Abstract and Applied Analysis

Volume 2014 (2014), Article ID 173461, 15 pages

http://dx.doi.org/10.1155/2014/173461

Research Article

Hierarchical Fixed Point Problems in Uniformly Smooth Banach Spaces

Lu-Chuan Ceng,¹ Ching-Feng Wen,² and Chin-Tzong Pang³

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

²Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

³Department of Information Management, Yuan Ze University, Chung-Li 32003, Taiwan

Received 4 November 2013; Accepted 18 December 2013; Published 22 January 2014

Academic Editor: Ngai-Ching Wong

Copyright © 2014 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 propose some relaxed implicit and explicit viscosity approximation methods for hierarchical fixed point problems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces. These relaxed viscosity approximation methods are based on the well-known viscosity approximation method and hybrid steepest-descent method. We obtain some strong convergence theorems under mild conditions.

1. Introduction

Let be a real Banach space and the unit sphere of ; that is, . Recall that is said to be smooth if the limit

exists for all ; in this case, is also said to have a Gâteaux differentiable norm. is said to have a uniformly Gâteaux differentiable norm if for each , the limit is attained uniformly for . Moreover, it is said to be uniformly smooth if this limit is attained uniformly for . The norm of is said to be the Fréchet differential if for each , this limit is attained uniformly for . In addition, we define a function called the modulus of smoothness of as follows:

It is known that is uniformly smooth if and only if .

Let be a real Banach space and let denote the normalized duality mapping from to   given by

where denotes the dual space of and denotes the generalized duality pairing. We use to denote the set of fixed points of the mapping . It is well known that if is smooth, then is single-valued and norm-to-weak* continuous, whereas if is a Banach space with a uniformly Gâteaux differentiable norm, then is single-valued and norm-to-weak* uniformly continuous on bounded subsets of . Further, if is a uniformly smooth Banach space, then is single-valued and norm-to-norm uniformly continuous on bounded subsets of . In what follows, we still denote by the single-valued normalized duality mapping.

Let be a nonempty closed convex subset of . Recall that a mapping is said to be -Lipschitzian if there exists a constant such that

In particular, if , then is said to be nonexpansive; that is,

We use the notation to indicate the weak convergence and the one to indicate the strong convergence.

Definition 1. Let be a mapping of into . Then is said to be

(i)accretive if for each , there exists such that

 where is the normalized duality mapping;(ii)-strongly accretive if for each , there exists such that

 for some ;(iii)pseudocontractive if for each , there exists such that

(iv)-strongly pseudocontractive if for each , there exists such that

 for some ;(v)-strictly pseudocontractive if for each , there exists such that

 for some .

In a real smooth Banach space we say that an operator is strongly positive [1] if there exists a constant with the property

where is the identity mapping.

Recently, the problem of convergence of implicit iterative algorithms to a common fixed point for a family of nonexpansive mappings and its extensions to Hilbert spaces or Banach spaces have been considered by many authors; see [1–9] and the references therein.

Yao et al. [10] introduced the following Halpern-type implicit iterative algorithm,

and proved a strong convergence theorem under suitable conditions.

On the other hand, let be a nonempty closed convex subset of a real Hilbert space and let be a nonlinear mapping. The classical variational inequality problem (VIP) is to find such that

If we assume that is the fixed point set of a nonexpansive mapping and is another nonexpansive mapping (not necessarily with fixed points), the problem (13) becomes the VIP of finding such that

introduced first by Moudafi and Maingé in [11], which is called hierarchical fixed point problem.

In particular, whenever , all elements of are solutions of VIP (14). If is a -contraction (i.e., for some ), the set of solutions of VIP (14) is a singleton and it is well known as a viscosity problem, which was first introduced by Moudafi [12] and then developed by several authors [13, 14].

Very recently, Cai and Bu [1] investigated a general hierarchical fixed point problem for a countable family of continuous pseudocontractions, which covers as a special case of the problem considered in [10]. For this purpose, they first established strong convergence of an implicit iterative scheme for solving a hierarchical fixed point problem for a continuous pseudocontractive mapping in a uniformly smooth Banach space.

In this paper, let be a nonempty closed convex subset of a uniformly smooth Banach space such that . Let be a nonexpansive mapping with and let be a fixed contractive mapping with contractive coefficient . Let be -strongly accretive and -strictly pseudocontractive with and let be a -strongly positive linear bounded operator. First of all, we introduce a relaxed implicit viscosity scheme for solving a hierarchical fixed point problem for a nonexpansive mapping :

where . It is proven that as converges strongly to a point , which is the unique solution in to the VIP:

On the other hand, let be a countable family of nonexpansive mappings from to itself such that . We propose a relaxed implicit viscosity iterative algorithm for solving a hierarchical fixed point problem for a countable family of nonexpansive mappings :

where , and are four sequences in . It is proven that under mild conditions converges strongly to a point , which is the unique solution in to the VIP:

Furthermore, we also propose a relaxed explicit viscosity iterative algorithm for solving another hierarchical fixed point problem for a countable family of nonexpansive mappings :

where and are two sequences in . It is proven that under appropriate assumptions, converges strongly to a point , which is the unique solution in to the VIP:

The above relaxed viscosity algorithms are based on the well-known viscosity approximation method (see, e.g., [4–6, 9]) and hybrid steepest-descent method (see, e.g., [14–17]). Our results extend, improve, supplement, and develop the recent results announced by many authors.

2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma 2 can be found in [18]. Lemma 3 is an immediate consequence of the subdifferential inequality of the function .

Lemma 2. Let be a sequence of nonnegative real numbers satisfying

where , and satisfy the following conditions:

(i);(ii);(iii).

Then .

Lemma 3. In a smooth Banach space , there holds the following inequality:

Let be a continuous linear functional on and . We write instead of . is said to be Banach limit if satisfies and for all . It is well known that for Banach limit the following holds:

(i)for all implies that ;(ii) for any fixed positive integer ;(iii) for all .

It is easy to see that there holds the following conclusion.

Lemma 4 (see [19]). Let . If , then there exists a subsequence of such that as .

Recall that a Banach space is said to satisfy Opial’s condition, if whenever is a sequence in which converges weakly to as , then

Lemma 5 (Demiclosedness principle; see [20, Theorem 10.3]). Let be a reflexive Banach space satisfying Opial’s condition, a nonempty closed convex subset of , and a nonexpansive mapping. Then the mapping is demiclosed on , where is the identity mapping; that is, if is a sequence of such that and , then .

The following lemma can be derived by the standard argument and hence its proof will be omitted.

Lemma 6. Let be a nonempty closed convex subset of a real smooth Banach space and let be a mapping.

(i)If is  -strongly accretive and -strictly pseudocontractive with , then nonexpansive and is Lipschitz continuous with constant ;(ii)If is  -strongly accretive and -strictly pseudocontractive with , then for any fixed is contractive with coefficient .

3. Relaxed Implicit Viscosity Scheme for Hierarchical Fixed Point Problem for a Nonexpansive Mapping

In this section, we introduce our relaxed implicit viscosity scheme for solving hierarchical fixed point problem for a nonexpansive mapping and show the strong convergence theorem. First, we list several useful and helpful lemmas.

Lemma 7 (see [21]). Let be a Banach space, a nonempty closed and convex subset of , and a continuous and strong pseudocontraction. Then has a unique fixed point in .

Lemma 8 (see [19]). Let be a sequence of nonnegative real numbers satisfying the property , where and such that (i)   and (ii)   . Then converges to zero as .

Lemma 9 (see [22]). Let be a nonempty closed convex subset of a real Banach space and a continuous pseudocontractive map. We denote . Then the following holds.

(i)The map is a nonexpansive self-mapping on .(ii)If  , then .

Lemma 10 (see [23]). Assume that is a strongly positive linear bounded operator on a smooth Banach space with coefficient and . Then .

We now state and prove our first result.

Theorem 11. Let be a nonempty closed convex subset of a uniformly smooth Banach space such that . Let be a nonexpansive mapping with and   -strongly accretive and -strictly pseudocontractive with . Let be a fixed contractive mapping with contractive coefficient . Let be a -strongly positive linear bounded operator with . Let be defined by

where . Then, as converges strongly to some fixed point of  , which is the unique solution in to the VIP:

Proof. First, we claim that . Indeed, it is known that strongly accretive constant and strictly pseudocontractive constant . Moreover, observe that

Let us show that the net is defined well. As a matter of fact, we define the mapping as follows:

Since , we may assume, without loss of generality, that for all , where . Utilizing Lemmas 6 and 10, we obtain that for each

It follows that for each , is a continuous and strongly pseudocontractive mapping with pseudocontractive coefficient . Hence, by Lemma 7 we know that there exists a unique fixed point in , denoted by , which uniquely solves the fixed point equation:

Let us show the uniqueness of the solution of VIP (25). Suppose both and are solutions to VIP (25). Then we have

Adding up the above two inequalities, we obtain

Note that

Taking into account , we have , and hence the uniqueness is proved. We use to denote the unique solution of VIP (25).

Next, we prove that is bounded. Indeed, we note that . Take a fixed arbitrarily. Utilizing Lemma 10 we deduce that for all

which immediately yields

Thus is bounded.

Assume that and as . Set and , and define by , where is a Banach limit on . Let

We see easily that is a nonempty closed convex subset of . Note that as . In terms of Lemma 9, we know that the mapping is nonexpansive and and , where denotes the identity operator. It follows that

which implies that ; that is, is invariant under . Since a uniformly smooth Banach space has the fixed point property for nonexpansive mapping, has a fixed point, say . Since is also a minimizer of over , we have that, for and ,

Since is uniformly smooth, we conclude that the duality mapping is norm-to-norm uniformly continuous on any bounded subset of . Letting , we find that the two limits above can be interchanged and obtain

On the other hand, we have

It follows that

Therefore,

Combining (38) and (41), we get

which leads to . Hence there exists a subsequence which is still denoted as such that as .

Next, we prove that solves VIP (25). Since

we can deduce that

Since is nonexpansive, is accretive. So, from the accretivity of , it follows that, for any fixed ,

This implies that

Now replacing with , letting , and noticing the boundedness of and the fact that for , we have that

That is, is a solution of VIP (25). Then . In summary, we infer that each cluster point of is equal to as . This completes the proof.

4. Relaxed Viscosity Algorithms for Hierarchical Fixed Point Problems for a Countable Family of Nonexpansive Mappings

In this section, we propose relaxed implicit and explicit viscosity algorithms for solving hierarchical fixed point problems for a countable family of nonexpansive mappings and show strong convergence theorems. For this purpose, we will use the following lemmas in the sequel.

Lemma 12 (see [24]). Let be a nonempty closed convex subset of a Banach space . Let be a sequence of mappings of into itself. Suppose that . Then, for each converges strongly to some point of . Moreover, let be a mapping of into itself defined by , for all . Then .

Lemma 13 (see [1, Lemma 2.6]). Let be a nonempty closed convex subset of a real Banach space which has uniformly Gateaux differentiable norm. Let be a continuous pseudocontractive mapping with and let be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient and Lipschitzian constant . Let be a -strongly positive linear bounded operator with coefficient . Assume that and that converges strongly to as , where is defined by . Suppose that is bounded and that . Then .

Theorem 14. Let be a nonempty closed convex subset of a uniformly smooth Banach space such that . Let be a countable family of nonexpansive mappings from to itself such that . Let be -strongly accretive and -strictly pseudocontractive with , and let be a fixed contractive mapping with contractive coefficient . Let be a -strongly positive linear bounded operator with . For arbitrarily given , let the sequence be generated iteratively by

where , and are four sequences in satisfying the following conditions:

(i);(ii), and .

Assume that for any bounded subset of , let be a mapping of into itself defined by for all , and suppose that . Then, converges strongly to a point of such that is a unique solution in to the VIP:

Proof. By condition (i), we may assume, without loss of generality, that . Since is a -strongly positive linear bounded operator on , from (11) we have

Observe that

It follows that

Next, we show that is well defined. For each , define a mapping by

For every , we have

where . Therefore, is a continuous strong pseudocontraction for each . By Lemma 7, we see that there exists a unique fixed point for each such that

That is, the sequence is well defined. Next, we prove that is bounded. Take a fixed arbitrarily. Taking into account , we may assume that there exists a constant such that for all . Then we have

which implies that

Therefore, we have

By induction, we get

Therefore, is bounded and so are the sequences ,. We observe that

which go together with condition (i) and , implying that

On the other hand, we have

Utilizing Lemma 12, we immediately derive

Let . Utilizing [1, Lemma 2.5] and Lemma 13, we conclude that converges strongly to and

Finally, we show that as . We observe that

which implies that

Furthermore, utilizing Lemma 3 from the last relation we have

We note that

Therefore, condition (ii) leads to . In addition, since , and , we get the following from (64)

Applying Lemma 2, we have as . This completes the proof.

Remark 15. Put and . Then , and satisfy conditions (i) and (ii) of Theorem 14. But we note that .

Remark 16. In the iterative scheme of Theorem 14, the first iterative step is a predictor step and the second iterative step is a corrector step. Hence our iteration process is the predictor-corrector method.

Remark 17. Theorem 14 extends and improves Theorem 3.1 of [10] to a great extent in the following aspects:

(i) is replaced by a fixed contractive mapping;(ii)one continuous pseudocontractive mapping (including nonexpansive mapping) is replaced by a countable family of nonexpansive mappings;(iii)condition is weakened to the one and as ;(iv)we add a strongly positive linear bounded operator and a strongly accretive and strictly pseudocontractive mapping in our iterative algorithm.

Theorem 18. Let be a nonempty closed convex subset of a uniformly smooth Banach space which has the weakly sequentially continuous duality mapping . Assume that . Let be a countable family of nonexpansive mappings from to itself such that . Let be -strongly accretive and -strictly pseudocontractive with , and let be a fixed contractive mapping with contractive coefficient . Let be a -strongly positive linear bounded operator with . For arbitrarily given , let the sequence be generated iteratively by

where and are two sequences in satisfying the following conditions:

(i) and ;(ii) or .

Assume that for any bounded subset of , let be a mapping of into itself defined by for all , and suppose that . Then, converges strongly to a point of such that is a unique solution in to the VIP:

Proof. First, since is a -strongly positive linear bounded operator on , from (11) we have

Let us show that is bounded. Indeed, since , without loss of generality, we may assume that . Take . Then it follows that , and

Hence we deduce the following that

By induction

This implies that is bounded and so are and .

Now we claim that

Indeed, first of all, (70) can be rewritten as follows:

Observe that

and hence

where for some (it is easy to see that is bounded due to the boundedness of ). Utilizing Lemma 2, we conclude that as from conditions (i)-(ii) and the property imposed on .

Next let us show that

Indeed, from (76), (77), and , it follows that

That is,

Also, it is clear that

By Lemma 12, we conclude from (82) and (83) that (80) holds. Let +. According to Theorem 11, we know that converges strongly to , which is the unique solution in to the VIP:

Further, let us show that

Indeed, take a subsequence of such that

Without loss of generality, we may assume that . Utilizing Lemma 5 we obtain from (80) that . Hence from (84) and (86) we get

As required, let us show that as .

As a matter of fact, we observe that

where and

It can be easily seen from (85) and conditions (i) and (ii) that

In terms of Lemma 8, we infer that as .

Finally, we provide an example to illustrate Theorem 18.

Example 19. Let with inner product and norm which are defined by

for all with and . Let . Clearly, is a nonempty closed convex subset of a uniformly smooth Banach space such that . Let be a countable family of nonexpansive mappings from to itself such that , for instance, putting with . Then and . It is clear that and are nonexpansive mappings with , and satisfies the assumption in Theorem 18. Let be -strongly accretive and -strictly pseudocontractive with , and be a fixed contractive mapping with contractive coefficient , for instance, putting ,  , and , we know that and that is a ()-strongly accretive and -strictly pseudocontractive mapping and is a -contraction with and . Let be a -strongly positive linear bounded operator with ; for instance, putting , we know that is a -strongly positive linear bounded operator with . In this case, from iterative scheme (70) in Theorem 18, we obtain that for any given ,

It can be readily seen that

We claim that converges to the unique point in if and . Indeed, observe that

Thus, we conclude from that converges to the unique point in . It is clear that is a unique solution in for the following variational inequality problem (VIP):

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002) for Lu-Chuan Ceng. This research was partially supported by a Grant from NSC for Chin-Feng Wen. This work was supported partly by the National Science Council of the Republic of China for Chin-Tzong Pang.

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