About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 153456, 16 pages
http://dx.doi.org/10.1155/2012/153456
Research Article

Some Algorithms for Finding Fixed Points and Solutions of Variational Inequalities

Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Received 1 December 2011; Accepted 9 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 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.

Abstract

We introduce new implicit and explicit algorithms for finding the fixed point of a k-strictly pseudocontractive mapping and for solving variational inequalities related to the Lipschitzian and strongly monotone operator in Hilbert spaces. We establish results on the strong convergence of the sequences generated by the proposed algorithms to a fixed point of a k-strictly pseudocontractive mapping. Such a point is also a solution of a variational inequality defined on the set of fixed points. As direct consequences, we obtain the unique minimum-norm fixed point of a k-strictly pseudocontractive mapping.

1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed convex subset of and a self-mapping on . We denote by the set of fixed points of . We recall that a mapping is said to be -strictly pseudocontractive if there exists a constant such that

Note that the class of -strictly pseudocontractive mappings includes the class of nonexpansive mappings on (i.e., ) as a subclass. That is, is nonexpansive if and only if is 0-strictly pseudocontractive. Recently, many authors have been devoting their studies to the problems of finding fixed points for -strictly pseudocontractive mappings; see [15] and the references therein.

Variational inequalities have been studied widely and are being used as a mathematical programming tool in modeling a wide class of problems arising in several branches of pure and applied sciences; see [68]. For general variational inequalities and extended general variational inequalities, we can refer to [913] and references therein.

A variational inequality (VI) is formulated as finding a point with the property where is a nonlinear mapping. It is well known that VI (1.2) is equivalent to the fixed point equation where and is the the metric projection of onto , which assigns, to each , the unique point in , denoted by , such that

Therefore, fixed point algorithms can be applied to solve VI (1.2). It is also well known that if is -Lipschitzian and -strongly monotone with constants (i.e., there exist such that and , resp.), then, for small enough , the mapping is a contractive mapping on and so the sequence of Picard iterates, given by , converges strongly to the unique solution of VI (1.2).

This sort of VI (1.2) where is -Lipschitzian and -strongly monotone and where solutions are sought from the set of fixed points of a nonexpansive mapping is originated from Yamada [14], who provided the hybrid method for solving VI (1.2). In order to find solutions of ceratin variational inequality problems defined on the set of fixed points of nonexpansive mappings, several iterative algorithms were studied by many authors; see [1524] and the references therein.

In this paper, we investigate the following variational inequality (VI) as a special form of VI (1.2), where the constraint set is the fixed points of a -strictly pseudocontractive mapping : finding a point with property where is a -strictly pseudocontractive mapping with for some , is a -Lipschitzian and -strongly monotone mapping with constants , and is an -Lipschitzian mapping with constant and . Indeed, variational inequalities of form (1.5) cover several topics recently considered in the literature, including monotone inclusions, convex optimization, and quadratic minimization over fixed point sets; see [24, 15, 18, 19, 21, 22, 24] and the references therein. For some iterative methods and some results related to our approach about VI (1.5), we can refer to [2531] and references therein.

The main purpose of the present paper is to further study the hierarchical fixed point approach to the VI of form (1.5). First, we introduce new implicit and explicit algorithms for finding the fixed point of the -strictly pseudocontractive mapping . Then, we establish results on the strong convergence of the sequences generated by the proposed algorithms to a fixed point of the mapping , which is also a solution of VI (1.5) defined on the set of fixed points of . As direct consequences, we obtain the unique minimum-norm fixed point of . Namely, we find the unique solution of the quadratic minimization problem: .

2. Preliminaries and Lemmas

Throughout this paper, when is a sequence in , (resp., ) denotes strong (resp., weak) convergence of the sequence to .

Let be a nonempty closed convex subset of a real Hilbert space . Recall that is called a contractive mapping with constant if there exists a constant such that .

For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of to . It is well known that is nonexpansive and that, for ,

In a Hilbert space , we have

We need the following lemmas for the proof of our main results.

Lemma 2.1. In a real Hilbert space , the following inequality holds:

Lemma 2.2 (see [32]). Let be a sequence of nonnegative real numbers satisfying where and satisfy the following conditions:(i),(ii),(iii) or .Then, .

Lemma 2.3 (see [33]). Let and be bounded sequences in a Banach space and a sequence in that satisfies the following condition: Suppose that   for all and Then, .

Lemma 2.4 (Demiclosedness principle [34]). Let be a nonempty closed convex subset of a real Hilbert space and a nonexpansive mapping with . If is a sequence in weakly converging to and converges strongly to , then ; in particular, if , then .

Lemma 2.5 (see [14, 16]). Let be a nonempty closed convex subset of a real Hilbert space . Assume that the mapping is monotone and weakly continuous along segments, that is, weakly as . Then, the variational inequality is equivalent to the dual variational inequality

Lemma 2.6 (see [5]). Let be a real Hilbert space and a closed convex subset of . If is a - strictly pseudocontractive mapping on , then the fixed point set is closed convex, so that the projection is well defined.

Lemma 2.7 (see [5]). Let be a Hilbert space, a closed convex subset of , and a - strictly pseudocontractive mapping. Define a mapping by for all . Then, as , is a nonexpansive mapping such that .

The following lemma can be easily proven, and, therefore, we omit the proof.

Lemma 2.8. Let be a real Hilbert space. Let be an -Lipschitzian mapping with constant and a -Lipschitzian and -strongly monotone mapping with constants . Then, for , That is, is strongly monotone with constant .

The following lemma is an improvement of Lemma  2.9 in [4] (see also [14]).

Lemma 2.9. Let be a real Hilbert space . Let be a -Lipschitzian and -strongly monotone mapping with constants . Let and . Then, is a contractive mapping with constant , where .

Proof. First we show that is strictly contractive. In fact, by applying the -Lipschitz continuity and -strongly monotonicity of and (2.3), we obtain, for , and so Now, noting that , from (2.12), we have, for , Hence, is a contractive mapping with constant .

3. Iterative Algorithms

Let be a real Hilbert space and a nonempty closed convex subset of . Let be a -strictly pseudocontractive mapping with for some , a -Lipschitzian and -strongly monotone mapping with constants , and an -Lipschitzian mapping with constant . Let and , where . Let be a mapping defined by and a metric projection of onto .

In this section, we introduce the following algorithm that generates a net in an implicit way: We prove the strong convergence of as to a fixed point of , which is a solution of the following variational inequality: We also propose the following explicit algorithm, which generates a sequence in an explicit way: where , and is an arbitrary initial guess, and we establish the strong convergence of this sequence to a fixed point of , which is also a solution of the variational inequality (3.2).

3.1. Strong Convergence of the Implicit Algorithm

Now, for , consider a mapping defined by It is easy to see that is a contractive mapping with constant . Indeed, by Lemma 2.9, we have Hence, has a unique fixed point, denoted by , which uniquely solves the fixed point equation (3.1).

We summarize the basic properties of .

Proposition 3.1. Let be a real Hilbert space and a nonempty closed convex subset of . Let be a -strictly pseudocontractive mapping with for some , a -Lipschitzain and -strongly monotone mapping with constants , and an -Lipschitzian mapping with constant . Let and , where . Let be a mapping defined by and a metric projection of onto . Let be defined via (3.1). Then,(i) is bounded for ,(ii),(iii) defines a continuous path from in .

Proof. (i) Let . Observing by Lemma 2.7, we have So, it follows that Hence, is bounded and so are , , and .
(ii) By the boundedness of and in (i), we have
(iii) Let , and calculate It follows that This shows that is locally Lipschitzian and hence continuous.

We establish the strong convergence of the net as , which guarantees the existence of solutions of the variational inequality (3.2).

Theorem 3.2. Let be a real Hilbert space and a nonempty closed convex subset of . Let be a -strictly pseudocontractive mapping with for some , a -Lipschitzain and -strongly monotone mapping with constants , and an -Lipschitzian mapping with constant . Let and , where . Let be a mapping defined by and a metric projection of onto . The net defined via (3.1) converges strongly to a fixed point of as , which solves the variational inequality (3.2), or, equivalently, one has .

Proof. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of . In fact, noting that and , it follows from Lemma 2.8 that That is, is strongly monotone for . Suppose that and both are solutions to (3.2). Then, we have Adding up (3.12) yields The strong monotonicity of implies that and the uniqueness is proved.
Next, we prove that as . To this end, set for all . Then, observing by Lemma 2.7, we have and for any Also it follows that Since is the metric projection from onto , we have, for given , It follows that By (3.15), this implies and hence Since is bounded as (by Proposition 3.1 (i)), we see that if is a subsequence in such that and , then, from (3.16), we have . By Proposition 3.1 (ii), . By Lemmas 2.4 and 2.7, . Therefore, we can substitute for in (3.20) to obtain Consequently, the weak convergence of to yields that strongly. Now we show that solves the variational inequality (3.2). Again, observe (3.20) and take the limit as to obtain Hence solves the following variational inequality: or the equivalent dual variational inequality (see Lemmas 2.5 and 2.8) Moreover, if is another subsequence in such that and , then we also have from (3.16). By the same argument, we can show that and solves the variational inequality (3.2); hence by uniqueness. In sum, we have shown that each cluster point of (at ) equals . Therefore as .
The variational inequality (3.2) can be rewritten as By reminding the reader of (2.2) and Lemma 2.6, this is equivalent to the fixed point equation

From Theorem 3.2, we can deduce the following result.

Corollary 3.3. Let be a real Hilbert space and a nonempty closed convex subset of . Let be a -strictly pseudocontractive mapping with for some . For each , let the net be defined by where is a mapping defined by and is a metric projection of onto . Then the net defined via (3.27) converges strongly, as , to the minimum-norm point .

Proof. In (3.20) with , , , , and , letting yields Equivalently, This obviously implies that It turns out that for all . Therefore, is minimum-norm point of .

3.2. Strong Convergence of the Explicit Algorithm

Now, using Theorem 3.2, we show the strong convergence of the sequence generated by the explicit algorithm (3.3) to a fixed point of , which is also a solution of the variational inequality (3.2).

Theorem 3.4. Let be a real Hilbert space and a nonempty closed convex subset of . Let be a -strictly pseudocontractive mapping with for some , a -Lipschitzain and -strongly monotone mapping with constants , and an -Lipschitzian mapping with constant . Let and , where . Let be a mapping defined by and a metric projection of onto . For any given , let be the sequence generated by the explicit algorithm (3.3), where and satisfy the following conditions:(C1),(C2),(C3).
Then, converges strongly to , which is the unique solution of the variational inequality (3.2).

Proof. First, from condition (C1), without loss of generality, we assume that and for . From now, we put and .
We divide the proof several steps.
Step 1. We show that for all and all . Indeed, let . Then, from Lemma 2.9, we have Thus, it follows that Using an induction, we have . Hence, is bounded, and so are , , , , and .Step 2. We show that . Indeed, from (3.3), we observe where . Thus, it follows that which implies, from condition (C1), that Hence, by Lemma 2.3, we have Consequently, from condition (C3), it follows that Step 3. We show that . Indeed, we have that is, This together with conditions (C1) and (C3) and Step 2 implies Step 4. We show that where with being defined by (3.1). (We note that, from Theorem 3.2, and is the unique solution of the variational inequality (3.2)). To show this, we can choose a subsequence of such that Since is bounded, there exists a subsequence of , which converges weakly to . Without loss of generality, we can assume that . Since by Step 3, we obtain by virtue of Lemma 2.4. From Lemma 2.7, we have . Therefore, from (3.2), it follows that We notice that, by condition (C1), Hence, from (3.43), we obtain Step 5. We show that , where with being defined by (3.1), and is the unique solution of the variational inequality (3.2). Indeed, we observe that Therefore, from the convexity of , (3.3), and Lemma 2.1, we have where , , , , and From conditions (C1), (C2), and (C3) and Step 4, it is easy to see that , and . Hence, by Lemma 2.2, we conclude that as . This completes the proof.

From Theorem 3.4, we can also deduce the following result.

Corollary 3.5. Let be a real Hilbert space and a nonempty closed convex subset of . Let be a -strictly pseudocontractive mapping with for some . For each , let the sequence be defined by where is a mapping defined by and is a metric projection of onto . If and satisfy conditions (C1), (C2), and (C3) in Theorem 3.4, then the sequence defined via (3.49) converges strongly, as , to the minimum-norm point .

Proof. VI (3.2) is reduced to the inequality This is equivalent to for all . It turns out that for all and is the minimum-norm point of .

Remark 3.6. We point out that our algorithms (3.1) and (3.3) are new ones different from those in the literature (see [24, 15, 18, 21, 22] and references therein).

4. Conclusion and Future Directions

In this paper, we have introduced new implicit and explicit algorithms for finding fixed points of a -strictly pseudocontractive mapping and for solving a certain variational inequality and have established strong convergence of the proposed algorithms to a fixed point of the mapping, which is a solution of a certain variational inequality, where the constraint set is the fixed points of the mapping. As direct consequences, we have considered the quadratic minimization problem on the set of fixed points of the mapping.

In forthcoming studies, we will consider implicit and explicit algorithms for solving some variational inequalities, where the constraint set is the common set of the set of fixed points of the mapping and the set of solutions of the equilibrium problem.

We hope that the ideas and techniques of this paper may stimulate further research in this field.

Acknowledgments

The author thanks the referees for their valuable comments and suggests, which improved the presentation of this paper, and for providing some recent related papers. This study was supported by research funds from Dong-A University.

References

  1. G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, vol. 67, no. 7, pp. 2258–2271, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. Y. J. Cho, S. M. Kang, and X. Qin, “Some results on k-pseudo-contractive mappings in Hilbert spaces,” Nonlinear Analysis, vol. 70, no. 5, pp. 1956–1964, 2009. View at Publisher · View at Google Scholar
  3. J. S. Jung, “Strong convergence of iterative methods for k-strictly pseudo-contractive mappings in Hilbert spaces,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3746–3753, 2010. View at Publisher · View at Google Scholar
  4. J. S. Jung, “Some results on a general iterative method for k-strictly pseudo-contractive mappings,” Fixed Point Theory and Applications, vol. 2011, article 24, 2011. View at Publisher · View at Google Scholar
  5. H. Zhou, “Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, vol. 69, no. 2, pp. 456–462, 2008. View at Publisher · View at Google Scholar
  6. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Problems, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1984.
  7. F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, NY, USA, 1995.
  8. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 88 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1980.
  9. M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis. In press.
  10. M. Aslam Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182–186, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative methods for solving nonconvex equilibrium variational inequalities,” Applied Mathematics & Information Sciences, vol. 6, no. 1, pp. 65–69, 2010.
  13. M. A. Noor, K. I. Noor, Z. Y. Huang, and E. Al-Said, “Implicit schemes for solving extended general nonconvex vatiational inequalities,” Journal of Applied Mathematics, vol. 2012, Article ID 646259, 10 pages, 2012. View at Publisher · View at Google Scholar
  14. I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of 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., vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001. View at Zentralblatt MATH
  15. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Some iterative methods for finding fixed points and for solving constrained convex minimization problems,” Nonlinear Analysis, vol. 74, no. 16, pp. 5286–5302, 2011. View at Publisher · View at Google Scholar
  16. X. Lu, H.-K. Xu, and X. Yin, “Hybrid methods for a class of monotone variational inequalities,” Nonlinear Analysis, vol. 71, no. 3-4, pp. 1032–1041, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. P.-E. Maingé and A. Moudafi, “Strong convergence of an iterative method for hierarchical fixed-point problems,” Pacific Journal of Optimization, vol. 3, no. 3, pp. 529–538, 2007. View at Zentralblatt MATH
  18. G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. G. Marino and H.-K. Xu, “Explicit hierarchical fixed point approach to variational inequalities,” Journal of Optimization Theory and Applications, vol. 149, no. 1, pp. 61–78, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. A. Moudafi and P.-E. Maingé, “Towards viscosity approximations of hierarchical fixed-point problems,” Fixed Point Theory and Applications, vol. 2006, Article ID 646259, 10 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis, vol. 73, no. 3, pp. 689–694, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. M. Tian, “A general iterative method based on the hybrid steepest descent scheme for nonexpansive mappings in Hilbert spaces,” in Proceedings of the International Conference on Computational Intelligence and Software Engineering (CiSE '10), pp. 1–4, December 2010. View at Publisher · View at Google Scholar
  23. H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. Y. Yao, R. Chen, and H.-K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis, vol. 72, no. 7-8, pp. 3447–3456, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. Y. Yao, Y. J. Cho, and Y.-C. Liou, “Hierarchical convergence of an implicit double-net algorithms for nonexpansive semiroups and variational inequalities problems,” Fixed Point Theory and Applications, vol. 2011, article 101, 2011.
  26. Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1506–1515, 2012. View at Publisher · View at Google Scholar
  27. Y. Yao, W. Jiang, and Y.-C. Liou, “Regularized methods for the split feasibility problem,” Abstract and Applied Analysis. In press. View at Publisher · View at Google Scholar
  28. Y. Yao, M. A. Noor, and Y.-C. Liou, “Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalites,” Abstract and Applied Analysis, vol. 2012, Article ID 817436, 9 pages, 2012. View at Publisher · View at Google Scholar
  29. Y. Yao, M. A. Noor, Y.-C. Liou, and S. M. Kang, “Iterative algorithms for general multivalued variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 768277, 10 pages, 2012. View at Publisher · View at Google Scholar
  30. Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In press. View at Publisher · View at Google Scholar
  31. Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters. In press. View at Publisher · View at Google Scholar
  32. H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. 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 · View at Google Scholar · View at Zentralblatt MATH
  34. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar