Existence and iterative approximation of solutions of generalized mixed equilibrium problems

Abstract

In this paper, we consider a generalized mixed equilibrium problem involving non-monotone set-valued mappings in real Hilbert space. We extend the notions of the Yosida approximation and its corresponding regularized operator given by Moudafi and Thera [A. Moudafi, M. Thera, Proximal and dynamical approaches to equilibrium problems, in: Lecture Notes in Econom. and Math. System, vol. 477, Springer-Verlag, Berlin, 2002, pp. 187–201] and discuss some of their properties. Further, we consider a generalized Wiener–Hopf equation problem and show its equivalence with the generalized mixed equilibrium problem. Using a fixed point formulation of the generalized Wiener–Hopf equation problem, we construct an iterative algorithm. Furthermore, we extend the notion of stability given by Harder and Hick [A.M. Harder, T.L. Hicks, Stability results for fixed-point iteration procedures, Math. Japonica 33 (5) (1998) 693–706]. We prove the existence of a solution and discuss the convergence and stability of the iterative algorithm for the generalized Wiener–Hopf equation problem. Since the generalized mixed equilibrium problems include variational inequalities as special cases, the results presented in this paper continue to hold for these problems.