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. ᮊ

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Abstract. We present new theorems on the existence of equilibria (or zeros) of convex as well as nonconvex set-valued maps defined on compact neighborhood retracts of normed spaces. The maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets. Among other things, we show that if K is a compact neighborhood retract with nontrivial Euler characteristic in a Banach space E , and Φ : K −→ 2 E is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency conditionthen there exists x 0 ∈ K such that 0 ∈ Φ(x 0 ). Here, T r K (x) denotes a new concept of retraction tangent cone to K at x suited for compact neighborhood retracts. When K is locally convex at x, T r K (x) coincides with the usual tangent cone of convex analysis.Special attention is given to neighborhood retracts having "lipschitzian behavior", called L−retracts below. This class of sets is very broad; it contains compact homeomorphically convex subsets of Banach spaces, epi-Lipschitz subsets of Banach spaces, as well as proximate retracts. Our results thus generalize classical theorems for convex domains, as well as recent results for nonconvex sets.

Abstract. The purpose of this note is to generalize the celebrated Ran and Reurings fixed point theorem to the setting of a space with a binary relation that is only transitive (and not necessarily a partial order) and a relation-complete metric. The arguments presented here are simple and straightforward. It is also shown that extensions by Rakotch and Hu-Kirk of Edelstein's generalization of the Banach contraction principle to local contractions on chainable complete metric spaces derive from the theorem of Ran-Reurings. Keywords and Phrases:Existence and uniqueness of fixed point; contraction and local contraction; transitive relation; monotonic chainability; monotoniccomplete metric. Mathematics Subject Classification: 47H10 Dedicated to Professor Andrzej Granas PreliminariesIn 1961, M. Edelstein [3] extended the Banach contraction principle by establishing that every uniform local contraction f : X −→ X of an ε-chainable complete metric space (X, d) has a unique fixed point. In 1962, E. Rakotch [8] refined Edelstein's result to a local contraction f of a complete metric space containing some rectifiable path (i.e., a path of finite length 1 ) joining a given point x 0 to f (x 0 ).Recall that a metric space (X, d) is said to be ε-chainable for some ε > 0, if ∀x, y ∈ X, ∃{u i } m i=0 a finite sequence in X such that: x = u 0 , u m = y and d(u i−1 , u i ) < ε for all i = 1, . . . , m. * e-mail address: hmechaie@brocku.ca 1 The length of (continuous) path γ : [0, 1] −→ X is l(γ) := sup{L(P ) : P ∈ P[0, 1]} where P[0, 1] is the collection of all finite partitions P = {0 = t 0 < t 1 < . . . < tn = 1} of [0, 1], and L(P ) = n i=1 d(γ(t i−1 ), γ(t i )).

International audienceWe 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