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Article

Agarwal, Ravi P. ; Balaj, Mircea ; O'Regan, Donal
An intersection theorem for set-valued mappings. (English). Applications of Mathematics, vol. 58 (2013), issue 3, pp. 269-278
MSC: 47H04, 47H10, 49J53 | MR 3066821 | Zbl 1275.47105 | DOI: 10.1007/s10492-013-0013-7
Keywords:
intersection theorem; fixed point; saddle point; equilibrium problem; complementarity problem
Summary:
Given a nonempty convex set $X$ in a locally convex Hausdorff topological vector space, a nonempty set $Y$ and two set-valued mappings $T\colon X\rightrightarrows X$, $S\colon Y\rightrightarrows X$ we prove that under suitable conditions one can find an $x\in X$ which is simultaneously a fixed point for $T$ and a common point for the family of values of $S$. Applying our intersection theorem we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems.
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