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

In this paper, we establish strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Our results extend and improve the recent ones announced by Matsushita and Takahashi [A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257-266], Matinez-yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400-2411], and many others.     

Strong convergence for maximal monotone operators, relatively quasi-nonexpansive mappings, variational inequalities and equilibrium problems

In this paper, we introduce a new hybrid iterative scheme for finding a common element of the set of zeroes of a maximal monotone operator, the set of fixed points of a relatively quasi-nonexpansive mapping, the sets of solutions of an equilibrium problem and the variational inequality problem in Banach spaces. As applications, we apply our results to obtain strong convergence theorems for a maximal monotone operator and quasi-nonexpansive mappings in Hilbert spaces and we consider a problem of finding a minimizer of a convex function.     

Strong convergence of projection algorithms for a family of relatively quasi-nonexpansive mappings and an equilibrium problem in Banach spaces

In this article, we introduce two kinds of new hybrid projection algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinitely countable family of relatively quasi-nonexpansive mappings in a Banach space. Our main results improve and extend the result obtained by Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), pp. 2400–2411] and the corresponding results.     

Strong convergence theorems for generalized asymptotically quasi-nonexpansive mappings

The purpose of this paper is to prove strong convergences of a modified implicit iteration process to a common fixed point for a finite family of generalized asymptotically quasi-nonexpansive mappings. The results presented in this paper improve and extend Shahzad and Zegeye’s corresponding results (Shahzad and Zegeye in Appl. Math. Comput. 189:1058–1065, 2007)     

Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings

The purpose of this article is to prove strong convergence theorems for common fixed points of two closed hemi-relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems, the monotone hybrid algorithms are presented and are used to approximate the common fixed points. Finally, a new simplified hybrid algorithm has been proposed and relative convergence theorem has been proved by using the new method for proofs. The results of this article modify and improve the results of Matsushita, Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257–266] and the results of Plubtieng, Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space, J. Approx. Theory 149 (2007) 103–115], and many others.     

Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces

The paper proposes two new iterative methods for solving pseudomonotone equilibrium problems involving fixed point problems for quasi-\(\phi \)-nonexpansive mappings in Banach spaces. The proposed algorithms combine the extended extragradient method or the linesearch method with the Halpern iteration. The strong convergence theorems are established under standard assumptions imposed on equilibrium bifunctions and mappings. The results in this paper have generalized and enriched existing algorithms for equilibrium problems in Banach spaces.     

Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces

In this paper, we introduce two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then we study the strong and weak convergence of the sequences.     

Strong convergence theorems for infinite family of relatively quasi nonexpansive mappings and systems of equilibrium problems

In this paper, we construct a new iterative scheme by hybrid method for approximation of common element of set of common fixed points of countably infinite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. Then, we prove strong convergence of the scheme to a common element of the two sets. Furthermore, we apply our results to solve convex minimization problem. Our results extend important recent results.     

Strong convergence theorems for variational inequality, equilibrium and fixed point problems with applications

In this paper, we introduce a new iterative scheme for finding a common element of the set of common solutions of a finite family of equilibrium problems with relaxed monotone mappings, of the set of common solutions of a finite family of variational inequalities and of the set of common fixed points of an infinite family of nonexpansive mappings in a Hilbert space. Strong convergence for the proposed iterative scheme is proved. As an application, we solve a multi-objective optimization problem using the result of this paper. Our results improve and extend the corresponding ones announced by others.     

Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings

In this paper, we prove strong convergence theorems to a zero of monotone mapping and a fixed point of relatively weak nonexpansive mapping. Moreover, strong convergence theorems to a point which is a fixed point of relatively weak nonexpansive mapping and a solution of a certain variational problem are proved under appropriate conditions.     

Strong convergence theorems of common elements for equilibrium problems and fixed point problems in banach spaces

In this paper,we consider hybrid algorithms for finding common elements of the set of common fixed points of two families quasi-φ-non-expansive mappings and the set of solutions of an equilibrium problem.We establish strong convergence theorems of common elements in uniformly smooth and strictly convex Banach spaces with the property (K).     

Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems

In this paper, we propose a new composite iterative method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings. Our results improve and extend the corresponding ones announced by many others.     

Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications

The purpose of this article is to prove strong convergence theorems for common fixed points of two countable families of weak relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems, the monotone hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article modify and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257-266] and the results of Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space, J. Approx. Theory 149 (2007) 103-115] and the results of Su et al. [Y. Su, Z. Wang and H. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009) 5616-5628], and many others.     

Convergence theorems of shrinking projection methods for equilibrium problem, variational inequality problem and a finite family of relatively quasi-nonexpansive mappings

We introduce a W-mapping for a finite family of relatively quasi-nonexpansive mappings and construct an iterative scheme for finding a common element of the solution set of equilibrium problem, the solution set of the variational inequality problem for an inverse-strongly-monotone operator and set of common fixed points of a finite family of relatively quasi-nonexpansive mappings. Strong convergence theorems are presented in a 2-uniformly convex and uniformly smooth Banach space. Our results generalize and extend relative results.     

Strong convergence theorems for variational inequalities and relatively weak nonexpansive mappings

In this paper, we introduce an iterative sequence for finding a common element of the set of fixed points of a relatively weak nonexpansive mapping and the set of solutions of a variational inequality in a Banach space. Our results extend and improve the recent ones announced by Li (J Math Anal Appl 295:115–126, 2004), Jianghua (J Math Anal Appl 337:1041–1047, 2008), and many others.     

Strong convergence of composite iterative methods for equilibrium problems and fixed point problems

We introduce a new composite iterative scheme by viscosity approximation method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping. Our results substantially improve the corresponding results of Takahashi and Takahashi [A. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506-515]. Essentially a new approach for finding solutions of equilibrium problems and the fixed points of nonexpansive mappings is provided.     

Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems

In this paper, a new iterative scheme based on the extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz continuous mapping is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Applications to optimization problems are given.     

On the hybrid projection method for fixed point and equilibrium problems

In this paper, fixed point and equilibrium problems are considered based on a hybrid projection method. Strong convergence theorems of common elements are established in the framework of real Hilbert spaces.     

Generalized equilibrium and fixed point problems for Bregman relatively nonexpansive mappings in Banach spaces

In this paper, we introduce and study a hybrid iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem for a Bregman relatively nonexpansive mapping in reflexive Banach spaces. We prove that the sequences generated by the hybrid iterative algorithm converge strongly to a common solution of these problems. Further, we give some consequences of the main result. Finally, we discuss a numerical example to demonstrate the applicability of the iterative algorithm.