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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 681348, 19 pages
http://dx.doi.org/10.1155/2012/681348
Research Article

Strong Convergence Theorems for Zeros of Bounded Maximal Monotone Nonlinear Operators

1Mathematics Institute, African University of Science and Technology, Abuja, Nigeria
2Department of Mathematics, Gaston Berger University, Saint Louis, Senegal

Received 20 November 2011; Accepted 19 January 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 C. E. Chidume and N. Djitté. 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

An iteration process studied by Chidume and Zegeye 2002 is proved to converge strongly to a solution of the equation where A is a bounded m-accretive operator on certain real Banach spaces E that include spaces The iteration process does not involve the computation of the resolvent at any step of the process and does not involve the projection of an initial vector onto the intersection of two convex subsets of E, setbacks associated with the classical proximal point algorithm of Martinet 1970, Rockafellar 1976 and its modifications by various authors for approximating of a solution of this equation. The ideas of the iteration process are applied to approximate fixed points of uniformly continuous pseudocontractive maps.

1. Introduction

Consider the following problem: where is a real Hilbert space and is a maximal monotone operator (defined below) on . One of the classical algorithms for approximating a solution of (1.1), assuming existence, is the so-called proximal point algorithm introduced by Martinet [1] and studied further by Rockafellar [2] and a host of other authors. Specifically, given , an approximation of a solution of (1.1), the proximal point algorithm generates the next iterate by solving the following equation: where is a regularizing parameter. If the sequence is bounded from above, then the resulting sequence of proximal point iterates converges weakly to a solution of (1.1), provided that a solution exists (Rockafellar [2]).

Rockafellar then posed the following question.(Q1) Does the proximal point algorithm always converge strongly?

This question was resolved in the negative by Güler [3] who produced a proper closed convex function in the infinite-dimensional Hilbert space for which the proximal point algorithm converges weakly but not strongly. This naturally raises the following question.(Q2) Can the proximal point algorithm be modified to guarantee strong convergence?

Before we comment on this question, we make the following observation. The proximal point algorithm (1.2) is not at all convenient to use in any possible application. This is because at each step of the iteration process, one has to compute and this is certainly not convenient. Consequently, while thinking of modifications of the proximal point algorithm that will guarantee strong convergence, the following question is, perhaps, more important than Q2.(Q3)Can an iteration process be developed which will not involve the computation of at each step of the iteration process and which will guarantee strong convergence to a solution of (1.1)?

With respect to Q2, Solodov and Svaiter [4] were the first to propose a modification of the proximal point algorithm, which guarantees strong convergence in a real Hilbert space. Their algorithm is as follows.

Algorithm. Choose any and . At iteration , having , choose , and find an inexact solution of , with tolerance . Define Take The authors themselves noted ([4], p.195) that “ at each iteration, there are two subproblems to be solved ”: (i) find an inexact solution of the proximal point algorithm, and (ii) find the projection of onto , the intersection of the two halfspaces. They also acknowledged that these two subproblems constitute a serious drawback in their algorithm. This method of Solodov and Svaiter is the so-called -method which has been studied by various authors.

For more on the computation of an approximate solution and the projection step, both of which must be performed at each step, the reader may consult [4].

Kamimura and Takahashi [5], extended the work of Solodov and Svaiter [4] to the framework of Banach spaces that are both uniformly convex and uniformly smooth.

Xu [6] noted that “ Solodov and Svaiter’s algorithm, though strongly convergent, does need more computing time due to the projection in the second subproblem ”.

He then proposed another modification of the proximal point algorithm, which does not increase computing time by much compared to the algorithm of Solodov and Svaiter.

Xu [7] proposed and studied the following algorithm: He proved that (1.5) converges strongly provided that the sequences and of real numbers and the sequence of errors are chosen appropriately. He argued that once has been calculated, the calculation of the mean is much easier than the projection of onto mentioned earlier, and so his algorithm seems simpler than that of Solodov and Svaiter [4]. But the algorithm (1.5) of Xu still has the serious setback associated with the classical proximal point algorithm: the computation of at each step of the iteration process.

Lehdili and Moudafi [8] considered the technique of the proximal map and the Tikhonov regularization to introduce the so-called Prox-Tikhonov method which generates the sequence by the following algorithm: where , is viewed as a Tikhonov regularization of and . Using the notation of variational distance, Lehdili and Moudafi [8] proved convergence theorems for the algorithm (1.6) and its perturbed version, under appropriate conditions on the sequences and .

It is known that Tikhonov regularization is not generally effective if an appropriate regularization parameter is not chosen, especially for ill-posed problems. For example, in order to use the discrepancy principle, it is necessary to have information about the noise. Also, in the case of generalized cross validation, efficient implementation for Tikhonov regularization requires computing the singular value decomposition of the matrix, which, for large-scale problems, may be formidable.

Xu [6] studied the algorithm (1.6). He used the technique of nonexpansive mappings to get convergence theorems for the perturbed version of the algorithm (1.6), under much relaxed conditions on the sequences and . Here again, the algorithm (1.6) has the drawback of the classical proximal point algorithm: has to be computed at each step.

Another modification of the proximal point algorithm, perhaps the most significant, which yields strong convergence, is implicitly contained in the following theorem of Reich.

Theorem 1.1 (Reich [9]). Let be a -uniformly smooth real Banach space. Let be accretive with , and suppose that satisfies the range condition: . Let , be the resolvent of , and assume that is nonempty. Then for each , .

We first make the following observations about this theorem.(i)The -uniformly smooth real Banach spaces include the spaces, . In particular, all Hilbert spaces and spaces, , are 2-uniformly smooth.(ii)Any -accretive operator satisfies the range condition. The converse is not necessarily true. Hence, range condition is weaker than -accretive.

Now, let be a sequence in such that as . Set . Then, using the notation of Theorem 1.1, define By the theorem, for arbitrary , exists, call it say, and . We can now define the sequence as follows: as a modification of the proximal point algorithm, which yields strong convergence to a solution of the equation , assuming existence. Clearly, (1.8) is easier to use and requires less computation time than any of (1.2), (1.5), or (1.6), or using the -method of Solodov and Svaiter.

We have seen, in response to Q2, that all modifications of the classical proximal point algorithm to obtain strong convergence so far studied inherited the drawback of the algorithm: the computation of at each step of the process.

We remark, however, that the proximal point algorithm can be still useful in some special cases. For example, the algorithm was recently successfully used in signal processing and in image restoration where the proximal mappings are fairly evaluated.

We now turn our attention to the consideration of the more important question Q3. It is our purpose in this paper to prove that an iteration process studied by Chidume and Zegeye [10] converges strongly to a solution of the equation where is a bounded -accretive (defined below) operator even on certain real Banach spaces much more general than Hilbert spaces. The iteration process will not involve the computation of at any stage and will not involve the computation of two convex subsets at each step and the projection of an initial vector to their intersection.

Before we prove our convergence theorems, we need the following definitions and preliminaries.

Let be a real normed linear space with dual . For , we denote by the generalized duality mapping from to defined by where denotes the generalized duality pairing. is denoted by and is the so-called normalized duality map. If is strictly convex, then is single-valued (see, e.g., Xu [11]). A mapping with domain and range in is said to be strongly -accretive if, for any , there exist and a strictly increasing function with such that The mapping is called generalized -accretive if, for any , there exist and a strictly increasing function with such that It is well known that the class of generalized -accretive mappings includes the class of strongly -accretive operators as a special case (one sets for all ).

Let . The mapping is called strongly quasi-accretive if for all , there exists such that is called strongly -quasi-accretive if, for all , there exists such that Finally, is called generalized -quasi-accretive if, for all , there exists such that It is well known that the class of generalized -quasi-accretive mappings is the largest class (among those defined above) for which the equation has a unique solution.

A mapping with domain and range in is called accretive if and only if for all , the following inequality is satisfied: As a consequence of a result of Kato [12], it follows that is accretive if and only if for each , there exists such that It follows from inequality (1.15) that is accretive if and only if is expansive, and consequently its inverse exists and is nonexpansive (i.e., as a mapping from into , where denotes the range of . The range of needs not be the whole of . This leads to the following definitions. A mapping is said to be -accretive if is accretive and the range of is all of for some . The operator is said to satisfy the range condition if for all . It can be shown that if for some , then it holds for all . We prove this important fact, which will be used in the sequel in Lemma 2.5. Hence, -accretive condition implies range condition. In Hilbert space, accretive operators are called monotone.

Let be a real Banach space and a map. Assume that the equation has a solution. Iterative methods for approximating such a solution have been of interest to numerous researchers in nonlinear operator theory. Bruck [13] considered an iteration process of the Mann-type and proved that the sequence of the process converges strongly to a solution of the inclusion in a real Hilbert space where is a maximal monotone map, provided the initial vector is taken in a neighbourhood of a solution of this inclusion. Chidume [14] extended this result to spaces, . These results of Bruck and Chidume are not convenient in any possible application because the neighbourhood of a solution in which the initial vector must be chosen is not known precisely. Other early results involved mappings that are Lipschitz and strongly accretive (see, e.g., Browder and Petryshyn [15, 16], Chidume [17, 18], Chidume and Moore [19, 20], Deng [21, 22], Zhou [23], Zhou and Jia [24], Weng [25], Xu and Yin [26], Xu and Roach [27] and a host of other authors).

Numerous papers were later published, extending these results to the class of mappings that are Lipschitz and strongly -accretive (see, e.g., Chang et. al [28], Zhou [23] and the references contained therein). Recently, some authors have proved convergence theorems for the solution of , where is assumed to be a Lipschitz and generalized -accretive map.

A few papers have recently been published establishing convergence theorems for the solution of , where is a uniformly continuous and generalized -quasi-accretive map (see, e.g., Gu [29], Chang et al. [30], and the references contained in them). Related results on general variational inequalities can be found in the paper of Aslam Noor [31].

Recall that the class of generalized -quasi accretive maps is the largest class for which the equation has a unique solution. Most papers have been devoted to this. It is well known that for the general accretive operator , the solution of this equation, whenever it exists, is generally not unique. In this case, the technique used in approximating a solution of the equation when it exists and is unique does not carry over to the case when it exists and is not unique.

The accretive mappings were introduced independently in 1967 by Browder [32] and Kato [12]. Interest in such mappings stems mainly from their firm connection with the existence theory for nonlinear equations of evolution in real Banach spaces. It is known (see, e.g., Zeidler [33]) that many physically significant problems can be modelled in terms of an initial-value problem of the form where is an accretive map on an appropriate real Banach space. Typical examples of such evolution equations are found in models involving the heat, wave, or Schrödinger equations (see [34]). Observe that in the model (1.17), if the solution is independent of time (i.e., at the equilibrium state of the system), then and (1.17) reduces to whose solutions then correspond to the equilibrium state of the system described by (1.17). To approximate a solution of (1.18), Browder converted (1.18) to a fixed point problem as follows. He called an operator pseudocontractive if is accretive and is the identity map on the space. Then, is a solution of (1.18) if and only if it is a fixed point of . Consequently, a lot of effort has been devoted to approximating fixed points of pseudocontractive maps.

Let be a nonempty subset of . We reiterate that a mapping is called pseudocontractive if is accretive. Consequently, solutions of (when they exist) for accretive operators correspond to fixed points of pseudocontractions . Every nonexpansive map is Lipschitz pseudocontractive.

In the late 1960s and early 1970s, the well-known Mann iteration process [35] was successfully employed to approximate fixed points of nonexpansive maps under suitable assumptions. All attempts to apply it to approximate fixed points of Lipschitz pseudocontractions proved abortive. In 1974, Ishikawa [36] introduced an iteration scheme involving two parameters and for approximating a fixed point of a Lipschitz pseudocontraction defined on a compact convex subset of a real Hilbert space. He proved strong convergence of the sequence generated by his scheme. An example of and satisfying his condition is (see, e.g., Berinde [37], Chidume [38], Ishikawa [36]). It is still an open question whether or not this theorem of Ishikawa can be extended to real Banach spaces more general than Hilbert spaces. Since 1974, three other iteration methods have been introduced and studied and have been succesfully employed to approximate fixed points of Lipschitz pseudocontractive mappings in certain Banach spaces more general than Hilbert spaces.

One of these three iteration processes was introduced and studied by Schu [39]. The recursion formula studied involved the use of two real sequences which are required to have the so-called Property (A).

Let for all nonnegative integers . Then, ( is said to have Property (A) if and only if the following conditions hold:(i) is decreasing and is strictly increasing;(ii)there is a sequence , strictly increasing such that(a)(b), (c).

Schu proved his convergence theorem in real Hilbert spaces. Chidume [40] extended it to real Banach spaces possessing weakly sequentially continuous duality maps (e.g., spaces, ). However, it is known that spaces, , do not possess this property.

Another iteration scheme for approximating fixed points of Lipschitz pseudocontractive mappings was implicitly introduced by Bruck [13], who actually applied the scheme, still in real Hilbert spaces, to approximate a solution of the inclusion , where is an - accretive operator. The recursion formula studied by Bruck involved two real sequences and which are required to be acceptably paired.

Two real sequences and are called acceptably paired if they satisfy the following conditions: is decreasing, , and there exists a strictly increasing sequence of positive integers such that(i),(ii),(iii).

An example of such sequences given in Bruck [13] is The idea of sequences with Property (A) or acceptably paired are due to Halpern [41]. Reich [9, 42] also studied the recursion formula studied by Bruck for Lipschitz accretive operators on real uniformly convex Banach spaces with a duality map that is weakly sequentially continuous at zero, but with and not necessarely being acceptably paired.

Motivated by the papers of Reich [9, 42], Chidume and Zegeye [10] introduced and studied a perturbation of the Mann recurrence relation (see Theorem CZ below) to approximate zeros of Lipschitz accretive maps in real Banach spaces much more general than real Hilbert spaces. They proved the following theorem.

Theorem CZ (Chidume and Zegeye [10]). Let be a reflexive real Banach space with a uniformly Gâteaux differentiable norm. Let be a Lipschitz accretive operator and let . Suppose that every nonempty closed convex and bounded subset of has the fixed point property for nonexpansive self-mappings. Let a sequence be generated from arbitrary by where and are sequences in satisfying the following conditions:(1),(2),(3). Then, converges strongly to a solution of .

An example of sequences and satisfying Theorem CZ is , , and .

It is clear that these parameters are much simpler than the requirement that they have Property (A) or are acceptably paired.

It is our purpose in this paper to provide affirmative answer to Q3 when is bounded and satisfies the range condition. We show that the iteration process (1.20) converges strongly even in real Banach spaces much more general than Hilbert spaces to a solution of the equation (assuming existence) for arbitrary initial vector . These spaces include spaces, . Moreover, it is clear that the recursion formula (1.20) is simpler than that of the proximal point algorithm and involves direct applications of . Furthermore, the regularization parameters are easily chosen at the begining of the iteration process. Computation of is not required at any stage, and computation of two convex sets and projection of onto their intersection will not be required. Finaly, we derive some applications of our theorems to approximate fixed points of uniformly continuous pseudocontractions. We achieve this by means of an incisive result recently proved by C. E. Chidume and C. O. Chidume [43].

2. Preliminaries

Let be a real normed linear space of dimension . The modulus of smoothness of is defined by If there exist a constant and a real number , such that , then is said to be -uniformly smooth. Typical examples of such spaces are the , , and spaces for , where In the sequel will we will need the following results.

Theorem 2.1 (Xu [11]). Let and be a real Banach space. Then the following are equivalent.(i) is -uniformly smooth.(ii)There exists a constant such that for all ,

For the remainder of this paper, and will denote the constants appearing in Theorem 2.1.

Lemma 2.2. Let be a real normed linear space. Then, the following inequality holds:

Lemma 2.3 (Xu [7]). Let be a sequence of nonnegative real numbers satisfying the following relations: where (i) ; (ii) ; (iii) . Then, as .

Lemma 2.4 (C. E. Chidume and C. O. Chidume [43]). Let and be real normed linear spaces and let be a uniformly continuous map. For arbitrary and fixed , let Then is bounded.

Since this result is yet to appear, we reproduce its short proof here.

Proof. By the uniform continuity of and by taking , there exists such that for all , For , let be arbitrary. Choose fixed such that . Set Then, By the uniform continuity of , . Furthermore, Hence, is bounded.

We now prove the following lemma, which will be used in the sequel.

Lemma 2.5. For , let be a -uniformly smooth real Banach space and be a map with . Suppose that is -accretive, that is, (i) such that . (ii) for some . Then satisfies the range condition, that is, for all .

Proof. Assume that there exists some such that . It is known that since is accretive, the map is invertible and, moreover, is nonexpansive, that is,
Claim. for any . Indeed, let and , we solve the equation We observe that solves (2.12) if and only if or equivalently Observing that , it follows from the Banach fixed point theorem that (2.14) has a unique solution. This proves the claim.Since is -accretive, for some . Using the claim, it follows that for any . By induction, we have that for any and for any . So, the conclusions follows.

3. Approximation of Zeros of Bounded -Accretive Operators

We now prove the following theorem.

Theorem 3.1. Let be a 2-uniformly smooth real Banach space, and let be a bounded -accretive map. For arbitrary , define the sequence iteratively by where and are sequences in satisfying the following conditions:(1), and is decreasing;(2);(3). Suppose that the equation has a solution. Then, there exists a constant such that if , converges strongly to a solution of the equation .

Proof. Let be a solution of the equation . There exists sufficiently large such that . Define . Since is bounded, it follows that is bounded. So, Let where in Theorem 2.1 with .Step 1. We prove that is bounded. Indeed, it suffices to show that is in for all . The proof is by induction. By construction, . Suppose that for some . We prove that .
Using inequality (2.3) of Theorem 2.1 with , we have Using the fact that is accretive, we obtain Therefore, we have the following estimates: Thus, So, using the induction assumption, the fact that and the condition , we obtain Therefore, . Thus by induction, is bounded.Step 2. We prove that converges to a solution of . Since is -accretive, using Lemma 2.5 and Theorem 1.1, there exists a unique sequence in satisfying the following properties: Indeed, applying Theorem 1.1, with , then the sequence , with has the desired properties.Claim. as . Since and are bounded and is bounded, there exists some positive constant such that Using (23) and the fact that is accretive, we have Therefore, Using again the fact that is accretive, we obtain From (i), observing that it follows that By Lemma 2.2, we have Using Schwartz’s inequality, we obtain: Using (3.12), (3.15), (3.17), and the fact that and are bounded, we have, for some positive constant . Thus, by Lemma 2.3, . Using (ii), it follows that and . This completes the proof.

Corollary 3.2. Let be a 2-uniformly smooth real Banach space, and let be a uniformly continuous -accretive map. For arbitrary , define the sequence iteratively by where and are sequences in satisfying the following conditions:(1) and is decreasing;(2);(3). Suppose that the equation has a solution. Then, there exists a constant such that if , converges strongly to a solution of the equation .

Proof. Since is uniformly continuous, then, by Lemma 2.4, is bounded. So the result follows from Theorem 3.1.

Corollary 3.3. Let be real Hilbert space, and let be a bounded maximal monotone operator. For arbitrary , define the sequence iteratively by where and are sequences in satisfying the following conditions:(1) and is decreasing;(2);(3). Suppose that the equation has a solution. Then, there exists a constant such that if , converges strongly to a solution of the equation .

Corollary 3.4. Let be real Hilbert space, and let be a uniformly continuous maximal monotone operator. For arbitrary , define the sequence iteratively by where and are sequences in satisfying the following conditions:(1) and is decreasing;(2);(3). Suppose that the equation has a solution. Then, there exists a constant such that if converges strongly to a solution of the equation .

Corollary 3.5. Let (or ) space, (), and let be a bounded -accretive map. For arbitrary , define the sequence iteratively by where and are sequences in satisfying the following conditions:(1) and is decreasing;(2);(3). Suppose that the equation has a solution. Then, there exists a constant such that if , converges strongly to a solution of the equation .

Corollary 3.6. Let (or ) space, (), and let be a uniformly continuous -accretive map. For arbitrary , define the sequence iteratively by where and are sequences in satisfying the following conditions:(1) and is decreasing;(2);(3). Suppose that the equation has a solution. Then, there exists a constant such that if , converges strongly to a solution of the equation .

4. Approximation of Fixed Points of Uniformly Continuous Pseudocontractive Operators

We will make use of the following result.

Lemma 4.1 (Reich [9], Morales and Jung [44], Takahashi and Ueda [45]). Let be closed convex subset of a reflexive Banach space with a uniformly Gâteaux differentiable norm. Let be continuous pseudocontractive mapping with . Suppose that every closed convex and bounded subset of has the fixed point property for nonexpansive self-mappings. Then for , the path , satisfying, , converges strongly to a fixed point of as , where is the unique sunny nonexpansive retract from onto .

We now prove the following theorem.

Theorem 4.2. Let be a nonempty closed convex subset of a 2-uniformly smooth real Banach space . Let be a uniformly continuous pseudocontractive map with . Let a sequence be generated from arbitrary by where and are sequences in satisfying the following conditions:(1);(2) and is decreasing;(3);(4). Then, converges strongly to a fixed point of .

Proof. Since is a 2-uniformly smooth real Banach space, it has uniformly Gâteaux differentiable norm and every closed bounded convex nonempty subset of has the fixed point property for nonexpansive self-mappings. Set . Then is uniformly continuous and accretive. Now, using the fact is uniformly continuous and pseudocontractive, it follows from Lemma 4.1 that there exists a unique sequence , with , satisfying the following properties: Since is uniformly continuous, then from Lemma 2.4 it is bounded. As in the proof of Theorem 3.1, converges strongly to , with . So, converges strongly to a fixed point of .

Corollary 4.3. Let be a nonempty closed convex subset of a 2-uniformly smooth real Banach space . Let be a Lipschitz pseudocontractive map with . Let a sequence be generated from arbitrary by where and are sequences in satisfying the following conditions:(1);(2); and is decreasing;(3);(4). Then, converges strongly to a fixed point of .

Corollary 4.4. Let (or ) space, (), and let be a nonempty closed convex subset of . Let be a uniformly continuous pseudocontractive map with . Let a sequence be generated from arbitrary by where and are sequences in satisfying the following conditions:(1);(2) and is decreasing;(3);(4). Then, converges strongly to a fixed point of .

Remark 4.5. Addition of bounded error terms to any of the recursion formulas studied in this paper yields no further generalizations.

Remark 4.6. Real sequences that satisfy the hypotheses of Theorems 3.1 are and with , and . We verify that these choices satisfy in particular, the first part of condition (3) of Theorem 3.1. In fact, using the fact that , for and , we have as .

Acknowledgment

The authors thank the referees for useful comments.

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