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We give a generalized version of the well-known Borsuk’s antipodal fixed point theorem for a large class of antipodally approachable condensing or compact set-valued maps defined on closed subsets of locally convex topological vector spaces. These results contain corresponding results obtained in the literature for compact set-valued maps with convex values.

Mann&rsquo, s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fr&eacute, chet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann&rsquo, s iteration process and its variants.

In this paper, we give a uniform approach for generalized convexity by using the concept of L-convexity defined by Ben El-Mechaiekh et al. (J Math Anal Appl 222:138–150, 1998). We prove that the generalized notion of L-space contains well-known generalized convex spaces defined in the literature in topological vector spaces as well as several generalized convexity structures defined on metric spaces. In this context, we give a generalized version of the Fan–Knaster–Kuratowski–Mazurkiewicz Principle (FKKM Principle) in L-spaces and a Browder-Fan type theorem about the existence of fixed points for open lower section set-valued maps defined in an L-space. As an application, we prove the existence of equilibria for an abstract economy with an infinite number of agents.

International audience; We present in this paper a generalized version of the celebrated Knaster–Kuratowski–Mazurkiewicz–Fan's principle on the intersection of a family of closed sets subject to a classical geometric condition and a weakened compactness condition. The fixed point formulation of this generalized principle extends the Browder–Fan fixed point theorem to set-valued maps of non-compact convex subsets of topological vector spaces.

The purpose of this paper is to extend Himmelberg's fixed-point theorem, replacing the usual convexity in topological vector spaces with an abstract topological notion of convexity that generalizes classical convexity as well as several metric convexity structures found in the literature. We prove the existence, under weak hypotheses, of a fixed point for a compact approachable map, and we provide sufficient conditions under which this result applies to maps whose values are convex in the abstract sense mentioned above.