### Fixed points, intersection theorems, variational inequalities, and equilibrium theorems.

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Generalized and unified versions of coincidence or maximal element theorems of Fan, Yannelis and Prabhakar, Ha, Sessa, Tarafdar, Rim and Kim, Mehta and Sessa, Kim and Tan are obtained. Our arguments are based on our recent works on a broad class of multifunctions containing composites of acyclic maps defined on convex subsets of Hausdorff topological vector spaces.

Let $X$ be a uniformly convex Banach space, $D\subset X$, $f:D\to X$ a nonexpansive map, and $K$ a closed bounded subset such that $\overline{\text{co}}\phantom{\rule{0.166667em}{0ex}}K\subset D$. If (1) ${f|}_{K}$ is weakly inward and $K$ is star-shaped or (2) ${f|}_{K}$ satisfies the Leray-Schauder boundary condition, then $f$ has a fixed point in $\overline{\text{co}}\phantom{\rule{0.166667em}{0ex}}K$. This is closely related to a problem of Gulevich [Gu]. Some of our main results are generalizations of theorems due to Kirk and Ray [KR] and others.

Using our own generalization [7] of J.C. Bellenger’s theorem [1] on the existence of maximizable u.s.cq̇uasiconcave functions on convex spaces, we obtain extended versions of the existence theorem of H. Ben-El-Mechaiekh [2] on zeros for multifunctions with non-compact domains, the coincidence theorem of S.H. Kum [5] for upper hemicontinuous multifunctions, and the Ky Fan type fixed point theorems due to E. Tarafdar [13].

In this paper, a set-valued mapping with G-KKM property is defined and a generalization of minimax theorem for set-valued maps with G-KKM property on generalized convex space is established. As a consequence of this results we verify the coincidence theorem for set-valued maps with G-KKM property on G-convex space. Finally, we apply our results to the best approximation problem and fixed point problem.

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