%0 Journal Article
%T A Wiener-Hopf Dynamical System for Mixed Equilibrium Problems
%A Farhat Suhel
%A S. K. Srivastava
%A Suhel Ahmad Khan
%J International Journal of Mathematics and Mathematical Sciences
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/102578
%X We suggest and analyze dynamical systems associated with mixed equilibrium problems by using the resolvent operator technique. We show that these systems have globally asymptotic property. The concepts and results presented in this paper extend and unify a number of previously known corresponding concepts and results in the literature. 1. Introduction Equilibrium problems theory has emerged as an interesting and fascinating branch of applicable mathematics. This theory has become a rich source of inspiration and motivation for the study of a large number of problems arising in economics, optimization, and operation research in a general and unified way. There are a substantial number of papers on existence results for solving equilibrium problems based on different-relaxed monotonicity notions and various compactness assumptions; see, for example, [1¨C6]. In 2002, Moudafi [5] considered a class of mixed equilibrium problems which includes variational inequalities as well as complementarity problems, convex optimization, saddle point problems, problems of finding a zero of a maximal monotone operator, and Nash equilibria problems as special cases. He studied sensitivity analysis and developed some iterative methods for mixed equilibrium problems. In recent years, much attention has been given to consider and analyze the projected dynamical systems associated with variational inequalities and nonlinear programming problems, in which the right-hand side of the ordinary differential equation is a projection operator. Such types of the projected dynamical system were introduced and studied by Dupuis and Nagurney [7]. Projected dynamical systems are characterized by a discontinuous right-hand side. The discontinuity arises from the constraint governing the question. The innovative and novel feature of a projected dynamical systems is that the set of stationary points of dynamical system correspond to the set of solution of the variational inequality problems. It has been shown in [8¨C14] that the dynamical systems are useful in developing efficient and powerful numerical technique for solving variational inequalities and related optimization problems. Xia and Wang [13], Zhang and Nagurney [14], and Nagurney and Zhang [11] have studied the globally asymptotic stability of these projected dynamical systems. Noor [15¨C17] has also suggested and analyzed similar resolvent dynamical systems for variational inequalities. It is worth mentioning that there is no such type of the dynamical systems for mixed equilibrium problems. In this paper, we show that such type of
%U http://www.hindawi.com/journals/ijmms/2014/102578/