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

Abstract

In this paper, by introducing a new concept of the ( f, g, h)-quasimonotonicity and applying the maximal monotonicity of bifunctions and KKM technique, we show the existence results of solutions for quasi mixed equilibrium problems when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. Next, we also obtain a result of optimal control to a minimization problem. Our main results can be applied to the problems of evolution equations, differential inclusions and hemivariational inequalities.

Inverse problems for nonlinear quasi-variational hemivariational inequalities with application to obstacle problems of elliptic type

“…Migórski et al [33 , H ϕ ]) and the condition g(u, v ) + g(v , u ) ≥ 0 is not necessary satisfied (cf. Liu et al [35,Theorem 3.3] ).…”

“…Moreover, we would like to mention that the regularized optimization method for inverse problems of inequalities is closely related to optimal control problems governed by inequalities. Recent researches on this field can be found in literature such as Liu and Zeng [34] , Liu et al [35] , Peng and Kunisch [36] , Peng [37] , Peng et al [38] , Sofonea [39] and Xiao and Sofonea [40] .…”

Painlevé–Kuratowski convergence of the solution sets for controlled systems of fuzzy vector quasi-optimization problems with application to controlling traffic networks under uncertainty

“…Although the theory and computational techniques, optimal control problem for variational and hemivariational inequalities have been studied for quite some time now, it seems that there are still many unanswered questions and many interesting ideas are to be discovered. see, for example, Patrone (1977), Peng and Kunisch (2018), Shi (1988), Sofonea (2019), Zhou et al (2006), Zeng et al (2021), Zeng and Vilches (2020), , Liu et al (2019). Recently, Hung (2021) introduced a class of the controlled systems of fuzzy mixed quasi-hemivariational inequalities of the Minty type.…”

Existence and generic stability conditions of equilibrium points to controlled systems for n-player multiobjective generalized games using the Kakutani–Fan–Glicksberg fixed-point theorem

“…Research papers focused on optimal control for variational inequalities, see for example [14,15,18,19,26] and references therein. In 2019, Liu et al [11] studied the class of the controlled systems of quasi-mixed scalar equilibrium problems, then they established existence results for solutions to these problems by using a concept of the ( f , g, h)-quasimonotonicity and the KKM technique. Recently, Hung [9] established the Levitin-Polyak wellposedness for a class of the controlled systems of fuzzy mixed quasi-hemivariational inequalities of the Minty type.…”