Approximating Common Fixed Points of Bregman Weakly Relatively Nonexpansive Mappings in Banach Spaces

Pang, Chin-Tzong; Naraghirad, Eskandar

doi:https://doi.org/10.1155/2014/743279

Journal of Function Spaces

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Research Article | Open Access

Volume 2014 | Article ID 743279 | https://doi.org/10.1155/2014/743279

Approximating Common Fixed Points of Bregman Weakly Relatively Nonexpansive Mappings in Banach Spaces

Chin-Tzong Pang¹and Eskandar Naraghirad^2,3

Academic Editor: Yakov Alber

Received08 Sept 2013

Accepted01 Dec 2013

Published03 Feb 2014

Abstract

Using Bregman functions, we introduce a new hybrid iterative scheme for finding common fixed points of an infinite family of Bregman weakly relatively nonexpansive mappings in Banach spaces. We prove a strong convergence theorem for the sequence produced by the method. No closedness assumption is imposed on a mapping , where is a closed and convex subset of a reflexive Banach space . Furthermore, we apply our method to solve a system of equilibrium problems in reflexive Banach spaces. Some application of our results to the problem of finding a minimizer of a continuously Fréchet differentiable and convex function in a Banach space is presented. Our results improve and generalize many known results in the current literature.

1. Introduction

Throughout this paper, we denote the set of real numbers and the set of positive integers by and , respectively. Let be a Banach space with the norm and the dual space . For any , we denote the value of at by . Let be a sequence in ; we denote the strong convergence of to as by and the weak convergence by . The modulus of convexity of is denoted by for every with . A Banach space is said to be uniformly convex if for every . Let . The norm of is said to be Gâteaux differentiable if, for each , the limit exists. In this case, is called smooth. If the limit (2) is attained uniformly for all , then is called uniformly smooth. The Banach space is said to be strictly convex if whenever and . It is well known that is uniformly convex if and only if is uniformly smooth. It is also known that if is reflexive, then is strictly convex if and only if is smooth; for more details, see [1–3].

Let be a nonempty subset of . Let be a mapping. We denote the set of fixed points of by ; that is, . A mapping is said to be nonexpansive if for all . A mapping is said to be quasinonexpansive if and for all and . The mapping is called closed, if for any sequence with and , then we have . Let be a nonexpansive mapping. Recall that the Mann-type [4] iteration is given by the following formula: Here, is a sequence of real numbers in satisfying some appropriate conditions. A more general iteration scheme is the Halpern [5] iteration given by where the sequences and satisfy some appropriate conditions. Numerous results have been proved on Mann's and Halpern's iterations for nonexpansive mappings in Hilbert and Banach spaces (see, e.g., [6–11]).

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 [12] is defined and denoted by where . Let be a nonempty, closed, and convex subset of smooth Banach space and let be a mapping from into itself. A point is said to be an asymptotic fixed point [13] 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 .

Following Matsushita and Takahashi [14], a mapping is said to be relatively nonexpansive if the following conditions are satisfied:(1) is nonempty;(2), ;(3).

The mapping is called relatively weak quasinonexpansive [15, 16] if the following conditions are satisfied:(1) is nonempty;(2), .

In 2005, Matsushita and Takahashi [14] proved the following strong convergence theorem for relatively nonexpansive mappings in a Banach space.

Theorem 1. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty, closed, and convex subset of , let be a relatively nonexpansive mapping from into itself, and let be a sequence of real numbers such that and . Suppose that is given by Then converges strongly to .

In 2010, Plubtieng and Ungchittrakool [17] proved the following strong convergence theorem for relatively nonexpansive mappings in a Banach space.

Theorem 2. Let be a uniformly convex and uniformly smooth Banach space and let and be two nonempty, closed, and convex subsets of such that . Let be a sequence of relatively nonexpansive mappings from into such that is nonempty and let be a sequence defined as follows: where satisfies either(a) for all and or(b).
Suppose that for any bounded subset of , there exists an increasing, continuous, and convex function such that , and . Let be a mapping from into defined by for all and suppose that . Then , , and converge strongly to .

In 2010, Cai and Hu [15] proved the following strong convergence theorem for a finite family of closed relatively weak quasinonexpansive mappings in a Banach space.

Theorem 3. Let be a nonempty, closed, and let convex subset of uniformly convex and uniformly smooth Banach space and let be a finite family of closed relatively weak quasinonexpansive mappings from into itself with . Assume that is uniformly continuous for all . Let be a sequence generated by the following algorithm Let and be sequences in such that and . Then converge strongly to as .

1.1. Some Facts about Gradients

For any convex function , we denote the domain of by . For any and any , we denote by the right-hand derivative of at in the direction ; that is, The function is said to be Gâteaux differentiable at if exists for any . In this case, coincides with , the value of the gradient of at . The function is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function is said to be Fréchet differentiable at if this limit is attained uniformly in . The function is Fréchet differentiable at (see, e.g., [18, p. 13] or [19, p. 508]) if for all , there exists such that implies 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-weak* continuous (see, e.g., [18, Proposition ]). Also, it is known that if is Fréchet differentiable, then is norm-to-norm continuous (see, [19, p. 508]). The function is said to be strongly coercive if It is also said to be bounded if is bounded for each bounded subset of . Finally, is said to be uniformly Fréchet differentiable on a subset of if the limit (9) is attained uniformly for all and .

1.2. Some Facts about Legendre Functions

Let be a reflexive Banach space. For any proper, lower semicontinuous, and convex function , the conjugate function of is defined by It is well known that for all . It is also known that is equivalent to Here, is the subdifferential of [20, 21]. We also know that if is a proper, lower semicontinuous, and convex function, then is a proper, weak* lower semicontinuous, and convex function; see [2] for more details on convex analysis.

The function is called Legendre if it satisfies the following conditions:(i) is both locally bounded and single-valued on its domain;(ii) is locally bounded on its domain and is strictly convex on every convex subset of dom .

For more details, we refer to [22].

If is a reflexive Banach space and is a Legendre function, then in view of [23];

Examples of Legendre functions are given in [22, 24]. One important and interesting Legendre function is , where the Banach space is smooth and strictly convex and, in particular, a Hilbert space.

1.3. Some Facts about Bregman Distances

Let be a Banach space and let be the dual space of . Let be a convex and Gâteaux differentiable function. Then the Bregman distance [25, 26] corresponding to is the function defined by It is clear that for all . In that case when is a smooth Banach space, setting for all , we obtain that for all and hence for all .

Let be a Banach space and let be a nonempty, convex, and subset of . Let be a convex and Gâteaux differentiable function. Then, we know from [27, 28] that for and , Furthermore, if is a nonempty, closed, and convex subset of a reflexive Banach space and is a strongly coercive Bregman function, then for each , there exists a unique such that The Bregman projection from onto is defined by for all . It is also well known that has the following property [27]: for all and (see [18] for more details). Let be a reflexive Banach space, let be a strongly coercive Bregman function, and let be the Bregman distance corresponding to . Then, is convex and Gâteaux differentiable [29]. Let be the function defined by for , where is the gradient of . We know from [28] that for all . We have from the definition of that In particular, Indeed, there exist such that , and . Therefore,

1.4. Some Facts about Uniformly Convex Functions

Let be a Banach space and let for all . Then a function is said to be uniformly convex on bounded subsets of ([29, pp. 203, 221]) if for all , where is defined by for all . The function is called the gauge of uniform convexity of . The function is also said to be uniformly smooth on bounded subsets of ([29, pp. 207, 221]) if for all , where is defined by for all .

1.5. Some Facts about Resolvents

Let be a Banach space with the norm and the dual space . 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 [30, 31] if whenever . It is also said to be maximal monotone [20] if its graph is not contained in the graph of any other monotone operator on . If is maximal monotone, then we can show that the set is closed and convex. Let be a reflexive Banach space with the dual space and 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 [32]). It is well known that for each (for more details, see, e.g., [1, 33]).

1.6. Some Facts about Bregman Quasinonexpansive Mappings

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 quasinonexpansive, if and Nontrivial examples of such mappings are given in [34].

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 weakly relatively nonexpansive if the following conditions are satisfied:(1) is nonempty;(2), , ;(3).It is clear that any Bregman relatively nonexpansive mapping is a Bregman quasinonexpansive mapping. It is also obvious that every Bregman relatively nonexpansive mapping is a Bregman weakly relatively nonexpansive mapping, but the converse is not true in general. Indeed, for any mapping , we have . If is Bregman relatively nonexpansive, then . It is easy to verify that any closed mapping is a Bregman weakly relatively nonexpansive mapping. To this end, let be a sequence of such that and as . This implies that as . From the closedness of , we conclude that . Below we show that there exists a Bregman weakly relatively nonexpansive mapping which is neither a Bregman relatively nonexpansive mapping nor a closed mapping.

Example 4. 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 Then and is a Bregman weakly relatively nonexpansive mapping which is not a Bregman relatively nonexpansive mapping; see [35] for more details. Now, we prove that is not a closed mapping. Indeed, let for all in . Then as , (since for all ), but .

An example of a Bregman quasinonexpansive mapping which is neither a Bregman relatively nonexpansive mapping nor a Bregman weakly relatively nonexpansive mapping can be found in [35].

In this paper, we investigate the problem of finding zeros of mappings ; that is, find such that Recently, Sabach [36] proved the following two strong convergence theorems for the products of finitely many resolvents of maximal monotone operators in a reflexive Banach space.

Theorem 5. Let be a reflexive Banach space and let , be maximal monotone operators such that . Let be a Legendre function that is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of . Let be a sequence defined by the following iterative algorithm: If, for each , , and the sequences of errors satisfy , then each such sequence converges strongly to as .

Theorem 6. Let be a reflexive Banach space and let , be maximal monotone operators such that . Let be a Legendre function that is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of . Let be a sequence defined by the following iterative algorithm: If, for each , , and the sequences of errors satisfy , then each such sequence converges strongly to as .

The approximation of fixed points of Bregman nonexpansive type mappings via Bregman distances has been studied in the last ten years and much intensively in the last five years. For some recent articles on the existence and the construction of fixed points for Bregman nonexpansive type mappings, we refer the readers to [36–40].

But it is worth mentioning that, in all the above results for Bregman nonexpansive type mappings, the assumption is imposed on the map or the closedness of is required. So, the following question arises naturally in a Banach space setting.

Question 1. Is it possible to obtain strong convergence of modified Mann's type schemes to a common fixed point of an infinite family of Bregman quasinonexpansive mappings without imposing the closedness assumption, the uniformly continuity assumption, or the assumption on the mapping ?

In this paper, using Bregman functions, we introduce a new hybrid iterative scheme for finding common fixed points of an infinite family of Bregman weakly relatively nonexpansive mappings in Banach spaces. We prove a strong convergence theorem for the sequence produced by the method. No closedness assumption is imposed on a mapping , where is a closed and convex subset of a reflexive Banach space . Consequently, the above question is answered in the affirmative in reflexive Banach space setting. Furthermore, we apply our method to solve a system of equilibrium problems in reflexive Banach spaces. Some application of our results to the problem of finding a minimizer of a continuously Fréchet differentiable and convex function in a Banach space is presented. Our results improve and generalize many known results in the current literature; see, for example, [8, 11–14, 41–48].

2. Preliminaries

In this section, we begin by recalling some preliminaries and lemmas which will be used in the sequel.

Definition 7 (see [19]). Let be a Banach space. The 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 and .

The following lemma follows from Butnariu and Iusem [18] and Zălinescu [29].

Lemma 8. Let be a reflexive Banach space and let be 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 and ;(4) is Gâteaux differentiable and .

We know the following two results; see [29, Proposition ].

Theorem 9. Let be a reflexive Banach space and let be a convex function which is bounded on bounded subsets of . Then the following assertions are equivalent:(1) is strongly coercive and uniformly convex on bounded subsets of ;(2) is bounded on bounded subsets and uniformly smooth on bounded subsets of ;(3) is Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of .

Theorem 10. Let be a reflexive Banach space and let be a continuous convex function which is strongly coercive. Then the following assertions are equivalent:(1) is bounded on bounded subsets and uniformly smooth on bounded subsets of ;(2) is Fréchet differentiable and is uniformly norm-to-norm continuous on bounded subsets of ;(3) is strongly coercive and uniformly convex on bounded subsets of .

Let be a Banach space and let be a convex and Gâteaux differentiable function. Then the Bregman distance [48] (see also [25, 26]) satisfies the three point identity that is In particular, it can be easily seen that Indeed, by letting in (39) and taking into account that , we get the desired result.

Lemma 11 (see [39]). Let be a Banach space and let be a Gâteaux differentiable function which is uniformly convex on bounded subsets of . Let and be bounded sequences in . Then the following assertions are equivalent:(1);(2).

The following result was first proved in [49] (see also [19, 38]).

Lemma 12. Let be a reflexive Banach space let be a strongly coercive Bregman function and the function defined by Then the following assertions hold:(1) for all and ;(2) for all and .

The following lemma which is a generalization of Lemma 3.2 in [50] plays a key role in our results.

Lemma 13 (see [17]). Let be a subset of a real Banach space and let be a family of mappings from into . Suppose that for any bounded subset of , there exists a continuous increasing function such that and , where , for all . Then, for each , converges strongly to some point of . Moreover, let the mapping be defined by Then, .

Lemma 14. Let be a Banach space and let be a convex function which is uniformly convex on bounded subsets of . Let be a constant, , , let be the gauge of uniform convexity of , and let be the gauge of uniform convexity of , respectively. Then(i)for any and (ii)for any (iii)if, in addition, is bounded on bounded subsets and uniformly convex on bounded subsets of , then, for any , and (iv)if, in addition, is bounded on bounded subsets, uniformly convex, and uniformly smooth on bounded subsets of , then, for any ,

Proof. In view of (24), we get (i). Let us prove (ii). If and , then we obtain Letting in the above inequality, we arrive at This implies that (iii) Let , , and . Then
(iv) Since is uniformly smooth on bounded subsets of , is uniformly convex on bounded subsets of . Then, in view of (i), there exists a continuous, strictly increasing, and convex function such that for all and all . If , then we obtain Letting in the above inequality, we conclude that This implies that which completes the proof.

Lemma 15 (see [35]). Let be a Banach space, let be a constant, and let be a continuous and convex function which is uniformly convex on bounded subsets of . Then for all , , , and with , where is the gauge of uniform convexity of .

Lemma 16 (see [51]). Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a subsequence such that and the following properties are satisfied by all (sufficiently large) numbers : In fact, .

Lemma 17 (see [52–54]). Let be a sequence of nonnegative real numbers satisfying the inequality where and satisfy the following conditions:(i) and , or equivalently, ;(ii);(ii)'. Then, .

3. Strong Convergence Theorems

In this section, we prove a strong convergence theorem concerning the approximation of fixed point of Bregman weak relatively nonexpansive mappings in a reflexive Banach space. We start with the following simple lemma which has been proved in [33].

Lemma 18. Let be a reflexive Banach space and let be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, closed, and convex subset of . Let be a Bregman quasinonexpansive mapping. Then is closed and convex.

Lemma 19. Let be a reflexive Banach space and let be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let be an infinite family of Bregman quasinonexpansive mappings from into itself such that . Let the mapping be defined by Then, is a Bregman quasinonexpansive mapping.

Proof. Let and be fixed. Then we have that is a bounded sequence in . The function is bounded on bounded subsets of and, thus, is also bounded on bounded subsets of (see, e.g., [18, Proposition ] for more details). This implies that the sequence is bounded in . Since is uniformly norm-to-norm continuous on any bounded subset of , we obtain Thus, is a Bregman quasinonexpansive mapping, which completes the proof.

Theorem 20. Let be a reflexive Banach space and let be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of . Let be a nonempty, closed and convex subset of and let be an infinite family of Bregman weak relatively nonexpansive mappings from into itself such that . Suppose in addition that , where is the identity mapping on . Let be a sequence generated by where is the gradient of . Let be a sequence in such that .
Suppose that for any bounded subset of , there exists an increasing, continuous, and convex function such that , and . Let be a mapping from into defined by for all and suppose that . Then , , and converge strongly to .

Proof. We divide the proof into several steps.
Step 1. We prove that is closed and convex for each .
It is clear that is closed and convex. Let be closed and convex for some . For , we see that is equivalent to It could easily be seen that is closed and convex. Therefore, is closed and convex for each .
Step 2. We claim that for all .
It is obvious that . Assume now that for some . Employing Lemma 12, for any , we obtain This proves that and hence for all .
Step 3. We prove that , , , and are bounded sequences in .
In view of (18), we conclude that This implies that the sequence is bounded and hence there exists such that In view of Lemma 8(3), we conclude that the sequence is bounded. Since is an infinite family of Bregman weak relatively nonexpansive mappings from into itself, we have for any that This, together with Definition 7 and the boundedness of , implies that the sequence is bounded.
Step 4. We show that for some , where .
From Step 3 it follows that is bounded. By the construction of , we conclude that and for any positive integer . This, together with (18), implies that In view of (16), we conclude that It follows from (68) that the sequence is bounded and hence there exists such that In view of (67), we conclude that This proves that is an increasing sequence in and hence by (69) the limit exists. Letting in (67), we deduce that . In view of Lemma 11, we obtain that as . This means that is a Cauchy sequence. Since is a Banach space and is closed and convex, we conclude that there exists such that Now, we show that . In view of (67), we obtain Since , we conclude that This, together with (72), implies that It follows from Lemma 11, (72), and (74) that In view of (71), we get From (71) and (76), it follows that Since is uniformly norm-to-norm continuous on any bounded subset of , we obtain Applying Lemma 11, we derive that It follows from the three point identity (see (39)) that as .
The function is bounded on bounded subsets of and, thus, is also bounded on bounded subsets of (see, e.g., [18, Proposition ] for more details). This implies that the sequences , , and are bounded in .
In view of Theorem 10(3), we know that and is strongly coercive and uniformly convex on bounded subsets. Let and be the gauge of uniform convexity of the conjugate function . We prove that for any , Let us show (81). For any given , in view of the definition of the Bregman distance (see (13)), (9), Lemma 14, we obtain In view of (80), we obtain In view of (81) and (83), we conclude that as . From the assumption , we get Therefore, from the property of we deduce that Since is uniformly norm-to-norm continuous on bounded subsets of , we arrive at From the boundedness of , it follows that there exists a bounded subset of such that . Let for all . In view of Lemma 19, is a Bregman quasinonexpansive mapping. On the other hand, we have Since is an increasing, continuous, and convex function, we have Exploiting Lemma 13 and (87), we obtain By the properties of , we conclude that This, together with Lemma 19 and (71), implies that .
Finally, we show that . From , we conclude that Since for each , we obtain Letting in (93), we deduce that In view of (16), we have , which completes the proof.

Remark 21. Theorem 20 improves Theorems 1, 2, and 3 in the following aspects.(1)For the structure of Banach spaces, we extend the duality mapping to more general case, that is, a convex, continuous, and strongly coercive Bregman function which is bounded on bounded subsets, and uniformly convex and uniformly smooth on bounded subsets.(2)For the mappings, we extend the mapping from a relatively nonexpansive mapping to a countable family of Bregman weak relatively nonexpansive mappings. We remove the assumption on the mapping and extend the result to a countable family of Bregman weak relatively nonexpansive mappings, where is the set of asymptotic fixed points of the mapping .(3)For the algorithm, we remove the set in Theorem 1.(4)Theorem 20 extends and improves Theorem 3.1 in [17]. We note that the proof of Theorem 3.3 (lines 24-25) in [17] is not valid in our discussion.(5)We note also that the main result of the paper cannot be deduced from the results of [35].

We end this section with the following simple example in order to support Theorem 20.

Example 22. Let , and be as in Example 4. We define a countable family of mappings by for all and . It is clear that for all . Choose ; then, for any , If , then we have Therefore, is a Bregman quasinonexpansive mapping. Next, we claim that is a Bregman weak relatively nonexpansive mapping. Indeed, for any sequence such that and as , 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. This implies that for all . It follows from that and hence . Since , we conclude that is a Bregman weak relatively nonexpansive mapping. It is clear that . Thus is a countable family of Bregman weak relatively nonexpansive mappings. Next, we show that is not a countable family of Bregman relatively nonexpansive mappings. In fact, though and as for all . Therefore, for all . This implies that . Let for all . It is easy to see that In view of Example 4, we obtain that is a Bregman weak relatively nonexpansive mapping with . Let be a bounded subset of . Then there exists such that . Let be the gauge of uniform convexity of . Then, in view of Lemma 14, we obtain On the other hand, for any , we have This implies that Furthermore, we have It is clear that, for any , is not continuous. Finally, it is obvious that the family satisfies all the aspects of the hypothesis of Theorem 20.

4. Equilibrium Problems

Let be a nonempty, closed, and convex of a reflexive Banach space . Let be a bifunction. Consider the following equilibrium problem [55]. Find such that For solving the equilibrium problem, let us assume that satisfies the following conditions: for all ; is monotone; that is, for all ;for each , the function is upper semicontinuous;for each , the function is convex and lower semicontinuous.

The set of solutions of problem (104) is denoted by .

Let be a Legendre function. The resolvent of a bifunction [36] is the operator , defined by for all . We also define the mapping in the following way:

Lemma 23 (see [36, 56]). Let be a reflexive Banach space and let be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let be a bifunction satisfying (A1)–(A4) and . Then, the following statements hold:(1);(2) is single-valued;(3) is a Bregman firmly nonexpansive mapping [57]; that is, for all , (4)the set of fixed points of is the solution of the corresponding equilibrium problem; that is, ;(5) is a closed and convex subset of ;(6).

Lemma 24 (see [36]). Let be a reflexive Banach space and let be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be a nonempty, closed, and convex subset of and let be a bifunction satisfying (A1)–(A4) and . Then, the following statements hold:(1);(2) is a maximal monotone operator;(3).

In this section, we propose a new Halpern-type iterative scheme for finding common zeros of an infinite family of maximal monotone operators and prove the following strong convergence theorem in a Banach space.

Theorem 25. Let be a reflexive Banach space and let be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of . For any , let be a maximal monotone operator such that . Let , be sequences in satisfying the following control conditions:(a);(b);(c);(d).
Let be a sequence generated by where is the gradient of and is a constant. Then the sequence defined in (108) converges strongly to as .

Proof. We divide the proof into several steps.
Let . For every , we denote by the resolvent . Therefore,
Step 1. We prove that , and are bounded sequences in . We first show that is bounded. Let be fixed. In view of Lemma 12 and (108), we have This implies that By induction, we obtain for all . It follows from (112) that the sequence is bounded and hence there exists such that In view of Lemma 11(3), we get that the sequence is bounded. Since is an infinite family of Bregman relatively nonexpansive mappings from into itself, we conclude that This, together with Definition 7 and the boundedness of , implies that the sequence is bounded. The function is bounded on bounded subsets of and therefore is also bounded on bounded subsets of (see, e.g., [18, Proposition ] for more details). This, together with Step 1, implies that the sequences , , and are bounded in . In view of Theorem 10(3), we obtain that dom and is strongly coercive and uniformly convex on bounded subsets of . Let and be the gauge of uniform convexity of the conjugate function .
Step 2. We prove that for any Let us show (115). For each , in view of the definition of Bregman distance (see (15)), Lemmas 13, 14, 15, and (110), we obtain It then follows from Lemma 12 and (115) that Let . Then we get from (117) that In view of Lemma 12 and (115), we obtain We will show that as by considering two possible cases on the sequence .
Case 1. If is eventually decreasing, then there exists such that is decreasing and hence is convergent. Thus, we have as . This, together with condition (c) and (118), implies that Therefore, from the property of , we deduce that Since is uniformly norm-to-norm continuous on bounded subsets of , we arrive at Since, for any , is a Bregman relatively nonexpansive mapping, there exists a subsequence of converging weakly to some such that This, together with (13), implies that In view of Lemma 11 and (122), we obtain that This implies that as . Observe also that as . In view of Lemma 11 and (119), (126), and (127), we conclude that Moreover, from (124) and (128), we deduce that Thus we have the desired result by Lemma 17.
Case 2. If is not eventually decreasing, then there exists a subsequence of such that for all . Applying Lemma 16, we can find a nondecreasing sequence such that , for all . This, together with (118), implies that for all . Then, by conditions (a) and (c), we get By the same argument, as in Case 1, we arrive at It follows from (119) that Since , it follows that In particular, since , we obtain In view of (135), we deduce that This, together with (136), implies that On the other hand, we have for all which implies that as . Thus, we have as .

In the following, we propose a new Halpern-type iterative scheme for finding common solutions of a system of equilibrium problems in a reflexive Banach space and obtain a strong convergence theorem.

Theorem 26. Let be a reflexive Banach space and let be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of . Let be an infinite family of nonempty, closed, and convex subsets of . For any , let be a bifunction that satisfies conditions (A1)–(A4) such that , where is the set of solutions to the equilibrium problem (104). Let , be sequences in satisfying the following control conditions:(a);(b);(c);(d).
Let be a sequence generated by where is the gradient of and is a constant. Then, the sequence defined in (140) converges strongly to as .

Remark 27. (1) We propose a new type of Halpern iterative scheme for finding common zeros of an infinite family of maximal monotone operators in a reflexive Banach space. This scheme has an advantage that we do not use any projection which creates some difficulties in a practical calculation of the iterative sequence.
(2) In Theorem 25, we present a strong convergence theorem for an infinite family of maximal monotone operators with a new algorithm and new control conditions. We remove the sets and in Theorems 5 and 6.
(3) Theorem 20 improves and extends the corresponding results of [45, 46] from one maximal monotone operator in Hilbert spaces to more general an infinite family of maximal monotone operators in Banach spaces.
(4) Theorem 20 improves and extends the corresponding result of [47] from two maximal monotone operators in Hilbert spaces to more general an infinite family of maximal monotone operators in Banach spaces.
(5) Theorems 25 and 26 improve and generalize Theorems 5 and 6, respectively.

5. Applications

In this section, we consider the problem of finding a minimizer of a continuously Fréchet differentiable and convex function in a Banach space.

Theorem 28. Let be a reflexive Banach space and let be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of . Let be an infinite family of continuously Fréchet differentiable and convex functions on such that the gradient of , is continuous and monotone for each . Assume that . Let , be sequences in satisfying the following control conditions:(a);(b);(c);(d).Let be a sequence generated by Then the sequence defined in (141) converges strongly to as .

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank the editor and the referees for sincere evaluation and constructive comments, which improved the paper considerably.

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Copyright © 2014 Chin-Tzong Pang and Eskandar Naraghirad. 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.

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