Minimax Theorems for Set-Valued Mappings under Cone-Convexities

and Applied Analysis 3 Lemma 2.4 see 9, Lemma 3.1 . Let X, Y, and Z be three topological spaces. Let Y be compact, F : X × Y ⇒ Z a set-valued mapping, and the set-valued mapping T : X ⇒ Z defined by T x ⋃ y∈Y F ( x, y ) , ∀x ∈ X. 2.2 a If F is upper semi-continuous on X × Y, then T is upper semi-continuous on X. b If F is lower semi-continuous onX, so is T. Lemma 2.5 see 9, Lemma 3.2 . Let Z be a Hausdorff topological vector space, F : Z ⇒ R a set-valued mapping with nonempty compact values, and the functions p, q : Z → R defined by p z maxF z and q z minF z . a If F is upper semi-continuous, so is p. b If F is lower semi-continuous, so is p. c If F is continuous, so are p and q. We shall use the following nonlinear scalarization function to establish our results. Definition 2.6 see 6, 10 . Let k ∈ intC and v ∈ Z. The Gerstewitz function ξkv : Z → R is defined by ξkv u min{t ∈ R : u ∈ v tk − C}. 2.3 We present some fundamental properties of the scalarization function. Proposition 2.7 see 6, 10 . Let k ∈ intC and v ∈ Z. The Gerstewitz function ξkv : Z → R has the following properties: a ξkv u < r ⇔ u ∈ v rk − intC; b ξkv u ≤ r ⇔ u ∈ v rk − C; c ξkv u 0 ⇔ u ∈ v − ∂C, where ∂C is the topological boundary of C; d ξkv u > r ⇔ u / ∈ v rk − C; e ξkv u ≥ r ⇔ u / ∈ v rk − intC; f ξkv · is a convex function; g ξkv · is an increasing function, that is, u2 − u1 ∈ intS ⇒ ξkv u1 < ξkv u2 ; h ξkv · is a continuous function. Theorem 2.8 Fan-Browder fixed-point theorem see 12 . Let X be a nonempty compact convex subset of a Hausdorff topological vector space and let T : X ⇒ X be a set-valued mapping with nonempty convex values and open fibers, that is, T−1 y {x ∈ X : y ∈ T x } is open for all y ∈ X. Then, T has a fixed point. 4 Abstract and Applied Analysis 3. Cone-Convexities In this section, we present different kinds of cone-convexities for set-valued mappings and give some relations among them. Some examples of such set-valuedmappings are also given. Definition 3.1. Let X be a nonempty convex subset of a topological vector space W. A setvalued mapping F : X ⇒ Z is said to be a above -C-convex 4 resp., above-C-concave 5 on X if for all x1, x2 ∈ X and all λ ∈ 0, 1 , F λx1 1 − λ x2 ⊂ λF x1 1 − λ F x2 − C, ( resp., λF x1 1 − λ F x2 ⊂ F λx1 1 − λ x2 − C ) ; 3.1 b below-C-convex 13 resp., below-C-concave 9, 13 on X if for all x1, x2 ∈ X and all λ ∈ 0, 1 , λF x1 1 − λ F x2 ⊂ F λx1 1 − λ x2 C ( resp., F λx1 1 − λ x2 ⊂ λF x1 1 − λ F x2 C ) ; 3.2 c above-C-quasi-convex resp., below-C-quasiconcave 7, Definition 2.3 on X if the set LevF≤ z : {x ∈ X : F x ⊂ z − C} ( resp., LevF≥ z : {x ∈ X : F x ⊂ z C} ) , 3.3 is convex for all z ∈ Z; d above-properly C-quasiconvex resp., above-properly C-quasiconcave 6 on X if for all x1, x2 ∈ X and all λ ∈ 0, 1 , either F λx1 1 − λ x2 ⊂ F x1 − C ( resp., F x1 ⊂ F λx1 1 − λ x2 − C ) 3.4 or F λx1 1 − λ x2 ⊂ F x2 − C ( resp., F x2 ⊂ F λx1 1 − λ x2 − C ) ; 3.5 e below-properly C-quasiconvex 7 resp., below-properly C-quasiconcave on X if for all x1, x2 ∈ X and all λ ∈ 0, 1 , either F x1 ⊂ F λx1 1 − λ x2 C ( resp., F λx1 1 − λ x2 ⊂ F x1 C ) 3.6 Abstract and Applied Analysis 5and Applied Analysis 5