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.     

Fixed point results for multimaps in CAT(0) spaces

Common fixed point results for families of single-valued nonexpansive or quasi-nonexpansive mappings and multivalued upper semicontinuous, almost lower semicontinuous or nonexpansive mappings are proved either in CAT(0) spaces or R-trees. It is also shown that the fixed point set of quasi-nonexpansive self-mapping of a nonempty closed convex subset of a CAT(0) space is always nonempty closed and convex.     

Products of resolvents and multivalued hybrid mappings in CAT(0) spaces

In this article, we introduce and investigate the concept of multivalued hybrid mappings in CAT(0) spaces by using the concept of quasilinearization. Also, we present a new iterative algorithm involving products of Moreau-Yosida resolvents for finding a common element of the set of minimizers of a finite family of convex functions and a common fixed point of two multivalued hybrid mappings in CAT(0) spaces.     

Banach空间中单值非扩张映射和多值非扩张映射公共不动点混合迭代逼近研究

引入了两个单值非扩张映射与两有限族多值非扩张映射新的混合迭代,在Banach空间中,研究了两个单值非扩张映射与两有限族多值非扩张映射在新迭代下关于公共不动点的收敛性,改进和推广了已有文献相关的结果.     

Common fixed point results in CAT(0) spaces

Let X be a complete CAT(0) space, T be a generalized multivalued nonexpansive mapping, and t be a single valued quasi-nonexpansive mapping. Under the assumption that T and t commute weakly, we shall prove the existence of a common fixed point for them. In this way, we extend and improve a number of recent results obtained by Shahzad (2009) [7] and [12], Shahzad and Markin (2008) [6], and Dhompongsa et al. (2005) [5].     

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.     

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 theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method

Strong convergence theorems are obtained for a finite family of nonexpansive mappings and semigroups by the hybrid method.     

Towards Lim

The paper contains an elegant extension of the Nadler fixed point theorem for multivalued contractions (see Theorem 21). It is based on a new idea of the α-step mappings (see Definition 17) being more efficient than α-contractions. In the present paper this theorem is a tool in proving some fixed point theorems for “nonexpansive” mappings in the bead spaces (metric spaces that, roughly speaking, are modelled after convex sets in uniformly convex spaces). More precisely the mappings are nonexpansive on a set with respect to only one point - the centre of this set (see condition (4)). The results are pretty general. At first we assume that the value of the mapping under consideration at this central point looks “sharp” (see Definition 6). This idea leads to a group of theorems (based on Theorem 7). Their proofs are compact and the theorems, in particular, are natural extensions of the classical results for (usual) nonexpansive mappings. In the second part we apply the idea of Lim to investigate the regular sequences and here the proofs are based on our extension of Nadler's Theorem. In consequence we obtain some fixed point theorems that generalise the classical Lim Theorem for multivalued nonexpansive mappings (see e.g. Theorem 26).     

Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups

Strong convergence theorems are obtained for a finite family of asymptotically nonexpansive mappings and semigroups by the modified Mann method.     

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     

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.     

A new iteration process for finite families of generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings

A new iteration process is introduced and proved to converge strongly to a common fixed point for a finite family of generalized Lipschitz nonlinear mappings in a real reflexive Banach space E$E$ with a uniformly Gâteaux differentiable norm if at least one member of the family is pseudo-contractive. It is also proved that a slight modification of the process converges to a common zero for a finite family of generalized Lipschitz accretive operators defined on E$E$. Results for nonexpansive families are obtained as easy corollaries. Finally, the new iteration process and the method of proof are of independent interest.     

Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces

First, we consider a strongly continuous semigroup of nonexpansive mappings defined on a closed convex subset of a complete CAT(0) space and prove a convergence of a Mann iteration to a common fixed point of the mappings. This result is motivated by a result of Kirk (2002) and of Suzuki (2002). Second, we obtain a result on limits of subsequences of Mann iterations of multivalued nonexpansive mappings on metric spaces of hyperbolic type, which leads to a convergence theorem for nonexpansive mappings on these spaces.     

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.     

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.     

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.     

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.     

Noncyclic mappings and best proximity pair results in Hilbert and uniformly convex Banach spaces

A new class of noncyclic mappings, called generalized noncyclic relatively nonexpansive, is introduced and used to study the existence of best proximity pairs in the setting of uniformly convex Banach spaces. We also obtain a weak convergence theorem for noncyclic relatively nonexpansive mappings in the setting of Hilbert spaces.     

Existence of solutions for generalized variational inequality problems in Banach spaces

In this paper, we prove the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over compact convex subsets in a reflexive Banach space with a Fréchet differentiable norm. Moreover, we give some conditions that guarantee the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over unbounded closed convex subsets. The result obtained in this paper improves and extends the recent ones announced by Yu and Yang [J. Yu, H. Yang, Existence of solutions for generalized variational inequality problems, Nonlinear Anal., 71 (2009) e2327-e2330] and many others.