Existence and Strong Convergence Theorems for Generalized Mixed Equilibrium Problems of a Finite Family of Asymptotically Nonexpansive Mappings in Banach Spaces

We first prove the existence of solutions for a generalized mixed equilibrium problem under the new conditions imposed on the given bifunction and introduce the algorithm for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of finite family of asymptotically nonexpansive mappings. Next, the strong convergence theorems are obtained, under some appropriate conditions, in uniformly convex and smooth Banach spaces. The main results extend various results existing in the current literature.

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STRONG CONVERGENCE OF A HYBRID ITERATION FOR A GENERALIZED MIXED EQUILIBRIUM PROBLEM AND A BREGMAN TOTALLY QUASI-ASYMPTOTICALLY NONEXPANSIVE MAPPING IN BANACH SPACES

Tạp chí Khoa học ◽

10.54607/hcmue.js.18.9.3159(2021) ◽

2021 ◽

Vol 18 (9) ◽

pp. 1620

Author(s):

Nguyễn Trung Hiếu

Keyword(s):

Nonexpansive Mapping ◽

Strong Convergence ◽

Equilibrium Problem ◽

Banach Spaces ◽

Asymptotically Nonexpansive Mapping ◽

Generalized Mixed Equilibrium Problem ◽

Mixed Equilibrium Problem ◽

Asymptotically Nonexpansive ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

Mục đích của nghiên cứu này là kết hợp khoảng cách Bregman với phương pháp chiếu thu hẹp để giới thiệu một dãy lặp lai ghép mới cho bài toán cân bằng hỗn hợp tổng quát và ánh xạ tựa tiệm cận không giãn hoàn toàn Bregman. Sau đó, với những điều kiện thích hợp, chúng tôi chứng minh rằng dãy lặp được đề xuất hội tụ mạnh đến hình chiếu Bregman của điểm xuất phát lên giao của tập nghiệm bài toán cân bằng hỗn hợp tổng quát và tập điểm bất động của ánh xạ tựa tiệm cận không giãn hoàn toàn Bregman trong không gian Banach phản xạ. Định lí này cải tiến kết quả trong (Alizadeh & Moradlou, 2016) từ ánh xạ lai ghép tổng quát và bài toán cân bằng trong không gian Hilbert sang ánh xạ tựa tiệm cận không giãn hoàn toàn Bregman và bài toán cân bằng hỗn hợp tổng quát trong không gian Banach phản xạ. Kết quả được áp dụng cho bài toán cân bằng hỗn hợp tổng quát và ánh xạ tựa tiệm cận không giãn Bregman trong không gian Banach phản xạ. Đồng thời, một ví dụ được đưa ra để minh họa cho dãy lặp được đề xuất.

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A strong convergence theorem for finding a common fixed point of a finite family of Bregman nonexpansive mappings in Banach spaces which solves a generalized mixed equilibrium problem

Bollettino dell Unione Matematica Italiana ◽

10.1007/s40574-016-0059-z ◽

2016 ◽

Vol 9 (4) ◽

pp. 421-434

Author(s):

Vahid Darvish

Keyword(s):

Fixed Point ◽

Strong Convergence ◽

Banach Spaces ◽

Convergence Theorem ◽

Nonexpansive Mappings ◽

Finite Family ◽

Generalized Mixed Equilibrium Problem ◽

Strong Convergence Theorem ◽

Mixed Equilibrium Problem ◽

Generalized Mixed Equilibrium

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

Symmetry ◽

10.3390/sym11050722 ◽

2019 ◽

Vol 11 (5) ◽

pp. 722 ◽

Cited By ~ 2

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.

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An iterative process for a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems in reflexive Banach spaces

10.29007/2mn6 ◽

2022 ◽

Author(s):

Trung Hieu Nguyen

Keyword(s):

Equilibrium Problem ◽

Banach Spaces ◽

Equilibrium Problems ◽

Nonexpansive Mappings ◽

Finite Family ◽

Reflexive Banach Spaces ◽

Generalized Mixed Equilibrium Problems ◽

Asymptotically Nonexpansive ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

The equilibrium problem and its generalizations had a great influence in the development of some branches of pure and applied sciences. The equilibrium problems theory provides a natural and novel approach for some problems arising in nonlinear analysis, physics and engineering, image reconstruction, economics, finance, game theory and optimization. In recent times, there were many methods in order to solve the equilibrium problem and its generalizations. Some authors proposed many iterative methods and studied the convergence of such iterative methods for equilibrium problems and nonexpansive mappings in the setting of Hilbert spaces and Banach spaces. Note that a generalized mixed equilibrium problem is a generalization of an equilibrium problem and a Bregman totally quasi-asymptotically nonexpansive mapping is a generalization of a nonexpansive mapping in reflexive Banach spaces. The purpose of this paper is to combine the parallel method with the Bregman distance and the Bregman projection in order to introduce a new parallel hybrid iterative process which is to find common solutions of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems. After that, we prove that the proposed iteration strongly converges to the Bregman projection of initial element on the intersection of common fixed point set of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and the solution set of a system of generalized mixed equilibrium problems in reflexive Banach spaces. As application, we obatin some strong convergence results for a Bregman totally quasi-asymptotically nonexpansive mapping and a generalized mixed equilibrium problem in reflexive Banach spaces. These results are extensions and improvements to the main results in [7, 8]. In addition, a numerical example is provided to illustrate for the obtained result.

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Strong convergence theorems for a generalized mixed equilibrium problem and variational inequality problems

Fixed Point Theory and Applications ◽

10.1186/1687-1812-2013-65 ◽

2013 ◽

Vol 2013 (1) ◽

Cited By ~ 1

Author(s):

Jae Ug Jeong

Keyword(s):

Variational Inequality ◽

Strong Convergence ◽

Equilibrium Problem ◽

Variational Inequality Problems ◽

Generalized Mixed Equilibrium Problem ◽

Mixed Equilibrium Problem ◽

Convergence Theorems ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

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A New Hybrid Algorithm for a Pair of Quasi--Asymptotically Nonexpansive Mappings and Generalized Mixed Equilibrium Problems in Banach Spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2011/956852 ◽

2011 ◽

Vol 2011 ◽

pp. 1-22

Author(s):

Jinhua Zhu ◽

Shih-Sen Chang

Keyword(s):

Strong Convergence ◽

Nonexpansive Mappings ◽

Convex Banach Space ◽

Generalized Mixed Equilibrium Problem ◽

Common Element ◽

Strong Convergence Theorem ◽

Asymptotically Nonexpansive Mappings ◽

Asymptotically Nonexpansive ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

The purpose of this paper is, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of quasi--asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

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Strong Convergence Theorems for a Generalized Mixed Equilibrium Problem and a Family of Total Quasi--Asymptotically Nonexpansive Multivalued Mappings in Banach Spaces

Abstract and Applied Analysis ◽

10.1155/2012/147915 ◽

2012 ◽

Vol 2012 ◽

pp. 1-16 ◽

Cited By ~ 1

Author(s):

J. F. Tan ◽

S. S. Chang

Keyword(s):

Equilibrium Problem ◽

Infinite Family ◽

Convex Banach Space ◽

Generalized Mixed Equilibrium Problem ◽

Common Element ◽

Multivalued Mappings ◽

Mixed Equilibrium Problem ◽

Asymptotically Nonexpansive ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

The main purpose of this paper is by using a hybrid algorithm to find a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of common fixed points for a infinite family of total quasi--asymptotically nonexpansive multivalued mapping in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results announced by some authors.

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On a Hybrid Method for Generalized Mixed Equilibrium Problem and Fixed Point Problem of a Family of Quasi--Asymptotically Nonexpansive Mappings in Banach Spaces

Fixed Point Theory and Applications ◽

10.1155/2010/157278 ◽

2010 ◽

Vol 2010 (1) ◽

pp. 157278 ◽

Cited By ~ 6

Author(s):

Min Liu ◽

Shih-sen Chang ◽

Ping Zuo

Keyword(s):

Fixed Point ◽

Banach Spaces ◽

Hybrid Method ◽

Nonexpansive Mappings ◽

Fixed Point Problem ◽

Generalized Mixed Equilibrium Problem ◽

Point Problem ◽

Mixed Equilibrium Problem ◽

Asymptotically Nonexpansive ◽

Generalized Mixed Equilibrium

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New Generalized Mixed Equilibrium Problem with Respect to Relaxed Semi-Monotone Mappings in Banach Spaces

Journal of Applied Mathematics ◽

10.1155/2012/790592 ◽

2012 ◽

Vol 2012 ◽

pp. 1-25

Author(s):

Rabian Wangkeeree ◽

Pakkapon Preechasilp

Keyword(s):

Equilibrium Problem ◽

Banach Spaces ◽

Projection Algorithm ◽

Generalized Mixed Equilibrium Problem ◽

Common Element ◽

Strong Convergence Theorem ◽

Monotone Mappings ◽

Mixed Equilibrium Problem ◽

Generalized Mixed Equilibrium ◽

Mixed Equilibrium

We introduce the new generalized mixed equilibrium problem with respect to relaxed semimonotone mappings. Using the KKM technique, we obtain the existence of solutions for the generalized mixed equilibrium problem in Banach spaces. Furthermore, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the fixed point set of an asymptotically nonexpansive mapping. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main results extend various results existing in the current literature.

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