Suppose now X is a locally convex topological vector space and X* its dual space.
(b) Let A;B[member of] WCC (X) with A [not equal to] B: We may assume without loss of generality that there exists some a [member of] A such that a [member of] B: Since B is a [T.sub.w]-compact subset of the locally convex topological vector space (X;[T.sub.w]) ; it follows from the Hahn-Banach Separation Theorem that there exists some x* [member of] X* such that sup Rex* (b) < Rex* (a): Let [delta] = Rex* (a) [mathematical expression not reproducible] (b) > 0: We have Rex* (a) Rex* (b) [greater than or equal to] [delta] for all b [member of] B which in turn implies that
Suppose X is a locally convex topological vector space equipped with original topology T as well as weak topology [T.sub.w]; and (WCC (X) ;[T.sub.w]) is the corresponding hyperspace.
Let K be a non-empty compact, convex subset of a locally convex topological vector space X: Then K has an extreme point in K:
Let X be a nonempty compact convex subset of a Hausdorff
locally convex topological vector space, Z be a nonempty set, [rho] bea relation on [2.sup.Z] and T : X [??] X, F : X x X [??] Z and C : X [??] Z be three mappings satisfying conditions (ii), (iii) and (iv) in Theorem 1.
Our applications are mainly on acyclic polyhedra,
locally convex topological vector spaces, admissible (in the sense of Klee) convex sets, and almost convex or Klee approximable sets in topological vector spaces.
Their topics include contraction mappings, fixed point theorems in partially ordered sets, topological fixed point theorems, variational and quasivariational inequalities in topological vectors spaces and generalized games, best approximations and fixed point theorems for set-valued mappings in topological vector spaces, degree theories for set-valued mappings, and nonexpansive types of mappings and fixed-point theorems in
locally convex topological vector spaces. Throughout they consider various aspects of fixed points, minimax inequalities, end points, variational inequalities, equilibrium analysis in economics and related topics, the result being remarkably balanced in theory and application.