Common fixed point theorems for set-valued mappings in normed spaces

Abstract

Let \(\Phi \) be the class of all real functions \(\varphi : [0, \infty [ \times [0, \infty [ \rightarrow [0, \infty [\) that satisfy the following condition: there exists \(\alpha \in ]0, 1[ \text { such that } \varphi ((1 - \alpha ) r, \alpha r) 0\). In this paper, we show that if X is a nonempty compact convex subset of a real normed vector space, any two closed set-valued mappings \(T, S: X \rightrightarrows X\), with nonempty and convex values, have a common fixed point whenver there exists a function \(\varphi \in \Phi \) such that $$\begin{aligned} \Vert y - u\Vert \le \varphi (\Vert y - x\Vert , \Vert u - x\Vert ), \; \text { for all } x\in X, y\in T(x), u\in S(x). \end{aligned}$$ Next, we prove that the same conclusion holds when at least one of the set-valued mappings is lower semicontinuous with nonempty closed and convex values. Our common fixed point theorems turn out to be useful for a unitary treatment of several problems from optimization and nonlinear analysis (quasi-equilibrium problems, quasi-optimization problems, constrained fixed point problems, quasi-variational inequalities).