  cmt(1 t)kx yk B: Then, by Lemma 2.4 and (2.6), [t + m(1 t)](tG(x) + m(1 t)G(y) + cmt(1 t)kx yk B) [t + m(1 t)]G(tx + m(1 t)y); or better, tG(x) + m(1 t)G(y) + cmt(1 t)kx yk B  G(tx + m(1 t)y):  88 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales Example 2.1. The set-valued function F : [0; 1]  R ! n(R); defined by F (x) = [0; 1] is m-convex ([3, Example 2.2]). Moreover, the function 1 1 2 2 G: [0; 1]  R ! cc(R) given by G(x) = x ; 1 x , is such that 2 2 F (x) = [0; 1] = G(x) + x [1; 1] : Hence, from Proposition 2.5, G is a strongly m-convex function with modulus 1=2. Note that this fact agrees with Example 1.2. 3. More results We finish the paper with this section, in which some properties of the union, intersection and sum of strongly m-convex set-valued functions are shown same as a Jensen type inclusion for this class of functions. Proposition 3.1. Let F ; F : D ! n(Y ) be two strongly m-convex func- 1 2 tions with modulus c; such that (3.1) F (x)  F (x) (or F (x)  F (x)) 1 2 2 1 for each x 2 D. Then the union function ([3, Definition 2.18]) of F and F 1 2 is also strongly m-convex function with modulus c. Proof. It is straightforward from assumption (3.1). The following example shows that the condition (3.1) can not be omitted. Example 3.1. In Example 1.2 was shown that the functions F; G: [0; 1] ! 1 2 1 2 1 2 1 2 n(R) defined by F (x) = [ x ; 1 x ] and G(x) = [ x 1; x ]; are strongly 2 2 2 2 m-convex with modulus . Nevertheless, the function F [ G is not, since it is not m-convex (Remark 1.5). We may notice that its graph (Figure 3) clearly is not an m-convex set ([3, Theorem 2.10]). For any nonempty subsets A; B; C; D of a linear space and any scalar, the following properties hold: (A\ B) = ( A)\ ( B), A\ B + C \ D  (A + C)\ (B + D), If A  B and C  D; then A\ C  B \ D, with these in mind, proof of following result comes out. Strong m-convexity of set-valued functions 89 0:5 0:5 Figure 3. Graph of F [ G Proposition 3.2. Let F ; F : D ! n(Y ) be two set-valued functions, such 1 2 that F is strongly m-convex with modulus c and F is strongly m-convex 1 1 2 with modulus c : Then the intersection function ([3, Definition 2.18]) F \ F 2 1 2 is strongly m-convex with modulus c; where c = minfc ; c g: 1 2 Proof. Let x; y 2 D and t 2 [0; 1]: From Lemma 2.1(1) it follows that if 2 2 c = minfc ; c g; then cmt(1t)kxyk B  c mt(1t)kxyk B\c mt(1 1 2 1 2 t)kx yk B: Hence, t(F \ F )(x) + m(1 t)(F \ F )(y) + cmt(1 t)kx yk B 1 2 1 2 t[F (x)\ F (x)] + m(1 t)[F (y)\ F (y)] 1 2 1 2 2 2 + c mt(1 t)kx yk B \ c mt(1 t)kx yk B 1 2 = tF (x)\ tF (x) + m(1 t)F (y)\ m(1 t)F (y) 1 2 1 2 2 2 + c mt(1 t)kx yk B \ c mt(1 t)kx yk B 1 2 [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B] 1 1 1 \ [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B] 2 2 2 F (tx + m(1 t)y)\ F (tx + m(1 t)y) 1 2 = (F \ F )(tx + m(1 t)y): 1 2 90 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales Proposition 3.3. Let F ; F : D ! n(Y ) be two strongly m-convex func- 1 2 tions with modulus c and c ; respectively. Then the sum function ([3, Defini- 1 2 tion 2.18]) F + F is strongly m-convex with modulus c + c : 1 2 1 2 Proof. If x; y 2 D and t 2 [0; 1]; then t(F + F )(x) + m(1 t)(F + F )(y) + (c + c )mt(1 t)kx yk B 1 2 1 2 1 2 = [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B] 1 1 1 + [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B] 2 2 2 F (tx + m(1 t)y) + F (tx + m(1 t)y) 1 2 = (F + F )(tx + m(1 t)y): 1 2 Proposition 3.4. Let F : D ! n(Y ) and F : D ! n(Z) be two strongly 1 2 m-convex functions with modulus c and c ; respectively. Then the Cartesian 1 2 product function ([3, Definition 2.19]) F F is strongly m-convex with mod- 1 2 ulus c, where c = minfc ; c g, B , B are the closed unit balls in Y and Z, 1 2 Y Z and B = f(y; z) 2 Y  Z : maxfkyk;kzkg  1g  B  B . Y Z Proof. Let x; y 2 D and t 2 [0; 1]. Because c  c ; c , Lemma 2.1(1) 1 2 implies 2 2 cmt(1 t)kx yk B  c mt(1 t)kx yk B Y 1 Y (3.2) : 2 2 cmt(1 t)kx yk B  c mt(1 t)kx yk B Z 2 Z Taking into account (3.2) and properties of Cartesian product ([3]), 2 2 [cmt(1 t)kx yk B ] [cmt(1 t)kx yk B ] Y Z 2 2 [c mt(1 t)kx yk B ] [c mt(1 t)kx yk B ]: 1 Y 2 Z Then, t(F  F )(x) + m(1 t)(F  F )(y) + cmt(1 t)kx yk B 1 2 1 2 t[F (x) F (x)] + m(1 t)[F (y) F (y)] 1 2 1 2 + cmt(1 t)kx yk (B  B ) Y Z = tF (x) tF (x) + m(1 t)F (y) m(1 t)F (y) 1 2 1 2 2 2 + cmt(1 t)kx yk B  cmt(1 t)kx yk B Y Z Strong m-convexity of set-valued functions 91 tF (x) tF (x) + m(1 t)F (y) m(1 t)F (y) 1 2 1 2 2 2 + c mt(1 t)kx yk B  c mt(1 t)kx yk B 1 Y 2 Z = [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B ] 1 1 1 Y [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B ] 2 2 2 Z F (tx + m(1 t)y) F (tx + m(1 t)y) 1 2 = (F  F )(tx + m(1 t)y): 1 2 We finish the work by presenting a Jensen type inclusion for strongly m- convex set-valued functions, for the discrete case. Thereon, we simplify the no- 0; if i 6= j; tation by employing the well-known Delta of Kronecker  = ij 1; if i = j: Theorem 3.5. Let t ; : : : ; t be positive real numbers (n  2) such that 1 n T = t 2 (0; 1]. If F : D  X ! n(Y ) is a strongly m-convex function n i i=1 with modulus c; then n n i1 X X X 2 1 1 i1 k1 m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=1 i=2 k=1 i1 F m t x ; i i i=1 for all x ; : : : ; x 2 D. 1 n Proof. The proof runs by induction on n. For n = 2, 2 2 i1 X X X 2 1 1 i1 k1 m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=1 i=2 k=1 = t F (x ) + mt F (x ) + cm kt x T x k B 1 1 2 2 1 1 1 2 T T 1 2 = t F (x ) + mt F (x ) + cm kt x t x k B 1 1 2 2 1 1 1 2 t (t + t ) 1 1 2 h i t t t t 1 2 1 2 = (t + t ) F (x ) + m F (x ) + cm kx x k B 1 2 1 2 1 2 t + t t + t (t + t ) 1 2 1 2 1 2 t t 1 2 (t + t )F x + m x ; 1 2 1 2 t + t t + t 1 2 1 2 92 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales where the last inclusion results from the strong m-convexity of F . From Re- mark 1.5 and [3, Proposition 2.11] we obtain the following inclusion t t 1 2 (t + t )F x + m x  F (t x + mt x ) 1 2 1 2 1 1 2 2 t + t t + t 1 2 1 2 i1 = F m t x : i i i=1 We assume now the result is true for n. So for n + 1, let t ; : : : ; t be 1 n+1 n+1 positive real numbers with T = t 2 (0; 1], and x ; : : : ; x 2 D. n+1 i 1 n+1 i=1 Then, n+1 n+1 i1 X X X 2 1 1 i1 k1 m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=1 i=2 k=1 = t F (x ) + mt F (x ) + cm kt x t x k B 1 1 2 2 1 1 1 2 T T 1 2 n+1 n+1 i1 X X X 1 1 i1 k1 + m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=3 i=3 k=1 h i t t t t 1 2 1 2 = (t + t ) F (x ) + m F (x ) + cm kx x k B 1 2 1 2 1 2 t + t t + t (t + t ) 1 2 1 2 1 2 n+1 n+1 i1 X X X 2 1 1 i1 k1 + m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=3 i=3 k=1 n+1 t t 1 2 i1 (t + t )F x + m x + m t F (x ) 1 2 1 2 i i t + t t + t 1 2 1 2 i=3 n+1 i1 X X 2 k1 + cm m t x T x B k k i1 i T T i1 i i=3 k=1 t t 1 2 = (t + t )F x + m x + m t F (x ) 1 2 1 2 i+1 i+1 t + t t + t 1 2 1 2 i=2 n i X X 2 i+1 k1 + cm m t x T x B k k i i+1 T T i i+1 i=2 k=1 t t 1 2 = (t + t )F x + m x + m t F (x ) 1 2 1 2 i+1 i+1 t + t t + t 1 2 1 2 i=2 n i X X 2 i+1 k1 + cm t x + mt x + m t x T x B 1 1 2 2 k k i i+1 T T i i+1 i=2 k=3 Strong m-convexity of set-valued functions 93 t t 1 2 = (t + t )F x + m x + m t F (x ) 1 2 1 2 i+1 i+1 t + t t + t 1 2 1 2 i=2 t t t i+1 1 2 + cm (t + t ) x + m x 1 2 1 2 T T t + t t + t i i+1 1 2 1 2 i=2 i1 (k+1)1 + m t x T x B: k+1 k+1 i i+1 k=2 Now we set t + t ; if i = 1; 1 2 t = t ; if i 2 f2; : : : ; ng; i+1 and t t 1 2 x + m x ; if i = 1; 1 2 t + t t + t x = i 1 2 1 2 x ; if i 2 f2; : : : ; ng; i+1 then T = t + t + + t = t + t + + t := T : With this in mind n+1 1 2 n+1 1 2 n n the latter expression can be rewritten as n n i1 X X X 2 k1 t F (x ) + m t F (x ) + cm m t x T x B 1 1 i i k k i1 i T T i1 i i=2 i=2 k=1 or better, n n i1 X X X 1 1 i1 k1 (3.3) m t F (x ) + cm m t x T x B; i i k k i1 i T T i1 i i=1 i=2 k=1 where t ; : : : ; t > 0 with T = t 2 (0; 1] and x ; : : : ; x 2 D: Therefore, 1 n n i 1 n i=1 i1 by using the inductive hypothesis, (3.3) is a subset of F m t x : i i i=1 In conclusion, n+1 n+1 i1 X X X 2 1 1 i1 k1 m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=1 i=2 k=1 n n+1 X X 1 1 i1 i1 F m t x = F m t x i i i i i=1 i=1 and the result is true for n + 1 as well.  94 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales References [1] T. Lara, N. Merentes, Z. Páles, R. Quintero, and E. Rosales, On m-convexity on real linear spaces, UPI J. Math. Biostat. 1 (2018), no. 2, JMB8, 16 pp. [2] T. Lara, N. Merentes, R. Quintero, and E. Rosales, On strongly m-convex funtions, Math. Æterna 5 (2015), no. 3, 521–535. [3] T. Lara, N. Merentes, R. Quintero, and E. Rosales, On m-convexity of set-valued func- tions, Adv. Oper. Theory 4 (2019), no. 4, 767–783. [4] H. Leiva, N. Merentes, K. Nikodem, and J.L. Sánchez, Strongly convex set-valued maps, J. Global Optim. 57 (2013), no. 3, 695–705. [5] G. Toader, Some generalizations of the convexity, in: I. Muruşciac and W.W. Breckner (eds.), Proceedings of the Colloquium on Approximation and Optimization, Univ. Cluj- Napoca, Cluj-Napoca, 1985, pp. 329–338. [6] G. Toader, On a generalization of the convexity, Mathematica (Cluj) 30(53) (1988), no. 1, 83–87. Teodoro Lara Departamento de Física y Matemáticas Universidad de los Andes Núcleo “Rafael Rangel” Trujillo Venezuela e-mail: tlara@ula.ve Nelson Merentes Universidad Central de Venezuela Escuela de matemáticas Caracas Venezuela e-mail: nmerucv@gmail.com Roy Quintero Department of Mathematical Sciences Northern Illinois University DeKalb USA e-mail: rquinterocontreras@niu.edu Edgar Rosales Departamento de Física y Matemáticas Universidad de los Andes Núcleo “Rafael Rangel” Trujillo Venezuela e-mail: edgarr@ula.ve http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Mathematicae Silesianae de Gruyter

Strong m-Convexity of Set-Valued Functions

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Annales Mathematicae Silesianae 37 (2023), no. 1, 82–94 DOI: 10.2478/amsil-2023-0003 Teodoro Lara , Nelson Merentes, Roy Quintero, Edgar Rosales Abstract. In this research we introduce the concept of strong m-convexity for set-valued functions defined on m-convex subsets of real linear normed spaces, a variety of properties and examples of these functions are shown, an inclusion of Jensen type is also exhibited. 1. Introduction In this research we introduce the notion of a strongly m-convex set-valued function, which represents a generalization of the usual concept of m-convexity for the real case that can be found in [3] and references therein. The idea of this new approach involves the concepts of strong convexity and m-convexity of set-valued functions. This is the main reason for which we start off by recalling both definitions. Along this paper X; Y will denote any real normed linear spaces, D an m-convex subset of X ([1]), B the closed unit ball in Y and n(Y ) the family of all nonempty subsets of Y . Received: 11.10.2022. Accepted: 24.01.2023. Published online: 07.02.2023. (2020) Mathematics Subject Classification: 26A51, 52A30. Key words and phrases: m-convex set, strongly m-convex set-valued function, Jensen type inclusion, normed space. 2023 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/). Strong m-convexity of set-valued functions 83 Definition 1.1 ([4]). Let c > 0. A set-valued function F : D ! n(Y ) is called strongly convex with modulus c if it satisfies the inclusion tF (x) + (1 t)F (y) + ct(1 t)kx yk B  F (tx + (1 t)y); for all x; y 2 D and t 2 [0; 1]. Definition 1.2 ([3]). Let m 2 [0; 1]. A set-valued function F : D ! n(Y ) is called m-convex if the inclusion tF (x) + m(1 t)F (y)  F (tx + m(1 t)y); holds for all x; y 2 D and t 2 [0; 1]. Our first definition runs as follows: Definition 1.3. Let c > 0 and m 2 [0; 1]. A set-valued function F : D ! n(Y ) is called strongly m-convex with modulus c if (1.1) tF (x) + m(1 t)F (y) + cmt(1 t)kx yk B  F (tx + m(1 t)y); for any x; y 2 D, t 2 [0; 1]. Remark 1.4. Notice that (1.1) is equivalent to mtF (x) + (1 t)F (y) + cmt(1 t)kx yk B  F (mtx + (1 t)y); with x; y; t as before. Remark 1.5. If a set-valued function F is strongly m-convex with mod- ulus c, then it is also m-convex. It follows immediately from the fact that 0 2 B. The converse in the foregoing remark is not true. Namely, we have the following. Example 1.1. The set-valued function F : [0; 1]  R ! n(R); given by F (x) = [0; x]; is m-convex ([3, Example 2.17]). But for all x; y; t 2 [0; 1] tF (x)+m(1 t)F (y) + cmt(1 t)kx yk B 2 2 = cmt(1 t)kx yk ; tx + m(1 t)y + cmt(1 t)kx yk ; 84 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales while that F (tx + m(1 t)y) = [0; tx + m(1 t)y]; so F can not be a strongly m-convex function. Example 1.2. If b > 0 and f; g : [0; b] ! R are two real functions, f and g being strongly m-convex with the same modulus ([2]) and f  g on [0; b], it is not difficult to verify (by reasoning as in Example 2.2 from [3]) that the set-valued functions F ; F ; F : [0; b]  R ! n(R) given by 1 2 3 F (x) = [f(x); g(x)]; F (x) = [f(x); +1); F (x) = (1; g(x)] 1 2 3 are strongly m-convex (with the same modulus). So, for example, functions f ; g : [0; 1] ! R defined as f (x) = 0 and g (x) = 1 are clearly m- 1 1 1 1 1 1 2 2 convex ([5, 6]), while functions f(x) = x , g(x) = 1 x are such that 2 2 f and g are strongly m-convex with modulus c = ; moreover f  g on [0; 1]. Consequently the set-valued function F : [0; 1] ! n(R) defined by 1 2 1 2 1 F (x) = x ; 1 x is strongly m-convex with modulus , and so is G(x) = 2 2 2 1 1 2 2 x 1; x . The graphs of F and G are shown in Figures 1 and 2, re- 2 2 spectively. 0:5 0:5 Figure 1. Graph of F Figure 2. Graph of G 2. Results In this section we present some set-properties of the unit ball B. At the same time, a characterization of the family of all the strongly m-convex functions is given and illustrate with an interesting example. We begin with Strong m-convexity of set-valued functions 85 a lemma related to two well-known properties of convexity whose proofs are omitted. Lemma 2.1. (1) If 0   ; then B  B: 1 2 1 2 (2) If  0; then ( + )B = B + B: 1 2 1 2 1 2 Proposition 2.2. A set-valued function F : D ! n(Y ) is strongly m- convex with modulus c if and only if (2.1) tF (A )+m(1t)F (A )+cmt(1t)kA A k B  F (A +m(1t)A ) 1 2 1 2 1 2 for all A ; A  D and t 2 [0; 1]; where F (A ) = fF (x) : x 2 A g (i = 1; 2) 1 2 i i and kA A k = inffkx yk : x 2 A ; y 2 A g: 1 2 1 2 Proof. ()) Let A ; A be two fixed but arbitrary subsets of D and z 2 1 2 tF (A ) + m(1 t)F (A ) + cmt(1 t)kA A k B: Then 1 2 1 2 (2.2) z 2 tF (a) + m(1 t)F (b) + cmt(1 t)kA A k B 1 2 for some a 2 A and b 2 A . Since 0  kA A k  ka bk, 0  cmt(1 1 2 1 2 2 2 t)kA A k  cmt(1 t)ka bk and from Lemma 2.1(1), the inclusion 1 2 2 2 cmt(1 t)kA A k B  cmt(1 t)ka bk B takes place. Hence, 1 2 (2.3) tF (a) + m(1 t)F (b) + cmt(1 t)kA A k B 1 2 tF (a) + m(1 t)F (b) + cmt(1 t)ka bk B: Furthermore, since ta + m(1 t)b 2 tA + m(1 t)A ; it is clear that 1 2 (2.4) F (ta + m(1 t)b)  F (tA + m(1 t)A ): 1 2 So, (2.1) follows from (2.2), (2.3), the strong m-convexity of F and (2.4). (() Let x; y 2 D and t 2 [0; 1]: The strong m-convexity with modu- lus c of F is obtained by considering in (2.1) the singletons A = fxg and A = fyg. Proposition 2.3. Let b 2 Rnf0g and D = [minf0; bg; maxf0; bg]  R: If F : D ! n(Y ) is strongly m-convex with modulus c, and 0 < n  m < 1; then F is strongly n-convex with modulus c. 86 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales Proof. If b < 0, then D = [b; 0]: Let t 2 [0; 1] and x; y 2 D with x  y. n 2 n So, x y  x y  0 and therefore, kx yk  x y : Since F is m m strongly m-convex with modulus c; F is m-convex (Remark 1.5). Thus, from [3, Proposition 2.11], Lemma 2.1(1), and the strong m-convexity of F , tF (x)+n(1 t)F (y) + cnt(1 t)kx yk B n n = tF (x) + m(1 t) F (y) + cmt(1 t) kx yk B m m n n tF (x) + m(1 t)F y + cmt(1 t) x y B m m F (tx + n(1 t)y): And for y < x, kx yk  x y , hence ntF (x)+(1 t)F (y) + cnt(1 t)kx yk B n n = mt F (x) + (1 t)F (y) + cmt(1 t) kx yk B m m n n mtF x + (1 t)F (y) + cmt(1 t) x y B m m F (ntx + (1 t)y); where the last inclusion arises from the strong m-convexity of F and Re- mark 1.4. If b > 0, D = [0; b] and the proof runs in a similar way, this time for x  y, 2 n we obtainkxyk  x y , and the result follows from Remark 1.4; while 2 n for y < x, kx yk  x y and the conclusion follows from (1.1). For the next proposition, X is a real inner product space, cc(Y ) denotes the subfamily of n(Y ) of all convex closed sets. We also recall the cancellation law of Rådström ([4]): Lemma 2.4. Let A; B; C be subsets of X such that A + C  B + C: If B is convex closed and C is nonempty bounded, then A  B: Proposition 2.5. If F : D  X ! n(Y ) is m-convex, c > 0; and there exists a function G: D ! cc(Y ) such F (x) = G(x) + ckxk B for all x 2 D; then G is strongly m-convex with modulus c. Strong m-convexity of set-valued functions 87 Proof. Let x; y 2 D and t 2 [0; 1]: By the m-convexity of F; 2 2 t[G(x) + ckxk B] + m(1 t)[G(y) + ckyk B] G(tx + m(1 t)y) + cktx + m(1 t)yk B; which in turn implies, multiplying by t+m(1t) and applying Lemma 2.1(1), (2.5) [t + m(1 t)](tG(x) + m(1 t)G(y)) 2 2 + [t + m(1 t)](ctkxk B + cm(1 t)kyk B) [t + m(1 t)]G(tx + m(1 t)y) + cktx + m(1 t)yk B; or 2 2 [t + m(1 t)](tkxk + m(1 t)kyk ) 2 2 = mt(1 t)kx yk +ktx + m(1 t)yk : So, by this equality, (2.5), and Lemma 2.1(2), we obtain 2 2 [t+m(1t)](tG(x)+m(1t)G(y))+cmt(1t)kxyk B+cktx+m(1t)yk B [t + m(1 t)]G(tx + m(1 t)y) + cktx + m(1 t)yk B: On the other hand, Lemma 2.1(1) implies 2 2 (2.6) [t + m(1 t)]cmt(1 t)kx yk B  cmt(1 t)kx yk B: Then, by Lemma 2.4 and (2.6), [t + m(1 t)](tG(x) + m(1 t)G(y) + cmt(1 t)kx yk B) [t + m(1 t)]G(tx + m(1 t)y); or better, tG(x) + m(1 t)G(y) + cmt(1 t)kx yk B  G(tx + m(1 t)y):  88 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales Example 2.1. The set-valued function F : [0; 1]  R ! n(R); defined by F (x) = [0; 1] is m-convex ([3, Example 2.2]). Moreover, the function 1 1 2 2 G: [0; 1]  R ! cc(R) given by G(x) = x ; 1 x , is such that 2 2 F (x) = [0; 1] = G(x) + x [1; 1] : Hence, from Proposition 2.5, G is a strongly m-convex function with modulus 1=2. Note that this fact agrees with Example 1.2. 3. More results We finish the paper with this section, in which some properties of the union, intersection and sum of strongly m-convex set-valued functions are shown same as a Jensen type inclusion for this class of functions. Proposition 3.1. Let F ; F : D ! n(Y ) be two strongly m-convex func- 1 2 tions with modulus c; such that (3.1) F (x)  F (x) (or F (x)  F (x)) 1 2 2 1 for each x 2 D. Then the union function ([3, Definition 2.18]) of F and F 1 2 is also strongly m-convex function with modulus c. Proof. It is straightforward from assumption (3.1). The following example shows that the condition (3.1) can not be omitted. Example 3.1. In Example 1.2 was shown that the functions F; G: [0; 1] ! 1 2 1 2 1 2 1 2 n(R) defined by F (x) = [ x ; 1 x ] and G(x) = [ x 1; x ]; are strongly 2 2 2 2 m-convex with modulus . Nevertheless, the function F [ G is not, since it is not m-convex (Remark 1.5). We may notice that its graph (Figure 3) clearly is not an m-convex set ([3, Theorem 2.10]). For any nonempty subsets A; B; C; D of a linear space and any scalar, the following properties hold: (A\ B) = ( A)\ ( B), A\ B + C \ D  (A + C)\ (B + D), If A  B and C  D; then A\ C  B \ D, with these in mind, proof of following result comes out. Strong m-convexity of set-valued functions 89 0:5 0:5 Figure 3. Graph of F [ G Proposition 3.2. Let F ; F : D ! n(Y ) be two set-valued functions, such 1 2 that F is strongly m-convex with modulus c and F is strongly m-convex 1 1 2 with modulus c : Then the intersection function ([3, Definition 2.18]) F \ F 2 1 2 is strongly m-convex with modulus c; where c = minfc ; c g: 1 2 Proof. Let x; y 2 D and t 2 [0; 1]: From Lemma 2.1(1) it follows that if 2 2 c = minfc ; c g; then cmt(1t)kxyk B  c mt(1t)kxyk B\c mt(1 1 2 1 2 t)kx yk B: Hence, t(F \ F )(x) + m(1 t)(F \ F )(y) + cmt(1 t)kx yk B 1 2 1 2 t[F (x)\ F (x)] + m(1 t)[F (y)\ F (y)] 1 2 1 2 2 2 + c mt(1 t)kx yk B \ c mt(1 t)kx yk B 1 2 = tF (x)\ tF (x) + m(1 t)F (y)\ m(1 t)F (y) 1 2 1 2 2 2 + c mt(1 t)kx yk B \ c mt(1 t)kx yk B 1 2 [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B] 1 1 1 \ [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B] 2 2 2 F (tx + m(1 t)y)\ F (tx + m(1 t)y) 1 2 = (F \ F )(tx + m(1 t)y): 1 2 90 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales Proposition 3.3. Let F ; F : D ! n(Y ) be two strongly m-convex func- 1 2 tions with modulus c and c ; respectively. Then the sum function ([3, Defini- 1 2 tion 2.18]) F + F is strongly m-convex with modulus c + c : 1 2 1 2 Proof. If x; y 2 D and t 2 [0; 1]; then t(F + F )(x) + m(1 t)(F + F )(y) + (c + c )mt(1 t)kx yk B 1 2 1 2 1 2 = [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B] 1 1 1 + [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B] 2 2 2 F (tx + m(1 t)y) + F (tx + m(1 t)y) 1 2 = (F + F )(tx + m(1 t)y): 1 2 Proposition 3.4. Let F : D ! n(Y ) and F : D ! n(Z) be two strongly 1 2 m-convex functions with modulus c and c ; respectively. Then the Cartesian 1 2 product function ([3, Definition 2.19]) F F is strongly m-convex with mod- 1 2 ulus c, where c = minfc ; c g, B , B are the closed unit balls in Y and Z, 1 2 Y Z and B = f(y; z) 2 Y  Z : maxfkyk;kzkg  1g  B  B . Y Z Proof. Let x; y 2 D and t 2 [0; 1]. Because c  c ; c , Lemma 2.1(1) 1 2 implies 2 2 cmt(1 t)kx yk B  c mt(1 t)kx yk B Y 1 Y (3.2) : 2 2 cmt(1 t)kx yk B  c mt(1 t)kx yk B Z 2 Z Taking into account (3.2) and properties of Cartesian product ([3]), 2 2 [cmt(1 t)kx yk B ] [cmt(1 t)kx yk B ] Y Z 2 2 [c mt(1 t)kx yk B ] [c mt(1 t)kx yk B ]: 1 Y 2 Z Then, t(F  F )(x) + m(1 t)(F  F )(y) + cmt(1 t)kx yk B 1 2 1 2 t[F (x) F (x)] + m(1 t)[F (y) F (y)] 1 2 1 2 + cmt(1 t)kx yk (B  B ) Y Z = tF (x) tF (x) + m(1 t)F (y) m(1 t)F (y) 1 2 1 2 2 2 + cmt(1 t)kx yk B  cmt(1 t)kx yk B Y Z Strong m-convexity of set-valued functions 91 tF (x) tF (x) + m(1 t)F (y) m(1 t)F (y) 1 2 1 2 2 2 + c mt(1 t)kx yk B  c mt(1 t)kx yk B 1 Y 2 Z = [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B ] 1 1 1 Y [tF (x) + m(1 t)F (y) + c mt(1 t)kx yk B ] 2 2 2 Z F (tx + m(1 t)y) F (tx + m(1 t)y) 1 2 = (F  F )(tx + m(1 t)y): 1 2 We finish the work by presenting a Jensen type inclusion for strongly m- convex set-valued functions, for the discrete case. Thereon, we simplify the no- 0; if i 6= j; tation by employing the well-known Delta of Kronecker  = ij 1; if i = j: Theorem 3.5. Let t ; : : : ; t be positive real numbers (n  2) such that 1 n T = t 2 (0; 1]. If F : D  X ! n(Y ) is a strongly m-convex function n i i=1 with modulus c; then n n i1 X X X 2 1 1 i1 k1 m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=1 i=2 k=1 i1 F m t x ; i i i=1 for all x ; : : : ; x 2 D. 1 n Proof. The proof runs by induction on n. For n = 2, 2 2 i1 X X X 2 1 1 i1 k1 m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=1 i=2 k=1 = t F (x ) + mt F (x ) + cm kt x T x k B 1 1 2 2 1 1 1 2 T T 1 2 = t F (x ) + mt F (x ) + cm kt x t x k B 1 1 2 2 1 1 1 2 t (t + t ) 1 1 2 h i t t t t 1 2 1 2 = (t + t ) F (x ) + m F (x ) + cm kx x k B 1 2 1 2 1 2 t + t t + t (t + t ) 1 2 1 2 1 2 t t 1 2 (t + t )F x + m x ; 1 2 1 2 t + t t + t 1 2 1 2 92 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales where the last inclusion results from the strong m-convexity of F . From Re- mark 1.5 and [3, Proposition 2.11] we obtain the following inclusion t t 1 2 (t + t )F x + m x  F (t x + mt x ) 1 2 1 2 1 1 2 2 t + t t + t 1 2 1 2 i1 = F m t x : i i i=1 We assume now the result is true for n. So for n + 1, let t ; : : : ; t be 1 n+1 n+1 positive real numbers with T = t 2 (0; 1], and x ; : : : ; x 2 D. n+1 i 1 n+1 i=1 Then, n+1 n+1 i1 X X X 2 1 1 i1 k1 m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=1 i=2 k=1 = t F (x ) + mt F (x ) + cm kt x t x k B 1 1 2 2 1 1 1 2 T T 1 2 n+1 n+1 i1 X X X 1 1 i1 k1 + m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=3 i=3 k=1 h i t t t t 1 2 1 2 = (t + t ) F (x ) + m F (x ) + cm kx x k B 1 2 1 2 1 2 t + t t + t (t + t ) 1 2 1 2 1 2 n+1 n+1 i1 X X X 2 1 1 i1 k1 + m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=3 i=3 k=1 n+1 t t 1 2 i1 (t + t )F x + m x + m t F (x ) 1 2 1 2 i i t + t t + t 1 2 1 2 i=3 n+1 i1 X X 2 k1 + cm m t x T x B k k i1 i T T i1 i i=3 k=1 t t 1 2 = (t + t )F x + m x + m t F (x ) 1 2 1 2 i+1 i+1 t + t t + t 1 2 1 2 i=2 n i X X 2 i+1 k1 + cm m t x T x B k k i i+1 T T i i+1 i=2 k=1 t t 1 2 = (t + t )F x + m x + m t F (x ) 1 2 1 2 i+1 i+1 t + t t + t 1 2 1 2 i=2 n i X X 2 i+1 k1 + cm t x + mt x + m t x T x B 1 1 2 2 k k i i+1 T T i i+1 i=2 k=3 Strong m-convexity of set-valued functions 93 t t 1 2 = (t + t )F x + m x + m t F (x ) 1 2 1 2 i+1 i+1 t + t t + t 1 2 1 2 i=2 t t t i+1 1 2 + cm (t + t ) x + m x 1 2 1 2 T T t + t t + t i i+1 1 2 1 2 i=2 i1 (k+1)1 + m t x T x B: k+1 k+1 i i+1 k=2 Now we set t + t ; if i = 1; 1 2 t = t ; if i 2 f2; : : : ; ng; i+1 and t t 1 2 x + m x ; if i = 1; 1 2 t + t t + t x = i 1 2 1 2 x ; if i 2 f2; : : : ; ng; i+1 then T = t + t + + t = t + t + + t := T : With this in mind n+1 1 2 n+1 1 2 n n the latter expression can be rewritten as n n i1 X X X 2 k1 t F (x ) + m t F (x ) + cm m t x T x B 1 1 i i k k i1 i T T i1 i i=2 i=2 k=1 or better, n n i1 X X X 1 1 i1 k1 (3.3) m t F (x ) + cm m t x T x B; i i k k i1 i T T i1 i i=1 i=2 k=1 where t ; : : : ; t > 0 with T = t 2 (0; 1] and x ; : : : ; x 2 D: Therefore, 1 n n i 1 n i=1 i1 by using the inductive hypothesis, (3.3) is a subset of F m t x : i i i=1 In conclusion, n+1 n+1 i1 X X X 2 1 1 i1 k1 m t F (x ) + cm m t x T x B i i k k i1 i T T i1 i i=1 i=2 k=1 n n+1 X X 1 1 i1 i1 F m t x = F m t x i i i i i=1 i=1 and the result is true for n + 1 as well.  94 Teodoro Lara, Nelson Merentes, Roy Quintero, Edgar Rosales References [1] T. Lara, N. Merentes, Z. Páles, R. Quintero, and E. Rosales, On m-convexity on real linear spaces, UPI J. Math. Biostat. 1 (2018), no. 2, JMB8, 16 pp. [2] T. Lara, N. Merentes, R. Quintero, and E. Rosales, On strongly m-convex funtions, Math. Æterna 5 (2015), no. 3, 521–535. [3] T. Lara, N. Merentes, R. Quintero, and E. Rosales, On m-convexity of set-valued func- tions, Adv. Oper. Theory 4 (2019), no. 4, 767–783. [4] H. Leiva, N. Merentes, K. Nikodem, and J.L. Sánchez, Strongly convex set-valued maps, J. Global Optim. 57 (2013), no. 3, 695–705. [5] G. Toader, Some generalizations of the convexity, in: I. Muruşciac and W.W. Breckner (eds.), Proceedings of the Colloquium on Approximation and Optimization, Univ. Cluj- Napoca, Cluj-Napoca, 1985, pp. 329–338. [6] G. Toader, On a generalization of the convexity, Mathematica (Cluj) 30(53) (1988), no. 1, 83–87. Teodoro Lara Departamento de Física y Matemáticas Universidad de los Andes Núcleo “Rafael Rangel” Trujillo Venezuela e-mail: tlara@ula.ve Nelson Merentes Universidad Central de Venezuela Escuela de matemáticas Caracas Venezuela e-mail: nmerucv@gmail.com Roy Quintero Department of Mathematical Sciences Northern Illinois University DeKalb USA e-mail: rquinterocontreras@niu.edu Edgar Rosales Departamento de Física y Matemáticas Universidad de los Andes Núcleo “Rafael Rangel” Trujillo Venezuela e-mail: edgarr@ula.ve

Journal

Annales Mathematicae Silesianae – de Gruyter

Published: Mar 1, 2023

Keywords: m -convex set; strongly m -convex set-valued function; Jensen type inclusion; normed space; 26A51; 52A30

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