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.     

Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in CAT(0) spaces

We establish △-convergence and strong convergence theorems for an iterative process for a finite family of generalized nonexpansive multivalued mappings in a CAT(0) space. Moreover, we present a fixed point theorem for a pair consisting of a finite family of generalized nonexpansive single valued mappings, and a generalized nonexpansive multivalued mapping in CAT(0) spaces.     

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.     

Strong convergence theorem for nonexpansive semigroups in Hilbert space

The purpose of this paper is to prove the strong convergence of a method combining the descent method and the hybrid method in mathematical programming for finding a point in the common fixed point set of a semigroup of nonexpansive mappings in Hilbert space.     

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.     

Fixed point theorems for nonexpansive maps in Banach spaces

In this work, we present some new versions of fixed point theorems for nonexpansive maps and 1-set contractions defined on closed, convex, not necessarily bounded subsets of Banach spaces. Our proofs rely on a compactness result for an approximate fixed point set. The Kuratowski measure of noncompactness is used throughout. To illustrate the results obtained, some applications to Banach algebras and Hammerstein integral equations are provided.     

Right Bregman nonexpansive operators in Banach spaces

We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.     

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.     

Fixed point solutions of generalized equilibrium problems for nonexpansive mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a generalized equilibrium problem in a real Hilbert space. Then, strong convergence of the scheme to a common element of the two sets is proved. As an application, problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem is solved. Moreover, solution is given to the problem of finding a common element of fixed points set of nonexpansive mappings and the set of solutions of a variational inequality problem.     

Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications

In this paper, we prove a strong convergence theorem by the hybrid method for a countable family of relatively nonexpansive mappings in a Banach space. We also establish a new control condition for the sequence of mappings {T_n} which is weaker than the control condition in Lemma 3.1 of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350-2360]. Moreover, we apply our results for finding a common fixed point of two relatively nonexpansive mappings in a Banach space and an element of the set of solutions of an equilibrium problem in a Banach space, respectively. Our results are applicable to a wide class of mappings.     

Strong convergence theorems for finitely many nonexpansive mappings and applications

Let E$E$ be a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. Recently, Takahashi and Shimoji [W. Takahashi, K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Modelling 32 (2000) 1463–1471] introduced an iterative scheme given by finitely many nonexpansive mappings in E$E$ and proved weak convergence theorems which are connected with the problem of image recovery. In this paper we introduce a new iterative scheme which includes their iterative scheme as a special case. Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gâteaux differentiable and which has a weakly continuous duality mapping, we prove strong convergence theorems which are connected with the problem of image recovery. Using the established results, we consider the problem of finding a common fixed point of finitely many nonexpansive mappings.     

Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.     

New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces

Recently, Ceng, Guu and Yao introduced an iterative scheme by viscosity-like approximation method to approximate the fixed point of nonexpansive mappings and solve some variational inequalities in Hilbert space (see Ceng et al. (2009) [9]). Takahashi and Takahashi proposed an iteration scheme to solve an equilibrium problem and approximate the fixed point of nonexpansive mapping by viscosity approximation method in Hilbert space (see Takahashi and Takahashi (2007) [12]). In this paper, we introduce an iterative scheme by viscosity approximation method for finding a common element of the set of a countable family of nonexpansive mappings and the set of an equilibrium problem in a Hilbert space. We prove the strong convergence of the proposed iteration to the unique solution of a variational inequality.     

The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces

A recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique. The advantage of this technique is that one can find a particular solution to the associated problems, and in most cases this particular solution solves some variational inequality. In this paper, we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a Banach space which is reflexive and has a weakly continuous duality map. Both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved.     

Fixed point results for a generalized nonexpansive map in uniformly convex metric spaces

In this paper, the existence of a unique fixed point of a map satisfying a very general contractive condition on a suitable subset of a uniformly convex metric space is proved. This fixed point is approximated by averaging Krasnosel’skii iterations of a generalized nonexpansive map. Our results substantially improve and extend several known results existing in the literature.     

A fixed point theorem for pointwise eventually nonexpansive mappings in nearly uniformly convex Banach spaces

The purpose of this paper is to prove the existence of a fixed point for a pointwise eventually nonexpansive mapping in a nearly uniformly convex Banach space. This provides an affirmative answer to a question given by Kirk and Xu [W.A. Kirk, Hong-Kun Xu, Asymptotic pointwise contraction, Nonlinear Anal. 69 (2008), 4706-4712].     

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.     

Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces

Viscosity approximation methods for a family of finite nonexpansive mappings are established in Banach spaces. The main theorems extend the main results of Moudafi [Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55] and Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] to the case of finite mappings. Our results also improve and unify the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159], Browder [Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Archiv. Ration. Mech. Anal. 24 (1967) 82–90], Cho et al. [Some control conditions on iterative methods, Commun. Appl. Nonlinear Anal. 12 (2) (2005) 27–34], Ha and Jung [Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990) 330–339], Halpern [Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961], Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], Jung et al. [Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl. 2005 (2) (2005) 125–135], Jung and Kim [Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc. 34 (1) (1997) 93–102], Lions [Approximation de points fixes de contractions, C.R. Acad. Sci. Ser. A-B, Paris 284 (1977) 1357–1359], O’Hara et al. [Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], Reich [Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (12) (1997) 3641–3645], Takahashi and Ueda [On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546–553], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992) 486–491], Xu [Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2) (2002) 240–256], and Zhou et al. [Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press] among others.     

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.