A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces

In this paper, we prove strong convergence theorems by the hybrid method for a family of hemi-relatively nonexpansive mappings in a Banach space. Our results improve and extend the corresponding results given by Qin et al. [Xiaolong Qin, Yeol Je Cho, Shin Min Kang, Haiyun Zhou, Convergence of a modified Halpern-type iteration algorithm for quasi-?-nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051-1055], and at the same time, our iteration algorithm is different from the Kimura and Takahashi algorithm, which is a modified Mann-type iteration algorithm [Yasunori Kimura, Wataru Takahashi, On a hybrid method for a family of relatively nonexpansive mappings in Banach space, J. Math. Anal. Appl. 357 (2009) 356-363]. In addition, we succeed in applying our algorithm to systems of equilibrium problems which contain a family of equilibrium problems.     

Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces

In an infinite-dimensional Hilbert space, the normal Mann’s iteration algorithm has only weak convergence, in general, even for nonexpansive mappings. In order to get a strong convergence result, we modify the normal Mann’s iterative process for an infinite family of nonexpansive mappings in the framework of Banach spaces. Our results improve and extend the recent results announced by many others.     

A composite iterative algorithm for common fixed points for a finite family of nonexpansive mappings and applications

This paper is concerned with a new composite iteration approximating to common fixed points for a finite family of nonexpansive mappings in Banach spaces which have a uniformly Gâteaux differentiable norm. Utilizing the iterative algorithm, we obtain the strong convergence theorems for a finite family of nonexpansive mappings. Furthermore, the problem of image recovery is considered in the above result. Our results extend and improve the corresponding results.     

Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings

In this paper we propose a new implicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive mappings. We establish some convergence theorems for this implicit iteration scheme. In particular, necessary and sufficient conditions for strong convergence of this implicit iteration scheme were obtained.     

Halpern’s iteration in Banach spaces

We prove strong convergence theorems for a sequence which is generated by Halpern’s iteration. We also apply our result for finding zeros of an accretive operator. Our result improves the recent result of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350-2360] by removing some assumptions on the parameters. Finally we discuss the new sufficient condition studied by Song [Y. Song, A new sufficient condition for the strong convergence of Halpern type iterations. Appl. Math. Comput. 198 (2) (2008) 721-728; Y. Song, New strong convergence theorems for nonexpansive nonself-mappings without boundary conditions. Comput. Math. Appl. 56 (6) (2008) 1473-1478] and correct the main result of Song and Chai [Y. Song, X. Chai, Halpern iteration for firmly type nonexpansive mappings, Nonlinear Anal. 71 (10) (2009) 4500-4506].     

Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces

In this paper, we continue to study weak convergence problems for the implicit iteration process for a finite family of Lipschitzian continuous pseudocontractions in general Banach spaces. The results presented in this paper improve and extend the corresponding ones of Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773], Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81], Chen et al. [R. Chen, Y.S. Song, H.Y. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006) 701–709] and others.     

A modified halpern-type iteration algorithm for totally quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to modify the Halpern-type iteration algorithm for total quasi-?-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results of [X.L. Qin, Y.J. Cho, S.M. Kang, H. Y. Zhou, Convergence of a modified Halpern-type iterative algorithm for quasi-?-nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051-1055], [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3906], [C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400-2411] and others.     

A note on the paper “A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings”

In this note, we will modify several gaps in Chidume and Ofoedu [C.E. Chidume, E.U. Ofoedu, A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings, Nonlinear Anal. (2006), in press (doi:10.1016/j.na.2006.05.012)].     

Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces

In this paper, we introduce a modified Ishikawa iterative process for approximating a fixed point of nonexpansive mappings in Hilbert spaces. We establish some strong convergence theorems of the general iteration scheme under some mild conditions. The results improve and extend the corresponding results of many others.     

Convergence to common fixed points of a finite family of nonexpansive mappings

We study approximation of common fixed points of a finite family of nonexpansive mappings and introduce an iterative scheme defined by using a convex combination of mappings. Strong convergence of the iteration is obtained under several types of control conditions.Received: 28 May 2004     

Strong convergence of the CQ method for fixed point iteration processes

Strong convergence theorems are obtained for the CQ method for an Ishikawa iteration process, a contractive-type iteration process for nonexpansive mappings, and the proximal point algorithm for maximal monotone operators in Hilbert spaces.     

Strong convergence of modified Mann iterations for a countable family of nonexpansive mappings

In this paper we propose a new modified Mann iteration for computing common fixed points of nonexpansive mappings in a Banach space. We give certain different control conditions for the modified Mann iteration. Then, we prove strong convergence theorems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces. These results improve and extend results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], Yao, et al. [Y. Yao, R. Chen and J. Yao, Strong convergence and certain control conditions for modified Mann iteration, Nonlinear Anal. 68 (2008) 1687–1693], Qin and Su [X. Qin, Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 329 (2007) 415–424], and many others.     

Weak and strong convergence theorems for countable Lipschitzian mappings and its applications

We use Mann’s iteration and the hybrid method in mathematical programming to obtain weak and strong convergence to common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings.     

Strong convergence and certain control conditions for modified Mann iteration

In this paper we propose a new modified Mann iteration for computing fixed points of nonexpansive mappings in a Banach space setting. This new iterative scheme combines the modified Mann iteration introduced by Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] and the viscosity approximation method introduced by Moudafi [A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55]. We give certain different control conditions for the modified Mann iteration. Strong convergence in a uniformly smooth Banach space is established.     

Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces

In this paper, we consider an Ishikawa type iteration process with errors to approximate the fixed point of two uniformly quasi-Lipschitzian mappings in convex metric spaces. Some results have been obtained which generalize and unify many important known results in recent literature.     

The common minimal-norm fixed point of a finite family of nonexpansive mappings

Consider a finite family of nonexpansive mappings which are defined on a closed convex subset of a Hilbert space H. Suppose the set of common fixed points of this family is nonempty. We then address the problem of finding the minimum-norm element of this common fixed point set. We introduce both cyclic and parallel iteration methods to find this minimal-norm element.     

A new iterative algorithm for common solutions of a finite family of accretive operators

The purpose in this paper is to prove a theorem of strong convergence to a common solution for a finite family of accretive operators in a strictly convex Banach space by means of a new iterative algorithm, which is a generalization and extension of the results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], and Zegeye and Shahzad [H. Zegeye, N. Shahzad, Strong convergence theorems for a common zero of a finite family of m-accretive mappings, Nonlinear Anal. 66 (2007) 1161–1169]. Further using the result, the theorem of strong convergence to a common fixed point is discussed for a finite family of pseudocontractive mappings under certain conditions.     

On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces

We prove a strong convergence theorem for multivalued nonexpansive mappings which includes Kirk’s convergence theorem on CAT(0) spaces. The theorem properly contains a result of Jung for Hilbert spaces. We then apply the result to approximate a common fixed point of a countable family of single-valued nonexpansive mappings and a compact valued nonexpansive mapping.     

Implicit iteration process for a finite family of asymptotically hemi-contractive mappings in Banach spaces

The purpose of this paper is to study the strong convergence of a process of implicit iteration to a common fixed point for a finite family of asymptotically hemi-contractive mappings. Our results extend a recent result of M.O. Osilike and B.G. Akuchu [Common fixed points of a finite family of asymptotically pseudocontractive maps, Fixed Point Theory Appl. 2 (2004) 81–88] from Hilbert spaces to p$p$-uniformly convex Banach spaces with p>1$p>1$.     

Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space

In this paper, to find a common fixed point of a family of nonexpansive mappings, we introduce a Halpern type iterative sequence. Then we prove that such a sequence converges strongly to a common fixed point of nonexpansive mappings. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings and the problem of finding a zero of an accretive operator.