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

Eskandar Naraghirad,¹ Ngai-Ching Wong,² and Jen-Chih Yao³

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

Thus we have

Since converges, together with the control condition (74), we have

Therefore, from the property of we deduce that

Since is uniformly norm-to-norm continuous on bounded subsets of (see, e.g., [14]), we arrive at

Theorem 21. 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 with . Let and be two sequences in satisfying the control conditions . Let be a sequence generated by the algorithm (71). Then, there exists a subsequence of which converges weakly to a fixed point of as .

Proof. It follows from Theorem 20 that is bounded and . Since is reflexive, then there exists a subsequence of such that as . In view of Proposition 12, we conclude that and the desired conclusion follows.

The construction of fixed points of nonexpansive mappings via Halpern's algorithm [24] has been extensively investigated recently in the current literature (see, e.g., [2] and the references therein). Numerous results have been proved on Halpern's iterations for nonexpansive mappings in Hilbert and Banach spaces (see, e.g., [11, 25, 26]).

Before dealing with the strong convergence of a Halpern-type iterative algorithm, we need the following lemmas.

Lemma 22 (see [27]). Let be a sequence in with a subsequence such that for all in . Then there exists another subsequence such that for all (sufficiently large) number one have

In fact, one can set .

Lemma 23 (see [28]). Let be a sequence of nonnegative real numbers satisfying

where and satisfy the following conditions:

(i) and , or, equivalently, ,(ii), or(iii).

Then, .

Theorem 24. 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 with . Let and be two sequences in satisfying the following control conditions:

(a);(b);(c).

Let be a sequence generated by

Then the sequence defined in (85) converges strongly to as .

Proof. We divide the proof into several steps. In view of Lemma 10, we conclude that is closed and convex. Set

Step  1. We prove that and are bounded sequences in .

We first show that is bounded. Let be fixed. In view of Lemma 8 and (85), we have

This, together with (71), implies that

By induction, we obtain

for all in . It follows from (89) that the sequence is bounded and hence there exists such that

In view of Definition 4, we deduce that the sequence is bounded. Since is a Bregman quasi-nonexpansive mapping from into itself, we conclude that

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, together with Step  1, implies that the sequences , , and are bounded in . In view of Proposition 7, we obtain 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 .

Step  2. We prove that

For each in , in view of the definition of Bregman distance (4), Lemma 8, and (30), we obtain

In view of Lemma 8 and (92), we obtain

Let

It follows from (94) that

Let

Then for all in . In view of Lemma 8 and (92) we obtain

Step  3. We show that as .

Case  1. If there exists in such that is non-increasing, then is convergent. Thus, we have as . This, together with (96) and conditions (a) and (c), implies that

Therefore, from the property of we deduce that

Since (Lemma 5) is uniformly norm-to-norm continuous on bounded subsets of (see, e.g., [14]), we arrive at

On the other hand, we have

This, together with Lemma 2 and (101), implies that

Similarly, we have

In view of Lemma 2 and (101), we conclude that

Since is bounded, together with (33) we can assume that there exists a subsequence of such that (Proposition 12) and

We thus conclude

The desired result follows from Lemmas 2 and 23 and (98).

Case  2. Suppose there exists a subsequence of such that

for all in . By Lemma 22, there exists a nondecreasing sequence of positive integers such that ,

This, together with (96), implies that

Then, by conditions (a) and (c), we get

By the same argument, as in Case  1, we arrive at

It follows from (98) that

Since , we have that

In particular, since , we obtain

In view of (112), we deduce that

This, together with (113), implies that

On the other hand, we have for all in . This ensures that as by Lemma 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was partially supported by Grant NSC 102-2115-M-110-002-MY2 (Ngai-Ching Wang), and by Grant NSC 102-2111-E-037-004-MY3 (Jen-Chin Yao).

References

W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.

S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.

Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.

D. van Dulst, “Equivalent norms and the fixed point property for nonexpansive mappings,” The Journal of the London Mathematical Society, vol. 25, no. 1, pp. 139–144, 1982.

J.-P. Gossez and E. Lami Dozo, “Some geometric properties related to the fixed point theory for nonexpansive mappings,” Pacific Journal of Mathematics, vol. 40, pp. 565–573, 1972.

G. Chen and M. Teboulle, “Convergence analysis of a proximal-like minimization algorithm using Bregman functions,” SIAM Journal on Optimization, vol. 3, no. 3, pp. 538–543, 1993.

L. M. Bregman, “The relation method of finding the common point of convex sets and its application to the solution of problems in convex programming,” USSR Computational Mathematics and Mathematical Physics, vol. 7, pp. 200–217, 1967.

Y. Censor and A. Lent, “An iterative row-action method for interval convex programming,” Journal of Optimization Theory and Applications, vol. 34, no. 3, pp. 321–353, 1981.

F. Kohsaka and W. Takahashi, “Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces,” Archiv der Mathematik, vol. 91, no. 2, pp. 166–177, 2008.

W. Takahashi, N.-C. Wong, and J.-C. Yao, “Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces,” Journal of Fixed Point Theory and Applications, vol. 11, no. 1, pp. 159–183, 2012.

W. Takahashi and G.-E. Kim, “Approximating fixed points of nonexpansive mappings in Banach spaces,” Mathematica Japonica, vol. 48, no. 1, pp. 1–9, 1998.

D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, vol. 40, Kluwer Academic, Dordrecht, The Netherlands, 2000.

F. Kohsaka and W. Takahashi, “Proximal point algorithms with Bregman functions in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 3, pp. 505–523, 2005.

C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, USA, 2002.

E. Naraghirad and J.-C. Yao, “Bregman weak relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, p. 2013, article 141, 2013.

Y.-Y. Huang, J.-C. Jeng, T.-Y. Kuo, and C.-C. Hong, “Fixed point and weak convergence theorems for point-dependent -hybrid mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, article 105, 2011.

R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pacific Journal of Mathematics, vol. 17, pp. 497–510, 1966.

R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacific Journal of Mathematics, vol. 33, pp. 209–216, 1970.

D. Butnariu and E. Resmerita, “Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces,” Abstract and Applied Analysis, vol. 2006, Article ID 84919, 39 pages, 2006.

E. Naraghirad, W. Takahashi, and J.-C. Yao, “Generalized retraction and fixed point theorems using Bregman functions in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 13, no. 1, pp. 141–156, 2012.

S. Reich and S. Sabach, “Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 299–314, Springer, New York, NY, USA, 2010.

W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.

N. Hussain, E. Naraghirad, and A. Alotaibi, “Existence of common fixed points using Bregman nonexpansive retracts and Bregman functions in Banach spaces,” Fixed Point Theory and Applications, p. 2013, article 113, 2013.

B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967.

W. Nilsrakoo and S. Saejung, “Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 14, pp. 6577–6586, 2011.

T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.

P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.

H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003.