Equilibrium Problems with Applications to Eigenvalue Problems

Abstract

In this paper, we consider equilibrium problems and introduce the concept of (S) + condition for bifunctions. Existence results for equilibrium problems with the (S) + condition are derived. As special cases, we obtain several existence results for the generalized nonlinear variational inequality studied by Ding and Tarafdar [12], and the generalized variational inequality studied by Cubiotti and Yao [11], respectively. Finally, applications to a class of eigenvalue problems are given. In particular, we derive an existence result for this class of eigenvalue problems where the parameter does not need to be restricted to bounded intervals and the operator is not needed to be bounded.

An Iterative Algorithm for Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities

“…Proof of (6). First, we note that x n 1 P C I − α n A y n ; then for all x ∈ C, we have x n 1 − I − α n A y n , x n 1 − x ≤ 0.…”

Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings

“…Numerous problems in physics, optimization and economics reduce to find a solution of (1.1) (see [1][2][3][4]). Let T : C C be a mapping.…”

WEAK AND STRONG CONVERGENCE THEOREMS FOR AN ASYMPTOTICALLY k-STRICT PSEUDO-CONTRACTION AND A MIXED EQUILIBRIUM PROBLEM

“…The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems and the equilibrium problems as special cases; see, e.g., [1,3,4,10,22]. Some methods have been proposed to solve the equilibrium problems and the mixed equilibrium problems, see, e.g., [2,5,6,7,10,12,14,15,16,17,18,19,20,21].…”