The topological versions of KKM theorem and Fan's matching theorem with applications

Topological intersection theorems of set-valued mappings without closed graphs and their applications

“…Finally, some topological variational inequalities are also given. Our results improve and unify many corresponding results in the literature (e.g., see Chang et al [1], Jo6 [6], Geraghty and Lin [2], Uomornik [14], Kindler [9] and so on).…”

“…We also remark that Theorem 1 not only improves corresponding results of Theorem 1 of Chang et al [1], but it also shows that the condition 'X is a W-space' is superfluous which was given by Chang et al [1] (we recall that a topological space is called to be a W-space (e.g., see Chang et al When X is a non-empty subset of a vector space, the following theorem can be shown to follow from Theorem 1. However we provide a direct proof here.…”

“…Let X and Y be both topological spaces. Suppose that F : X 2 y is a set-valued mapping with non-empty compact values such that (1) for each x,y E X, there exists a continuous mapping u~,y : [0,1] ---* X with ux,v(0)= x, ux,y(1)= y and F((ux,y(t)) C F(ux,y(ti)) U F(ux,v(t2)) for each t E [tl,t2] C [0,1] (resp., the set {x E X: F(z) C F(x) U F(y)} is connected); (2) for each A • ~(X), if the set N~:eAF(x) is non-empty, then NxeAF(x) is connected; (3) for each y • Y, the set F-X(y) = { x • X: y E F(x)} is closed (resp.,…”

“…Let X and Y be both topological spaces. Suppose that F : X ---, 2 y is a set-valued mapping with non-empty compact values such that (1) for each z, y E X, there exists a continuous mapping u~,u : [0,1] ---, X with uz,u(0) = x, u~,y(1)= y and F((u~,y(t)) C F(uz,u(tl)) U F(u:~,y(t2)) for each t E [tl,t2] C [0, 1];…”

Minimax Inequalities for Vector-Valued Mappings onW-Spaces

“…In this paper, sightly modifying the topological KKM theorem of Chang w x et al 8,9 , we obtain a generalized section theorem and a generalized fixed point theorem on W-spaces, and establish some minimax inequalities for vector-valued mappings in Hausdorff topological vector spaces with closed pointed convex cones.…”