Nonlinear Evolution Equations by a Ky Fan Minimax Inequality Approach

Abstract

Nonlinear evolution equations appear in various fields of sciences including mechanics, physics, engineering, and material sciences. Essential functional methods for the treatment of both the linear as well as nonlinear evolution equations are based on the theory of spectral methods, maximal monotone operators, fixed point theorems, and concept of \(C_0\)-semigroups of linear mappings along with the Leray-Schauder degree theory. Recently, the solvability for the evolution equations of nonlinear type has been considered using the Ky Fan minimax inequality. This approach is quite new and different as compared to the traditional approaches. In 1972, Ky Fan (Inequalities. Academic, New York, pp. 103–113, 1972) put forward his pioneering result concerning the existence of solutions for an inequality of minimax type, which is nowadays called as “equilibrium problem” in literature. This kind of model has shown to be a cornerstone result of nonlinear analysis and has gained much interest in the past because it has been used in several contexts such as physics, chemistry, economics, engineering, and so on. This work aims to present a review of recently obtained results on the use of the equilibrium problem theory in the study of nonlinear (implicit) evolution equations. Along with that, we discuss the problem with initial value condition as well as periodic and anti-periodic solutions.