On the Krein-Milman theorem in CAT(κ) spaces

Let κ > 0 and (X, ρ) be a complete CAT(κ) space whose diameter smaller than ... It is shown that if K is a nonempty compact convex subset of X, then K is the closed convex hull of its set of extreme points. This is an extension of the Krein-Milman theorem to the general setting of CAT(κ) spaces.

Existence Solutions of Vector Equilibrium Problems and Fixed Point of Multivalued Mappings

LetKbe a nonempty compact convex subset of a topological vector space. In this paper-sufficient conditions are given for the existence ofx∈Ksuch thatF(T)∩VEP(F)≠∅, whereF(T)is the set of all fixed points of the multivalued mappingTandVEP(F)is the set of all solutions for vector equilibrium problem of the vector-valued mappingF. This leads us to generalize and improve some existence results in the recent references.

An application of KKM-map principle

The following theorem is proved and several fixed point theorems and coincidence theorems are derived as corollaries. LetCbe a nonempty convex subset of a normed linear spaceX,f:C→Xa continuous function,g:C→Ccontinuous, onto and almost quasi-convex. Assume thatChas a nonempty compact convex subsetDsuch that the setA={y∈C:‖g(x)−f(y)‖≥‖g(y)−f(y)‖ for all x∈D}is compact.Then there is a pointy0∈Csuch that‖g(y0)−f(y0)‖=d(f(y0),C).

Generalizations of F. E. Browder's sharpened form of the schauder fixed point theorem

Let E be a Hausdorff topological vector space, let K be a nonempty compact convex subset of E and let f, g: K → 2E be upper semicontinuous such that for each x ∈ K, f(x) and g(x) are nonempty compact convex. Let Ω ⊂ 2E be convex and contain all sets of the form x − f(x), y − x + g(x) − f(x), for x, y ∈ K. Suppose p: K × Ω →, R satisfies: (i) for each (x, A) ∈ K × Ω and for ε > 0, there exist a neighborhood U of x in K and an open subset set G in E with A ⊂ G such that for all (y, B) ∈ K ×Ω with y ∈ U and B ⊂ G, | p(y, B) - p(x, A)| < ε, and (ii) for each fixed X ∈ K, p(x, ·) is a convex function on Ω. If p(x, x − f(x)) ≤ p(x, g(x) − f(x)) for all x ∈ K, and if, for each x ∈ K with f(x) ∩ g(x) = ø, there exists y ∈ K with p(x, y − x + g(x) − f(x)) < p(x, x − f(x)), then there exists an x0 ∈ K such that f(x0) ∩ g(x0) ≠ ø. Another coincidence theorem on a nonempty compact convex subset of a Hausdorff locally convex topological vector space is also given.