Journal of Applied Mathematics

Volume 2012 (2012), Article ID 634927, 15 pages

http://dx.doi.org/10.1155/2012/634927

## Hybrid Algorithms of Nonexpansive Semigroups for Variational Inequalities

^{1}Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China^{2}Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

Received 19 March 2012; Accepted 30 April 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Peixia Yang 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

Two hybrid algorithms for the variational inequalities over the common fixed points set of nonexpansive semigroups are presented. Strong convergence results of these two hybrid algorithms have been obtained in Hilbert spaces. The results improve and extend some corresponding results in the literature.

#### 1. Introduction

Let be a real Hilbert space and a nonempty closed convex subset of . Recall that a mapping is called nonexpansive if for every . A family of mappings from into itself is called a nonexpansive semigroup on if it satisfies the following conditions:(i) for all ,(ii) for all ,(iii) for all ,(iv)for all is continuous.We denote by the set of all common fixed points of , that is, . It is known that is closed and convex.

Approximation of fixed points of nonexpansive mappings has been considered extensively by many authors, see, for instance, [1–18]. Nonlinear ergodic theorem for nonexpansive semigroups have been researched by some authors, see, for example, [19–23]. Our main purpose in the present paper is devoted to finding the common fixed points of nonexpansive semigroups.

Let be a nonlinear operator. The variational inequality problem is formulated as finding a point such that Now it is well known that VI problem is an interesting problem and it covers as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance. Several numerical methods including the projection and its variant forms have been developed for solving the variational inequalities and related problems, see [24–41].

It is clear that the is equivalent to the fixed point equation where is the projection of onto the closed convex set and is an arbitrarily fixed constant. So, fixed point methods can be implemented to find a solution of the provided satisfies some conditions and is chosen appropriately. The fixed point formulation (1.3) involves the projection , which may not be easy to compute, due to the complexity of the convex set . In order to reduce the complexity probably caused by the projection , Yamada [24] (see also [42]) recently introduced a hybrid steepest-descent method for solving the .

Assume that is an -strongly monotone and -Lipschitzian mapping with on . An equally important problem is how to find an approximate solution of the if any. A great deal of effort has been done in this problem; see [43, 44].

Take a fixed number such that . Assume that a sequence of real numbers in satisfies the following conditions:(C1),(C2),(C3).

Starting with an arbitrary initial guess , one can generate a sequence by the following algorithm: Yamada [24] proved that the sequence generated by (1.4) converges strongly to the unique solution of the . Xu and Kim [30] proved the strong convergence of to the unique solution of the if satisfies conditions (C1), (C2), and (C4): , or equivalently, . Recently, Yao et al. [25] presented the following hybrid algorithm: where is a -Lipschitzian and -strongly monotone operator on and is a -mapping. It is shown that the sequences and defined by (1.5) converge strongly to , which solves the following variational inequality: Very recently, Wang [26] proved that the sequence generated by the iterative algorithm (1.5) converges to a common fixed point of an infinite family of nonexpansive mappings under some weaker assumptions.

Motivated and inspired by the above works, in this paper, we introduce two hybrid algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces. We prove that the presented algorithms converge strongly to a common fixed point of . Such common fixed point is the unique solution of some variational inequality in Hilbert spaces.

#### 2. Preliminaries

In this section, we will collect some basic concepts and several lemmas that will be used in the next section.

Suppose that is a real Hilbert space with inner product and norm . For the sequence in , we write to indicate that the sequence converges weakly to . means that converges strongly to . We denote by the weak -limit set of , that is Let be a nonempty closed convex subset of a real Hilbert space . A mapping is called -Lipschitzian if there exists a positive constant such that is said to be -strongly monotone if there exists a positive constant such that The following equalities are well known: for all and (see [45]).

In the sequel, we will make use of the following well-known lemmas.

Lemma 2.1 (see [46]). *Let be a nonempty bounded closed convex subset of and let be a nonexpansive semigroup on . Then, for any ,
*

Lemma 2.2 (see [47]). * Assume that is a nonexpansive mapping. If has a fixed point, then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here, is the identity operator of . *

Lemma 2.3 (see [27]). * Let be a real sequence satisfying . Assume that and are bounded sequences in Banach space , which satisfy the following condition: . If , then .*

Lemma 2.4 (see [48]). * Let be a -Lipschitzian and -strongly monotone operator on a Hilbert space with and . Then, is a contraction with contraction coefficient . *

Lemma 2.5 (see [49]). * Let be a sequence of nonnegative real numbers satisfying
**
where and satisfy the following conditions:*(i)* and ,*(ii)*,
*(iii)*. **Then, .*

#### 3. Main Results

In this section we will show our main results.

Theorem 3.1. *Let be a real Hilbert space. Let be a nonexpansive semigroup such that . Let be a -Lipschitzian and -strongly monotone operator on with . Let be a continuous net of positive real numbers such that . Putting , for each , let the net be defined by the following implicit scheme:
**
Then, as , the net converges strongly to a fixed point of , which is the unique solution of the following variational inequality:
*

*Proof. * First, we note that the net defined by (3.1) is well defined. We define a mapping
It follows that
Obviously, is a contraction. Indeed, from Lemma 2.4, we have
for all . So it has a unique fixed point. Therefore, the net defined by (3.1) is well defined.

We prove that is bounded. Taking and using Lemma 2.4, we have
It follows that
Observe that
Thus, (3.7) and (3.8) imply that the net is bounded for small enough . Without loss of generality, we may assume that the net is bounded for all . Consequently, we deduce that is also bounded.

On the other hand, from (3.1), we have
This together with Lemma 2.1 implies that
Let be a sequence such that as . Put . Since is bounded, without loss of generality, we may assume that converges weakly to a point . Noticing (3.10), we can use Lemma 2.2 to get .

Again, from (3.1), we have
Therefore,
It follows that
Thus, implies that .

Again, from (3.12), we obtain
It is clear that , , and . We deduce immediately from (3.14) that
which is equivalent to its dual variational inequality
That is, is a solution of the variational inequality (3.2).

Suppose that and both are solutions to the variational inequality (3.2); then
Adding up (3.17) and the last inequality yields

The strong monotonicity of implies that and the uniqueness is proved. Later, we use to denote the unique solution of (3.2).

Therefore, by uniqueness. In a nutshell, we have shown that each cluster point of equals . Hence as . This completes the proof.

Next we introduce an explicit algorithm for finding a solution of the variational inequality (3.2).

*Algorithm 3.2. *For given arbitrarily, define a sequence iteratively by
where and are sequences in and is a sequence in .

Theorem 3.3 3.3. *Let be a real Hilbert space. Let be a -Lipschitzian and -strongly monotone operator on with . Let be a nonexpansive semigroup with . Assume that*(i)* and ,*(ii)* and ,*(iii)*, for some .**Then, the sequences and generated by (3.19) converge strongly to if and only if , where solves the variational inequality (3.2). *

*Proof. *The necessity is obvious. We only need to prove the sufficiency. Suppose that . First, we show that is bounded. In fact, letting , we have
From condition (i), without loss of generality, we can assume that for all . By (3.19) and Lemma 2.4, we have
where .

Then, from (3.20) and (3.21), we obtain
Since , we have by induction
where . Hence, is bounded. We also obtain that , and are all bounded.

Define for all . Observe that
Next, we estimate . As a matter of fact, we have
where . From (3.24) and (3.25), we have
Namely,
Since and , we get
Consequently, by Lemma 2.3, we deduce . Therefore,

Next, we claim that . Observe that
Note that
It follows that
By Lemma 2.1, (3.30), and (3.32), we derive

Next, we show that , where and is defined by . Since is bounded, there exists a subsequence of that converges weakly to . It is clear that . From Lemma 2.2, we have . Hence, by Theorem 3.1, we have

Finally, we prove that converges strongly to . From (3.19), we have
where and . Obviously, we can see that and . Hence, all conditions of Lemma 2.5 are satisfied. Therefore, we immediately deduce that the sequence converges strongly to .

Observe that

Consequently, it is clear that converges strongly to . From and Theorem 3.1, we get that is the unique solution of the variational inequality
This completes the proof.

#### Acknowledgments

Y. Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Y.-C. Liou was partially supported by NSC 100-2221-E-230-012. R. Chen was supported in part by NSFC 11071279.

#### References

- 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, pp. 82–90, 1967. View at Google Scholar - B. Halpern, “Fixed points of nonexpanding maps,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 957–961, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-B. Baillon, “Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert,”
*Comptes Rendus de l'Académie des Sciences*, vol. 280, no. 22, pp. A1511–A1514, 1975. View at Google Scholar · View at Zentralblatt MATH - H. Brézis and F. E. Browder, “Nonlinear ergodic theorems,”
*Bulletin of the American Mathematical Society*, vol. 82, no. 6, pp. 959–961, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Brézis and F. E. Browder, “Remarks on nonlinear ergodic theory,”
*Advances in Mathematics*, vol. 25, no. 2, pp. 165–177, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 - 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 · View at Google Scholar · View at Zentralblatt MATH - S.-S. Chang, “Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 323, no. 2, pp. 1402–1416, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P.-E. Maingé, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 1, pp. 469–479, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 · View at Google Scholar · View at Zentralblatt MATH - L.-C. Zeng and J.-C. Yao, “Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 64, no. 11, pp. 2507–2515, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, R. Chen, and Y. C. Liou, “A unified implicit algorithm for solving the triplehierarchical constrained optimization problem,”
*Mathematical and Computer Modelling*, vol. 55, pp. 1506–1515, 2012. View at Google Scholar - Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,”
*Optimization Letters*, vol. 6, no. 4, pp. 621–628, 2012. View at Publisher · View at Google Scholar - Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert spaces,”
*Fixed Point Theory and Applications*, vol. 2011, article 79, 2011. View at Google Scholar - Y. Yao and J.-C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,”
*Applied Mathematics and Computation*, vol. 186, no. 2, pp. 1551–1558, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Plubtieng and R. Wangkeeree, “Strong convergence of modified Mann iterations for a countable family of nonexpansive mappings,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 9, pp. 3110–3118, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. J. Cho and X. Qin, “Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces,”
*Journal of Computational and Applied Mathematics*, vol. 228, no. 1, pp. 458–465, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Petruşel and J.-C. Yao, “Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 4, pp. 1100–1111, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Cianciaruso, G. Marino, and L. Muglia, “Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces,”
*Journal of Optimization Theory and Applications*, vol. 146, no. 2, pp. 491–509, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Zegeye and N. Shahzad, “Strong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 1, pp. 325–329, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. Buong, “Strong convergence theorem for nonexpansive semigroups in Hilbert space,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 12, pp. 4534–4540, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. T.-M. Lau and W. Takahashi, “Fixed point properties for semigroup of nonexpansive mappings on Fréchet spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 11, pp. 3837–3841, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. T.-M. Lau, H. Miyake, and W. Takahashi, “Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 67, no. 4, pp. 1211–1225, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, M. A. Noor, and Y.-C. Liou, “A new hybrid iterative algorithm for variational inequalities,”
*Applied Mathematics and Computation*, vol. 216, no. 3, pp. 822–829, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Wang, “Convergence and weaker control conditions for hybrid iterative algorithms,”
*Fixed Point Theory and Applications*, vol. 2011, article 3, 2011. View at Google Scholar - T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2005, no. 1, pp. 103–123, 2005. View at Google Scholar · View at Zentralblatt MATH - G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” vol. 258, pp. 4413–4416, 1964. View at Google Scholar · View at Zentralblatt MATH
- P.-L. Lions, “Approximation de points fixes de contractions,” vol. 284, no. 21, pp. A1357–A1359, 1977. View at Google Scholar · View at Zentralblatt MATH
- 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 - J. T. Oden,
*Qualitative Methods on Nonlinear Mechanics*, Prentice-Hall, Englewood Cliffs, NJ, USA, 1986. - E. Zeidler,
*Nonlinear Functional Analysis and Its Applications: Variational Methods and Optimization*, Springer, New York, NY, USA, 1985. - P. Jaillet, D. Lamberton, and B. Lapeyre, “Variational inequalities and the pricing of American options,”
*Acta Applicandae Mathematicae*, vol. 21, no. 3, pp. 263–289, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao and Y.-C. Liou, “Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces,”
*Analele Stiintifice ale Universitatii Ovidius Constanta*, vol. 19, no. 1, pp. 331–346, 2011. View at Google Scholar · View at Zentralblatt MATH - Y. Yao, Y. C. Liou, and S. M. Kang, “Two-step projection methods for a systemof variational inequality problems in Banach spaces,”
*Journal of Global Optimization*. In press. - Y. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalities,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 817436, 9 pages, 2012. View at Publisher · View at Google Scholar - 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 768272, 10 pages, 2012. View at Publisher · View at Google Scholar - 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 · View at Google Scholar - L.-C. Ceng, A. Petruşel, S. Szentesi, and J.-C. Yao, “Approximation of common fixed points and variational solutions for one-parameter family of Lipschitz pseudocontractions,”
*Fixed Point Theory*, vol. 11, no. 2, pp. 203–224, 2010. View at Google Scholar · View at Zentralblatt MATH - L. C. Ceng, S. Schaible, and J. C. Yao, “Approximate solutions of variational inqualities on sets of common fixed points of a one-parameter semigroup of nonexpansive mappings,”
*Journal of Optimization Theory and Applications*, vol. 143, no. 2, pp. 245–263, 2009. View at Publisher · View at Google Scholar - L.-C. Ceng, Q. H. Ansari, and J. C. Yao, “On relaxed viscosity iterative methods for variational inequalities in Banach spaces,”
*Journal of Computational and Applied Mathematics*, vol. 230, no. 2, pp. 813–822, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,”
*Numerical Functional Analysis and Optimization*, vol. 19, no. 1-2, pp. 33–56, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Glowinski,
*Numerical Methods for Nonlinear Variational Problems*, Scientific Computation, Springer, New York, NY, USA, 1984. - 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 - W. Takahashi,
*Introduction to Nonlinear and Convex Analysis*, Yokohama Publishers, Yokohama, Japan, 2009. - K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,”
*Journal of Mathematical Analysis and Applications*, vol. 279, no. 2, pp. 372–379, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Geobel and W. A. Kirk,
*Topics in Metric Fixed Point Theory*, vol. 28 of*Cambridge Studies in Advanced Mathematics*. - S. Wang and C. Hu, “Two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 852030, 12 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - H.-K. Xu, “A regularization method for the proximal point algorithm,”
*Journal of Global Optimization*, vol. 36, no. 1, pp. 115–125, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH