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

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.

On Mann-Type Subgradient-like Extragradient Method with Linear-Search Process for Hierarchical Variational Inequalities for Asymptotically Nonexpansive Mappings

We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint.

Iterative approximation of split feasibility problem in real Hilbert spaces

In this paper, we propose an iterative algorithm for finding solution of split feasibility problem involving a λ−strictly pseudo-nonspreading map and asymptotically nonexpansive semigroups in two real Hilbert spaces. We prove weak and strong convergence theorems using the sequence obtained from the proposed algorithm. Finally, we applied our result to solve a monotone inclusion problem and present a numerical example to support our result.

Periodic Points of Modular Firmly Mappings in the Variable Exponent Sequence Spaces ℓp(·)

In this work, we investigate the existence of periodic points of mappings defined on nonconvex domains within the variable exponent sequence spaces ℓp(·). In particular, we consider the case of modular firmly nonexpansive and modular firmly asymptotically nonexpansive mappings. These kinds of results have never been obtained before.

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

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.

Convergence of a Three-step Iteration Scheme to the Common Fixed Points of Mixed-Type Total Asymtotically Nonexpansive Mappings in Uniformly Convex Banach Spaces

We propose a three-step iteration scheme of hybrid mixed-type for three total asymptotically nonexpansive self mappings and three total asymptotically nonexpansive nonself mappings. In addition, we establish some weak convergence theorems of the scheme to the common fixed point of the mappings in uniformly convex Banach spaces. Our results extend and generalize numerous results currently in literature.

The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

The generalized viscosity implicit rules of nonexpansive asymptotically mappings in Hilbert spaces are considered. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. An application of it in the convex minimization problem is considered. The results presented in this paper improve and extend some recent corresponding results in the literature.

Mixed Type Algorithms for Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

In this paper, a new mixed type iteration process for approximating a common fixed point of two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings is constructed. We then establish a strong convergence theorem under mild conditions in a uniformly convex hyperbolic space. The results presented here extend and improve some related results in the literature.

MODIFIED PROXIMAL POINT ALGORITHM FOR MINIMIZATION AND FIXED POINT PROBLEM IN CAT(0) SPACES

In this paper, we study modified-type proximal point algorithm for approximating a common solution of a lower semi-continuous mapping and fixed point of total asymptotically nonexpansive mapping in complete CAT(0) spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved.