Strong convergence theorems for equilibrium problem and bregman totally quasi-asymptotically nonexpansive mapping in banach spaces

Zhu, Shining; Jian-hua, Huang

doi:10.1016/s0252-9602(16)30079-0

Cited by 4 publications

(3 citation statements)

References 34 publications

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“…Definition 1.19. (Zhu & Huang, 2016, Definition 2.10). Let X be a Banach space,  be a nonempty subset of , X : S    be a mapping.…”

Section: Thenmentioning

confidence: 99%

Strong Convergence of a Hybrid Iteration for a Generalized Mixed Equilibrium Problem and a Bregman Totally Quasi-Asymptotically Nonexpansive Mapping in Banach Spaces

Hiếu¹

2021

Tạp chí Khoa học

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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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“…Definition 1.19. (Zhu & Huang, 2016, Definition 2.10). Let X be a Banach space,  be a nonempty subset of , X : S    be a mapping.…”

Section: Thenmentioning

confidence: 99%

Strong Convergence of a Hybrid Iteration for a Generalized Mixed Equilibrium Problem and a Bregman Totally Quasi-Asymptotically Nonexpansive Mapping in Banach Spaces

Hiếu¹

2021

Tạp chí Khoa học

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“…In 2016, Zhu and Huang [17] proposed the iterative methods for finding the common solutions of the equilibrium problems and fixed points for Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces. Furthermore, they proved and obtained the strong convergence theorems.…”

Section: Introductionmentioning

confidence: 99%

Strong convergence theorems for mixed equilibrium problems and uniformly Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces

Jantakarn¹,

Kaewcharoen²

2019

J. Nonlinear Sci. Appl.

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“…We can mention, here, vector variational inequalities, vector optimization problems, or Debreu‐type equilibrium problems. For further relevant information, the reader is referred to the following recent publications available in our bibliography …”

Section: Introductionmentioning

confidence: 99%

Existence of solutions for systems of generalized quasi‐equilibrium problems

Patriche

2017

Math Methods in App Sciences

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In this paper, first of all, we consider a generalized game in choice form with 2 constraints and its corresponding equilibrium in choice. We assert new conditions under which the equilibrium in choice exists. As a consequence, we establish the existence of the equilibrium for generalized abstract economies.Then, we apply the obtained theorems to prove the existence of solutions for systems of quasi-equilibrium problems. We do this by considering new hypotheses for the properties of the involved correspondences. This approach leads us to results which differ a lot from the ones existing in literature. KEYWORDS equilibrium in choice, generalized game in choice form, quasi-equilibrium problem Math Meth Appl Sci. 2018;41:803-817.wileyonlinelibrary.com/journal/mma 804 PATRICHE the approached topic. New techniques of proof are developed by defining new correspondences which are requested to verify the local intersection property. Further, a selection lemma and the Brouwer fixed-point theorem are appealed. Two versions of equilibrium, slightly different, are discussed: weak equilibrium and strong equilibrium. As direct applications, we prove that the equilibrium exists for generalized abstract economies in which the preference correspondences do not have convex values, and their upper or lower sections do not verify topological properties. This study is a continuation of the author's research work, concerning the existence of equilibrium, as it can be seen, for instance, in Hervés-Beloso and Patriche. [10][11][12] Secondly, we study the existence of the solutions for some systems of vector quasi-equilibrium problems. We emphasize the importance of research studies on this topic, justified by the fact that it is, in fact, a unified model of other several problems. We can mention, here, vector variational inequalities, vector optimization problems, or Debreu-type equilibrium problems. For further relevant information, the reader is referred to the following recent publications available in our bibliography. [13][14][15][16][17][18][19][20][21][22][23][24][25][26] We define the notion of "weak solution," and we show its existence by considering an associate generalized abstract economy model which admits weak equilibrium. In this way, we obtain results which are totally new in literature, and here, we refer to the hypotheses, as well as to the tools of demonstration. We show that very slight continuity properties of the involved correspondences can also lead to the existence of the solutions for various classes of equilibrium problems.The rest of the paper is organized as follows: In Section 2, the systems of vector quasi-equilibrium problems are presented, and some new definitions for correspondences are introduced. Section 3 contains results concerning the existence of equilibrium. In Section 4, the existence of solutions for systems of vector quasi-equilibrium problems is proved. The concluding remarks are presented in Section 5.

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Strong convergence theorems for equilibrium problem and bregman totally quasi-asymptotically nonexpansive mapping in banach spaces

Cited by 4 publications

References 34 publications

Strong Convergence of a Hybrid Iteration for a Generalized Mixed Equilibrium Problem and a Bregman Totally Quasi-Asymptotically Nonexpansive Mapping in Banach Spaces

Strong Convergence of a Hybrid Iteration for a Generalized Mixed Equilibrium Problem and a Bregman Totally Quasi-Asymptotically Nonexpansive Mapping in Banach Spaces

Strong convergence theorems for mixed equilibrium problems and uniformly Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces

Existence of solutions for systems of generalized quasi‐equilibrium problems

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