Abstract
A class of implicit multivalued vector equilibrium problems is studied. By using the generalized Fan-Browder fixed point theorem, some existence results of solutions for the implicit multivalued vector equilibrium problems are obtained under some suitable assumptions. Moreover, a stability result of solutions for the implicit multivalued vector equilibrium problems is derived. These results extend and unify some recent results for implicit vector equilibrium problems, multivalued vector variational inequality problems, and vector variational inequality problems.
Highlights
Let X be a Hausdorff topological vector space, K a nonempty subset of X, and f : K × K → R a function
The scalar equilibrium problem consists in finding x ∈ K such that f x, y ≥ 0, ∀y ∈ K. This problem provides a unifying framework for many important problems, such as, optimization problems, variational inequality problems, complementary problems, minimal inequality problems, and fixed point problems, and has been widely applied to study the problems arising in economics, mechanics, and engineering science see 1
By using the fixed point theorem of Chowdhury and Tan 27, we obtain some existence results of solutions for the implicit multivalued vector equilibrium problems in the setting of topological vector spaces
Summary
Let X be a Hausdorff topological vector space, K a nonempty subset of X, and f : K × K → R a function. In 3, 4 , the authors studied the following implicit vector equilibrium problems for short, IVEP : to find x ∈ K such that f g x , y ∈/ − int C x , ∀y ∈ K, 1.6 where g : K → K is a vector-valued map and C : K → 2Y is a multivalued map such that, for all x ∈ K, C x is a closed convex cone in Y with int C x / ∅. By using the fixed point theorem of Chowdhury and Tan 27 , we obtain some existence results of solutions for the implicit multivalued vector equilibrium problems in the setting of topological vector spaces. These results extend and unify some recent results for implicit vector equilibrium problems, multivalued vector variational inequality problems, and vector variational inequality problems
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