Mixed Equilibrium Problems and Anti-periodic Solutions for Nonlinear Evolution Equations

Abstract

By using some new developments in the theory of equilibrium problems, we study the existence of anti-periodic solutions for nonlinear evolution equations associated with time-dependent pseudomonotone and quasimonotone operators in the topological sense. More precisely, we establish new existence results for mixed equilibrium problems associated with pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are therefore applied to study the existence of anti-periodic solutions for nonlinear evolution equations in the setting of reflexive 123 J Optim Theory Appl Banach spaces. This new approach leads us to improve and unify most of the recent results obtained in this direction.

Existence Results and Optimal Control for a Class of Quasi Mixed Equilibrium Problems Involving the (f, g, h)-Quasimonotonicity

“…To establish our results, we introduce a new concept of stable ( f, g, h)-quasimonotonicity, and use the properties of the maximal monotonicity of bifunctions and KKM technique. Our results extend and improve some results in [10,24,31,33,34] in many respects. The ( f, g, h)-quasimonotonicity depends on f, g, h which is more general than usual monotonicity.…”

“…Definition 2.5 A bifunction h : K × K → R with h(x, x) = 0 for all x ∈ K is said to be maximal monotone if for every x ∈ K and for every convex function ψ : K → R with ψ(x) = 0, we have For more details about the maximal monotonicity, we refer [1,8,10]. Next, we introduce the concept of ( f, g, h)-quasimonotonicity which is useful for establishing the existence theorems for the main results.…”

“…The ( f, g, h)-quasimonotonicity depends on f, g, h which is more general than usual monotonicity. Furthermore, our problem (2.1) is more general than that considered in [10], which involves a multi-valued mapping, and the conditions of g, h are different from the conditions in [10]. Moreover, the problems of hemivariational inequalities in [24,31,33,34] are also our special cases.…”

“…Therefore, since the equilibrium problem unifies at least all the above mentioned problems in a common formulation and many techniques and methods established in order to solve one of them may be extended, it is worth to find out suitable ways to solve them efficiently. For more details, we refer to [2][3][4]7,10,19].…”

“…In [10], Chadli, Ansari and Yao studied the existence of solutions for mixed equilibrium problems described by a bifunction as the sum of a monotone and maximal monotone bifunction and a bifunction which is, respectively, pseudomonotone and quasimonotone in topological sense. The concept of maximal monotonicity for bifunctions is in fact an extension to equilibrium problems of the corresponding one of nonlinear operators which can be seen in [1,8].…”