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

  • Mircea Balaj
  • Mohamed A. KhamsiEmail author
Original Paper
  • 51 Downloads

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 \(\alpha \in ]0, 1[ \text { such that } \varphi ((1 - \alpha ) r, \alpha r) < r,\; \text { for all } r > 0\). 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).

Keywords

Common fixed point Hyperconvex metric space Multivalued mapping Quasi-equilbrium problem Quasi-optimization problem 

Mathematics Subject Classification

47H10 49J53 

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Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OradeaOradeaRomania
  2. 2.Department of Mathematical SciencesUniversity of Texas at El PasoEl PasoUSA
  3. 3.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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