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Journal of Function Spaces
Volume 2016, Article ID 5973468, 9 pages
http://dx.doi.org/10.1155/2016/5973468
Research Article

Iterative Approximations for Zeros of Sum of Accretive Operators in Banach Spaces

Department of Mathematics, University of Nigeria, Nsukka, Nigeria

Received 27 October 2015; Revised 13 December 2015; Accepted 24 December 2015

Academic Editor: Adrian Petrusel

Copyright © 2016 Yekini Shehu. 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

We study the approximation of zero for sum of accretive operators using a modified Mann type forward-backward splitting algorithm and obtain strong convergence of the sequence generated by our scheme to the zero of sum of accretive operators in uniformly convex real Banach spaces which are also uniformly smooth. Our result is new and complements many recent and important results in this direction in the literature.

1. Introduction and Preliminaries

Let be a real Banach space. The modulus of convexity is defined as is called uniformly convex if for any and -uniformly convex if there is so that for any . The modulus of smoothness is defined by is called uniformly smooth if and -uniformly smooth if there is so that for any . Hilbert spaces, spaces, , and the Sobolev spaces, , , are -uniformly smooth. Hilbert spaces are 2-uniformly smooth whileIt is shown in [1] that there is no Banach space which is -uniformly smooth with . It is obvious that -uniformly smooth Banach space must be uniformly smooth.

The normalized duality mapping is defined byIt is well known that is single-valued and norm-to-norm uniformly continuous on each bounded subsets of if is a real smooth and uniformly convex Banach space (please see [2]). In the sequel, we will denote by the single-valued normalized duality mapping. If is reduced to the Hilbert space , then is the identity mapping.

Let be a mapping on . We will use to denote its set of fixed points; that is, . Furthermore, we say that is a fixed contraction if there exists a constant such that, for all , we have . The mapping is said to be nonexpansive if . Thus, is nonexpansive if .

Let be a nonempty, closed, and convex subset of and let be a mapping of onto . Then is said to be sunny [3] if , for all and . A mapping of into is said to be a retraction [3] if . If a mapping is a retraction, then for every , where is the range of . In [4], it was shown that if is uniformly smooth and if is the fixed point set of a nonexpansive mapping from into itself, then there is a unique sunny nonexpansive retraction from onto . We know that, in a uniformly smooth Banach space, a retraction is sunny and nonexpansive, if and only ifA set-valued operator , with domain and range , is said to be accretive if, for all and every ,Equivalently, by Lemma of Kato [5] we know that is accretive if and only if, for each , there exists such thatFurthermore, an accretive operator is said to be -accretive if the range for all . Given and , we say that an accretive operator is -inverse strongly accretive (-isa) of order if, for each , there exists such thatWhen , we simply say -isa, instead of -isa of order 2; that is, is -isa if, for each , there exists such thatLet be a real uniformly convex Banach space which is also uniformly smooth. Suppose is an -accretive operator and an -inverse strongly accretive. In this paper, we will consider the following inclusion problem (assuming that the solution exists): find such thatProblem (10) is a very general format for certain concrete problems in machine learning, image processing and linear inverse problem, and many nonlinear problems such as convex programming, variational inequalities, split feasibility problem, and minimization problem.

It is well known that a forward-backward splitting method (please see, e.g., [69]) is the classical method for solving problem (10). This forward-backward splitting method is given by andwhere . This method generalizes the proximal point algorithm (please see [1014]) and the gradient method (see, e.g., [15]). There have been many works concerning the problem of finding zero points of the sum of two monotone operators (in Hilbert spaces) and accretive operators (in Banach spaces). For more details, please, see [1620] and the references contained therein. Motivated by the results of López et al. [21], Cholamjiak [22] and Wei and Shi [23] have recently extended the results of López et al. [21] from Halpern-type forward-backward method to viscosity-type forward-backward method with fixed contraction in -uniformly smooth Banach spaces which are also uniformly convex.

Remark 1. We note that for the class of inverse strongly accretive of order coincides with that of inverse strongly accretive. Thus, in the case of , (8) is not equivalent to (9). Furthermore, for , inverse strongly accretive operators of order   do represent a subclass of inverse strongly accretive operators (see [24]). In light of above comments, the class of inverse strongly accretive operators of order considered in the results of Cholamjiak [22], López et al. [21], and Wei and Shi [23] is a subclass of inverse strongly accretive operators we will consider in this paper.
Recently, Wei and Duan [25] studied the problem of finding zeros of the sum of finitely many -accretive operators and finitely many -inversely strongly accretive operators in a real smooth and uniformly convex Banach space (i.e., find such that ). They proved the following strong convergence result.

Theorem 2. Let be a real smooth and uniformly convex Banach space and let be a nonempty, closed, and convex sunny nonexpansive retraction of , and let be the sunny nonexpansive retraction of onto . Let be a fixed contractive mapping with coefficient ,  and let   be a strongly positive linear bounded operator with coefficient . Suppose that the duality mapping is weakly sequentially continuous at zero, and . Let be -accretive operator and let be -inversely strongly accretive operator, where . Suppose that, for and ,Let be generated by, for . Suppose , and are three sequences in and satisfying the following conditions:(i), as ;(ii);(iii) and , for and ;(iv).If  , then converges strongly to a point , which is the unique solution of the following variational inequality: for all ,

We observe that Theorem 2 extends the results of López et al. [21] from -uniformly smooth Banach spaces which are also uniformly convex to uniformly smooth and uniformly convex Banach spaces under the condition that the normalized duality mapping is weakly sequentially continuous at zero. We also observe that this result is not applicable in , . This leads to this natural question.

Question 1. Can we obtain strong convergence result for approximation of zero of sum of accretive operators in uniformly smooth Banach spaces which are also uniformly convex without assuming that the normalized duality mapping is weakly sequentially continuous at zero or ?

Our interest in this paper is to answer the above question in the affirmative. That is, we establish strong convergence result for approximation of zero of sum of accretive operators in uniformly smooth Banach spaces which are also uniformly convex without assuming that the normalized duality mapping is weakly sequentially continuous at zero or . To achieve this, apply generalized forward-backward method which involves viscosity approximation method with fixed contraction and weakly contractive mappings.

In the sequel, we will assume that is an -accretive operator. Then we can define, for each , the resolvent by , and the Yosida approximation by . Denote by the zero set of ; that is, . It is well known that is single-valued and nonexpansive, and for every , where .

In what follows we will make use of the following lemmas.

Lemma 3 (see Maingé [26]). Let be a real sequence that does not decrease at infinity, in the sense that there exists a subsequence so that for all .
For every define an integer sequence asThen as and for all

Lemma 4 (see Xu [27]). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that(i)(ii) or Then .

Lemma 5 (see Xu [28]). Let be a nonempty closed convex subset of a uniformly smooth Banach space and let be a nonexpansive mapping with a fixed point. Let be a fixed contraction with coefficient . If there exists a bounded sequence such that and exists, where is defined by and solves the variational inequalitythen

Lemma 6. Let be a real Banach space. Then, for all , one has

Lemma 7 (see López et al. [21]). Let be a real Banach space. Let be an -accretive operator and let be an -inverse strongly accretive mapping on . Then one has(i)for , ,(ii)for and .

Lemma 8 (see Suantai and Cholamjiak [29]). Let be a real Banach space with Fréchet differentiable norm. For , let be defined for byThen, andfor all .

Remark 9. In a real Hilbert space, we see that for .

In the result of Suantai and Cholamjiak [29], the authors assumed that for . This naturally leads to this important question.

Question 2. What uniformly smooth Banach spaces (except Hilbert spaces) satisfy the assumption for ? In particular, do spaces, , satisfy it?

In , , we know thatThen in (21) is estimated by for .

In our more general setting, throughout this paper, we will assume that where is the function appearing in (21).

By following the same method of proof as contained in the proof of Lemma of [21], we have the following lemma.

Lemma 10. Let be a real uniformly convex with Fréchet differentiable norm. Assume that is a single-valued -inverse strongly accretive mapping on . Then, given , there exists a continuous, strictly increasing and convex function with such that, for all ,where .

We will adopt the following notation in this paper: means that strongly.

2. Main Results

We first prove the following lemma in a real Banach space with Fréchet differentiable norm.

Lemma 11. Let be a real Banach space with Fréchet differentiable norm. Let be an -accretive operator and let be an -inverse strongly accretive mapping. Assume that . Let be a fixed contraction with coefficient . Let be a sequence of positive real numbers and suppose that and are sequences in . Let be a sequence generated by ,where , , . Then is bounded.

Proof. For each , let and . Then, for all , we haveThus, is nonexpansive for all . Furthermore,Thus, , . So,Now, using (29), we haveHence, is bounded.

Using the idea of the proof partly taken from [18, 26, 30, 31], we state and prove the following strong convergence theorem.

Theorem 12. Let be a real uniformly convex Banach space which is also uniformly smooth. Let be an -accretive operator and let be an -inverse strongly accretive. Assume that . Let be a fixed contraction with coefficient . Let be a sequence of positive real numbers and suppose that and are sequences in satisfying the following conditions:(i);(ii);(iii);(iv).Let the sequence be generated by (26). Then converges strongly to , where is the unique sunny nonexpansive retraction of onto ; that is, solves the variational inequality

Proof. Let . Then, using Lemma 6 in (26), we have thatIt follows from (32) and Lemma 10 that there exists a continuous, strictly increasing, and convex function with such thatNow, using (26), we haveLet , , and . Then, (34) becomesWe now show that , by considering two possible cases.
Case 1. Suppose that is eventually decreasing. That is, there exists such that is decreasing for . This implies that is convergent. It then follows from (35) thatThis implies thatConsequently,Since , there exists such that , Then, by Lemma 7, we haveAlso, from (26), we obtainSince , it follows from Lemma 5 thatEquivalently, . We observe further thatAgain from (26), we have that Hence,From (36), we have thatObserve that since . Taking in (45), we haveand thus must converge to zero.
Case 2. Suppose is not eventually decreasing. Hence, we can find a subsequence of such that , . Then we can define a subsequence as in Lemma 3 so thatThis implies from (36) thatand thus , . In a similar way to Case , we haveEquivalently, which by (48) implies that . By repeating the same arguments in Case , we can show thatUsing the definition of , we haveThis together with (47) implies thatConsequently, we have that , and so , .

If the mapping maps every point in to a fixed element, then we have the following result.

Corollary 13. Let be a real uniformly convex Banach space which is also uniformly smooth. Let be an -accretive operator and let be an -inverse strongly accretive. Assume that . Let be a fixed element in . Let be a sequence of positive real numbers and suppose that and are sequences in satisfying the following conditions:(i);(ii);(iii);(iv).For a fixed element , let the sequence be generated by ,where . Then converges strongly to , where is the unique sunny nonexpansive retraction of onto ; that is, solves the variational inequality

Recall that a mapping is said to be weakly contractive ifwhere is a continuous and strictly increasing function such that is positive on and . Let for , where . Then we see that the weakly contractive mapping is a contraction with constant . The next lemma gives existence of unique fixed point for weakly contractive mapping.

Lemma 14 (see [32, Theorem ]). Let be a complete metric space and a weakly contractive mapping on . Then, has a unique fixed point in . Moreover, for converges strongly to .

The following lemma will be used in the next theorem.

Lemma 15 (see [33, 34]). Let and be two sequences of nonnegative real numbers and a sequence of positive numbers satisfying the following conditions:(i);(ii).Let the recursive inequality where is a continuous and strictly increasing function such that is positive on and . Then, .

We now consider the viscosity approximation method with weakly contractive mapping in the following theorem.

Theorem 16. Let be a real uniformly convex Banach space which is also uniformly smooth. Let be an -accretive operator and let be an -inverse strongly accretive. Assume that . Let be a weakly contractive mapping. Let be a sequence of positive real numbers and suppose that and are sequences in satisfying the following conditions:(i);(ii);(iii);(iv).Let be a sequence generated by ,Then converges strongly to , where is the unique sunny nonexpansive retraction of onto ; that is, solves the variational inequality

Proof. We first observe that is a weakly contractive mapping. Indeed, ,Then Lemma 14 assures that there exists a unique element such that . Now, define an iterative scheme as follows:Let be the sequence generated by (60). Then, Corollary 13 with a constant assures that converges strongly to . Now, for , we obtainTake ; we have the following recursive inequality:Since , it follows from Lemma 15 that , . Furthermore,This completes the proof.

Remark 17. All the results in this paper can still be obtained if error terms are added to our iterative scheme (26). Thus, addition of error terms to our iterative scheme (26) leads to no further generalization of the results obtained in this paper.

Remark 18. We make the following comments which highlight our contributions in this paper (please see also [31]): (1)The results of Wei and Shi [23] do not hold in spaces with . Here in this paper, our results hold, in particular, in spaces with . Furthermore, the condition assumed in the results of Wei and Shi [23] is dispensed with in our results. Thus, our results improve on the results of Wei and Shi [23].(2)The condition assumed in the results of Wei and Duan [25] is dispensed with in our results. Also, the conditionfor all and for all assumed in the results of Wei and Duan [25], is dispensed with in our results. Hence, our results improved on the results of Wei and Duan [25].(3)Our results extend the results of [2123, 25] from -uniformly smooth Banach spaces to uniformly smooth Banach spaces. By Remark 1, our results also extend the results of [2123, 25] from the class of inverse strongly accretive operators of order to the more general class of inverse strongly accretive operators.(4)In our result in this paper, we consider the viscosity approximation method with weakly contractive mapping for solving (10), which extends all the previous results concerning viscosity approximation method with fixed contraction for solving (10) in -uniformly smooth Banach spaces to uniformly smooth Banach spaces. Our result in this paper also complements the results of Shehu and Cai [31].

Remark 19. We specifically clarify the differences between the results of this paper and the results obtained in [23, 25]:(1)We state here that the results obtained in Wei and Duan [25] and Wei and Shi [23] do not hold in spaces with because the duality mapping is assumed to be weakly sequentially continuous but here in this paper, our results are applicable, in particular, in spaces with because the weak sequential continuity of the duality mapping is dispensed with.(2)The choice of sequence of positive real numbers withdoes not satisfy the extra condition assumed in Wei and Duan [25] and Wei and Shi [23].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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