# Some Characterizations for a Family of Nonexpansive Mappings and Convergence of a Generated Sequence to Their Common Fixed Point

- Yasunori Kimura
^{1}Email author and - Kazuhide Nakajo
^{2}

**2010**:732872

**DOI: **10.1155/2010/732872

© Y. Kimura and K. Nakajo. 2010

**Received: **7 October 2009

**Accepted: **19 October 2009

**Published: **27 October 2009

## Abstract

Motivated by the method of Xu (2006) and Matsushita and Takahashi (2008), we characterize the set of all common fixed points of a family of nonexpansive mappings by the notion of Mosco convergence and prove strong convergence theorems for nonexpansive mappings and semigroups in a uniformly convex Banach space.

## 1. Introduction

Let be a nonempty bounded closed convex subset of a Banach space and a nonexpansive mapping; that is, satisfies for any , and consider approximating a fixed point of . This problem has been investigated by many researchers and various types of strong convergent algorithm have been established. For implicit algorithms, see Browder [1], Reich [2], Takahashi and Ueda [3], and others. For explicit iterative schemes, see Halpern [4], Wittmann [5], Shioji and Takahashi [6], and others. Nakajo and Takahashi [7] introduced a hybrid type iterative scheme by using the metric projection, and recently Takahashi et al. [8] established a modified type of this projection method, also known as the shrinking projection method.

for each , where is the closure of the convex hull of , is the generalized projection onto , and is a sequence in with as . Then, he proved that converges strongly to . Matsushita and Takahashi [10] considered a sequence generated by

for each , where is the metric projection onto and is a sequence in with as . They proved that converges strongly to .

In this paper, motivated by these results, we characterize the set of all common fixed points of a family of nonexpansive mappings by the notion of Mosco convergence and prove strong convergence theorems for nonexpansive mappings and semigroups in a uniformly convex Banach space.

## 2. Preliminaries

Throughout this paper, we denote by a real Banach space with norm . We write to indicate that a sequence converges weakly to . Similarly, will symbolize strong convergence. Let be the family of all strictly increasing continuous convex functions satisfying that . We have the following theorem [11, Theorem ] for a uniformly convex Banach space.

Theorem 2.1 (Xu [11]).

for all and .

Bruck [12] proved the following result for nonexpansive mappings.

Theorem 2.2 (Bruck [12]).

for all , , with and nonexpansive mapping of into .

Let be a sequence of nonempty closed convex subsets of a reflexive Banach space . We denote the set of all strong limit points of by , that is, if and only if there exists such that converges strongly to and that for all . Similarly the set of all weak subsequential limit points by ; if and only if there exist a subsequence of and a sequence such that converges weakly to and that for all . If satisfies that , then we say that converges to in the sense of Mosco and we write . By definition, it always holds that . Therefore, to prove , it suffices to show that

One of the simplest examples of Mosco convergence is a decreasing sequence with respect to inclusion. The Mosco limit of such a sequence is . For more details, see [13].

Suppose that is smooth, strictly convex, and reflexive. The normalized duality mapping of is denoted by , that is,

for . In this setting, we may show that is a single-valued one-to-one mapping onto . For more details, see [14–16].

Let be a nonempty closed convex subset of a strictly convex and reflexive Banach space . Then, for an arbitrarily fixed , a function has a unique minimizer . Using such a point, we define the metric projection by for every . The metric projection has the following important property: if and only if and for all .

In the same manner, we define the generalized projection [17] for a nonempty closed convex subset of a strictly convex, smooth, and reflexive Banach space as follows. For a fixed , a function has a unique minimizer and we define by this point. We know that the following characterization holds for the generalized projection [17, 18]: if and only if and for all .

Tsukada [19] proved the following theorem for a sequence of metric projections in a Banach space.

Theorem 2.3 (Tsukada [19]).

Let be a reflexive and strictly convex Banach space and let be a sequence of nonempty closed convex subsets of . If exists and nonempty, then, for each , converges weakly to , where is the metric projection onto a nonempty closed convex subset of . Moreover, if has the Kadec-Klee property, the convergence is in the strong topology.

On the other hand, Ibaraki et al. [20] proved the following theorem for a sequence of generalized projections in a Banach space.

Theorem 2.4 (Ibaraki et al. [20]).

Let be a strictly convex, smooth, and reflexive Banach space and let be a sequence of nonempty closed convex subsets of . If exists and nonempty, then, for each , converges weakly to , where is the generalized projection onto a nonempty closed convex subset of . Moreover, if has the Kadec-Klee property, the convergence is in the strong topology.

Kimura [21] obtained the further generalization of this theorem by using the Bregman projection; see also [22].

Theorem 2.5 (Kimura [21]).

Let be a nonempty closed convex subset of a reflexive Banach space and let be a Bregman function on ; that is, (i) is lower semicontinuous and strictly convex; (ii) is contained by the interior of the domain of ; (iii) is Gâteaux differentiable on ; (iv) the subsets and of are both bounded for all and , where for all and . Let be a sequence of nonempty closed convex subsets of such that exists and is nonempty. Suppose that is sequentially consistent; that is, for any bounded sequence of and of the domain of , implies . Then, the sequence of Bregman projections converges strongly to for all .

We note that the generalized duality mapping coincides with if the function is defined by for . In this case, the Bregman projection with respect to becomes the generalized projection for any nonempty closed convex subset of .

## 3. Main Results

Let be a nonempty closed convex subset of and let be a sequence of mappings of into itself such that . We consider the following conditions.

(I)For every bounded sequence in , implies , where is the set of all weak cluster points of ; see [23–25].

(II)for every sequence in and , and imply .

We know that condition (I) implies condition (II). Then, we have the following results.

Theorem 3.1.

Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and let be a family of nonexpansive mappings of into itself with . Let for each , where . Then, the following are equivalent:

(i) satisfies condition (I);

(ii)for every with as , .

Proof.

and define by for each . Suppose that a subsequence of converges weakly to . Then since for all and , we have ; that is, condition (I) holds.

For a sequence of mappings satisfying condition (II), we have the following characterization.

Theorem 3.2.

Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and let be a family of nonexpansive mappings of into itself with . Let and for each , where . Then, the following are equivalent:

(i) satisfies condition (II);

(ii)for every with as , .

Proof.

Let . We have for all . As in the proof of Theorem 3.1, we get . By condition (II), we obtain , which implies . Hence we have .

for each , we have . Letting for each , we have for every and as , which implies . Hence (i) holds, which is the desired result.

Remark 3.3.

In Theorem 3.2, it is obvious by definition that is a decreasing sequence with respect to inclusion. Therefore, conditions and are also equivalent to

for every with as , ,

where is the Painlevé-Kuratowski limit of ; see, for example, [13] for more details.

In the next section, we will see various types of sequences of nonexpansive mappings which satisfy conditions (I) and (II).

## 4. The Sequences of Mappings Satisfying Conditions (I) and (II)

First let us show some known results which play important roles for our results.

Theorem 4.1 (Browder [1]).

Let be a nonempty closed convex subset of a uniformly convex Banach space and a nonexpansive mapping on with . If converges weakly to and converges strongly to , then is a fixed point of .

Theorem 4.2 (Bruck [26]).

Theorems 4.3, 4.5(i), 4.6–4.9 show the examples of a family of nonexpansive mappings satisfying condition (I). Theorems 4.5(ii), 4.11, and 4.12 show those satisfying condition (II).

Theorem 4.3.

Let be a nonempty closed convex subset of a uniformly convex Banach space and let be a nonexpansive mapping of into itself with . Let for all . Then, is a family of nonexpansive mappings of into itself with and satisfies condition (I).

Proof.

This is a direct consequence of Theorem 4.1.

Remark 4.4.

In the previous theorem, if is bounded, then is guaranteed to be nonempty by Kirk's fixed point theorem [27].

Let be a Banach space and a set-valued operator on . is called an accretive operator if for every and with and .

Let be an accretive operator and . We know that the operator has a single-valued inverse, where is the identity operator on . We call the resolvent of and denote it by . We also know that is a nonexpansive mapping with for any , where . For more details, see, for example, [15].

We have the following result for the resolvents of an accretive operator by [25]; see also [15, Theorem ], and [16, Theorem ] .

Theorem 4.5.

Let be a nonempty closed convex subset of a uniformly convex Banach space and let be an accretive operator with and . Let for every , where for all . Then, is a family of nonexpansive mappings of into itself with and the following hold:

(i)if , then satisfies condition (I),

(ii)if there exists a subsequence of such that , then satisfies condition (II).

Proof.

It is obvious that is a nonexpansive mapping of into itself and for all .

For (i), suppose and let be a bounded sequence in such that . By [25, Lemma ], we have . Using Theorem 4.1 we obtain .

Let us show (ii). Let be a subsequence of with and let be a sequence in and such that and . As in the proof of (i), we get and .

Such a mapping is called the W-mapping generated by and . We have the following result for the W-mapping by [29, 30]; see also [25, Lemma ].

Theorem 4.6.

Let be a nonempty closed convex subset of a uniformly convex Banach space and let be a family of nonexpansive mappings of into itself with . Let be a sequence of real numbers such that for every with and let be the W-mapping generated by and . Let for every . Then, is a family of nonexpansive mappings of into itself with and satisfies condition (I).

Proof.

As in the proof of [30, Theorem ], we get for each . Using Theorem 4.1 we obtain .

We have the following result for a convex combination of nonexpansive mappings which Aoyama et al. [31] proposed.

Theorem 4.7.

Let be a nonempty closed convex subset of a uniformly convex Banach space and let be a family of nonexpansive mappings of into itself such that . Let be a family of nonnegative numbers with indices with such that

(i) for every ,

(ii) for each ,

and let for all , where for some with . Then, is a family of nonexpansive mappings of into itself with and satisfies condition (I).

Proof.

for , where . Since for all and , we get and hence for each . Therefore, using Theorem 4.1 we obtain .

Let be a nonempty closed convex subset of a Banach space and let be a semigroup. A family is said to be a nonexpansive semigroup on if

(i)for each , is a nonexpansive mapping of into itself;

(ii) for every .

We denote by the set of all common fixed points of , that is, . We have the following result for nonexpansive semigroups by [25, Lemma ]; see also [32, 33].

Theorem 4.8.

Let be a nonempty closed convex subset of a uniformly convex Banach space and let be a semigroup. Let be a nonexpansive semigroup on such that and let be a subspace of such that contains constants, is -invariant (i.e., ) for each , and the function belongs to for every and . Let be a sequence of means on such that as for all and let for each . Then, is a family of nonexpansive mappings of into itself with and satisfies condition (I).

Proof.

It is obvious that is a family of nonexpansive mappings of into itself. By [25, Lemma ], we have . Let be a bounded sequence in such that . Then we get for every . Using Theorem 4.1 we have .

Let be a nonempty closed convex subset of a Banach space . A family of mappings of into itself is called a one-parameter nonexpansive semigroup on if it satisfies the following conditions:

(i) for all ;

(ii) for every ;

(iii) for each and ;

(iv)for all , is continuous.

We have the following result for one-parameter nonexpansive semigroups by [25, Lemma ].

Theorem 4.9.

for all and . Then, is a family of nonexpansive mappings of into itself satisfying that and condition (I).

Remark 4.10.

If is bounded, then is guaranteed to be nonempty; see [34].

Proof.

It is obvious that is a family of nonexpansive mappings of into itself. By [25, Lemma ], we have . Let be a bounded sequence in such that . We get for every . Hence, using Theorem 4.1 we have .

Motivated by the idea of [23, page 256], we have the following result for nonexpansive mappings.

Theorem 4.11.

Let be a nonempty closed convex subset of a uniformly convex Banach space and let be a countable index set. Let be an index mapping such that, for all , there exist infinitely many satisfying . Let be a family of nonexpansive mappings of into itself satisfying and let for all . Then, is a family of nonexpansive mappings of into itself with and satisfies condition (II).

Proof.

It is obvious that . Let be a sequence in and such that and . Fix . There exists a subsequence of such that for all . Thus we have . Therefore, using Theorem 4.1 for every and hence we get .

From Theorem 4.11, we have the following result for one-parameter nonexpansive semigroups.

Theorem 4.12.

Let be a nonempty closed convex subset of a uniformly convex Banach space and let be a one-parameter nonexpansive semigroup on such that . Let for every with and as and for all , where is an an index mapping satisfing, for all , there exist infinitely many such that . Then, is a family of nonexpansive mappings of into itself with and satisfies condition (II).

Remark 4.13.

If is bounded, it is guaranteed that . See [34].

Proof.

We have by [35, Lemma ]; see also [36]. By Theorem 4.11, we obtain the desired result.

## 5. Strong Convergence Theorems

Throughout this section, we assume that is a nonempty bounded closed convex subset of a uniformly convex Banach space and is a family of nonexpansive mappings of into itself with . Then, we know that is closed and convex.

We get the following results for the metric projection by using Theorems 2.3, 3.1, and 3.2.

Theorem 5.1.

for each , where such that as , and is the metric projection onto . If satisfies condition (I), then converges strongly to .

Theorem 5.2.

for each , where such that as . If satisfies condition (II), then converges strongly to .

On the other hand, we have the following results for the Bregman projection by using Theorems 2.5, 3.1, and 3.2.

Theorem 5.3.

for each , where such that as and is the Bregman projection onto . If satisfies condition (I), then converges strongly to .

Theorem 5.4.

for each , where such that as . If satisfies condition (II), then converges strongly to .

In a similar fashion, we have the following results for the generalized projection by using Theorems 2.4, 3.1, and 3.2.

Theorem 5.5.

for each , where such that as and is the generalized projection onto . If satisfies condition (I), then converges strongly to .

Theorem 5.6.

for each , where with as . If satisfies condition (II), then converges strongly to .

Combining these theorems with the results shown in the previous section, we can obtain various types of convergence theorems for families of nonexpansive mappings.

## 6. Generalization of Xu's and Matsushita-Takahashi's Theorems

At the end of this paper, we remark the relationship between these results and the convergence theorems by Xu [9] and Matsushita and Takahashi [10] mentioned in the introduction.

for every from basic properties of and . Therefore, for each theorem we have

and therefore . Consequently, by using Theorems 2.3 and 2.4, we obtain the following results generalizing the theorems of Xu, and Matsushita and Takahashi, respectively.

Theorem 6.1.

for each , where is a sequence in with as . Then, converges strongly to .

Theorem 6.2.

for each , where is a sequence in with as . Then, converges strongly to .

## Declarations

### Acknowledgment

The first author is supported by Grant-in-Aid for Scientific Research no. 19740065 from Japan Society for the Promotion of Science. This work is Dedicated to Professor Wataru Takahashi on his retirement.

## Authors’ Affiliations

## References

- Browder FE:
**Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces.***Archive for Rational Mechanics and Analysis*1967,**24:**82–90.MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Strong convergence theorems for resolvents of accretive operators in Banach spaces.***Journal of Mathematical Analysis and Applications*1980,**75**(1):287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleMATHGoogle Scholar - Takahashi W, Ueda Y:
**On Reich's strong convergence theorems for resolvents of accretive operators.***Journal of Mathematical Analysis and Applications*1984,**104**(2):546–553. 10.1016/0022-247X(84)90019-2MathSciNetView ArticleMATHGoogle Scholar - Halpern B:
**Fixed points of nonexpanding maps.***Bulletin of the American Mathematical Society*1967,**73:**957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar - Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar - Shioji N, Takahashi W:
**Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces.***Proceedings of the American Mathematical Society*1997,**125**(12):3641–3645. 10.1090/S0002-9939-97-04033-1MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.***Journal of Mathematical Analysis and Applications*2003,**279**(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar - Takahashi W, Takeuchi Y, Kubota R:
**Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(1):276–286. 10.1016/j.jmaa.2007.09.062MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Strong convergence of approximating fixed point sequences for nonexpansive mappings.***Bulletin of the Australian Mathematical Society*2006,**74**(1):143–151. 10.1017/S0004972700047535MathSciNetView ArticleMATHGoogle Scholar - Matsushita S-Y, Takahashi W:
**Approximating fixed points of nonexpansive mappings in a Banach space by metric projections.***Applied Mathematics and Computation*2008,**196**(1):422–425. 10.1016/j.amc.2007.06.006MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**Inequalities in Banach spaces with applications.***Nonlinear Analysis: Theory, Methods & Applications*1991,**16**(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleMATHGoogle Scholar - Bruck RE Jr.:
**On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces.***Israel Journal of Mathematics*1981,**38**(4):304–314. 10.1007/BF02762776MathSciNetView ArticleMATHGoogle Scholar - Beer G:
*Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications*.*Volume 268*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xii+340.View ArticleMATHGoogle Scholar - Goebel K, Reich S:
*Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 83*. Marcel Dekker, New York, NY, USA; 1984:ix+170.Google Scholar - Takahashi W:
*Nonlinear Functional Analysis, Fixed Point Theory and Its Applications*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Takahashi W:
*Convex Analysis and Approximation of Fixed Points, Mathematical Analysis Series*.*Volume 2*. Yokohama Publishers, Yokohama, Japan; 2000:iv+280.MATHGoogle Scholar - Alber YI:
**Metric and generalized projection operators in Banach spaces: properties and applications.**In*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*.*Volume 178*. Marcel Dekker, New York, NY, USA; 1996:15–50.Google Scholar - Reich S:
**Constructive techniques for accretive and monotone operators.**In*Applied Nonlinear Analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978)*. Academic Press, New York, NY, USA; 1979:335–345.Google Scholar - Tsukada M:
**Convergence of best approximations in a smooth Banach space.***Journal of Approximation Theory*1984,**40**(4):301–309. 10.1016/0021-9045(84)90003-0MathSciNetView ArticleMATHGoogle Scholar - Ibaraki T, Kimura Y, Takahashi W:
**Convergence theorems for generalized projections and maximal monotone operators in Banach spaces.***Abstract and Applied Analysis*2003,**2003**(10):621–629. 10.1155/S1085337503207065MathSciNetView ArticleMATHGoogle Scholar - Kimura Y:
**On Mosco convergence for a sequence of closed convex subsets of Banach spaces.**In*Proceedings of the International Symposium on Banach and Function Spaces, 2004, Kitakyushu, Japan*Edited by: Kato M, Maligranda L. 291–300. - Resmerita E:
**On total convexity, Bregman projections and stability in Banach spaces.***Journal of Convex Analysis*2004,**11**(1):1–16.MathSciNetMATHGoogle Scholar - Bauschke HH, Combettes PL:
**A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces.***Mathematics of Operations Research*2001,**26**(2):248–264. 10.1287/moor.26.2.248.10558MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Shimoji K, Takahashi W:
**Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces.***Taiwanese Journal of Mathematics*2006,**10**(2):339–360.MathSciNetMATHGoogle Scholar - Nakajo K, Shimoji K, Takahashi W:
**Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces.***Journal of Nonlinear and Convex Analysis*2007,**8**(1):11–34.MathSciNetMATHGoogle Scholar - Bruck RE Jr.:
**Properties of fixed-point sets of nonexpansive mappings in Banach spaces.***Transactions of the American Mathematical Society*1973,**179:**251–262.MathSciNetView ArticleMATHGoogle Scholar - Kirk WA:
**A fixed point theorem for mappings which do not increase distances.***The American Mathematical Monthly*1965,**72:**1004–1006. 10.2307/2313345MathSciNetView ArticleMATHGoogle Scholar - Takahashi W:
**Weak and strong convergence theorems for families of nonexpansive mappings and their applications.***Annales Universitatis Mariae Curie-Skłodowska. Sectio A*1997,**51**(2):277–292.MathSciNetMATHGoogle Scholar - Takahashi W, Shimoji K:
**Convergence theorems for nonexpansive mappings and feasibility problems.***Mathematical and Computer Modelling*2000,**32**(11–13):1463–1471.MathSciNetView ArticleMATHGoogle Scholar - Kimura Y, Takahashi W:
**Weak convergence to common fixed points of countable nonexpansive mappings and its applications.***Journal of the Korean Mathematical Society*2001,**38**(6):1275–1284.MathSciNetMATHGoogle Scholar - Aoyama K, Kimura Y, Takahashi W, Toyoda M:
**Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(8):2350–2360. 10.1016/j.na.2006.08.032MathSciNetView ArticleMATHGoogle Scholar - Atsushiba S, Shioji N, Takahashi W:
**Approximating common fixed points by the Mann iteration procedure in Banach spaces.***Journal of Nonlinear and Convex Analysis*2000,**1**(3):351–361.MathSciNetMATHGoogle Scholar - Shioji N, Takahashi W:
**Strong convergence theorems for asymptotically nonexpansive semigroups in Banach spaces.***Journal of Nonlinear and Convex Analysis*2000,**1**(1):73–87.MathSciNetMATHGoogle Scholar - Bruck RE Jr.:
**A common fixed point theorem for a commuting family of nonexpansive mappings.***Pacific Journal of Mathematics*1974,**53:**59–71.MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Shimoji K, Takahashi W:
**Strong convergence theorems of Browder's type for families of nonexpansive mappings in Hilbert spaces.***International Journal of Computational and Numerical Analysis and Applications*2004,**6**(2):173–192.MathSciNetMATHGoogle Scholar - Suzuki T:
**On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces.***Proceedings of the American Mathematical Society*2003,**131**(7):2133–2136. 10.1090/S0002-9939-02-06844-2MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.