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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 272867, 14 pages
http://dx.doi.org/10.1155/2014/272867
Research Article

Applications of Bregman-Opial Property to Bregman Nonspreading Mappings in Banach Spaces

1Department of Mathematics, Yasouj University, Yasouj 75918, Iran
2Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
3Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 29 October 2013; Accepted 6 December 2013; Published 16 January 2014

Academic Editor: Jinlu Li

Copyright © 2014 Eskandar Naraghirad et al. 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.

Abstract

The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, nonspreading mappings. Unfortunately, not every Banach space shares the Opial property. However, every Banach space has a similar Bregman-Opial property for Bregman distances. In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading mappings and investigate the Mann and Ishikawa iterations for these mappings. We establish weak and strong convergence theorems for Bregman nonspreading mappings.

1. Introduction

Let be a (real) Banach space with norm and dual space . For any in , we denote the value of in at by . When is a sequence in , we denote the strong convergence of to by and the weak convergence by . Let be a nonempty subset of . Let be a map. We denote by the set of fixed points of . We call the map (i)nonexpansive if for all in ,(ii)quasi-nonexpansive if and for all in and in .

The nonexpansivity plays an important role in the study of the Ishikawa iteration, given by where the sequences and satisfy some appropriate conditions. When all , Ishikawa iteration (1) reduces to the classical Mann iteration. Construction of fixed points of nonexpansive mappings via Mann's and Ishikawa’s algorithms [1] has been extensively investigated in the literature (see, e.g., [2] and the references therein).

A powerful tool in deriving weak or strong convergence of iterative sequences is due to Opial [3]. A Banach space is said to satisfy the Opial property [3] if for any weakly convergent sequence in with weak limit we have for all in with . It is well known that all Hilbert spaces, all finite dimensional Banach spaces, and the Banach spaces () satisfy the Opial property. However, not every Banach space satisfies the Opial property; see, for example, [4, 5].

Working with the Bregman distance , the following Bregman-Opial-like inequality holds for every Banach space : whenever . See Lemma 11 for details. The Bregman-Opial property suggests introducing the notions of Bregman nonexpansive-like mappings and developing fixed point theorems and convergence results for the Ishikawa iterations for these mappings.

We recall the definition of Bregman distances. Let be a strictly convex and Gâteaux differentiable function on a Banach space . The Bregman distance [6] (see also [7, 8]) corresponding to is the function defined by It follows from the strict convexity of that for all in . However, might not be symmetric and might not satisfy the triangular inequality.

When is a smooth Banach space, setting for all in , we have that for all in . Here is the normalized duality mapping from into . Hence, reduces to the usual map as If is a Hilbert space, then .

Let be strictly convex and Gâteaux differentiable, and let be nonempty. A mapping is said to be(i)Bregman nonexpansive if (ii)Bregman quasi-nonexpansive if and (iii)Bregman skew quasi-nonexpansive if and (iv)Bregman nonspreading if

It is obvious that every Bregman nonspreading map with is Bregman quasi-nonexpansive. Bregman nonspreading mappings include, in particular, the class of nonspreading functions studied by Takahashi and his coauthors (see, e.g., [9, 10]), which is defined with the map in (5).

Let us give an example of a Bregman nonspreading mapping with nonempty fixed point set, which is not quasi-nonexpansive.

Example 1. Let be defined by . The associated Bregman distance is given by Define by We have . Plainly, is neither nonexpansive nor continuous.
However, is Bregman nonspreading. To see this, we define by Consider the following three possible cases.

Case 1. If , then we have and hence

Case 2. If and , then we have , , and hence

Case 3. If , then we have and hence Thus we have for all in and hence is a Bregman nonspreading mapping.

In Section 2, we collect and study some basic ties of Bregman distances. In Section 3, utilizing the Bregman-Opial property, we present some fixed point theorems. In Sections 4 and 5, we investigate weak and strong convergence of the Ishikawa and Bregman-Ishikawa iterations for Bregman nonspreading mappings. Our results improve and generalize some known results in the current literature; see, for example, [11].

2. Bregman Functions and Bregman Distances

Let be a (real) Banach space, and let . For any in , the gradient   is defined to be the linear functional in such that The function is said to be Gâteaux differentiable at if is well defined, and is Gâteaux differentiable if it is Gâteaux differentiable everywhere on . We call Fréchet differentiable at (see, e.g., [12, page 13] or [13, page 508]) if, for all , there exists such that The function is said to be Fréchet differentiable if it is Fréchet differentiable everywhere.

Let be the closed unit ball of a Banach space . A function is said to be(i)strongly coercive if (ii)locally bounded if is bounded for all ;(iii)locally uniformly smooth on ([14, pp. 207, 221]) if the function , defined by satisfies (iv)locally uniformly convex on (or uniformly convex on bounded subsets of ([14, pp. 203, 221])) if the gauge   of uniform convexity of , defined by satisfies

For a locally uniformly convex map , we have for all in and for all in .

Let be a Banach space and a strictly convex and Gâteaux differentiable function. By (4), the Bregman distance satisfies that [6] In particular,

Lemma 2 (see [15]). Let be a Banach space and a Gâteaux differentiable function which is locally uniformly convex on . Let and be bounded sequences in . Then the following assertions are equivalent: (1),(2).

The following Bregman-Opial-like inequality has been proved in [16].

Lemma 3 (see [16]). Let be a Banach space and let be a strictly convex and Gâteaux differentiable function. Suppose is a sequence in such that for some in . Then for all in the interior of dom  with .

We call a function   lower semicontinuous if is closed for all in . For a lower semicontinuous convex function , the subdifferential   of is defined by for all in . It is well known that is maximal monotone [17, 18]. For any lower semicontinuous convex function , the conjugate function   of is defined by It is well known that We also know that if is a proper lower semicontinuous convex function, then is a proper weak* lower semicontinuous convex function. Here, saying is proper we mean that .

The following definition is slightly different from that in Butnariu and Iusem [12].

Definition 4 (see [13]). Let be a Banach space. A function is said to be a Bregman function if the following conditions are satisfied: (1) is continuous, strictly convex, and Gâteaux differentiable;(2)the set is bounded for all in and .

The following lemma follows from Butnariu and Iusem [12] and Zǎlinescu [14].

Lemma 5. Let be a reflexive Banach space and a strongly coercive Bregman function. Then(1) is one-to-one, onto, and norm-to-weak* continuous;(2) if and only if ;(3) is bounded for all    in    and ; (4) is Gâteaux differentiable, and .

The following two results follow from [14, Proposition ].

Proposition 6. Let be a reflexive Banach space and let be a convex function which is locally bounded. The following assertions are equivalent:(1) is strongly coercive and locally uniformly convex on ;(2) and is locally bounded and locally uniformly smooth on ;(3) is Fréchet differentiable, and is uniformly norm-to-norm continuous on bounded subsets of .

Proposition 7. Let be a reflexive Banach space and a continuous convex function which is strongly coercive. The following assertions are equivalent: (1) is locally bounded and locally uniformly smooth on ;(2) is Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of ;(3) is strongly coercive and locally uniformly convex on .

Lemma 8 (see [13, 19]). Let be a reflexive Banach space, a strongly coercive Bregman function, and the function defined by The following assertions hold: (1) for all in and in ,(2) for all in and in .

It also follows from the definition that is convex in the second variable , and

Let be a Banach space and let be a nonempty convex subset of . Let be a strictly convex and Gâteaux differentiable function. Then, we know from [20] that, for in and in , we have Further, if is a nonempty, closed, and convex subset of a reflexive Banach space and is a strongly coercive Bregman function, then, for each in , there exists a unique in such that The Bregman projection from onto defined by has the following property: See [12] for details.

Let be a reflexive Banach space and let be a lower semicontinuous, strictly convex, and Gâteaux differentiable function. Let be a nonempty, closed, and convex subset of and let be a bounded sequence in . For any in , we set The Bregman asymptotic radius of relative to is defined by The Bregman asymptotic center of relative to is the set

Proposition 9. Let be a nonempty, closed, and convex subset of a reflexive Banach space , and let be strictly convex, Gâteaux differentiable, and locally bounded on . If is a bounded sequence of , then is a singleton.

Proof. In view of the definition of Bregman asymptotic radius, we may assume that converges weakly to in . By Lemma 3, we conclude that .

3. Fixed Point Theorems

Lemma 10 (see [21]). Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be strictly convex, continuous, strongly coercive, Gâteaux differentiable, and locally bounded on . Let be a Bregman quasi-nonexpansive mapping. Then is closed and convex.

Lemma 11. Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be a strictly convex and Gâteaux differentiable function. Let be a Bregman nonspreading mapping. Then

Proof. Let . In view of (24), we have This, together with (24), implies that

Proposition 12 (demiclosedness principle). Let be a nonempty subset of a reflexive Banach space . Let be a strictly convex, Gâteaux differentiable, and locally bounded function. Let be a Bregman nonspreading mapping. If in and , then . That is, is demiclosed at zero, where is the identity mapping on .

Proof. Since converges weakly to and , both the sequences and are bounded. Since is uniformly norm-to-norm continuous on bounded subsets of (see, e.g., [14]), we arrive at In view of Lemma 2, we deduce that . Set By Lemma 11, for all in , This implies From the Bregman-Opial-like property, we obtain .

Let be the Banach lattice of bounded real sequences with the supremum norm. It is well known that there exists a bounded linear functional on such that the following three conditions hold:(1)if and for every in , then ;(2)if for every in , then ;(3) for all in .

Here, denotes the sequence in . Such a functional is called a Banach limit and the value of at in is denoted by . Therefore, condition means . If satisfies conditions and , we call a mean on . See, for example, [22].

To see some examples of those mappings satisfying all the stated hypotheses in the following result, we refer the reader to [23].

Theorem 13 (see [23]). Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded and locally uniformly convex on . Let be a mapping. Let be a bounded sequence of and let be a mean on . Suppose that Then has a fixed point in .

Corollary 14. Let be a nonempty, bounded, closed, and convex subset of a reflexive Banach space . Let be strictly convex, continuous, strongly coercive, Gâteaux differentiable function, locally bounded, and locally uniformly convex on . Let be a Bregman nonspreading mapping. Then has a fixed point.

Proof. Let a Banach limit on and be such that is bounded. For any in we have This implies that Thus we have It follows from Theorem 13 that .

4. Weak and Strong Convergence Theorems for Bregman Nonspreading Mappings

In this section, we prove weak and strong convergence theorems concerning Bregman nonspreading mappings in a reflexive Banach space.

Lemma 15. Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be a strictly convex and Gâteaux differentiable function. Let be a Bregman skew quasi-nonexpansive mapping with a nonempty fixed point set . Let and be two sequences defined by (1) such that and are arbitrary sequences in . Then the following assertions hold:(1), for all in and ,(2) exists for any in .

Proof. Let . In view of (23), we have Consequently, This implies that is a bounded and nonincreasing sequence for all in . Thus we have that exists for any in .

Theorem 16. Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be strictly convex, Gâteaux differentiable, locally bounded, and locally uniformly convex on . Let be a Bregman nonspreading and Bregman skew quasi-nonexpansive mapping. Let and be sequences in , and let be a sequence with in defined by (1). (a)If is bounded and , then the fixed point set .(b)Assume . Then is bounded.(i) when and .(ii) when either(1) and or(2) and .

Proof. Assume that is bounded and . Consequently, there is a bounded subsequence of such that . Since is uniformly norm-to-norm continuous on bounded subsets of (see, e.g., [14]), In view of Proposition 9, we conclude that for some in . Let It follows from Lemma 11 that This implies From the Bregman-Opial-like property, we obtain .
Let and let . It follows from Lemma 15 that exists and hence is bounded. This implies that the sequence is bounded too. Let . In view of (23), we obtain a continuous, strictly increasing, and convex function with such that Consequently, we conclude that It follows that From the property of we deduce that In the same manner, we also obtain that Since is uniformly norm-to-norm continuous on bounded subsets of (see, e.g., [14]), we arrive at On the other hand, from (1) we get Assuming first . By (60) we see that Since is Bregman nonspreading, in view of (24), (25), and (62), we obtain When , we conclude that In view of Lemma 2, we have that
Finally, we assume and instead. By (59) we have subsequences and of and , respectively, such that Replacing with the finite number , and dealing with the subsequences and in (60) and (62). Passing to a further subsequence if necessary, we will arrive at the desired conclusion with (66) that . Hence, . The other case can be argued similarly.

Theorem 17. Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be strictly convex, Gâteaux differentiable, locally bounded, and locally uniformly convex on . Let be a Bregman nonspreading and Bregman skew quasi-nonexpansive mapping with . Let and be sequences in , and let be a sequence with in defined by (1). Assume that and . Then converges weakly to a fixed point of .

Proof. It follows from Theorem 16 that is bounded and . Since is reflexive, then there exists a subsequence of such that as . By Proposition 12, . We claim that as . If not, then there exists a subsequence of such that converges weakly to some in with . In view of Proposition 12 again, we conclude that . By Lemma 15, exists for all in . Thus we obtain by the Bregman-Opial-like property that This is a contradiction. Thus we have , and the desired assertion follows.

Theorem 18. Let be a nonempty, compact, and convex subset of a reflexive Banach space . Let be strictly convex, Gâteaux differentiable, locally bounded, and uniformly convex on bounded sets. Let be a Bregman nonspreading and Bregman skew quasi-nonexpansive mapping. Let and be sequences in . Assume that either and or and . Let be a sequence with in defined by (1). Then converges strongly to a fixed point of .

Proof. By Corollary 14, we see that the fixed point set of is nonempty. In view of Theorem 16, we obtain that is bounded and . By the compactness of , there exists a subsequence of such that converges strongly to some in . In view of Lemma 2 we deduce that . We can even assume that , and in particular, is bounded. Since is uniformly norm-to-norm continuous on bounded subsets of (see, e.g., [14]), Let . In view of Lemma 11, we obtain for all in .
It follows that . Thus we have . In view of Lemmas 15 and 2, we conclude that . Therefore, is the strong limit of the sequence .

5. Bregman-Ishikawa’s Type Iteration for Bregman Nonspreading Mappings

We propose the following Bregman-Ishikawa's type iteration. Let be a reflexive Banach space and let be a strictly convex and Gâteaux differentiable function. Let be a nonempty, closed, and convex subset of . Let be a Bregman nonspreading mapping such that the fixed point set is nonempty. Let and be two sequences defined by where and are arbitrary sequences in .

Lemma 19. Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be a strongly coercive Bregman function. Let be a Bregman quasi-nonexpansive mapping. Let and be two sequences defined by (71) such that and are arbitrary sequences in . Then the following assertions hold:(1) for all in and ,(2) exists for any in .

Proof. Let . In view of Lemma 8 and (71), we conclude that Consequently, using (35) we have This implies that is a bounded and nonincreasing sequence for all in . Thus we have that exists for any in .

Theorem 20. Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be a strongly coercive Bregman function which is locally bounded, locally uniformly convex, and locally uniformly smooth on . Let be a Bregman nonspreading mapping. Let and be two sequences in satisfying the control condition: Let be a sequence generated by algorithm (71). Then the following are equivalent. (1)There exists a bounded sequence such that .(2)The fixed point set .

Proof. The implication follows similarly as in the first part of the proof of Theorem 16.
For the implication , we assume . The boundedness of the sequences and follows from Lemma 19 and Definition 4. Since is a Bregman quasi-nonexpansive mapping, for any in , we have This, together with Definition 4 and the boundedness of , implies that is bounded.
The function is bounded on bounded subsets of and therefore is also bounded on bounded subsets of (see, e.g., [12, Proposition ] for more details). This implies that the sequences , , , and are bounded in .
In view of Proposition 7, we have that and is strongly coercive and uniformly convex on bounded subsets of . Let and let be the gauge of uniform convexity of the conjugate function .
Claim. For any in and in ,
Let . For each in , it follows from the definition of Bregman distance (4), Lemma 8, (23), and (71) that In view of Lemma 8 and (76), we obtain