Generalized R-KKM Theorems And Collectively Fixed Point Theorems In Topological Spaces And Their Applications

Recently, Deng[56] and author introduced generalized .R-KKM mapping in topological space. By using the notion of generalized R-KKM mapping, we prove some new generalized R-KKM theorems on topological spaces without any convex structure. These theorems improve, unify, and generalize many well known reults. The main results are the following theorems.Theorem 2.1.1. Let X be a nonempty set and Y be a Hausdorff topological space. Let T : X 4 2Y be a set-valued mapping such that T(x) is nonempty and compactly closed in Y for each (i) If T is a generalized il-KKM mapping, then for each N = where is the continuous mapping in touch with N in definition of generalized .R-KKM mapping.(ii) If the family has finite intersection property, then T is a generalized R-KKM mapping.Theorem 2.1.2. Let X be a nonempty set and Y be a Hausdorff topological space. Let T : X 2Y be a set-valued mapping such that T(x) is nonempty and compactly closed in Y for each x X. Suppose that is compact for some M (X). Then if and only if T is a generalized .R-KKM mapping.Theorem 2.1.3. Let X be a nonempty set and Y be a compact Hausdorff topological space. Let T : X - 2Y be a generalized .R-KKM mapping with nonempty compactly closed values. If there exists a nonempty compact subset K of Y and a mapping S : X - 2Y, suchthat for each where is the continuous mapping in touch with N in definition of generalized i?-KKM mapping, then Theorem 2.1.4. Let X be a nonempty set and V be a Hausdorff topological space. Let T : X -2Y be a generalized R-KKM mapping with nonempty transfer compactly closed values. If is compact for some M (X), then Theorem 2.1.5. Let X be a nonempty set and V be a compact Hausdorff topological space. Let T : X - 2Y be a generalized R-KKM mapping with nonempty transfer compactly closed values. If there exits a nonempty compact subset K of Y and a mapping such that for each andAs applications, we have the following new minimax inequalities and saddle point theorem.Theorem 2.2.1. Let X be a nonempty set, Y be a Hausdorff topological space, and A R. Let f,g: be two mappings satisfying(i) for each (x,y) G X x Y, f(x,y) < g{x,y),(ii) f{x,y) is A-transfer compactly lower semi-continuous in its second variable,(iii) g(x, y) is A-generalized R-diagonally quasiconcave in x and(a) If there exists M G (X) such that compact subset of Y, then there exists a yo G Y such that for all x X.(b) Let Y be compact. If there exists a nonempty compact subset K of Y and a mapping such that for each where is the continuous mapping in touch with N in definition of A-generalized R-diagonally quasiconcave, then there exists a y0 K such that /(x, yo) < A for all x G X.Theorem 2.2.2. Let X and Y be two Hausdorff topological spaces. Let be a mapping satisfying(i) f(x,y) is 0-transfer compactly upper semicontinuous in its first argument x and 0-transfer compactly lower semicontinuous in its second argument y,(ii) f(x,y) is 0-generalized R-diagonally quasiconcave in its first argument x and 0-generalized R-diagonally quasiconvex in its second argument y,(iii) there exist M (X) and N e (Y) such that compact in Y and is compact in X. Then / has a saddle point (X,Y) such thatIn particular, we haveBy using the technique of a continuous partition of unity and Tychonoff's fixed point theorem, we establish some new collectively fixed point theorems for a family of set-valued mappings denned on the product spaces of noncompact Hausdorff topological spaces which have no convex structure. We have the following theorems.Theorem 3.1.1. Let be a family of Hausdorff topological spaces where / is an (finite or infinite) index set. Let X = be two set-valued mappings satisfying the following conditions:(i) Gi is a .D-convex mapping with respect to Fi;(ii) Fi satisfies one of Conditions (i)-(v) in Lemma 1.3.2;(iii) There exists a nonempty subset Xi0 of Xi such that the set is empty or compact in X, where denotes the complement of (iv) For each Ni (Fi(X)), there is a c...