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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 424875, 10 pages

http://dx.doi.org/10.1155/2014/424875
Research Article

On Solutions of Variational Inequality Problems via Iterative Methods

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana

Received 12 May 2014; Revised 24 June 2014; Accepted 30 June 2014; Published 4 August 2014

Academic Editor: Adrian Petrusel

Copyright © 2014 Mohammed Ali Alghamdi 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 investigate an algorithm for a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of γ-inverse strongly accretive mappings. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.

1. Introduction

Let be a subset of a real Hilbert space . Let be a nonlinear mapping. The variational inequality problem for and is to

The set of solutions of variational inequality problem is denoted by ; that is,

It is well known that variational inequality theory has emerged as an important tool in studying a wide class of numerous problems in variational inequalities, minimax problems, optimization, physics, and the Nash equilibrium problems in noncooperative games. Several numerical methods have been developed for solving variational inequalities and related optimization problems; see, for instance, [15] and the references therein.

A mapping is said to be -inverse strongly accretive (or -inverse strongly monotone) if there exists a positive real number such that

If is -inverse strongly accretive, then inequality (3) implies that is Lipschitzian with constant ; that is, , for all . If in (3) we have that , then is called accretive (or monotone).

Let be a closed and convex subset of a real Hilbert space . A mapping is called a contraction mapping if there exists such that for all . If , then is called nonexpansive. A mapping is called -strictly pseudocontractive of Browder-Petryshyn type [6] if and only if there exists such that is called pseudocontractive if

We note that inequalities (4) and (5) can be equivalently written as for some and respectively. We remark that is pseudocontractive if and only if is accretive. A point is a fixed point of if and we denote by the set of fixed points of ; that is, .

We observe that in a real Hilbert space a class of pseudocontractive mappings includes the class of -strictly pseudocontractive mappings and hence the classes of nonexpansive and contraction mappings.

Closely related to the variational inequality problems is the problem of finding fixed points of nonexpansive mappings, -strict pseudocontraction mappings or pseudocontractive mappings which is the current interest in functional analysis. Several researchers considered a unified approach that approximates a common point of fixed point of nonlinear problems and solutions of variational inequality problems and solutions of variational inequality problems; see, for example, [718] and the references therein.

In [19], Takahashi and Toyoda studied the problem of finding a common point of fixed points of a nonexpansive mapping and solutions of a variational inequality problem (1) by considering the following iterative algorithm: where is a sequence in , is a positive sequence, is a nonexpansive mapping, and is an -inverse strongly accretive mapping. They showed that the sequence generated by (8) converges weakly to some provided that the control sequences satisfy some restrictions.

Iiduka and Takahashi [20] reconsidered the common element problem via the following iterative algorithm: where is a nonexpansive mapping, is a -inverse-strongly accretive mapping, is a sequence in , and is a sequence in . They proved that the sequence strongly converges to some point .

Recently, Zegeye and Shahzad [21] investigated the problem of finding a common point of fixed points of a Lipschitz pseudocontractive mapping and solutions of a variational inequality problem for -inverse strongly accretive mapping by considering the following iterative algorithm: where is a metric projection from onto and are in satisfying certain conditions. Then, they proved that the sequence converges strongly to the minimum-norm point of .

A natural question arises whether we can obtain an iterative scheme which converges strongly to a common point of fixed points of a finite family of pseudocontractive mappings and solutions of a finite family of variational inequality problems for -inverse strongly accretive mappings or not.

It is our purpose in this paper to introduce an algorithm and prove that the algorithm converges strongly to a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of variational inequality problems for -inverse strongly accretive mappings. The results obtained in this paper improve and extend the results of Takahashi and Toyoda [19], Iiduka and Takahashi [20], and Zegeye and Shahzad [21], Theorem 3.2 of Yao et al. [22], and some other results in this direction.

2. Preliminaries

In what follows we will make use of the following lemmas.

Lemma 1. Letting be a real Hilbert space, the following identity holds:

Lemma 2 (see [23]). Let be a nonempty closed and convex subset of a real Hilbert space . Let be a -inverse strongly accretive mapping. Then, for , the mapping is nonexpansive.

Lemma 3 (see [24]). Let be a nonempty, closed, and convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be an accretive operator of into . Then for all ,

Lemma 4 (see [25]). Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be nonexpansive mappings such that . Let with . Then is nonexpansive and .

Lemma 5 (see [26]). Let be a convex subset of a real Hilbert space . Let . Then if and only if

Lemma 6 (see [27]). Let be a closed convex subset of a real Hilbert space and be a continuous pseudo-contractive mapping. Then, for , the mapping is nonexpansive(i) is a closed convex subset of ;(ii) is demiclosed at zero; that is, if is a sequence in such that and , as , then .

Lemma 7 (see [28]). Let be a real Hilbert space. Then for all and for such that the following equality holds:

Lemma 8 (see [29]). Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists an increasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :

In fact, is the largest number in the set such that the condition holds.

Lemma 9 (see [30]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and satisfying the following conditions: , and . Then, .

3. Main Result

For the rest of this paper, let , for some , and , for some , satisfy ( ) ; ( ) ; and ( ) .

Theorem 10. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be Lipschitz pseudocontractive mappings with Lipschitz constants , respectively. Let , for , be -inverse strongly accretive mappings. Let be a contraction with constant . Assume that is nonempty. Let a sequence be generated from an arbitrary by where and , for , for with and , for . Then, converges strongly to a point which is the unique solution of the variational inequality for all .

Proof. From Lemmas 2, 4, and 3 we get that is nonexpansive mapping with . Let . Then from (17), (5), and Lemma 7 we have that

Now, substituting (18) in (19) we get that

Moreover, from (17), Lemma 7, and Lipschitz property of we get that

Substituting (21) into (20) we obtain that

But, from the hypothesis we have that and hence inequality (22) gives that

But we have that

Substituting (25) into (24) we get that

Therefore, by induction we get that which implies that and hence are bounded.

Let . Then, from (17), Lemmas 1 and 7, and the methods used to get (22) we obtain that which implies that

But

Thus, substituting (31) in (30) we obtain that

Next, we consider two cases.

Case 1. Suppose that there exists such that is decreasing for all . Then, we get that is convergent. Thus, from (29) and (23) we have that

Furthermore, from (17) and (33) we obtain that and hence Lipschitz continuity of , (34), and (33) implies that

Thus, from (33) and (35) we have that

Therefore, , as , for all , and hence as , for all .

Now, since is bounded subset of , we can choose a subsequence of such that and . Then, from (37) and Lemma 6 we have that , for each . Hence, .

In addition, since is nonexpansive, from Lemma 6 we get that and hence by Lemmas 4 and 3 we obtain that , for each .

Therefore, by Lemma 5, we immediately obtain that

Then, it follows from (32), (38), and Lemma 9 that as . Consequently, .

Case 2. Suppose that there exists a subsequence of such that for all . Then, by Lemma 8, there exists a nondecreasing sequence such that , and for all . Now, from (29) and (23) we get that and as . Thus, following the method in Case 1, we obtain that , , and

Furthermore, from (32) and (40) we obtain that

Now, using the fact that and (41) we get that and this together with (32) implies that as . Since for all , we obtain that . Hence, from the above two cases, we can conclude that converges strongly to a point , which satisfies the variational inequality , for all . The proof is complete.

If, in Theorem 10, we assume that , a constant mapping, then we get the following corollary.

Corollary 11. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be Lipschitz pseudocontractive mappings with Lipschitz constants , respectively. Let , for , be -inverse strongly accretive mappings. Assume that is nonempty. Let a sequence be generated from an arbitrary by where , , for , for with , and , for . Then, converges strongly to a unique point satisfying , which is the unique solution of the variational inequality for all .

If, in Theorem 10, we assume that and , then we get the following corollary which is Theorem 3.1 of [21].

Corollary 12. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be Lipschitz pseudocontractive mappings with Lipschitz constant and an -inverse strongly accretive mapping. Let be a contraction with constant . Assume that is nonempty. Let a sequence be generated from an arbitrary by where and . Then, converges strongly to a point , which is the unique solution of the variational inequality for all .

If, in Theorem 10, we assume that are strictly pseudocontractive mappins, then we get the following corollary.

Corollary 13. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be -strictly pseudocontractive mappings and let , for , be an -inverse strongly accretive mappings. Let be a contraction with constant . Assume that is nonempty. Let a sequence be generated from an arbitrary by where , , for , for with , and , . Then, converges strongly to a point , which is the unique solution of the variational inequality for all .

If, in Theorem 10, we assume that are nonexpansive mapping, then we get the following corollary.

Corollary 14. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be nonexpansive mappings and let , for , be an -inverse strongly accretive mappings. Let be a contraction with constant . Assume that is nonempty. Let a sequence be generated from an arbitrary by where , , for , for with , , and . Then, converges strongly to point , which is the unique solution of the variational inequality for all .

We note that the method of proof of Theorem 10 provides the following theorem which is a convergence theorem for a minimum norm point of common fixed points of a finite family of Lipschitz pseudocontractive mappings and common solutions of a finite family of variational inequality problems for accretive mappings.

Theorem 15. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be Lipschitz pseudocontractive mappings with Lipschitz constants , respectively. Let , for , be -inverse strongly accretive mappings. Assume that is nonempty. Let a sequence be generated from an arbitrary by where , , for , for with , and , for . Then, converges strongly to a unique minimum norm point of i.e., , which is the unique solution of the variational inequality for all .

4. Numerical Example

Now, we give an example of two Lipschitz pseudocontractive mappings and two -inverse strongly accretive mappings satisfying Theorem 10 and some numerical experiment result to explain the conclusion of the theorem as follows.

Example 1. Let with absolute value norm. Let and let be defined by

Clearly, for we have that which show that both mappings are pseudocontractive. Next, we show that is Lipschitz with . If , then

If , then

If and , then

Thus, we get that is Lipschitz pseudocontractive with and which is not nonexpansive, since if we take and , we have that . Similarly, we can show that is Lipschitz pseudocontractive with and which is not nonexpansive.

Furthermore, for , let be defined by

Then we first show that is -inverse strongly accretive mapping with .

If , then

If and , we get that

If , then we get that and hence

Therefore, is -inverse strongly accretive mapping with and . Similarly, we can show that is -inverse strongly accretive mapping with and .

Note that we have .

Thus, taking , , , , and , we observe that conditions of Theorem 10 are satisfied and Scheme (17) provides the following Table 1 and Figures 1(a) and 1(b) for and , respectively.

We observe that the data provides strong convergence of the sequence to the common point of fixed points of both pseudocontractive mappings and solutions of both variational inequality problems for -inverse strongly accretive mappings.

tab1
Table 1
fig1
Figure 1

Remark 2. Theorem 10 provides an iteration scheme which converges strongly to a common point of fixed points of a finite family of Lipschitzian pseudocontractive mappings and solutions of a finite family of variational inequality problems in Hilbert spaces.

Remark 3. Theorem 10 improves Theorem 3.1 of Takahashi and Toyoda [19], Iiduka and Takahashi [20], and Zegeye and Shahzad [21] and Theorem 3.2 of Yao et al. [22] in the sense that our convergence is to a common point of fixed points of a finite family of Lipschitzian pseudocontractive mappings and solutions of a finite family of variational inequality problems.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 167/130/1434. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors also thank the referees for their valuable comments and suggestions, which improved the presentation of this paper.

References

  1. D. Kinderlehrer and G. Stampaccia, An Iteration to Variational Inequalities and Their Applications, Academic Press, New York, NY, USA, 1990.
  2. H. Zegeye and N. Shahzad, “A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 1, pp. 263–272, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. H. Zegeye and N. Shahzad, “Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces,” Optimization Letters, vol. 5, no. 4, pp. 691–704, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. H. Zegeye and N. Shahzad, “Strong convergence theorems for variational inequality problems and quasi-ϕ-asymptotically nonexpansive mappings,” Journal of Global Optimization, vol. 54, no. 1, pp. 101–116, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. H. Zegeye and N. Shahzad, “An iteration to a common point of solution of variational inequality and fixed point-problems in Banach spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 504503, 19 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  6. F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. L. C. Ceng, A. Petruşel, and J. C. Yao, “Composite viscosity approximation methods for equilibrium problem, variational inequality and common fixed points,” Journal of Nonlinear and Convex Analysis, vol. 15, no. 2, pp. 219–240, 2014. View at MathSciNet
  8. L. C. Ceng, A. Petrusel, M. M. Wong, and J. C. Yao, “Hybrid algorithms for solving variational inequalities, variational inclusions, mixed equilibria, and fixed point problems,” Abstract and Applied Analysis, vol. 2014, Article ID 208717, 22 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  9. L. Ceng, A. Petrusel, and M. Wong, “Strong convergence theorem for a generalized equilibrium problem and a pseudocontractive mapping in a Hilbert space,” Taiwanese Journal of Mathematics, vol. 14, no. 5, pp. 1881–1901, 2010. View at MathSciNet · View at Scopus
  10. D. R. Sahu, V. Colao, and G. Marino, “Strong convergence theorems for approximating common fixed points of families of nonexpansive mappings and applications,” Journal of Global Optimization, vol. 56, no. 4, pp. 1631–1651, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. D. R. Sahu and A. Petruşel, “Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 17, pp. 6012–6023, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. Y. Yao, Y. Liou, and N. Shahzad, “Construction of iterative methods for variational inequality and fixed point problems,” Numerical Functional Analysis and Optimization, vol. 33, no. 10, pp. 1250–1267, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Y. Yao, G. Marino, and L. Muglia, “A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality,” Optimization, vol. 63, no. 4, pp. 559–569, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Y. Yao and M. Postolache, “Iterative methods for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 155, no. 1, pp. 273–287, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. H. Zegeye and N. Shahzad, “A hybrid approximation method for equilibrium, variational inequality and fixed point problems,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 4, pp. 619–630, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. H. Zegeye and N. Shahzad, “Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems,” Optimization, vol. 63, no. 2, pp. 207–223, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  18. H. Zegeye and N. Shahzad, “Extragradient method for solutions of variational inequality problems in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 832548, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 3, pp. 341–350, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. H. Zegeye and N. Shahzad, “Solutions of variational inequality problems in the set of fixed points of pseudocontractive mappings,” Carpathian Journal of Mathematics, vol. 30, no. 2, pp. 257–265, 2014.
  22. Y. Yao, Y. C. Liou, and S. M. Kang, “Algorithms construction for variational inequalities,” Fixed Point Theory and Applications, vol. 2011, Article ID 794203, 12 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  23. Y. Cai, Y. Tang, and L. Liu, “Iterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert space,” Fixed Point Theory and Applications, vol. 2012, article 49, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. K. Aoyama, H. Iiduka, and W. Takahashi, “Weak convergence of an iterative sequence for accretive operators in Banach spaces,” Fixed Point Theory and Applications, vol. 2006, Article ID 35390, 13 pages, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  25. C. E. Chidume, H. Zegeye, and E. Prempeh, “Strong convergence theorems for a finite family of nonexpansive mappings in Banach spaces,” Communications on Applied Nonlinear Analysis, vol. 11, no. 2, pp. 25–32, 2004.
  26. Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
  27. Q. B. Zhang and C. Z. Cheng, “Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space,” Mathematical and Computer Modelling, vol. 48, no. 3-4, pp. 480–485, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. H. Zegeye and N. Shahzad, “Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings,” Computers and Mathematics with Applications, vol. 62, no. 11, pp. 4007–4014, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. P. E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. H. K. Xu, “Another control condition in an iterative method for nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 65, no. 1, pp. 109–113, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus