Abstract
We introduce a class of generalized vector quasivariational-like inequality problems in Banach spaces. We derive some new existence results by using KKM-Fan theorem and an equivalent fixed point theorem. As an application of our results, we have obtained as special cases the existence results for vector quasi-equilibrium problems, generalized vector quasivariational inequality and vector quasi-optimization problems. The results of this paper generalize and unify the corresponding results of several authors and can be considered as a significant extension of the previously known results.
Highlights
Let K be a nonempty subset of a space X and f : K × K → R be a bifunction
If we take f (x, y) = T(x), y − x, where T : K → X∗ and ·, · is the pairing between X and X∗ the equilibrium problem reduces to standard variational inequality, introduced and studied by Stampacchia [20] in 1964
In recent years this theory has become very powerful and effective tool for studying a wide class of linear and nonlinear problems arising in mathematical programming, optimization theory, elasticity theory, game theory, economics, mechanics, and engineering sciences
Summary
Let K be a nonempty subset of a space X and f : K × K → R be a bifunction. The equilibrium problem introduced and studied by Blum and Oettli [4] in 1994 is defined to be the problem of finding a point x ∈ K such that f (x, y) ≥ 0 for each y ∈ K. If we take f (x, y) = T(x), y − x , where T : K → X∗ (dual of X) and ·, · is the pairing between X and X∗ the equilibrium problem reduces to standard variational inequality, introduced and studied by Stampacchia [20] in 1964. In recent years this theory has become very powerful and effective tool for studying a wide class of linear and nonlinear problems arising in mathematical programming, optimization theory, elasticity theory, game theory, economics, mechanics, and engineering sciences.
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