On generalized implicit vector equilibrium problems in Banach spaces
Let X X and Y Y be real Banach spaces, K K be a nonempty convex subset of X X , and C:K?2Y C : K ? 2 Y be a multifunction such that for each u?K u ? K , C(u) C ( u ) is a proper, closed and convex cone with int C ( u ) ? 0? , where int C ( u ) denotes the interior of C(u) C ( u ) . Given the mappings T:K?2L(X,Y) T : K ? 2 L ( X , Y ) , A:L(X,Y)?L(X,Y) A : L ( X , Y ) ? L ( X , Y ) , f1:L(X,Y)KK?Y f 1 : L ( X , Y )  K  K ? Y , f2:KK?Y f 2 : K  K ? Y , and g:K?K g : K ? K , we introduce and consider the generalized implicit vector equilibrium problem: Find u??K u ? ? K such that for any v?K v ? K , there is s??Tu? s ? ? T u ? satisfying f 1 ( A s ? , v , g ( u ? ) ) + f 2 ( v , g ( u ? ) ) ? ? int C ( u ? ) . By using the KKM technique and the well-known Nadlers result, we prove some existence theorems of solutions for this class of generalized implicit vector equilibrium problems. Our theorems extend and improve the corresponding results of several authors.
generalized implicit vector equilibrium problem kkm technique generalized implicit vector variational inequality nadlers theorem hausdorff metric