Abstract
We study new types for minimax theorems with a couple of set-valued mappings, and we propose several versions for minimax theorems in topological vector spaces setting. These problems arise naturally from some minimax theorems in the vector settings. Both the types of scalar minimax theorems and the set minimax theorems are discussed. Furthermore, we propose three versions of minimax theorems for the last type. Some examples are also proposed to illustrate our theorems. MSC: 49J35, 58C06.
Highlights
1 Introduction and preliminaries Let X, Y be two nonempty sets in two Hausdorff topological vector spaces, respectively, Z be a Hausdorff topological vector space, C ⊂ Z a closed convex and pointed cone with apex at the origin and int C = ∅, this means that C is a closed set with nonempty interior and satisfies λC ⊆ C, ∀λ > ; C + C ⊆ C; and C ∩ (–C) = { }
Scalar minimax theorems and set minimax theorems for non-continuous set-valued mappings were first proposed by Lin et al [ ]
The set-valued mapping F : X × Y ⇒ R is continuous with nonempty compact values and satisfies the following conditions: (i) y → F(x, y) is above-R+-concave on Y for each x ∈ X; (ii) x → F(x, y) is above-naturally R+-quasi-convex for each y ∈ Y ; and (iii) for each y ∈ Y, there is an xy ∈ X such that max F(xy, y) ≤ max min F(x, y)
Summary
The set-valued mapping F : X × Y ⇒ R such that the sets y∈Y F(x, y), x∈X F(x, y) and F(x, y) are compact for all (x, y) ∈ X × Y , and they satisfy the following conditions: (i) y → F(x, y) is above-R+-concave on Y for each x ∈ X; (ii) x → F(x, y) is above-naturally R+-quasi-convex for each y ∈ Y , and (x, y) → F(x, y) is lower semi-continuous on X × Y ; and (iii) for each w ∈ Y , there is an xw ∈ X such that max F(xw, w) ≤ max min F(x, y).
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