scholarly journals Existence and Modification of Halpern-Mann Iterations for Fixed Point and Generalized Mixed Equilibrium Problems with a Bifunction Defined on the Dual Space

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Pongrus Phuangphoo ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problem ◽  
Dual Space ◽  
Fixed Point Problem ◽  
Generalized Mixed Equilibrium Problem ◽  
Point Problem ◽  
Mixed Equilibrium Problem ◽  
Existence Of A Solution ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We study and establish the existence of a solution for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space. Furthermore, we also modify Halpern-Mann iterations for finding a common solution of a generalized mixed equilibrium problem and a fixed point problem. Under suitable conditions of the purposed iterative sequences, the strong convergence theorems are established by using sunny generalized nonexpansive retraction in Banach spaces. Our results extend and improve various results existing in the current literature.

scholarly journals Strong Convergence of a System of Generalized Mixed Equilibrium Problem, Split Variational Inclusion Problem and Fixed Point Problem in Banach Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 722 ◽  
Author(s):  
Mujahid Abbas ◽  
Yusuf Ibrahim ◽  
Abdul Rahim Khan ◽  
Manuel de la Sen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Banach Spaces ◽  
Fixed Point Problem ◽  
Generalized Mixed Equilibrium Problem ◽  
Point Problem ◽  
Mixed Equilibrium Problem ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

The purpose of this paper is to introduce a new algorithm to approximate a common solution for a system of generalized mixed equilibrium problems, split variational inclusion problems of a countable family of multivalued maximal monotone operators, and fixed-point problems of a countable family of left Bregman, strongly asymptotically non-expansive mappings in uniformly convex and uniformly smooth Banach spaces. A strong convergence theorem for the above problems are established. As an application, we solve a generalized mixed equilibrium problem, split Hammerstein integral equations, and a fixed-point problem, and provide a numerical example to support better findings of our result.

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

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.

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

2018 ◽  
Vol 51 (1) ◽  
pp. 211-232 ◽  
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.

scholarly journals Modified Krasnoselski–Mann iterative method for hierarchical fixed point problem and split mixed equilibrium problem

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jong Kyu Kim ◽  
Prashanta Majee
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Equilibrium Problem ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Mixed Equilibrium Problem ◽  
Convergence Results ◽  
Hierarchical Fixed Point ◽  
Mixed Equilibrium ◽  
Split Mixed Equilibrium Problem

Abstract In this paper, we introduce a modified Krasnoselski–Mann type iterative method for capturing a common solution of a split mixed equilibrium problem and a hierarchical fixed point problem of a finite collection of k-strictly pseudocontractive nonself-mappings. Many of the algorithms for solving the split mixed equilibrium problem involve a step size which depends on the norm of a bounded linear operator. Since the computation of the operator norm is very difficult, we formulate our iterative algorithm in such a way that the implementation of the proposed algorithm does not require any prior knowledge of operator norm. Weak convergence results are established under mild conditions. We also establish strong convergence results for a certain class of hierarchical fixed point and split equilibrium problem. Our results generalize some important results in the recent literature.

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