Viscosity extragradient method with Armijo linesearch rule for pseudomonotone equilibrium problem and fixed point problem in Hilbert spaces

Abstract In this paper, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of a nonexpansive semigroup in real Hilbert spaces. The weak and strong convergence characteristics of the proposed algorithm are investigated by employing suitable control conditions in such a setting of spaces. As a consequence, we provide a simplified analysis of various existing results concerning the extragradient method in the current literature. We also provide a numerical example to strengthen the theoretical results and the applicability of the proposed algorithm.

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Iterative algorithms of common solutions for a hierarchical fixed point problem, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02645-4 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Yali Zhao ◽

Xin Liu ◽

Ruonan Sun

Keyword(s):

Fixed Point ◽

Variational Inequalities ◽

Equilibrium Problem ◽

Hilbert Spaces ◽

Fixed Point Problem ◽

Point Problem ◽

Split Equilibrium Problem ◽

System Of Variational Inequalities ◽

Hierarchical Fixed Point Problem ◽

Hierarchical Fixed Point

AbstractIn this paper, we suggest and analyze an iterative algorithm to approximate a common solution of a hierarchical fixed point problem for nonexpansive mappings, a system of variational inequalities, and a split equilibrium problem in Hilbert spaces. Under some suitable conditions imposed on the sequences of parameters, we prove that the sequence generated by the proposed iterative method converges strongly to a common element of the solution set of these three kinds of problems. The results obtained here extend and improve the corresponding results of the relevant literature.

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Strong Convergence Results of Split Equilibrium Problems and Fixed Point Problems

Journal of Mathematics ◽

10.1155/2021/6624321 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Li-Jun Zhu ◽

Hsun-Chih Kuo ◽

Ching-Feng Wen

Keyword(s):

Fixed Point ◽

Strong Convergence ◽

Equilibrium Problem ◽

Hilbert Spaces ◽

Equilibrium Problems ◽

Iterative Scheme ◽

Fixed Point Problem ◽

Point Problem ◽

Split Equilibrium Problem ◽

Convergence Results

In this paper, we investigate the split equilibrium problem and fixed point problem in Hilbert spaces. We propose an iterative scheme for solving such problem in which the involved equilibrium bifunctions f and g are pseudomonotone and monotone, respectively, and the operators S and T are all pseudocontractive. We show that the suggested scheme converges strongly to a solution of the considered problem.

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On split generalized mixed equilibrium and fixed point problems of an infinite family of quasi-nonexpansive multi-valued mappings in real Hilbert spaces

Asian-European Journal of Mathematics ◽

10.1142/s1793557122500826 ◽

2021 ◽

pp. 2250082

Author(s):

F. Akutsah ◽

H. A. Abass ◽

A. A. Mebawondu ◽

O. K. Narain

Keyword(s):

Fixed Point ◽

Equilibrium Problem ◽

Hilbert Spaces ◽

Infinite Family ◽

Fixed Point Problem ◽

Generalized Mixed Equilibrium Problem ◽

Point Problem ◽

Mixed Equilibrium Problem ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

In this paper, we study split generalized mixed equilibrium problem and fixed point problem in real Hilbert spaces with a view to analyze an iterative method for approximating a common solution of split generalized mixed equilibrium problem and fixed point problem of an infinite family of a quasi-nonexpansive multi-valued mappings. The iterative algorithm introduced in this paper is designed in such a way that it does not require the knowledge of the operator norm. We state and prove a strong convergence result of the aforementioned problems and also give application of our main result to split variational inequality problem. Our result complements and extends some related results in literature.

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A Unified Algorithm for Solving Split Generalized Mixed Equilibrium Problem, and for Finding Fixed Point of Nonspreading Mapping in Hilbert Spaces

Demonstratio Mathematica ◽

10.1515/dema-2018-0015 ◽

2018 ◽

Vol 51 (1) ◽

pp. 211-232 ◽

Cited By ~ 27

Author(s):

Lateef Olakunle Jolaoso ◽

Kazeem Olawale Oyewole ◽

Chibueze Christian Okeke ◽

Oluwatosin Temitope Mewomo

Keyword(s):

Fixed Point ◽

Equilibrium Problem ◽

Hilbert Spaces ◽

Fixed Point Problem ◽

Generalized Mixed Equilibrium Problem ◽

Point Problem ◽

Mixed Equilibrium Problem ◽

Common Solution ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

Abstract The purpose of this paper is to study a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces.We introduce a new iterative algorithm and prove its strong convergence for approximating a common solution of a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. Our algorithm is developed by combining a modified accelerated Mann algorithm and a viscosity approximation method to obtain a new faster iterative algorithm for finding a common solution of these problems in real Hilbert spaces. Also, our algorithm does not require any prior knowledge of the bounded linear operator norm. We further give a numerical example to show the efficiency and consistency of our algorithm. Our result improves and compliments many recent results previously obtained in this direction in the literature.

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