Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space

Plubtieng, Somyot; Ungchittrakool, Kasamsuk

doi:10.1016/j.jat.2007.04.014

Cited by 74 publications

(32 citation statements)

References 22 publications

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“…The equilibrium problem, introduced by Blum and Oettli [1] in 1994, has been attracting a growing attention of researchers; see, e.g., [2,3,4,5] and the references therein. Numerous problems in physics, optimization, and economics reduce to find a solution of the equilibrium problem.…”

Section: Introductionmentioning

confidence: 99%

Strong convergence theorems for equilibrium problems involving Bregman functions in Banach spaces

Naraghirad¹,

Timnak²

2017

Tamkang J. Math.

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Abstract. In this paper, using Bregman functions, we introduce new Halpern-type iterative algorithms for finding a solution of an equilibrium problem in Banach spaces. We prove the strong convergence of a modified Halpern-type scheme to an element of the set of solution of an equilibrium problem in a reflexive Banach space. This scheme has an advantage that we do not use any Bregman projection of a point on the intersection of closed and convex sets in a practical calculation of the iterative sequence. Finally, some application of our results to the problem of finding a minimizer of a continuously Fréchet differentiable and convex function in a Banach space is presented. Our results improve and generalize many known results in the current literature.

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Section: Introductionmentioning

confidence: 99%

Strong convergence theorems for equilibrium problems involving Bregman functions in Banach spaces

Naraghirad¹,

Timnak²

2017

Tamkang J. Math.

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“…(1) if E is smooth, then J is single-valued; (2) if E is strictly convex, then J is one-to-one; (3) if E is reflexive, then J is surjective; (4) if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E; (5) if E * is uniformly convex, then J is uniformly continuous on bounded subsets of E and J is singlevalued and also one-to-one (see [6,13,15,18]). …”

Section: Introductionmentioning

confidence: 99%

“…A mapping T from C into itself is called nonexpansive if T x − T y ≤ x − y for all x, y ∈ C and relatively nonexpansive ( [3,17,19,22]) if F (T ) = F (T ) and φ(p, T x) ≤ φ(p, x) for all x ∈ C and p ∈ F (T ). The asymptotic behavior of relatively nonexpansive mapping was studied in [3,13,15,18].…”

Section: Introductionmentioning

confidence: 99%

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Hybrid shrinking iterative solutions to convex feasibility problems and a system of generalized mixed equilibrium problems

Xiong¹

2016

J. Nonlinear Sci. Appl.

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The purpose of this paper is to propose a new hybrid shrinking iterative scheme for approximating common elements of the set of solutions to convex feasibility problems for countable families of weak relatively nonexpansive mappings of a set of solutions to a system of generalized mixed equilibrium problems. A strong convergence theorem is established in the framework of Banach spaces. The results extend those of other authors, in which the involved mappings consist of just finitely many ones.

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“…Such properties have a direct impact when the process is executed directly in the underlying infinite dimensional space. To improve the weak convergence of Krasnoselski-Mann iterative process, so called hybrid projections have been considered (see [14][15][16][17][18][19][20][21][22][23][24][25] for more details and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Convergence of algorithms for fixed points of generalized asymptotically quasi-ϕ-nonexpansive mappings with applications

Qin

Agarwal

Cho

et al. 2012

Fixed Point Theory Appl

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In this article, strong convergence of Krasnoselski-Mann iterative sequences and Halpern iterative sequences are investigated based on hybrid projection methods. Strong convergence theorems for common fixed points of a family of generalized asymptotically quasi-j-nonexpansive mappings are established in the framework of Banach spaces. Mathematics Subject Classification 2000: 47H09; 47J05; 47J25 Keywords: asymptotically quasi-j-nonexpansive mapping, asymptotically nonexpansive mapping, fixed point, generalized asymptotically quasi-j-nonexpansive mapping, generalized asymptotically quasi-nonexpansive mapping 1.IntroductionFixed point theory as an important branch of nonlinear analysis theory has been applied in the study of nonlinear phenomena. During the four decades, many famous existence theorems of fixed points were established; see, for example, [1][2][3][4][5]. However, from the standpoint of real world applications it is not only to know the existence of fixed points of nonlinear mappings, but also to be able to construct an iterative process to approximate their fixed points. The computation of fixed points is important in the study of many real world problems, including inverse problems; for instance, it is not hard to show that the split feasibility problem and the convex feasibility problem in signal processing and image reconstruction can both be formulated as a problem of finding fixed points of certain operators, respectively (see [6,7] for more details and the references therein).Recently, the study of the convergence of various iterative processes for solving various nonlinear mathematical models forms the major part of numerical mathematics. Among these iterative processes, Krasnoselski-Mann iterative process and Halpern iterative process are popular and hot. Let C be a nonempty, closed, and convex subset of a underlying space X, and T : C C a mapping. Halpern iterative process generates a sequence {x n } in the following manner:x 0 ∈ C, x n+1 = α n u + (1 − α n )Tx n , ∀n ≥ 0,(1:1)

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Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space

Cited by 74 publications

References 22 publications

Strong convergence theorems for equilibrium problems involving Bregman functions in Banach spaces

Strong convergence theorems for equilibrium problems involving Bregman functions in Banach spaces

Hybrid shrinking iterative solutions to convex feasibility problems and a system of generalized mixed equilibrium problems

Convergence of algorithms for fixed points of generalized asymptotically quasi-ϕ-nonexpansive mappings with applications

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