Abstract
We establish some coincidence theorems, generalized variational inequality theorems, and minimax inequality theorems for the family -KKM and the -mapping on -convex spaces.
Highlights
Introduction and preliminariesIn 1929, Knaster et al [1] had proved the well-known KKM theorem on n-simplex
Ansari et al [4] and Lin and Chen [5] studied the coincidence theorems for two families of set-valued mappings, and they gave some applications of the existence of minimax inequality and equilibrium problems
Throughout this paper, we assume that the set G-Co(A) is compact whenever A is a compact set
Summary
In 1929, Knaster et al [1] had proved the well-known KKM theorem on n-simplex. In 1961, Fan [2] had generalized the KKM theorem in the infinite-dimensional topological vector space. Let X be a nonempty almost G-convex subset of a G-convex space E which has a uniformity ᐁ and ᐁ has an open symmetric base family ᏺ, Y a nonempty set, and let f ,g : X × Y → be two real-valued functions. Let T,F : X → 2Y be two set-valued functions satisfying that for each finite subset A of X and for any V ∈ ᏺ, there exists a G-convex-inducing mapping hA,V : A → X such that. If the set-valued function T : X → 2Y satisfies the requirement that for any generalized G-KKM∗ mapping F with respect to T the family {F(x) | x ∈ X} has the finite intersection property, T is said to have the G-KKM∗ property. The class G-KKM∗(X, Y ) is defined to be the set {T : X → 2Y | T has the G-KKM∗ property}
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