The existence theorems of solutions for generalized vector-valued variational-like inequalities
The generalized vector-valued variational-like inequalities include as special cases several known variational inequalities. Let \(Y\) be an ordered Banach space with respect to a closed pointed convex cone with nonvoid interior; \(z_{0}\in A\) is called a vector maximal point of \(A\) if \(\{ z\in A:z_{0}\leq z\), \(z\neq z_{0}\} =\emptyset\), and the set of all vector maximal points of \(A\) is denoted by \(\max A\). Let \(X\) be a Banach space, \(K\subset X\) a closed convex nonvoid set, \( C:K\rightarrow 2^{Y}\) so that \(C(x)\) is a closed pointed convex cone with \(\text{int }C(x)\neq \emptyset .\) For \(\eta :K\times K\rightarrow K\) continuous and \(T:X\rightarrow 2^{L(X,Y)}\) nonvoid compact valued mapping, the problem of the existence of \(x_{\ast }\in K\) such that \[ \max \left\langle T(x_{\ast }),\eta (y,x_{\ast })\right\rangle \subset Y\backslash (-\text{int }C(x_{\ast })),\quad \text{for all }y\in K \] is considered. Using KKM theory, the authors prove the existence of solutions of the above generalized vector-valued variational-like inequality for \(T\) \(\eta\)-pseudomonotone and hemicontinuous.
generalized vector-valued variational-like inequality
scientific article; zbMATH DE number 1391543
scientific article