Abstract and Applied Analysis

Abstract and Applied Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 573156 | https://doi.org/10.1155/2009/573156

Jong Soo Jung, "Strong Convergence of Viscosity Iteration Methods for Nonexpansive Mappings", Abstract and Applied Analysis, vol. 2009, Article ID 573156, 17 pages, 2009. https://doi.org/10.1155/2009/573156

Strong Convergence of Viscosity Iteration Methods for Nonexpansive Mappings

Academic Editor: Simeon Reich
Received29 Dec 2008
Accepted07 Mar 2009
Published14 Jun 2009

Abstract

We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappings in a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for viscosity iterative scheme are given and strong convergence of viscosity iterative scheme to a solution of a ceratin variational inequality is established.

1. Introduction

Let be a real Banach space and let be a nonempty closed convex subset of . Recall that a mapping is a contraction on if there exists a constant such that We use to denote the collection of mappings verifying the above inequality. That is, Let be a nonexpansive mapping (recall that a mapping is nonexpansive if and let denote the set of fixed points of ; that is,

We consider the iterative scheme: for a nonexpansive mapping, and , As a special case of (1.1), the following iterative scheme: where are arbitrary (but fixed), has been investigated by many authors; see, for example, Browder [1], Chang [2], Cho et al. [3], Halpern [4], Lions [5], Reich [6, 7], Shioji and Takahashi [8], Wittmann [9], and Xu [10]. The authors above showed that the sequence generated by (1.2) converges strongly to a point in the fixed-point set under appropriate conditions on in Hilbert spaces or certain Banach spaces.

The viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping in a Hilbert space was proposed by Moudafi [11] in 2000. In 2004, Xu [12] extended Theorem  2.2 of Moudafi [11] for the iterative scheme (1.1) to a Banach space setting by using the following conditions on the sequence : For the iterative scheme (1.1) with generalized contractive mappings instead of contractions, we refer to [13].

In 2005, Kim and Xu [14] provided a simpler modification of Mann iterative scheme in a uniformly smooth Banach space as follows: where is an arbitrary (but fixed) element, and and are two sequences in (0,1). They proved that the sequence generated by (1.4) converges to a fixed point of under the following control conditions:

(i) (ii) (iii)

Recently, Yao et al. [15] considered the following modified Mann iterative scheme in a uniformly smooth Banach space as the viscosity approximation method: and proved strong convergence of the sequence generated by (1.5) under certain different control conditions on and . In particular, their results remove the condition imposed on .

Very recently, Qin et al. [16] proposed the composite Halpern type iterative scheme in a uniformly smooth Banach space as follows: and showed strong convergence of the sequence generated by (1.6) under the following control conditions:

(i) (ii) (iii)

In this paper, under the framework of a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of has the fixed point property for nonexpansive mappings, we consider a new composite iterative scheme for a nonexpansive mapping as the viscosity approximation method: for and the initial guess , where , and are sequences in . First, we prove under certain control conditions on the sequences , and different from those of Qin et al. [16] that the sequence generated by (IS) converges strongly to a fixed point of , which is a solution of a certain variational inequality. Next we study the composite iterative scheme (IS) with the weakly contractive mapping instead of the contractions. The main results develop and complement the corresponding results of [2, 3, 8, 9, 11, 12, 15, 16]. In particular, if for all in (IS), then (IS) reduces a new viscosity iterative scheme for finding a fixed point of :

2. Preliminaries and Lemmas

Let be a real Banach space with norm , and let be its dual. The value of at will be denoted by . When is a sequence in , then (resp., ) will denote strong (resp., weak) convergence of the sequence to .

The ( normalized) duality mapping from into the family of nonempty (by Hahn-Banach theorem) weak-star compact subsets of its dual is defined by for each [17].

The norm of is said to be Gâteaux differentiable (and is said to be smooth) if exists for each in its unit sphere . The norm is said to be uniformly Gâteaux differentiable if for , the limit is attained uniformly for . The space is said to have a uniformly Fréchet differentiable norm (and is said to be uniformly smooth) if the limit in (2.2) is attained uniformly for . It is known that is smooth if and only if each duality mapping is single-valued. It is also well-known that if has a uniformly Gâteaux differentiable norm, is uniformly norm-to- continuous on each bounded subsets of [17].

Let be a nonempty closed convex subset of . is said to have the fixed point property for nonexpansive mappings if every nonexpansive mapping of a bounded closed convex subset of has a fixed point in . Let be a subset of . Then a mapping is said to be a retraction from onto if for all . A retraction is said to be sunny if for all and with . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto . In a smooth Banach space , it is well-known [18, page 48] that is a sunny nonexpansive retraction from onto if and only if the following condition holds We need the following lemmas for the proof of our main results. Lemma 2.1 was also given in Jung and Morales [19], Lemma 2.2 is Lemma  2 of Suzuki [20] and Lemma 2.3 is essentially Lemma  2 of Liu [21] (also see [10]).

Lemma 2.1. Let be a real Banach space and let be the duality mapping. Then, for any given , one has for all .

Lemma 2.2. Let and be bounded sequences in a Banach space and let be a sequence in which satisfies the following condition: suppose that Then

Lemma 2.3. Let be a sequence of nonnegative real numbers satisfying where , , and satisfy the following conditions: (i) and or, equivalently, (ii) or (iii) . Then .

Recall that a mapping is said to be weakly contractive if where is a continuous and strictly increasing function such that is positive on and . As a special case, if for , where , then the weakly contractive mapping is a contraction with constant . Rhoades [22] obtained the following result for weakly contractive mapping.

Lemma 2.4 ([22, Theorem  2]). Let be a complete metric space and let be a weakly contractive mapping on . Then has a unique fixed point in . Moreover, for , converges strongly to .

The following Lemma was given in [23, 24].

Lemma 2.5. Let and be two sequences of nonnegative real numbers and let be a sequence of positive numbers satisfying the conditions:(i) ;(ii) .Let the recursive inequality, be given, where is a continuous and strict increasing function on with . Then .

3. Main Results

First, we study a strong convergence theorem for a viscosity iterative scheme for the nonexpansive mapping with the contraction.

For a nonexpansive mapping, and , defines a strict contraction mapping. Thus, by the Banach contraction mapping principle, there exists a unique fixed point satisfying

For simplicity we will write for provided no confusion occurs.

In 2006, the following result was given by Jung [25] (see also Xu [12] for the result in uniformly smooth Banach spaces).

Theorem 3 J (see [25]). Let be a reflexive Banach space having a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of and let be a nonexpansive mapping from into itself with Then defined by (R) converges strongly to a point in If we define by then is the unique solution of the variational inequality

Remark 3.1. In Theorem J, if is a constant, then (VI) become Hence by (2.3), reduces to the sunny nonexpansive retraction from to . Namely is a sunny nonexpansive retraction of .
Using Theorem J, we have the following result.

Theorem 3.2. Let be a reflexive Banach space having a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of and let be a nonexpansive mapping from into itself with .Let , and be sequences in which satisfy the conditions: (C1) , ;(C2) ;(C3) Let and the initial guess be chosen arbitrarily. Let be the sequence generated by If , then converges strongly to , where is the unique solution of the variational inequality

Proof. We note that by Theorem J, there exists the unique solution of the variational inequality Namely, where is defined by (R). We will show that .
We proceed with the following steps.
Step 1. We show that for all and all and so , , , , , and are bounded.
Indeed, let . Then, noting that
we have which yields that Using an induction, we obtain for all . Hence is bounded, and so are , , , , and .
Step 2. We show that and . Indeed, it follows from condition (C1) and (C2) that Also from , we get Step 3. We show that . To this end, set for . Then it follows from (C1) and (C3) that Define Observe that It follows from (3.13) that Since and are bounded, by (C1), (3.10), (3.11), and (3.14) we obtain that Hence by Lemma 2.2, we have It follows from (3.11) and (3.12) that Step 4. We show that . In fact, from (IS) it follows that So we have Thus, from condition (C3), Steps 2, and 3, we have Step 5. We show that . To prove this, let a subsequence of be such that
and for some . From Step 4, it follows that .
Now let , where . Then we can write
Putting by Step 4 and using Lemma 2.1, we obtain The last inequality implies It follows that where is a constant such that for all and . Taking the as in (3.26) and noticing the fact that the two limits are interchangeable due to the fact that is uniformly continuous on bounded subsets of from the strong topology of to the topology of , we have Indeed, letting , from (3.26) we have So, for any , there exists a positive number such that for any , Moreover, since as , the set is bounded and the duality mapping is uniformly continuous on bounded subset of , there exists such that, for any , Choose , we have for all and , which implies that Since , we have Since is arbitrary, we obtain that
Step 6. We show that . By using (IS), we have Applying Lemma 2.1 and (3.6), we obtain It then follows that where . Put From the condition (C1) and Step 5, it follows that , , and . Since (3.37) reduces to from Lemma 2.3 with , we conclude that . This completes the proof.

Corollary 3.3. Let be a uniformly smooth Banach space. Let , , , , , , , , , and be the same as in Theorem 3.2. Then the conclusion of Theorem 3.2 still holds.

Proof. Since is a uniformly smooth Banach space, is reflexive, the norm is uniformly Gâteaux differentiable, and every nonempty weakly compact convex subset of has the fixed point property for nonexpansive mappings. Thus the conclusion of Corollary 3.3 follows from Theorem 3.2 immediately.

Corollary 3.4. Let be a nonempty closed convex subset of a uniformly smooth Banach space . Let be a nonexpansive mapping with . Let , , and be three sequences in which satisfy the control conditions (C1)–(C3)   in Theorem 3.2. Then for the initial guess and , the sequence generated by (1.6) converges strongly to a fixed point of under the assumption .

Remark 3.5. (1) In general, the condition (C3) in Theorem 3.2 and the condition of Qin et al. [16, Theorem  2.1] are not comparable; neither of them implies the other. Theorem 3.2 (and Corollary 3.4) removes the conditions and imposed on the control parameters and of Qin et al. [16, Theorem  2.1].
(2) Theorem 3.2 (and Corollary 3.3) complements the corresponding results in Moudafi [11], Xu [12], and Yao et al. [15]. In particular, if in (IS), then (IS) in Theorem 3.2 reduces a new one for finding a fixed point of :
(3) Corollary 3.4 with in (IS) develops the corresponding results of Shioji and Takahashi [8], Wittmann [9] without the condition as well as the result of Chang [2] in which the condition was assumed.
Next, we consider the viscosity iterative scheme with the weakly contractive mappings instead of the contractions.

Theorem 3.6. Let be a reflexive Banach space having a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of and let be a nonexpansive mapping from into itself with . Let , and be sequences in which satisfy the conditions (C1)–(C3) in Theorem 3.2. Let be a weakly contractive mapping and let be chosen arbitrarily. Let be the sequence generated by If , then converges strongly to , where is a sunny nonexpansive retraction from onto .

Proof. It follows from Remark 3.1 that is the sunny nonexpansive retract of . Denote by the sunny nonexpansive retraction of onto . Then is a weakly contractive mapping of into itself. Indeed, Lemma 2.4 assures that there exists a unique element such that . Such a is an element of .
Now we define an iterative scheme as follows: Let be the sequence generated by (3.37). Then, by taking as in , Theorem 3.2 with a constant assures that converges strongly to as . For any , observe that Then we have Thus, for , we obtain the following recursive inequality: Since , it follows from Lemma 2.5 that . Hence This completes the proof.

Corollary 3.7. Let be a uniformly smooth Banach space. Let , , , , , , , , , and be the same as in Theorem 3.6. Then the conclusion of Theorem 3.6 still holds.

Remark 3.8. (1) Theorem 3.6 (and Corollary 3.7) develops and complements the corresponding results in Moudafi [11], Qin et al. [16], Shioji and Takahashi [8], Wittmann [9], Xu [10, 12], and Yao et al. [15].
(2) Even in the case of in Theorem 3.6, Theorem 3.6 appears to be independent of Theorem  5.6 of Wong et al. [13] in which the control conditions (C1) and were utilized. In fact, it appears to be unknown whether a reflexive and strictly convex space satisfies the fixed point property for nonexpansive mappings.
(3) The merits of our results in this paper are that fewer restrictions are imposed on the control parameters , , and . All of our results can be viewed as a supplement to the results obtained by Qin et al. [16], Kim and Xu [14], and Yao et al. [15].
(4) The conclusions of Theorems 3.2 and 3.6 still hold if is assumed to be strictly convex instead of having the fixed point property for nonexpansive mappings.

Acknowledgments

The author thanks the editor Professor Simeon Reich and the referees for their careful reading and valuable suggestions to improve the writing of this paper.

References

  1. F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, no. 1, pp. 82–90, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. S. S. Chang, “On Halpern's open question,” Acta Mathematica Sinica, vol. 48, no. 5, pp. 979–984, 2005. View at: Google Scholar | MathSciNet
  3. Y. J. Cho, S. M. Kang, and H. Zhou, “Some control conditions on iterative methods,” Communications on Applied Nonlinear Analysis, vol. 12, no. 2, pp. 27–34, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  4. B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, no. 6, pp. 957–961, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. P.-L. Lions, “Approximation de points fixes de contractions,” Comptes Rendus de l'Académie des Sciences. Série A-B, vol. 284, no. 21, pp. 1357–1359, 1977. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. S. Reich, “Approximating fixed points of nonexpansive mappings,” PanAmerican Mathematical Journal, vol. 4, no. 2, pp. 23–28, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641–3645, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486–491, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. 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 | MathSciNet
  11. 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 | MathSciNet
  12. 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 | MathSciNet
  13. N. C. Wong, D. R. Sahu, and J. C. Yao, “Solving variational inequalities involving nonexpansive type mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4732–4753, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51–60, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1687–1693, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. X. Qin, Y. Su, and M. Shang, “Strong convergence of the composite Halpern iteration,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 996–1002, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. View at: Zentralblatt MATH | MathSciNet
  18. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984. View at: Zentralblatt MATH | MathSciNet
  19. J. S. Jung and C. H. Morales, “The Mann process for perturbed m-accretive operators in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 46, no. 2, pp. 231–243, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  20. T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  21. L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  22. B. E. Rhoades, “Some theorems on weakly contractive maps,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 4, pp. 2683–2693, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  23. Ya. I. Alber and A. N. Iusem, “Extension of subgradient techniques for nonsmooth optimization in Banach spaces,” Set-Valued Analysis, vol. 9, no. 4, pp. 315–335, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  24. Ya. I. Alber, S. Reich, and J.-C. Yao, “Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2003, no. 4, pp. 193–216, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  25. J. S. Jung, “Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2536–2552, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2009 Jong Soo Jung. 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views474
Downloads499
Citations

Related articles