Mixed Equilibrium Problems and Existence of Solutions for Nonlinear Evolution Equations and Hemivariational Inequalities

Equilibrium Problems are among the most interesting and intensively studied classes of problems in nonlinear analysis and optimization. They are suitable and common format for investigating various applied problems arising in Economics, Mathematical Physics, Transportation, Communication Systems, Engineering and other fields. Moreover, equilibrium problems are closely related with other general problems in nonlinear analysis, such as fixed points, game theory, variational inequality, and optimization problems. As it has been shown from recent works, equilibrium problems have a great impact and influence in the development of several branches of pure and applied sciences, and collectively cover a vast range of applications. Very recently, nonlinear evolution equations and hemivariational inequalities have been studied by using the equilibrium problems. The theory of nonlinear evolutions equations provides an abstract framework for the functional analytic treatment of initial-boundary value problems for parabolic differential equations and inclusions. Nonlinear evolution equations, i.e., partial differential equations with time t as one of the independent variables, arise not only from many fields of mathematics, but also from other branches of science such as physics, mechanics and material science. For example, Navier-Stokes and Euler equations from fluid mechanics, nonlinear reaction-diffusion equations from heat transfers and biological sciences, nonlinear Klein-Gorden equations and nonlinear Schrdinger equations from quantum mechanics and Cahn-Hilliard equations from material science, to name just a few, are special examples of nonlinear evolution equations. Hemivariational inequality problem is a new type of inequality problem which was introduced by Panagiotopoulos in order to deal with problems in mechanics and engineering whose variational forms are such inequalities which express the principle of virtual work or power. We point out that most of the studies on the existence of solutions for nonlinear evolution equations and hemivariational inequalities rely on the theory of monotone operators and the Browders surjectivity result on pseudomonotone perturbation of maximal monotone operators. In this project, we intend to study how the theory of equilibrium problems can be applied to derive existence of solutions for nonlinear evolutions equations and hemivariational inequalities. The study will be based on some interesting new and some recent results established 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. We believe that this new approach can improve and unify most of the studies done on nonlinear evolution equations and hemivariational inequalities. Its interest lies in the fact that instead of transforming nonlinear evolutions equations and hemivariational inequalities into a form where the Browders surjectivity result on nonlinear operators can be applied, which is the method most used in literature, we shall use an equilibrium problem formulation. We are convinced that this new method can lead to many interesting results when applied to study time-dependent nonlinear evolutions equations and time-dependent evolution variational and hemivariational inequalities.