Weak Convergence Theorems for Bregman Relatively Nonexpansive Mappings in Banach Spaces

Pang, Chin-Tzong; Naraghirad, Eskandar; Wen, Ching-Feng

doi:https://doi.org/10.1155/2014/573075

Journal of Applied Mathematics

On this page

Abstract Introduction References Copyright Related Articles

Special Issue

Variational Inequalities and Vector Optimization 2014

View this Special Issue

Research Article | Open Access

Volume 2014 | Article ID 573075 | https://doi.org/10.1155/2014/573075

Weak Convergence Theorems for Bregman Relatively Nonexpansive Mappings in Banach Spaces

Chin-Tzong Pang,¹Eskandar Naraghirad,²and Ching-Feng Wen³

Academic Editor: Jen-Chih Yao

Received18 Feb 2014

Accepted26 Mar 2014

Published07 May 2014

Abstract

We study Mann type iterative algorithms for finding fixed points of Bregman relatively nonexpansive mappings in Banach spaces. By exhibiting an example, we first show that the class of Bregman relatively nonexpansive mappings embraces properly the class of Bregman strongly nonexpansive mappings which was investigated by Martín-Márques et al. (2013). We then prove weak convergence theorems for the sequences produced by the methods. Some application of our results to the problem of finding a zero of a maximal monotone operator in a Banach space is presented. Our results improve and generalize many known results in the current literature.

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 Mann iteration, given by where the sequence satisfies some appropriate conditions. Construction of fixed points of nonexpansive mappings via Mann’s algorithm [1] has been extensively investigated in the literature (see, e.g., [2] and the references therein).

Let be a strictly convex and Gâteaux differentiable function on a Banach space . The Bregman distance [3] (see also [4, 5]) 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 . For more details, we refer the readers to [6].

Let be a smooth, strictly convex, and reflexive Banach space and let be the normalized duality mapping of . Let be a nonempty, closed, and convex subset of . The generalized projection from onto is defined and denoted by where . Let be a nonempty, closed, and convex subset of a smooth Banach space ; let be a mapping from into itself. A point is said to be an asymptotic fixed point [7] of if there exists a sequence in which converges weakly to and . We denote the set of all asymptotic fixed points of by . A point is called a strong asymptotic fixed point of if there exists a sequence in which converges strongly to and . We denote the set of all strong asymptotic fixed points of by .

Let be a nonempty, closed, and convex subset of a reflexive Banach space . Let be a proper, lower semicontinuous, and convex function. Recall that a mapping is said to be Bregman quasi-nonexpansive, if and A mapping is said to be Bregman relatively nonexpansive if the following conditions are satisfied:(1) is nonempty;(2), ;(3).

A mapping is said to be Bregman weak relatively nonexpansive if the following conditions are satisfied:(1) is nonempty;(2), , ;(3).

A mapping is said to be Bregman strongly nonexpansive (BSNE) if the following conditions are satisfied:(1) is nonempty;(2), , ;(3);(4)for any bounded sequence and any we have

It is obvious that any Bregman strongly nonexpansive mapping is a Bregman relatively nonexpansive mapping, but the converse is not true in general. In the following, we show that there exists a Bregman relatively nonexpansive mapping which is not a Bregman strongly nonexpansive mapping.

Example 1. Let , where Let be a sequence defined by where for all . It is clear that the sequence converges weakly to . Indeed, for any , we have as . It is also obvious that for any with , sufficiently large. Thus, is not a Cauchy sequence. Let be an even number in and let be defined by It is easy to show that for all , where It is also obvious that Now, we define a mapping by It is clear that and for any If , then we have Therefore, is a Bregman quasi-nonexpansive mapping. Next, we claim that for any subsequence of , . If not, then there exists a subsequence of such that . This implies that , which is impossible. Now, we claim that is a Bregman relatively nonexpansive mapping. Indeed, for any sequence such that and as , since is not a Cauchy sequence, there exists a sufficiently large number such that , for any . If we suppose that there exists such that for infinitely many , then a subsequence would satisfy , so and which is impossible due to the fact that for all . This implies that for all . It follows from that and hence , which implies that . Since , we conclude that is a Bregman relatively nonexpansive mapping. Finally, we show that is not a Bregman strongly nonexpansive mapping. To this end, we consider the sequence defined by (8); then, we have This implies that On the other hand, we have which implies that Therefore, is not a Bregman strongly nonexpansive mapping.

We refer the readers to see some other examples of Bregman relatively nonexpansive mappings in [8].

A Banach space is said to satisfy the Opial property [9] 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. Working with the Bregman distance , the following Bregman Opial-like inequality holds for every Banach space such that is weakly sequentially continuous: whenever . See Lemma 3 for details. 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, quasinonexpansive mappings. The Bregman-Opial property suggests introducing the notions of Bregman nonexpansive-like mappings and developing fixed point theorems and convergence results for the Mann iterations for these mappings.

Let be a reflexive Banach space with the dual space and let be a set-valued mapping. We define the domain and range of by and , respectively. The graph of is denoted by . The mapping is said to be monotone [10] if whenever . It is also said to be maximal monotone [11] if its graph is not contained in the graph of any other monotone operators on . If is maximal monotone, then we can show that the set is closed and convex. Let be a proper, lower semicontinuous, and convex function. Let be a maximal monotone operator from to . For any , let the mapping be defined by The mapping is called the -resolvent of (see [12]). It is well known that for each (for more details, see, e.g., [13]).

Examples and some important properties of such operators are discussed in [14].

In this paper, using Bregman functions, we study Mann type iterative algorithms for finding fixed points of Bregman relatively nonexpansive mappings in Banach spaces. We prove weak convergence theorems for the sequences produced by the methods. Some application of our results to the problem of finding a zero of a maximal monotone operator in a Banach space is presented. Our results improve and generalize many known results in the current literature; see, for example, [15].

2. Properties of 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., [16, page 13] or [17, page 508]) if, for all , there exists such that The function is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function is Gâteaux differentiable, then is norm-to- continuous (see, e.g., [16, Proposition 1.1.10]). If is also Fréchet differentiable, then is norm-to-norm continuous (see, [17, page 508]).

Let be a Banach space, , and . A function is said to be(i)strongly coercive if (ii)locally bounded if is bounded for all ;(iii)locally uniformly smooth on ([18, pages 207, 221]) if the function , defined by satisfies (iv)locally uniformly convex on (or uniformly convex on bounded subsets of ([18, pages 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 (2), the Bregman distance satisfies [3] In particular,

Lemma 2 (see [8, 16]). 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).

In the following, derive an Opial-like inequality for the Bregman distance. For original Opial’s inequality, we refer the readers to Lemma 1 of [9].

Lemma 3. Let be a Banach space and let be a strictly convex and Gâteaux differentiable function such that is weakly sequentially continuous. Suppose that is a sequence in such that for some in . Then for all in the interior of with .

Proof. In view of the definition of Bregman distance (see (2)), we obtain Since as and is weakly sequentially continuous, we deduce that Taking into account that for , we obtain that which completes the proof.

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 [19, 20]. 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 lower semicontinuous convex function. Here, saying is proper, we mean that .

The following definition is slightly different from that in Butnariu and Iusem [16].

Definition 4 (see [17]). 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 [16] and Zălinescu [18].

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

The following two results follow from [18, Proposition 3.6.4].

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) 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 [17, 21]). Let be a reflexive Banach space, let be a strongly coercive Bregman function, and let be 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 and convex subset of . Let be a strictly convex and Gâteaux differentiable function. Then, we know from [22] that, for in and in , one has 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 [16] 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 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 .

Lemma 10 (see [23]). 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.

3. Weak Convergence Theorems for Bregman Relatively Nonexpansive Mappings

In this section, we prove weak convergence theorems concerning Bregman relatively nonexpansive mappings in a reflexive Banach space. We propose the following Bregman Mann’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 relatively nonexpansive mapping. Let be a sequence defined by where is an arbitrary sequence in .

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 quasi-nonexpansive mapping with a nonempty fixed point set . Let be a sequence defined by (49) such that is an arbitrary sequence in . Then the following assertions hold: (1) for all in and ;(2) exists for any in .

Proof. Let . In view of (49), we have This implies that is a bounded and nonincreasing sequence for all in . Thus we have that exists for any in .

Theorem 12. 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 relatively nonexpansive mapping. Let be a sequence in satisfying the control condition Let be a sequence generated by the algorithm (49). Then converges weakly to a fixed point of .

Proof. The boundedness of the sequence follows from Lemma 11 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., [16, Proposition 1.1.11] 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 (2), Lemma 8, (32), and (49) that Thus we have Since converges, together with the control condition (60), we have Therefore, from the property of we deduce that Since is uniformly norm-to-norm continuous on bounded subsets of (see, e.g., [18]), we arrive at Since is reflexive, then there exists a subsequence of such that as . Since is a Bregman relatively nonexpansive mapping, we deduce that . We claim that as . If not, then there exists a subsequence of such that converges weakly to some in with . This implies that . By Lemma 11, exists for all in . By the Bregman Opial-like property of , we obtain that This is a contradiction. Thus we have , and the desired assertion follows.

Corollary 13. Let be a reflexive Banach space and let be a strongly coercive Bregman function which is locally bounded, locally uniformly convex, and locally uniformly smooth on . Let be a Bregman relatively nonexpansive mapping. Let be a sequence in satisfying the control condition Let be a sequence generated by where is the right-hand derivative of . Then converges weakly to a fixed point of .

4. Applications (Approximating Zeros of Maximal Monotone Operators)

As an application of our main result, we include a concrete example in support of Theorem 12. Using Theorem 12, we obtain the following strong convergence theorem for maximal monotone operators.

Theorem 14. Let be a reflexive Banach space and a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of . Let be a maximal monotone operator from to such that . Let and be the -resolvent of . Let be an arbitrary sequence in which satisfies the control condition Let be a sequence generated by where is the right-hand derivative of . Then the sequence defined in (63) converges weakly to an element in as .

Proof. Letting , in Theorem 12, from (49), we obtain (63). We need only to show that satisfies all the conditions in Theorem 12. In view of [8, Lemma 3.2], we conclude that is a Bregman relatively nonexpansive mapping. Thus, we obtain where is the set of all strong asymptotic fixed points of . Therefore, in view of Theorem 12, we have the conclusions of Theorem 14. This completes the proof.

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 a grant from NSC.

References

W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. M. Brègman, “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.
View at: Google Scholar | MathSciNet
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178, pp. 313–318, Marcel Dekker, New York, NY, USA, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Naraghirad and J.-C. Yao, “Bregman weak relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2013, article 141, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Bregman monotone optimization algorithms,” SIAM Journal on Control and Optimization, vol. 42, no. 2, pp. 596–636, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
View at: MathSciNet
J. M. Borwein, S. Reich, and S. Sabach, “A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept,” Journal of Nonlinear and Convex Analysis, vol. 12, no. 1, pp. 161–184, 2011.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. Martín-Márquez, S. Reich, and S. Sabach, “Iterative methods for approximating fixed points of Bregman nonexpansive operators,” Discrete and Continuous Dynamical Systems: Series S, vol. 6, no. 4, pp. 1043–1063, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, vol. 40, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
View at: Publisher Site | MathSciNet
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.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, River Edge, NJ, USA, 2002.
View at: Publisher Site | MathSciNet
R. T. Rockafellar, “Characterization of the subdifferentials of convex functions,” Pacific Journal of Mathematics, vol. 17, pp. 497–510, 1966.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacific Journal of Mathematics, vol. 33, pp. 209–216, 1970.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
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, vol. 49, pp. 299–314, Springer, New York, NY, USA, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2014 Chin-Tzong Pang 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.

PDF Download Citation

Download other formats

Order printed copies

Views

1392

Downloads

1180

Citations