Abstract and Applied Analysis

Volume 2013 (2013), Article ID 590519, 12 pages

http://dx.doi.org/10.1155/2013/590519

Research Article

Bregman Distance and Strong Convergence of Proximal-Type Algorithms

Li-Wei Kuo¹ and D. R. Sahu²

¹Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan

²Department of Mathematics, Banaras Hindu University, Varanasi 221005, India

Received 31 October 2012; Revised 7 March 2013; Accepted 7 March 2013

Academic Editor: Dumitru Motreanu

Copyright © 2013 Li-Wei Kuo and D. R. Sahu. 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 purpose of this paper is to discuss some fundamental properties of Bregman distance, generalized projection operators, firmly nonexpansive mappings, and resolvent operators of set-valued monotone operators corresponding to a functional . We further study some proximal point algorithms for finding zeros of monotone operators and solving generalized mixed equilibrium problems in Banach spaces. Our results improve and extend some recent results concerning generalized projection operators corresponding to Bregman distance.

1. Introduction

In this paper, denotes a real Banach space with norm , and denotes the Banach dual of endowed with the dual norm . We write for the value of a functional in at in . As usual, and stand for the norm and weak convergence of a net to in , respectively.

A continuous strictly increasing function is said to be a gauge if

The mapping defined by

is called the duality mapping with gauge . In the special case where , the duality mapping is the classical normalized duality mapping. In the case , , the duality mapping is called the generalized duality mapping and it is given by

For a gauge , the function defined by

is a continuous convex strictly increasing differentiable function on with and . Therefore, has a continuous inverse function .

We recall the Bregman Distance and function studied in [1]. Let be a real smooth Banach space. The Bregman distance between and in is defined by

One can see from Lemma 3 that . In the case ,  , the distance is called the -Lyapunov functional studied in [2] and it is given by

Note that

is the Lyapunov functional. It is obvious that

See Brègman [3], Butnariu and Iusem [4], and Censor and Lent [5].

Let be a nonempty closed convex subset of a smooth Banach space . The generalized projection is defined by

The metric projection operator defined by

has been employed successfully in optimization, optimal control, approximation theory, and fixed point theory in the framework of Hilbert spaces. In such a framework, metric projections are nonexpansive (i.e., for all in ). However, this is no longer true in the framework of Banach spaces. Instead, the generalized projections are needed. In [6], Alber generalized the metric projection operator to generalized projection operators from Hilbert spaces to uniformly smooth Banach spaces. Many applications of the generalized projections in Banach spaces are discussed in the recent literature (see [7–12]).

Section 2 contains preliminaries. In Section 3, we study the fundamental properties of Bregman distance and -generalized projection operators, where is a proper, convex, lower semicontinuous function. In Section 4, we discuss -firmly nonexpansive mappings and -resolvent operators. In Section 5, we establish strong convergence of the proximal-projection methods for finding fixed points of -firmly nonexpansive mappings, zeros of (not necessarily maximal) monotone operators, and solutions of generalized mixed equilibrium problems in Banach spaces using -generalized projection operators . Here, we do not assume the maximality of monotone operators and the uniform smoothness of Banach spaces.

2. Preliminaries

Let be a set-valued operator. The set is called the effective domain of . The range of is defined by . The operator is said to be monotone if for any in , we have

A monotone operator is said to be maximal if the graph of is not a proper subset of the graph of another monotone operator. We know that if is a maximal monotone operator, then the zero set is closed and convex.

In the rest of this paper, by we always mean a gauge and by the corresponding function defined in (4). We list some properties of the duality mapping below (for more details see [13, 14]).

Proposition 1. Let be a real Banach space.

(i) is norm-to-weak* upper semicontinuous;(ii)for each in , the set is convex and weakly closed in ;(iii) and for all nonzero in ;(iv)there holds

(v) is maximal monotone;(vi)if is strictly convex, then is strictly monotone; that is,

(vii)if is strictly convex and reflexive, then is single-valued monotone and demicontinuous.

The following result is well known. We include a proof for completeness.

Lemma 2. If a Banach space has a uniformly Gâteaux differentiable norm, then is uniformly norm-to-weak* continuous on nonempty bounded subsets of to .

Proof. Suppose not, and there exist norm one vectors in and a constant such that , and , for all . For a fixed , define

Observe that

Hence,

Choose . By the uniform Gâteaux differentiability of the norm, if is large enough, both and are less than , and so . We arrive at a contradiction.

Together with Proposition 1, the conclusion in Lemma 2 also holds for .

Let be a Banach space and a function. The function is proper if . An element in is said to be a subgradient of a proper convex function at a point in if

The set (possibly empty)

of subgradients of at in , is called the subdifferential of at .

Lemma 3. Let be a smooth Banach space and the duality mapping with gauge . Then for ; that is,

The proof of the following result is straightforward.

Lemma 4. Let be a real Banach space and a gauge function.

(a) is a convex and continuous function on .(b) is strictly convex if and only if is strictly convex.

Lemma 5. Let be a real Banach space and a gauge function. Then the following assertions are equivalent:

(a) is uniformly convex;(b) is uniformly convex on the closed ball , where is arbitrarily given. That is, there exists a strictly increasing convex function with such that

  for all in and in .

Proof. First we note that the general case can be reduced to the case . Suppose it holds

We will show that (20) holds for any . Set and for . Then is still a gauge function, and let be the function corresponding to as defined in (4), . Let and and . Then . Applying (21) to , we get

which is exactly (20). A similar argument also shows that the case can be deduced from any other case .

Below, we assume and .

. Given in such that and . Setting in (20), we have

It turns out that

This verifies that is uniformly convex.

. Assume that is uniformly convex, which implies that is strictly convex by Lemma 4(b). Define a function on by setting , and for ,

where is the closed unit ball of .

Claim. Consider for .

Suppose on the contrary that for some . Then we can find sequences , in such that for all , and

Without loss of generality, we may assume that

It then follows from (26) that

The strict convexity of together with (28) implies that if . Since, on the other hand, by definition, . We therefore must have , which together with (28) implies that . If we set , , then for all ; moreover, from (27), we get

This contradicts the uniform convexity of , and verifies that for all .

It turns out from (25) that

for all in , with .

By the dyadic rational argument used in the proof of [15, Theorem 2.2], we can extend the inequality (30) to the case of a general convex combination of and , namely,

for in and in . Note that is increasing and continuous. By [16], the function can also be assumed to be convex (the convexity of is not needed in our argument throughout the rest of this paper however).

Lemma 6. Let be a real uniformly convex Banach space. Then there exists a strictly increasing convex function with such that

Proof. Since is the subdifferential of the functional , we have for in , in that

Let be the function that satisfies (20) with and assume that . Replacing with ,  , we obtain

Taking limit as , we get

Lemma 7 (see [17, Theorem 3.11, page 952]). Let be a reflexive Banach space and the duality map with gauge . Suppose that is maximal monotone. Then the operator is surjective.

3. Bregman Distance and Function

One can easily see that

Noticing that for in , the scalar function is coercive (see [18, Lemma 7.3(v)]).

Proposition 8. Let be a strictly convex smooth Banach space. Let . Then

Proof. See [18, Lemma 7.3(vi)]

Proposition 9. Let be a smooth and uniformly convex Banach space. Then there exists a strictly increasing convex function with such that

Proof. Let . By Lemma 6 we have

As in Butnariu et al. [19], we can prove the following proposition.

Proposition 10. Let be a smooth and uniformly convex Banach space. Let and be two sequences in such that . If is bounded, then .

Proof. Assume is bounded. From definition (5) we have

It follows that

Since is bounded, and as it follows from (41) that is bounded, too.

We may now assume that and both lie in the closed unit ball (otherwise consider the rescaled sequences and for a sufficiently small ). By Proposition 9, there exists a strictly increasing convex function with such that

Since and is strictly increasing, we immediately conclude that .

Proposition 11 (see [20, Lemma 3.1]). Let be a smooth Banach space. Let and be a sequence in such that is bounded. Then is bounded.

The statement in the following proposition is evident from the definition of (cf. [18, Lemma 7.3(ii)]).

Proposition 12. Let be a smooth Banach space. Then, for any fixed in , the scalar function is continuous, weakly lower semicontinuous, and convex on .

Let be a proper, convex, lower semicontinuous function. Define

Some of the following basic properties of the Bregman distance and function are known in the literature (see [18–21]).

The following proposition can be deduced from Butnariu and Kassay [21, Lemma 2.1].

Proposition 13. Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space . Let and let be a proper, convex, lower semicontinuous function with . Then there exists a unique element in such that

Bregman projections are thoroughly studied and used for iteration schemes such as sequential subspace methods or split feasibility problems successfully (see, [22–24]). The notion of -proximal mappings was introduced and studied in [1]. Recently, the notion of Moreau proximal mapping [25] is generalized by Butnariu and Kassay [21] as the proximal mapping relative to associated with a proper, convex, lower semicontinuous function . Using the idea of [1, 26], Proposition 13 allows us to extend generalized projections as follows.

Definition 14. In the setting of Proposition 13, we define the -generalized projection from onto by

In case and , we notice that coincides with . In case that is a Hilbert space and , denote by .

Applying the tools used in [1, 26, 27], we can establish the following results.

Proposition 15. Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space and let be a proper, convex, lower semicontinuous function with .

(i)Let and . Then the following assertions are equivalent:

(a),(b),  .

(ii)Given in , one has

Proposition 16. Let be a nonempty closed convex subset of a smooth and uniformly convex Banach space and a proper, convex, lower semicontinuous function with . Let and let be a bounded sequence in such that . If for all in , then .

Proof. Set . Then, by assumption, we have for all in . By the three-point identity (36), we have

Since is a bounded sequence in , there exists a subsequence of such that converges weakly to some element in . Note that

that is, . Using Proposition 15, we have

which implies that . It follows from Proposition 10 that . This implies that converges strongly to .

4. -Firmly Nonexpansive and -Resolvent Operators

Following [1], we study properties of -firmly nonexpansive mappings in Banach spaces.

Definition 17. Let be a smooth Banach space, a duality mapping with gauge function , and a nonempty subset of . An operator is called -firmly nonexpansive if

In the case of , inequality (50) reduces to

If satisfies condition (51), we call of firmly nonexpansive type. The class of firmly nonexpansive type operators is studied by Kohsaka and Takahashi [28]. When is a Hilbert space, inequality (50) reduces to the following inequality about firmly nonexpansive operators in the classical sense (see Goebel and Kirk [29]):

We now give useful characterizations of -firmly nonexpansive mappings which can be deduced from the Bregman distance (5).

Proposition 18. Let be a smooth Banach space and a nonempty closed convex subset of . Let be a -firmly nonexpansive mapping. Then

The geometry of the fixed point set of -firmly nonexpansive mappings is established in Reich and Sabach [30, Lemma 15.5] as follows.

Proposition 19. Let be a strictly convex smooth Banach space and a nonempty closed convex subset of . Let be a -firmly nonexpansive mapping. Then the set of fixed points of is closed and convex.

Let be a nonempty closed convex subset of a smooth Banach space and let be a mapping. A point in is an asymptotic fixed point of if contains a sequence such that and ; see [31]. We denote the set of asymptotic fixed points of by . A mapping is relatively -nonexpansive if the following conditions are satisfied:

(i) is nonempty; (ii) for all in and in ;(iii).

The class of relatively -nonexpansive mappings is larger than the class of relatively nonexpansive mappings (see [32]).

A mapping is -firmly quasinonexpansive if and

for all in and in . From the definition of the Bregman distance (5) and (54), the following proposition follows immediately.

Proposition 20. Let be a smooth Banach space, a nonempty subset of , and a -firmly quasinonexpansive mapping. Then

for all in and in .

The following supplements Reich and Sabach [30, Lemma 15.6].

Proposition 21. Let be a strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let be a nonempty closed convex subset of and let be a -firmly nonexpansive mapping. Then .

Proof. It is easy to see that . It remains to prove that . For this, suppose that . Then, there exists a sequence in such that and . We need to prove that . Using (53), we get

It is not hard to find that (56) is reduced to the relation

We note that . Indeed, for any norm one linear functional of , we have

Both terms in the right-hand side approach zero. Consequently,

Claim. Consider .

Since and , there is a constant such that all . It follows from the uniform norm to weak* continuity of on bounded subsets (Lemma 2); we have

Observe that

Moreover, . By the uniform continuity of on , we have

The claim is thus verified.

We obtain from (57) and the claim that

This is equivalent to

The strict monotonicity of the duality mapping implies that equality must hold. Namely, or .

The resolvent of an operator relative to a Gâteaux differentiable function is introduced and studied in [1]. We define -resolvent operators following [1, 18].

Definition 22. Let be a nonempty closed convex subset of a smooth Banach space and let be the duality mapping with gauge . Suppose that is an operator satisfying the range condition

For each , the -resolvent associated with operator is the operator defined by

For in and in , we have

If is maximal monotone, then, by Lemma 7, we see that condition (65) holds for .

Remark 23. For smooth and with , we have and . For ,   and this kind of resolvent operators is studied in the literature (see [28, 33]).

Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space . Let be a monotone operator satisfying the condition , where . Using the smoothness and strict convexity of , we obtain that is single-valued. The conditions ensure that is the single-valued -resolvent operator form into . In other words,

Following [18, 28], we have the following proposition.

Proposition 24. Let be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space and let be the duality mapping with gauge . Let be a monotone operator satisfying the condition , where is a positive real number. Let be a resolvent of , where

(a) is -firmly nonexpansive mapping from into ,(b).

5. Convergence Theorems

Let be a nonempty subset of a Banach space and a family of mappings from into with . Let be a family of mappings from into such that . We say the family has property with respect to the family if the following assertion holds:

If and the above condition holds, then we say has property .

Remark 25. If is a singleton, that is, , or for all , then always has property .

We now give some examples.

Example 26. Let be a nonempty closed convex subset of a Banach space and a nonexpansive mapping from into with . Assume in with for all . Define by for all in . Then has property () with respect to the family .

Proof. Let be a bounded net in such that as . Note that

and for all . Therefore, as .

The following example shows that the family of resolvent operators of a maximal monotone operator enjoys property .

Example 27. Let be a nonempty closed convex subset of a real Hilbert space and let be a monotone operator satisfying the following condition:

Let be a bounded net in such that as . Then as for each .

Proof. Let , . By Takahashi [34], we have

Using (72), we have

We now discuss the problem of finding common fixed points of a sequence of -firmly nonexpansive mappings. Our proximal-projection method is based on (a not necessarily Bregman distance) function . The proof is based on the technique in [20].

Theorem 28. Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Let be a gauge function, a nonempty closed convex subset of , and a proper, convex, lower semicontinuous function with . Let be a -firmly nonexpansive mapping and a sequence of -firmly nonexpansive self-mappings on such that and has property () with respect to . For in and with , define a sequence in as follows:

Then converges strongly to .

Proof. We proceed the proof in the following steps:

Step 1. is well defined.

Note that all are closed and convex. For in and in , we obtain from (54) that

It follows that and hence . Therefore, is well defined.

Step 2. is bounded.

Let . It follows from Proposition 15; we have

It follows that is bounded and hence from Proposition 11, we obtain that is bounded.

Step 3. Consider .

Note that . It follows from Proposition 15 that

This implies that

Therefore, the sequence (see (43)) is increasing. Note that is bounded by (76). It follows that exists. By (78), we obtain

One can see that . Using Proposition 10, we obtain that . Since , we have

Using (79), we obtain that

and hence . Note that is bounded. Then, one can see from Proposition 10 that is bounded and

This implies that

Since the family has property () with respect to , it follows from (83) that .

Step 4. The sequence converges strongly to .

Since is bounded, there exists a subsequence of such that . Hence . From Proposition 19, is closed and convex. The nonemptiness of implies that the generalized projection is well defined. Note that and is contained in ; we have

Therefore, we conclude from Proposition 16 that converges strongly to .

Theorem 29. Let be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Let be a gauge function, a nonempty closed convex subset of , and a proper, convex, lower semicontinuous function with . Let be a monotone operator with satisfying the following condition:

Let be a positive real number; for in and with , define a sequence in as follows:

Then converges strongly to .

Proof. Set . Note that is -firmly nonexpansive mapping from into and . Further, every singleton family enjoys property (). Therefore, Theorem 29 follows from Theorem 28.

Remark 30. Compared with other convergence theorems concerning proximal point algorithms in the literature (see, e.g., Agarwal et al. [35, Theorem 3.1]; Kamimura and Takahashi [26, Theorem 8]; Matsushita and Takahashi [32, Theorem 4.3]), Theorem 29 establishes a new proximal point algorithm for the problem of finding zeros of (not necessarily maximal) monotone operators in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm.

We now derive an interesting new result.

Corollary 31. Let be a nonempty closed convex subset of real Hilbert space , and let be a proper, convex, lower semicontinuous function with . Let be a monotone operator satisfying condition (71) such that . Let be a sequence in such that , for in , let with , and define a sequence in as follows:

Then converges strongly to .

Proof. Set for and for all in . Note that is a firmly nonexpansive mapping from into and . From (83), we have . Example 27 implies that has property (). It follows that . Therefore, Corollary 31 follows from Theorem 28.

Let be a nonempty, closed, and convex subset of a Banach space . Let be a bi-function, a nonlinear operator, and a real-valued function. We assume the following conditions are all satisfied.

(A1) for all in .(A2) is monotone; that is, for all in .(A3) for all in , .(A4) for all in , is convex and lower semicontinuous.

Blum and Oettli [36] studied the following equilibrium problem (EP).

Find in such that

The solution set of (88) is denoted by .

Following [37], we have the following lemma.

Lemma 32. Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space , and let be a gauge function. Let be a bi-function satisfying conditions (A1)–(A4), and let and . Then there exists in such that

We now consider the following generalized mixed equilibrium problem (GMEP): find in such that

The solution set of (90) is denoted by . The following auxiliary generalized mixed equilibrium problem is an important tool for finding the solution of GMEP (90).

Let . For a given point in , find in such that

The existence of a solution of the auxiliary mixed equilibrium problem (91) is guaranteed by [37].

Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space and let be a gauge function. Let be continuous and monotone, a bi-function satisfying conditions (A1)–(A4), and a lower semicontinuous and convex function. For , define the mapping as follows:

Lemma 33 (see [37]). One has the following.

(1) is single-valued. (2) is a -firmly nonexpansive mapping; that is, for all in ,

(4).(5) is closed and convex.(6) For all in and in , one has

The following theorem establishes the strong convergence of the proximal-projection method for solving generalized mixed equilibrium problems in the framework of uniformly convex Banach spaces.

Theorem 34. Let be a nonempty closed convex subset of a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Let be a gauge function and let be a proper, convex, lower semicontinuous function with . Let be continuous and monotone, a bi-function satisfying conditions (A1)–(A4), and a lower semicontinuous and convex function. Assume that GMEP. For in ,  , and with , define a sequence in as follows:

Then strongly converges to .

Proof. Note that is a -firmly nonexpansive mapping from into and . Therefore, Theorem 34 follows from Theorem 28.

Acknowledgments

This paper was initiated while D. R. Sahu was visiting the National Sun Yat-Sen University, Kaohsiung, Taiwan as a visiting professor. He would like to thank the Department of Applied Mathematics there for the warm hospitality. Professor Ngai-Ching Wong gave the encouragement and useful comments which were very important for this research. Both authors are also very grateful to the referees for their careful reading and many helpful suggestions. This research is partially supported by Taiwan NSC Grant 99-2115-M-110-007-MY3 and the National Sun Yat-Sen University.

References

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.

T. Bonesky, K. S. Kazimierski, P. Maass, F. Schöpfer, and T. Schuster, “Minimization of Tikhonov functionals in Banach spaces,” Abstract and Applied Analysis, vol. 2008, Article ID 192679, 19 pages, 2008.

L. M. Brègman, “A relaxation method of finding a 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.

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.

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.

Y. I. Alber, “Generalized projection operators in Banach spaces: properties and applications,” in Proceedings of the Israel Seminar, vol. 1 of Functional Differential Equation, pp. 1–21, Ariel, Israel, 1993.

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, Dekker, New York, NY, USA, 1996.

Y. Alber and S. Guerre-Delabriere, “On the projection methods for fixed point problems,” Analysis, vol. 21, no. 1, pp. 17–39, 2001.

Y. I. Alber and A. I. Notik, “On some estimates for projection operators in Banach spaces,” Communications on Applied Nonlinear Analysis, vol. 2, no. 1, pp. 47–55, 1995.

J. Li, “The generalized projection operator on reflexive Banach spaces and its applications,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55–71, 2005.

W. Song and Z. Cao, “The generalized decomposition theorem in Banach spaces and its applications,” Journal of Approximation Theory, vol. 129, no. 2, pp. 167–181, 2004.

Y. Su, H.-K. Xu, and X. Zhang, “Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 12, pp. 3890–3906, 2010.

R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, vol. 6 of Topological Fixed Point Theory and Its Applications, Springer, New York, NY, USA, 2009.

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1990.

T.-C. Lim, H. K. Xu, and Z. B. Xu, “Some inequalities and their applications to fixed point theory and approximation theory,” in Progress in Approximation Theory, pp. 609–624, Academic Press, New York, NY, USA, 1991.

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

J.-P. Penot and R. Ratsimahalo, “On the Yosida approximation of operators,” Proceedings of the Royal Society of Edinburgh Section A, vol. 131, no. 4, pp. 945–966, 2001.

H. H. Bauschke, J. M. Borwein, and P. L. Combettes, “Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces,” Communications in Contemporary Mathematics, vol. 3, no. 4, pp. 615–647, 2001.

D. Butnariu, A. N. Iusem, and E. Resmerita, “Total convexity for powers of the norm in uniformly convex Banach spaces,” Journal of Convex Analysis, vol. 7, no. 2, pp. 319–334, 2000.

S. Reich and S. Sabach, “Two strong convergence theorems for a proximal method in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 31, no. 1–3, pp. 22–44, 2010.

D. Butnariu and G. Kassay, “A proximal-projection method for finding zeros of set-valued operators,” SIAM Journal on Control and Optimization, vol. 47, no. 4, pp. 2096–2136, 2008.

F. Schöpfer, T. Schuster, and A. K. Louis, “Metric and Bregman projections onto affine subspaces and their computation via sequential subspace optimization methods,” Journal of Inverse and Ill-Posed Problems, vol. 16, no. 5, pp. 479–506, 2008.

F. Schöpfer, T. Schuster, and A. K. Louis, “An iterative regularization method for the solution of the split feasibility problem in Banach spaces,” Inverse Problems, vol. 24, no. 5, p. 20, 2008.

F. Schöpfer and T. Schuster, “Fast regularizing sequential subspace optimization in Banach spaces,” Inverse Problems, vol. 25, no. 1, p. 22, 2009.

J.-J. Moreau, “Fonctions convexes duales et points proximaux dans un espace hilbertien,” Comptes Rendus de l'Académie des Sciences, vol. 255, pp. 2897–2899, 1962.

S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.

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.

F. Kohsaka and W. Takahashi, “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces,” SIAM Journal on Optimization, vol. 19, no. 2, pp. 824–835, 2008.

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990.

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. 301–316, Springer, New York, NY, USA, 2011.

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 of Lecture Notes in Pure and Applied Mathematics, pp. 313–318, Dekker, New York, NY, USA, 1996.

S.-Y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

G. Kassay, “The proximal points algorithm for reflexive Banach spaces,” Universitatis Babeş-Bolyai. Studia. Mathematica, vol. 30, pp. 9–7, 1985.

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

R. P. Agarwal, Y. J. Cho, and X. Qin, “Generalized projection algorithms for nonlinear operators,” Numerical Functional Analysis and Optimization, vol. 28, no. 11-12, pp. 1197–1215, 2007.

E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

S.-S. Zhang, “Generalized mixed equilibrium problem in Banach spaces,” Applied Mathematics and Mechanics, vol. 30, no. 9, pp. 1105–1112, 2009.