Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space

Abstract

This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixed point problems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the proposed framework in coupling the fixed point iterations with the iterations of the proximal point method or the extragradient method, which are well-known methods for solving equilibrium problems. The strategy is to obtain the strong convergence from the weak convergence of the iterates without additional assumptions on the problem data. To achieve this aim, the solution set of the problem is outer approximated by a sequence of polyhedral subsets.

Hybrid projection methods for equilibrium problems with non‐Lipschitz type bifunctions

“…Because x n + 1 ∈ H n , we have , and from the definition of , we obtain . This together with implies that From , , and , one obtains From [, Theorem 5.1], we can conclude that the sequences , and are bounded. Lemma and a straightforward computation yield …”

“…Unlike in , the proposed algorithms have used the bifunction L with conditions B1 and B2. In this section, we report some numerical results for Algorithms – with different bifunctions L .…”

“…The mentioned methods, in general, are weakly convergent in infinite dimensional Hilbert spaces, see, for example, . So as to obtain the strong convergence, these methods are often incorporated with one or more other techniques, for instance, the viscosity approximation method or cutting‐hyperplane techniques , and lead to viscosity or hybrid projection methods. In recent years, strongly convergent algorithms for solving EPs as well as VIPs have received much attention by many authors, for instance, and the references cited there.…”

“…The mentioned methods, in general, are weakly convergent in infinite dimensional Hilbert spaces, see, for example, [28]. So as to obtain the strong convergence, these methods are often incorporated with one or more other techniques, for instance, the viscosity approximation method [29][30][31][32][33] or cutting-hyperplane techniques [34][35][36], and lead to viscosity or hybrid projection methods. In recent years, strongly convergent algorithms for solving EPs as well as VIPs have received much attention by many authors, for instance, [2-4, 8, 37-39] and the references cited there.In this paper, we focus on combining the extragradient-like method with cutting-hyperplane techniques for solving CSEP (1) in Hilbert spaces that can be conjunctive from the cyclic and parallel hybrid extragradient methods in [2][3][4][5].…”

Parallel Extragradient-Proximal Methods for Split Equilibrium Problems

“…The advantages of the extragradient method are that it is used for the class of pseudomonotone bifunctions and two optimization programs are solved at each iteration which seems to be numerically easier than non-linear inequality (1.5) in the proximal method, see for instance [28,31,33] and the references therein.…”