Abstract
A general vector‐valued variational inequality (GVVI) is considered. We establish the existence theorem for (GVVI) in the noncompact setting, which is a noncompact generalization of the existence theorem for (GVVI) obtained by Lee et al., by using the generalized form of KKM theorem due to Park. Moreover, we obtain the fuzzy extension of our existence theorem.
Highlights
Giannessi [1 imroduced a variational inequality for vector-valued mappings in a Euclidean space
Lee et al [7] have established the existence theorem of a variational inequality for a multifimction with vector values in a Banach space
Chang and Zhu [8] introduced the concept of variational inequalities for fuzzy mappings in locally convex Hausdorff topological vector spaces and investigated existence theorems for some kinds of variational inequalities for fazzy mappings, which were the extensions of some theorems in [9,10,11,12]
Summary
Giannessi [1 imroduced a variational inequality for vector-valued mappings in a Euclidean space. Chen et al [2,3,4,5,6] have intensively studied variational inequalities for vector-valued mappings in Banach spaces. Lee et al [7] have established the existence theorem of a variational inequality for a multifimction with vector values in a Banach space. Our motivation of this paper is to consider the noncompact cases of the existence theorems of variational inequalities for multifimctions with vector values or fuzzy mappings in Banach spaces obtained by Lee et al [7]. When T is a mapping fxom X into L(X,Y), (GVVD reduces to the following vector-valued variational inequality (VVI) considered by Chen et al [3,5,6]. Let X be a convex space, K a nonempty compact subset of X, and X 2x a KKM multifimction. Suppose that (1) for each X, G() is compactly closed; and (2) for each finite subset N ofX, there exists a compact convex subset Lv ofX such that N C LN and s fl {S() / s) c K
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