The present book is devoted to the systematic study of one of the most interesting and effective concepts of nonlinear analysis, the notion of equilibrium. The authors develop a unified approach to various equilibrium problems and consider their applications in game theory, mathematical economics, calculus of variations, optimization theory, and other fields.
At the very beginning, an overview on equilibrium problems is presented. After describing a basic equilibrium problem due to Ky Fan, the authors discuss some important special cases, including the convex minimization problem, the fixed point problem, the complementarity problem, the Nash equilibrium and minimax problems, and variational inequalities. Investigating the solvability of equilibrium problems, the authors consider the existence of solutions to vector- and set-valued problems, as well as problems given by the sum of two functions. While dealing with the well-posedness of equilibrium problems, the authors distinguish different kinds of well-posedness and study the relations between them.
Special attention is put to the applications of variational principles to the treatment of equilibrium problems, among which the Ekeland variational principle, metric regularity, linear openness, and the Aubin property are indicated. The authors consider applications to sensitivity of parametrized equilibrium problems, i.e., they study the dependence of the solution set of the equilibrium problem on parameters.
A separate chapter of the book is dedicated to the Nash equilibrium in such special cases as equilibrium for Perov contractions, systems of variational inequalities, applications to periodic problems. The use of the described methods in mathematical economics and variational inequalities is discussed in detail. The last chapter of the book is devoted to regularization and numerical methods for equilibrium problems, including the proximal point method, the Tikhonov regularization method, and the subgradient method.
In its significant part, the monograph is based on the authors’ original research. The book is a good introduction into one of the important parts of contemporary applied nonlinear analysis. It contains the explanations of the necessary mathematical tools and thus can be used as a manual for graduate and post-graduate students in theoretical and applied mathematics.