On \(m\)-convexity of set-valued functions. (English) Zbl 1468.26024

Lara, Teodoro; Merentes, Nelson; Quintero, Roy; Rosales, Edgar

Adv. Oper. Theory 4, No. 4, 767-783 (2019).

Let \(X\) and \(Y\) be real linear spaces, \(D\) a subset of \(X\) and \(m \in [0, 1]\). The set \(D\) is said to be \(m\)-convex if \(t x + m(1 - t)y \in D\), for all \(x, y \in X\) and each \(t \in [0,1]\). If \(D\) is \(m\)-convex, a nonempty-valued set-valued function \(F: D \rightrightarrows Y\) is called \(m\)-convex if \(t F(x) + m(1- t)F(y) \subseteq F(tx + m(1- t)y)\), for all \(x,y \in D\) and \(t \in [0,1]\). In this paper, some algebraic properties and several characterizations of this class of set-valued functions are given.

Reviewer: Mircea Balaj (Oradea)

MSC:

26E25	Set-valued functions
26A51	Convexity of real functions in one variable, generalizations
47H04	Set-valued operators
52A01	Axiomatic and generalized convexity

Keywords:

Jensen-type inclusion; \(m\)-convex set; \(m\)-convex set-valued function; sandwich theorem