Hierarchical Fixed Point Problems in Uniformly Smooth Banach Spaces

Ceng, Lu-Chuan; Wen, Ching-Feng; Pang, Chin-Tzong

doi:https://doi.org/10.1155/2014/173461

Abstract and Applied Analysis

On this page

Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

Special Issue

Recent Development in Fixed-Point Theory, Optimization, and their Applications

View this Special Issue

Research Article | Open Access

Volume 2014 | Article ID 173461 | https://doi.org/10.1155/2014/173461

Hierarchical Fixed Point Problems in Uniformly Smooth Banach Spaces

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

Academic Editor: Ngai-Ching Wong

Received04 Nov 2013

Accepted18 Dec 2013

Published22 Jan 2014

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.

References

G. Cai and S. Bu, “Approximation of common fixed points of a countable family of continuous pseudocontractions in a uniformly smooth Banach space,” Applied Mathematics Letters, vol. 24, no. 12, pp. 1998–2004, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
L. C. Ceng, S. Schaible, and J. C. Yao, “Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings,” Journal of Optimization Theory and Applications, vol. 139, no. 2, pp. 403–418, 2008.
View at: Publisher Site | Google Scholar
L. C. Ceng, P. Cubiotti, and J. C. Yao, “Strong convergence theorems for finitely many nonexpansive mappings and applications,” Nonlinear Analysis: Theory, Methods and Applications, vol. 67, no. 5, pp. 1464–1473, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
L. C. Ceng and J. C. Yao, “Relaxed viscosity approximation methods for fixed point problems and variational inequality problems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 10, pp. 3299–3309, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
L. C. Ceng and J. C. Yao, “Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 729–741, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. Zegeye, E. U. Ofoedu, and N. Shahzad, “Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3439–3449, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
T. Suzuki, “Moudafi's viscosity approximations with Meir-Keeler contractions,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 342–352, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Y. H. Wang and W. Xu, “Strong convergence of a modified iterative algorithm for hierarchical fixed point problems and variational inequalities,” Fixed Point Theory and Applications, vol. 2013, article 121, 9 pages, 2013.
View at: Publisher Site | Google Scholar
J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 509–520, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Y. Yao, Y. C. Liou, and R. Chen, “Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces,” Nonlinear Analysis: Theory, Methods and Applications, vol. 67, no. 12, pp. 3311–3317, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. Moudafi and P.-E. Maingé, “Towards viscosity approximations of hierarchical fixed-point problems,” Fixed Point Theory and Applications, vol. 2006, Article ID 95453, 10 pages, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis: Theory, Methods and Applications, vol. 73, no. 3, pp. 689–694, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
L. C. Ceng, S. M. Guu, and J. C. Yao, “A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces,” Computers and Mathematics with Applications, vol. 61, no. 9, pp. 2447–2455, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in banach spaces,” Numerical Functional Analysis and Optimization, vol. 29, no. 9-10, pp. 987–1033, 2008.
View at: Publisher Site | Google Scholar
I. Yamada, “The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
View at: Google Scholar
H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Goebel and W. A. Kirk, “Topics in metric fixed point theory,” in Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, 1990.
View at: Google Scholar | Zentralblatt MATH
K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, no. 4, pp. 365–374, 1974.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Y. Song and R. Chen, “Convergence theorems of iterative algorithms for continuous pseudocontractive mappings,” Nonlinear Analysis: Theory, Methods and Applications, vol. 67, no. 2, pp. 486–497, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
G. Cai and C. S. Hu, “Strong convergence theorems of a general iterative process for a finite family of λ_i-strict pseudo-contractions in q-uniformly smooth Banach spaces,” Computers and Mathematics with Applications, vol. 59, no. 1, pp. 149–160, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods and Applications, vol. 67, no. 8, pp. 2350–2360, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright

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.

PDF Download Citation

Download other formats

Order printed copies

Views

1048

Downloads

1175

Citations