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The objective of this research is to present a novel approach to enhance the extragradient algorithm's efficiency for finding an element within a set of fixed points of nonexpansive mapping and the set of solutions for equilibrium problems. Specifically, we focus on applications involving a pseudomonotone, Lipschitz-type continuous bifunction. Our main contribution lies in establishing a strong convergence theorem for this method, without relying on the assumption of lim n → ∞ ∥ x n + 1 − x n ∥ = 0 . Moreover, the main theorem can be applied to effectively solve the combination of variational inequality problem (CVIP). In support of our main result, numerical examples are also presented. [ABSTRACT FROM AUTHOR]

The aim of this paper is to modify proximal operators in Hilbert spaces. We introduce an intermixed algorithm with viscosity technique to find the solution of fixed point problem of two proximal operators in a real Hilbert space, utilizing the modified proximal operators. Under some mild conditions, a strong convergence theorem is established for the proposed algorithm. We also apply our main result to the split feasibility problem. Finally we provide numerical examples for supporting the main result. [ABSTRACT FROM AUTHOR]

In this article, we present a new iteration for approximating the solutions to generalized equilibrium problems, fixed point problems of k-strictly pseudononspreading mapping, and modifications of the system of variational inequality. The variational inequality problem and the general split feasibility problem were then solved using our primary theorem. Additionally, we provide a numerical example to support our primary theorem. [ABSTRACT FROM AUTHOR]

This study presents an innovative method for approximating solutions to the variational inequality and fixed-point problems. The proposed approach deviates from traditional methods by employing different conditions and techniques drawn from [18] [20] [21]. Uniquely, our work circumvents the utilization of a commonly used lemma (see [10]) that forms the basis for most proofs related to strong convergence theorems. As part of our investigation, we provide a comprehensive numerical example to substantiate our findings, thus enhancing the practical relevance and applicability of our research. [ABSTRACT FROM AUTHOR]

The purpose of this research is to study the generalized modification of the system of equilibrium problems (GMSEP) and a lemma is established to show the property of this problem. Then, we prove a strong convergence theorem for finding a common element of the set of the solutions of the fixed points problem and the set of the solutions of the GMSEP under some suitable conditions, in which α n + β n + δ n ≤ 1 , where { α n } , { β n } , { δ n } are coefficients in the main iteration. Moreover, we prove strong convergence theorems for finding solutions to the generalized equilibrium problem, the system of equilibrium problems, the variational inequality problem, the general system of variational inequality problems, and the minimization problem. Finally, we give two numerical examples, one of which shows the rate of convergence of the main iteration while the other shows the rate of convergence of the main iteration but the sum of coefficients equals 1. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modification of equilibrium problem (MEP) and new subgradient extragradient algorithm by using the concept of the set of solutions of the modified variational inequality problem introduced by [Kangtunyakarn A. A new iterative scheme for fixed-point problems of infinite family of κ i pseudo contractive mappings, equilibrium problem, variational inequality problems. J Optim Theory Appl. 2013;56:1543–1562.]. Then, we establish and prove weak and strong convergence theorem of the new subgradient extragradient algorithm for finding a common element of the set of solutions of the MEP and two sets of the variational inequality problems under some suitable conditions on α n and β n with α n + β n ≤ 1. Moreover, we apply our main theorem to prove weak and strong convergence theorems to solve the generalized equilibrium problem, the system of equilibrium problem, the variational inequality problem and the general system of variational inequality problems. Finally, we give two numerical examples to support our main result. [ABSTRACT FROM AUTHOR]