Strong Convergence of a New Hybrid Algorithm for Fixed Point Problems and Equilibrium Problems

Abstract

The paper considers the problem of finding a common solution of a pseudomonotone and Lipschitz-type equilibrium problem and a fixed point problem for a quasi nonexpansive mapping in a Hilbert space. A new hybrid algorithm is introduced for approximating a solution of this problem. The presented algorithm can be considered as a combination of the extragradient method (two-step proximal-like method) and a modified version of the normal Mann iteration. It is well known that the normal Mann iteration has the weak convergence, but in this paper we has obtained the strong convergence of the new algorithm under some mild conditions on parameters. Several numerical experiments are reported to illustrate the convergence of the algorithm and also to show the advantages of it over existing methods.

On inertial type self-adaptive iterative algorithms for solving pseudomonotone equilibrium problems and fixed point problems

“…In this section, we give some numerical examples to demonstrate the efficiency of our proposed algorithm. We compare our result with the algorithm 1.4 proposed by Hieu in [26]. Then T is a quasi-nonexpansive mapping, which is not nonexpansive [40].…”

“…In this paper, we introduced a modified inertial subgradient extragradient algorithm for the approximation of common solutions of pseudomonotone equilibrium problems and fixed point problems of quasi-nonexpansive mappings with the step sizes constructed such that the knowledge of the Lipschitz type constants of the equilibrium bifunction is not required in the analysis of the convergence result. We gave some numerical experiments to demonstrate the effectiveness and performance of our iterative algorithm by comparing with Hieu's algorithm [26]. We also demonstrated the effect of the inertial term by comparing our algorithm with its unaccelerated version.…”

“…where f is a pseudomonotone and Lipschitz-type continuous bifunction and T is a nonexpansive mapping. Recently, Hieu [26], in 2019, used the following algorithm to approximate a common solution of an equilibrium problem with a pseudomonotone and Lipschitz-type continuous bifunction f and a fixed point problem of a quasi-nonexpansive mapping…”

Regularization iterative method of bilevel form for equilibrium problems in Hilbert spaces

“…Other strongly convergent methods for solving problem (EP) in Hilbert spaces can be found, for example, in references. [17][18][19][20] In this paper, we present a different direction and introduce a new method of iterative regularization form to obtain the norm convergence for solving problem EP in a Hilbert space. Firstly, we study the Tikhonov regularization method for our original problem EP and establish some important properties of regularization solutions.…”

New extragradient methods with non-convex combination for pseudomonotone equilibrium problems with applications in Hilbert spaces

“…In the literatures, most of authors modify the hybrid projection methods or viscosity methods to obtain the strong convergence of the iterative algorithms for pseudomonotone equilibrium problems; see [6,[9][10][11][12][13][14][15][16][17][18]. In [19], a non-convex combination iterative algorithm for pseudomonotone equilibrium problem and fixed point problem was introduced and the strong convergence was proved. In this paper, we construct two new extragradient methods with non-convex combination to solve the pseudomonotone equilibrium problems in Hilbert space and prove the strong convergence for the constructed algorithms.…”