Abstract and Applied Analysis

Volume 2013 (2013), Article ID 854297, 14 pages

http://dx.doi.org/10.1155/2013/854297

Research Article

Implicit Relaxed and Hybrid Methods with Regularization for Minimization Problems and Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

Lu-Chuan Ceng,¹ Qamrul Hasan Ansari,2,3 and Ching-Feng Wen⁴

¹Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

²Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India

³Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

⁴Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80708, Taiwan

Received 1 February 2013; Accepted 11 March 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Lu-Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We first introduce an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in the setting of Hilbert spaces. The implicit relaxed method with regularization is based on three well-known methods: the extragradient method, viscosity approximation method, and gradient projection algorithm with regularization. We derive a weak convergence theorem for two sequences generated by this method. On the other hand, we also prove a new strong convergence theorem by an implicit hybrid method with regularization for the MP and the mapping . The implicit hybrid method with regularization is based on four well-known methods: the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm with regularization.

1. Introduction

In 1972, Goebel and Kirk [1] established that every asymptotically nonexpansive mapping defined on a nonempty closed convex bounded subset of a uniformly convex Banach space, that is, there exists a sequence such that and

has a fixed point in . It can be easily seen that every nonexpansive mapping is asymptotically nonexpansive, and every asymptotically nonexpansive mapping is uniformly Lipschitzian; that is, there exists a constant such that

Several researchers have weaken the assumption on the mapping . Bruck et al. [2] introduced the following concept of an asymptotically nonexpansive mapping in the intermediate sense.

Definition 1. Let be a nonempty subset of a normed space . A mapping is said to be asymptotically nonexpansive in the intermediate sense provided is uniformly continuous and

They also studied iterative methods for the approximation of fixed points of such mappings.

Recently, Kim and Xu [3] introduced the following concept of asymptotically -strict pseudocontractive mappings in setting of Hilbert spaces.

Definition 2. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping with sequence if there exists a constant and a sequence in with such that

They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with .

Very recently, Sahu et al. [4] considered the following concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 3. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence if there exist a constant and a sequence in with such that

Let

Then, (for all ), (), and (5) reduces to the relation

Whenever for all in (7), then is an asymptotically -strict pseudocontractive mapping with sequence .

Let be a convex and continuously Fréchet differentiable functional. We consider the following minimization problem:

We assume that the minimization problem (8) has a solution, and the solution set of this problem is denoted by .

To develop some new iterative methods for computing the approximate solutions of the minimization problem is one of the main areas of research in optimization and approximation theory. In the recent past, some study has also been done in the direction to suggest some iterative algorithms to compute the fixed point of a mapping which is also a solution of some minimization problem; for further detail, we refer to [5] and the references therein.

The main aim of this paper is to propose some iterative schemes for finding a common solution of fixed point set of an asymptotically -strict pseudocontractive mapping and the solution set of the minimization problem. In particular, we introduce an implicit relaxed algorithm with regularization for finding a common element of the fixed point set of an asymptotically -strict pseudocontractive mapping and the solution set of minimization problem (8). This implicit relaxed method with regularization is based on three well-known methods, namely, the extragradient method [6], viscosity approximation method, and gradient projection algorithm with regularization. We also propose an implicit hybrid algorithm with regularization for finding an element of . The implicit hybrid method with regularization is based on four well-known methods, namely, the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm with regularization. The weak and strong convergence results of these two algorithms are established, respectively.

2. Preliminaries

Throughout the paper, unless otherwise specified, we use the following assumptions, notations, and terminologies.

We assume that is a real Hilbert space whose inner product and norm are denoted by and , respectively, and is a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . Moreover, we use to denote the weak -limit set of the sequence , that is,

The metric (or nearest point) projection from onto is the mapping which assigns to each point the unique point satisfying the following property:

We mention some important properties of projections in the following proposition.

Proposition 4. For given and ,

(i),  for all  ;(ii),  for all  ;(iii),  for all  .

Consequently, is nonexpansive.

Definition 5. A mapping is said to be

(a)monotone if

(b)-strongly monotone if there exists a constant such that

(c)-inverse-strongly monotone (-ism) if there exists a constant such that

Obviously, if is -inverse-strongly monotone, then it is monotone and -Lipschitz continuous. It can be easily seen that if is nonexpansive, then is monotone. It is also easy to see that a projection mapping is -ism. The inverse strongly monotone (also known as cocoercive) operators have been applied widely in solving practical problems in various fields.

We need some facts and tools which are listed in the form of the following lemmas.

Lemma 6. Let be a real inner product space. Then,

Lemma 7 (see [7, Proposition 2.4]). Let be a bounded sequence in a reflexive Banach space . If , then .

Let be a nonlinear mapping. The classical variational inequality problem (VIP) is to find such that

The solution set of VIP is denoted by .

The theory of variational inequalities is a well-established subject in applied mathematics, nonlinear analysis, and optimization. For further details on variational inequalities, we refer to [8–13] and the references therein.

It is well known that the solution of a variational inequality can be characterized be a fixed point of a projection mapping. Therefore, by using Proposition 4(i), we have the following result.

Lemma 8. Let be a monotone mapping. Then,

Lemma 9. The following assertions hold:

(a) for all ;(b) for all and with [14];(c) if is a sequence in such that , then

Lemma 10 (see [4, Lemma 2.5]). For given points and given also a real number , the set

is convex (and closed).

Lemma 11 (see [4, Lemma 2.6]). Let be a nonempty subset of a Hilbert space and an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then,

for all and .

Lemma 12 (see [4, Lemma 2.7]). Let be a nonempty subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in such that and as . Then, as .

Lemma 13 (Demiclosedness Principle, [see [4, Proposition 3.1]]). Let be a nonempty closed convex subset of a Hilbert space and be a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then is demiclosed at zero in the sense that if is a sequence in such that and , then .

Lemma 14 (see [4, Proposition 3.2]). Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that . Then, is closed and convex.

To prove a weak convergence theorem by an implicit relaxed method with regularization for the minimization problem (8) and the fixed point problem of an asymptotically -strict pseudocontractive mapping in the intermediate sense, we need the following lemma due to Osilike et al. [15].

Lemma 15 (see [15, page 80]). Let , , and be sequences of nonnegative real numbers satisfying the inequality

If and , then exists. If, in addition, has a subsequence which converges to zero, then .

Corollary 16 (see [16, page 303]). Let and be two sequences of nonnegative real numbers satisfying the inequality

If converges, then exists.

Lemma 17 (see [17]). Every Hilbert space has the Kadec-Klee property; that is, given a sequence and a point , we have

It is well known that every Hilbert space satisfies Opial’s condition [18]; that is, for any sequence with , we have

A set-valued mapping is called monotone if for all and imply . A monotone mapping is maximal if its graph is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for all implies . Let be a monotone and -Lipschitz continuous mapping, and let be the normal cone to at , that is, . Define

It is known that in this case is maximal monotone, and if and only if ; see [19].

3. Weak Convergence Theorem

In this section, we will prove a weak convergence theorem for an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense and the set of solutions of the minimization problem (8) for a convex functional with -Lipschitz continuous gradient . This implicit relaxed method with regularization is based on the extragradient method, viscosity approximation method, and gradient projection algorithm (GPA) with regularization.

Theorem 18. Let be a nonempty bounded closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient , a -contraction with coefficient , and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that is nonempty. Let and be defined as in Definition 3. Let and be the sequences generated by

where is a sequence , , are sequences in , and , are sequences in satisfying the following conditions:

(i) and , for all for some ;(ii), , and ;(iii);(iv), for all and for some .

Then, the sequences , converge weakly to some point .

Remark 19. Observe that for all and all

By Banach contraction principle, we know that for each , there exists a unique such that

Also, observe that for all and all

Utilizing Banach contraction principle, for each , there exists a unique such that

Proof of Theorem 18. Note that the -Lipschitz continuity of the gradient implies that is -ism [20], that is,

Thus, is monotone and -Lipschitz continuous.

We divide the rest of the proof into several steps.

Step 1. exists for each .

Indeed, note that for all . Take arbitrarily. From Proposition 4(ii), monotonicity of , and , we have

Since and is -Lipschitz continuous, by Proposition 4(i), we have

So, we have

Therefore, from (33), and . By Lemma 9(b), we have

Since , , , and , from the boundedness of , it follows that

By Lemma 15, we have

Step 2. and .

Indeed, substituting (33) in (34), we get

which hence implies that

Since exists, , , , and , from the boundedness of , it follows that

Meantime, utilizing the arguments similar to those in (33), we have

Substituting (40) in (34), we get

which hence implies that

Since exists, , , , and , from the boundedness of it follows that

This together with implies that

Step 3.   and .

Indeed, from (33) and (34), we conclude that

which together with condition (i) yields

Since exists, , , , and , from the boundedness of it follows that

Note that

From the boundedness of , , , and , we deduce that

Furthermore, observe that

Utilizing Lemma 11, we have

for every . Hence, it follows from that . Thus, from (50) and , we get . Since , as , and is uniformly continuous, we obtain from Lemma 12 that as .

Step 4. .

Indeed, from the boundedness of , we know that . Take arbitrarily. Then, there exists a subsequence of such that converges weakly to . Note that is uniformly continuous and . Hence, it is easy to see that for all . By Lemma 13, we obtain . Furthermore, we show . Since and , we have and . Let

where is the normal cone to at . We have already mentioned that in this case the mapping is maximal monotone, and if and only if ; see [19] for more details. Let be the graph of and let . Then, we have and hence . So, we have for all . On the other hand, from and , we have

and hence

Therefore, from for all and , we have

Since (due to the Lipschitz continuity of ), (due to ), , and , we obtain as . Since is maximal monotone, we have and hence . Clearly, . Consequently, . This implies that .

Step 5. and converge weakly to the same point .

Indeed, it is sufficient to show that is a single-point set because as . Since , let us take two points arbitrarily. Then, there exist two subsequences and of such that and , respectively. In terms of Step 4, we know that . Meantime, according to Step 1, we also know that there exist both and . Let us show that . Assume that . From the Opial condition [18], it follows that

This leads to a contradiction. Thus, we must have . This implies that is a single-point set. Without loss of generality, we may write . Consequently, by Lemma 7, we obtain that . Since as , we also have

This completes the proof.

4. Strong Convergence Theorem

In this section, we establish a strong convergence theorem for an implicit hybrid method with regularization for finding a common element of the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense and the set of solutions of the MP (8) for a convex functional with -Lipschitz continuous gradient . This implicit hybrid method with regularization is based on the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm (GPA) with regularization.

Theorem 20. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient , a -contraction with coefficient , and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that is nonempty and bounded. Let and be defined as in Definition 3. Let , and be the sequences generated by

where , , , , , and

Assume that the following conditions hold:

(i) and , for all for some ;(ii) and ;(iii);(iv), for all and for some .

Then, the sequences , , and converge strongly to the same point .

Proof. Utilizing the condition for all and repeating the same arguments as in Remark 19, we can see that and are defined well. Note that the -Lipschitz continuity of the gradient implies that is -ism [20], that is,

Observe that

Hence, it follows that is -ism. It is clear that is closed and is closed and convex for every . As the defining inequality in is equivalent to the inequality

By Lemma 10, we also have that is convex for every . As , we have for all , and by Proposition 4(i), we get .

We divide the rest of the proof into several steps.

Step 1. , , and are bounded.

Indeed, take arbitrarily. Taking into account , for all , we deduce that

which implies that

Meantime, we also have

which hence implies that

Thus, from (64) and (66), it follows that

which hence implies that

Repeating the same arguments as in (33) and (34), we can obtain that

for every and hence . So, for every . Next, let us show by mathematical induction that is well-defined and for every . For , we have . Hence, we obtain . Suppose that is given and for some integer . Since is nonempty, is a nonempty closed convex subset of . So, there exists a unique element such that . It is also obvious that there holds for every . Since , we have for every and hence . Therefore, we obtain .

Step 2. and .

Indeed, let . From and , we have

for every . Therefore, is bounded. From (64), (66), and (70), we also obtain that , , and are bounded. Since and , we have

for every . Therefore, there exists . Since and , using Proposition 4(ii), we have

for every . This implies that

Since , we have

which implies that

Hence, we get

for every . From and , we conclude that as .

Step 3. , and .

Indeed, substituting (69) in (70), we get

which hence implies that

Since , , , , and , from the boundedness of , , , and , it follows that

Meantime, utilizing the arguments similar to those in (69), we have

Substituting (81) in (70), we get

which hence implies that

Since , , , , and , from the boundedness of , , , and , it follows that

This together with implies that

Since , we have . Then,

and hence . Furthermore, observe that

Utilizing Lemma 11, we have

for every . Hence, it follows from that . Thus, from (87) and , we get . Since , as , and is uniformly continuous, we obtain from Lemma 12 that as .

Step 4. .

Indeed, repeating the same arguments as in Step 4 of the proof of Theorem 18, we can derive the desired conclusion.

Step 5. , , and converge strongly to .

Indeed, take arbitrarily. Then, according to Step 4. Moreover, there exists a subsequence of such that . Hence, from , , and (71), we have

So, we obtain

From , we have due to the Kadec-Klee property of Hilbert spaces [17]. So, it is clear that . Since and , we have

As , we obtain by and . Hence, we have . This implies that . It is easy to see that and . This completes the proof.

Acknowledgments

In this research, first author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Profram Foundation of Ministry of Education of China (20123127110002). The research part of the second author was done during his visit to KFUPM, Dhahran, Saudi Arabia. The third author was partially supported by a Grant from the NSC 101-2115-M-037-001.

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