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Volume 2013 |Article ID 604369 | https://doi.org/10.1155/2013/604369

Li-Jun Zhu, "Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points of Nonexpansive Semigroups", Abstract and Applied Analysis, vol. 2013, Article ID 604369, 8 pages, 2013. https://doi.org/10.1155/2013/604369

Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points of Nonexpansive Semigroups

Accepted06 Feb 2013
Published03 Apr 2013

Abstract

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point such that , where is the set of the solutions of the following variational inequality: , where are two strongly positive bounded linear operators, is a -contraction, is a nonexpansive mapping, and is the fixed points set of a nonexpansive semigroup . We present a double-net convergence hierarchical to some elements in which solves the above hierarchical constrained variational inequalities.

1. Introduction

Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of . Recall that a self-mapping of is said to be contractive if there exists a constant such that for all . A mapping is called nonexpansive if

We denote by the set of fixed points of ; that is, . A bounded linear operator is called strongly positive on if there exists a constant such that

It is well known that the variational inequality for an operator, , over a nonempty, closed, and convex set, , is to find a point with the property

The set of the solutions of the variational inequality (3) is denoted by VI. If the mapping is a monotone operator, then we say that VI(3) is monotone. It is well known that if is Lipschitzian and strongly monotone, then for small enough , the mapping is a contraction on and so the sequence of Picard iterates, given by , converges strongly to the unique solution of the VI(3). This sort of VI(3) where is strongly monotone and Lipschitzian is originated from Yamada [1]. However, if is only monotone (not strongly monotone), then their iterative methods do not apply to VI.

Many practical problems such as signal processing and network resource allocation are formulated as the variational inequality over the set of the solutions of some nonlinear mappings (e.g., the fixed point set of nonexpansive mappings), and algorithms to solve these problems have been proposed. Iterative algorithms have been presented for the convex optimization problem with a fixed point constraint along with proof that these algorithms strongly converge to the unique solution of problems with a strongly monotone operator. The strong monotonicity condition guarantees the uniqueness of the solution. For some related works on the variational inequalities, please see [2â€“23] and the references therein. Particulary, the variational inequality problems over the fixed points of nonexpansive mappings have been considered. The reader can consult [16, 24]. On the other hand, we note that in the literature, nonlinear ergodic theorems for nonexpansive semigroups have been considered by many authors; see, for example, [25â€“32]. In this paper, we will consider a general variational inequality problem with the variational inequality constraint is the fixed points of nonexpansive semigroups.

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point such that where is the set of the solutions of the following variational inequality: where and are two strongly positive bounded linear operators, is a -contraction, is a nonexpansive mapping, and is the fixed points set of a nonexpansive semigroup . We present a double-net convergence hierarchical to some elements in which solves the above hierarchical constrained variational inequalities.

2. Preliminaries

Let be a nonempty closed convex subset of a real Hilbert space . The metric (or the nearest point) projection from onto is the mapping which assigns to each point the unique point satisfying the property

It is well known that is a nonexpansive mapping and satisfies

Moreover, is characterized by the following property:

Recall that a family of mappings of into itself is called a nonexpansive semigroup if it satisfies the following conditions:(S1) for all ;(S2) for all ;(S3) for all and ;(S4) for all , is continuous.

We denote by the set of fixed points of and by the set of all common fixed points of ; that is, . It is known that is closed and convex.

We need the following lemmas for proving our main results.

Lemma 1 (see [33]). Let be a nonempty bounded closed convex subset of a Hilbert space and let be a nonexpansive semigroup on . Then, for every ,

Lemma 2 (see [34]). Let be a closed convex subset of a real Hilbert space and let be a nonexpansive mapping. Then, the mapping is demiclosed. That is, if is a sequence in such that weakly and strongly, then .

Lemma 3 (see [35]). Let be a nonempty closed convex subset of a real Hilbert space . Assume that a mapping is monotone and weakly continuous along segments (i.e., weakly, as , whenever for ). Then the variational inequality is equivalent to the dual variational inequality

3. Main Results

Now we consider the following hierarchical variational inequality with the variational inequality constraint over the fixed points set of nonexpansive semigroups .

Problem 1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a -contraction with coefficient and let be a nonexpansive mapping. Let be a nonexpansive semigroup on and let be two strongly positive bounded linear operators with coefficients and , respectively. Let be a constant satisfying .
Now, our objective is to find such that where is the set of the solutions of the following variational inequality:

We observe that is strongly monotone and Lipschitz continuous. In fact, we have

Hence, the existence and the uniqueness of the solution to Problem 1 are guaranteed.

In order to solve the above hierarchical constrainted variational inequality, we present the following double net.

Algorithm 4. Set . Then, . For each , we define a double net implicitly by

Note that this implicit manner algorithm is well defined. In fact, we define the mapping

Note that this self-mapping is a contraction. As a matter of fact, we have Since , . Hence, is a contraction. Therefore, by Banachâ€™s Contraction Principle, has a unique fixed point which is denoted by .

Next we show the behavior of the net as and successively.

Theorem 5. Assume that . Then, for each fixed , the net defined by (15) converges in norm, as , to a solution . Moreover, as , the net converges in norm to the unique solution of Problem 1.

Proof. We first show that the sequence is bounded. Take . From (15), we have
Hence It follows that for each fixed , is bounded.
Next, we show that for all and consequently, as , the entire net converges in norm to .
For each fixed , we set . It is clear that for each fixed , . Notice that
Moreover, we observe that if , then
that is, is -invariant for all .
From (15), we deduce
Since is bounded, and are also bounded. Then, from Lemma 1, we deduce for all and fixed that
Set for all . We then have , and for any ,
Notice that
Thus, we have
So
Assume such that as . By (27), we obtain immediately that
Since is bounded, there exists a subsequence of such that converges weakly to a point . From (23) and Lemma 2, we get . We can substitute for in (28) to get
The weak convergence of to actually implies that strongly. This has proved the relative norm-compactness of the net as for each fixed .
In (28), we take the limit as to get
In particular, solves the following variational inequality:
Note that the mapping is monotone for all , since
By Lemma 3, (31) is equivalent to its dual VI:
Next we show that as , the entire net converges in norm to . We assume , where . Similarly, by the above proof, we deduce which solves the following variational inequality:
In (33), we take to get
In (34), we take to get
Adding up (35) and (36) yields
At the same time we note that
Therefore, by (37) and (38), we deduce Hence the entire net converges in norm to as .
As , the net converges to the unique solution of Problem 1.
In (33), we take any to deduce
By virtue of the monotonicity of and the fact that , we have
We can rewrite (40) as
It follows from (41) and (42) that
Hence
Therefore,
In particular,
which implies that is bounded.
We next prove that ; namely, if is a null sequence in such that weakly as , then . To see this, we use (33) to get
However, since is monotone,
Combining the last two relations yields
Letting as in (49), we get
The equivalent dual VI of (50) is
Namely, is a solution of VI(13); hence .
We further prove that , the unique solution of VI(12). As a matter of fact, we have by (45)
Therefore, the weak convergence to of right implies that in norm. Now we can let in (45) to get
which is equivalent to its dual VI
It turns out that solves VI(12). By uniqueness, we have . This is sufficient to guarantee that in norm, as . This completes the proof.

Corollary 6. For each , let be a double net defined by
for all . Then, for each fixed , the net defined by (55) converges in norm, as , to a solution . Moreover, as , the net converges in norm to which solves the following variational inequality: where is the set of the solutions of the following variational inequality:

Corollary 7. For each , let be a double net defined by
for all . Then, for each fixed , the net defined by (58) converges in norm, as , to a solution . Moreover, as , the net converges to the minimum norm solution of the following variational inequality:

Proof. In (55), we take and . Then (55) reduces to (58). Hence, the net defined by (58) converges in norm to which satisfies
This indicates that
Therefore, is the minimum norm solution of the VI(59). This completes the proof.

Acknowledgment

The author was supported in part by NSFC 71161001-G0105, NNSF of China (10671157 and 61261044), NGY2012097 and Beifang University of Nationalities scientific research project.

References

1. I. Yamada, â€śThe hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,â€ť in Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., pp. 473â€“504, Elsevier, Amsterdam, The Netherlands, 2001.
2. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, â€śRelaxed extragradient iterative methods for variational inequalities,â€ť Applied Mathematics and Computation, vol. 218, no. 3, pp. 1112â€“1123, 2011.
3. L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, â€śSome iterative methods for finding fixed points and for solving constrained convex minimization problems,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 74, no. 16, pp. 5286â€“5302, 2011. View at: Publisher Site | Google Scholar | MathSciNet
4. L.-C. Ceng, S.-M. Guu, and J.-C. Yao, â€śA general iterative method with strongly positive operators for general variational inequalities,â€ť Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1441â€“1452, 2010.
5. S.-S. Chang, H. W. Joseph Lee, and C. K. Chan, â€śA new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 70, no. 9, pp. 3307â€“3319, 2009.
6. S. S. Chang, H. W. Joseph Lee, C. K. Chan, and J. A. Liu, â€śA new method for solving a system of generalized nonlinear variational inequalities in Banach spaces,â€ť Applied Mathematics and Computation, vol. 217, no. 16, pp. 6830â€“6837, 2011.
7. Y. J. Cho, I. K. Argyros, and N. Petrot, â€śApproximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems,â€ť Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2292â€“2301, 2010.
8. Y. J. Cho, S. M. Kang, and X. Qin, â€śOn systems of generalized nonlinear variational inequalities in Banach spaces,â€ť Applied Mathematics and Computation, vol. 206, no. 1, pp. 214â€“220, 2008.
9. F. Cianciaruso, V. Colao, L. Muglia, and H.-K. Xu, â€śOn an implicit hierarchical fixed point approach to variational inequalities,â€ť Bulletin of the Australian Mathematical Society, vol. 80, no. 1, pp. 117â€“124, 2009.
10. V. Colao, G. L. Acedo, and G. Marino, â€śAn implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 7-8, pp. 2708â€“2715, 2009.
11. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, USA, 1984. View at: MathSciNet
12. P. Kumam, N. Petrot, and R. Wangkeeree, â€śExistence and iterative approximation of solutions of generalized mixed quasi-variational-like inequality problem in Banach spaces,â€ť Applied Mathematics and Computation, vol. 217, no. 18, pp. 7496â€“7503, 2011.
13. M. Aslam Noor, â€śSome developments in general variational inequalities,â€ť Applied Mathematics and Computation, vol. 152, no. 1, pp. 199â€“277, 2004.
14. M. A. Noor, â€śAn extragradient algorithm for solving general nonconvex variational inequalities,â€ť Applied Mathematics Letters, vol. 23, no. 8, pp. 917â€“921, 2010.
15. H.-K. Xu, â€śViscosity method for hierarchical fixed point approach to variational inequalities,â€ť Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 463â€“478, 2010.
16. I. Yamada, N. Ogura, and N. Shirakawa, â€śA numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems,â€ť in Inverse Problems, Image Analysis, and Medical Imaging, vol. 313 of Contemporary Mathematics, pp. 269â€“305, American Mathematical Society, Providence, RI, USA, 2002.
17. Y. Yao, Y. J. Cho, and R. Chen, â€śAn iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 7-8, pp. 3363â€“3373, 2009.
18. Y. Yao, R. Chen, and H.-K. Xu, â€śSchemes for finding minimum-norm solutions of variational inequalities,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 7-8, pp. 3447â€“3456, 2010.
19. Y. Yao, Y. J. Cho, and Y.-C. Liou, â€śIterative algorithms for hierarchical fixed points problems and variational inequalities,â€ť Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1697â€“1705, 2010.
20. Y. Yao and J.-C. Yao, â€śOn modified iterative method for nonexpansive mappings and monotone mappings,â€ť Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551â€“1558, 2007.
21. H. Zegeye, E. U. Ofoedu, and N. Shahzad, â€śConvergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings,â€ť Applied Mathematics and Computation, vol. 216, no. 12, pp. 3439â€“3449, 2010.
22. H. Zegeye and N. Shahzad, â€śA hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 74, no. 1, pp. 263â€“272, 2011.
23. L.-C. Zeng and J.-C. Yao, â€śStrong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,â€ť Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293â€“1303, 2006.
24. I. Yamada and N. Ogura, â€śHybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings,â€ť Numerical Functional Analysis and Optimization, vol. 25, no. 7-8, pp. 619â€“655, 2004.
25. N. Buong, â€śStrong convergence theorem for nonexpansive semigroups in Hilbert space,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 12, pp. 4534â€“4540, 2010.
26. R. Chen and H. He, â€śViscosity approximation of common fixed points of nonexpansive semigroups in Banach space,â€ť Applied Mathematics Letters, vol. 20, no. 7, pp. 751â€“757, 2007.
27. R. Chen and Y. Song, â€śConvergence to common fixed point of nonexpansive semigroups,â€ť Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 566â€“575, 2007.
28. F. Cianciaruso, G. Marino, and L. Muglia, â€śIterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces,â€ť Journal of Optimization Theory and Applications, vol. 146, no. 2, pp. 491â€“509, 2010.
29. A. T. M. Lau, â€śSemigroup of nonexpansive mappings on a Hilbert space,â€ť Journal of Mathematical Analysis and Applications, vol. 105, no. 2, pp. 514â€“522, 1985.
30. A. T.-M. Lau and W. Takahashi, â€śFixed point properties for semigroup of nonexpansive mappings on Fréchet spaces,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 70, no. 11, pp. 3837â€“3841, 2009.
31. A. T.-M. Lau and Y. Zhang, â€śFixed point properties of semigroups of non-expansive mappings,â€ť Journal of Functional Analysis, vol. 254, no. 10, pp. 2534â€“2554, 2008.
32. H. Zegeye and N. Shahzad, â€śStrong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 1, pp. 325â€“329, 2010.
33. T. Shimizu and W. Takahashi, â€śStrong convergence to common fixed points of families of nonexpansive mappings,â€ť Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71â€“83, 1997.
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 Site | MathSciNet
35. X. Lu, H.-K. Xu, and X. Yin, â€śHybrid methods for a class of monotone variational inequalities,â€ť Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 3-4, pp. 1032â€“1041, 2009.

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