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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 176283, 14 pages
http://dx.doi.org/10.1155/2012/176283
Research Article

Existence and Iterative Algorithm of Solutions for a System of Generalized Nonlinear Mixed Variational-Like Inequalities

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 29 December 2011; Accepted 16 January 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Yan-Mei Du. 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 and study a system of generalized nonlinear mixed variational-like inequality problems (SGNMVLIPs) in Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the SGNMVLIP. First, the existence of solutions of the auxiliary problems for the SGNMVLIP is shown. Second, an iterative algorithm for solving the SGNMVLIP is constructed by using this existence result. Finally, not only the existence of solutions of the SGNMVLIP is shown but also the convergence of iterative sequences generated by the algorithm is also proven. The technique and results presented in this paper generalize and unify the corresponding techniques and results given in the literature.

1. Introduction and Preliminaries

Variational inequality theory, which was introduced by Stampacchia [1] in 1964, is an important part of nonlinear analysis. Various kinds of iterative algorithms to solve the variational inequalities have been developed by many authors, see [28] and the references therein. Variational-like inequality introduced by Parida et al. [9] is an important generalization of the variational inequalities and has significant applications in nonconvex optimization. It is worth mentioning that the projection method cannot be extended for constructing iterative algorithms for variational-like inequalities. To overcome this drawback, one uses usually the auxiliary principle technique which deals with finding a suitble auxiliary problems for the original problem. Further, this auxiliary problem is used to construct an algorithm for solving the original problems. Glowinski et al. [10] introduced this technique and used it to study the existence of a solution of variational-like inequality. Later, many authors extended this technique to suggest and analyze a number of algorithms for solving various classes of variational inequalities (see [1119]).

Recently, the auxiliary principle technique was extended by Ding et al. [15] to study the existence and iterative algorithm of solutions of generalized strongly nonlinear mixed variational-like inequalities in Banach spaces. On the other hand, the auxiliary principle technique was also extended by Kazmi and Khan [16] who studied a system of generalized variational-like inequality problems in Hilbert spaces.

In this paper, we still extend the auxiliary principle technique to study a system of generalized nonlinear mixed variational-like inequality problems (SGNMVLIPs) in Banach spaces. At first, the existence of solutions of the auxiliary problems for the SGNMVLIP is shown. Next, an iterative algorithm for solving SGNMVLIP is constructed by using this existence and uniqueness result. Finally, we prove the existence of solutions of the SGNMVLIP and the convergence of the algorithm. These results improve and generalize many corresponding results given in [12, 13, 1618].

Throughout the paper unless otherwise stated, let be an index set. For each , let be a real Banach spaces with norm , let be the topological dual space of , and let be the generalized duality pairing between and . Let , be nonlinear mappings, then we consider the following system of generalized nonlinear mixed variational-like inequality problems (SGNMVLIPs): for given , find such that where for each , the bifunction is a real-valued nondifferential function with the following properties:(i) is linear in the first argument;(ii) is convex in the second argument;(iii) is bounded, that is, there exists a constant such that (iv), for all ;

Remark 1.1 (see [15]). (1) It follows from property (i), for any , . By property (iii), we have , and hence This shows that for any , .

(2) It follows from properties (iii) and (iv), for any , Therefore, we have This implies that is continuous with respect to the second argument.

Some Special Cases
(1) If for each , is a real Hilbert space and , then the SGNMVLIP (1.1) and (1.2) reduce to the problems: find such that The problem (1.7) has been studied by Kazmi and Khan [16].
(2) If index set , and for each , where are two single-valued mappings, then the SGNMVLIP (1.1) and (1.2) reduce to the problem: find such that The problem (1.8) with was introduced and studied by Ding [17] in reflexive Banach spaces.
In brief, for appropriate and suitable choice of the mappings , and the linear continuous functionals and , one can obtain a number of the known classes of variational inequalities as special cases from SGNMVLIP (1.1) and (1.2) (see [68]).
We need the following basic concepts, basic assumptions and basic results which will be used in the sequel.

Definition 1.2. Let be a nonempty subset of a Banach space with the dual space . Let and be two mappings, then(i) is said to be -strongly monotone if there exists a constant such that (ii) is said to be Lipschitz continuous if there exists a constant such that (iii) is said to be Lipschitz continuous if there exists a constant such that

Definition 1.3. Let be a nonempty subset of a Banach space with the dual space . Let , be two mappings, then is said to be(i)-Lipschitz continuous if there exist constants such that (ii)-strongly monotone in the first argument if there exists a constant such that (iii)-strongly monotone in the second argument if there exists a constant such that

Assumption 1.4. For each , the mapping satisfies the following conditions:(1), for all ;(2) is affine in the second argument;(3)for each fixed , the function is continuous from the weak topology to the weak topology.

Remark 1.5. It follows from Assumption 1.4 (1) that and for any . Moreover, we can prove that is also affine in the first argument by Assumption 1.4 (1) and (2).

Lemma 1.6 (see [20]). Let be a nonempty close convex subset of a Hausdorff linear topological space , and let be mappings satisfying the following conditions:(i)for each , and for each ;(ii)for each , is upper semicontinuous with respect to ;(iii)for each , the set is convex;(iv)there exists a nonempty compact set and such that for any .
Then, there exists an such that for any .

2. Auxiliary Problems and Algorithm

In this section, we introduce the auxiliary problems to study the SGNMVLIP (1.1) and (1.2), and we give an existence theorem for a solution of the auxiliary problems. By using the existence theorem, we construct the iterative algorithm for solving the SGNMVLIP (1.1) and (1.2).

For each , let be a single-valued mappings Given , we consider the following problems : find such that where is a constant. The problems are called the auxiliary problems for SGNMVLIP (1.1) and (1.2).

Remark 2.1. If for each , is a real Hilbert spaces, and are the identity mappings on , then the auxiliary problems reduce to Kazmi and Khan’s auxiliary problems in [16].

Theorem 2.2. For each , let be Lipschtiz continuous with constant , and let be -strongly monotone and Lipschtiz continuous with constant and , respectively. Let satisfy the properties (i)–(iv). If Assumption 1.4 holds, then the auxiliary problems have a unique solution.

Proof. For each , define the mappings by respectively.
We claim that the mappings satisfy all conditions of Lemma 1.6 in the weak topology. Indeed, since is -strongly monotone with constant and Remark 1.5, it is clear that and satisfy condition (i) of Lemma 1.6. Since the bifunction is convex in the second argument and is affine in the second argument, it follows from Assumption 1.4 (3) and Remark 1.1 (2) that is weakly upper semicontinuous. By Assumption 1.4 (1) and (2), and the property (ii) of , it is easy to prove that the set is convex, hence the conditions (ii) and (iii) of Lemma 1.6 hold.
Let and , then is a weakly compact subset of . For any , take . From Assumption 1.4 (1), Remark 1.1 (1), Lipschitz continuity of , and the -strongly monotonicity of , we have Therefore, the condition (iv) of Lemma 1.6 holds. By Lemma 1.6 there exists an such that for all , that is
For arbitrary and , let . Replacing by in (2.4) and utilizing Assumption 1.4 (1) and (2), Remark 1.5, and the property (ii) of , we obtain Hence, we derive Let , by the Lipschitz continuity of , we have Therefore, is a solution of the auxiliary problems .
Now, let be another solution of the auxiliary problems which is different from , then we have Taking in (2.7) and in (2.8) and adding these two inequalities, we obtain Since is -strongly monotone, we obtain and so . This completes the proof.

By virtue of Theorem 2.2, we now construct an iterative algorithm for solving the SGNMVLIP (1.1) and (1.2).

For given , from Theorem 2.2, we know that the auxiliary problems have a solution , that is, Again by Theorem 2.2, the auxiliary problems have a solution , that is, By induction, we can get the iterative algorithm for solving the SGNMVLIP (1.1) and (1.2) as follows.

Algorithm 2.3. For given , there exists a sequence such that where is a constant.

3. Existence of Convergence Theorem

In this section, we will prove not only that the sequence generated by Algorithm 2.3 converges strongly to , and also that is a solution of SGNMVLIP (1.1) and (1.2).

Theorem 3.1. For each , assume that the following conditions are satisfied:(1) is -strongly monotone and Lipschtiz continuous with constant and , respectively;(2) is Lipschtiz continuous with constant ;(3) is -Lipschtiz continuous;(4) is -strongly monotone in the th argument with constant ;(5) satisfies the properties (i)–(iv).
If Assumption 1.4 holds and there exists a constant such that then the sequence generated by Algorithm 2.3 converges strongly to , and is a solution of SGNMVLIP (1.1) and (1.2).

Proof. For any , it follows from Algorithm 2.3 that Taking in (3.3) and in (3.5), respectively, we get Adding (3.7) and (3.8), we obtain From conditions (1) and (2), we have From conditions (3) and (4), we obtain From condition (5) and Remark 1.1 (2), we have Therefore, from (3.9)–(3.12), we derive which implies where , .
Taking in (3.4) and in (3.6), respectively, we get Adding (3.15), we obtain From conditions (1) and (2), we have From conditions (3) and (4), we obtain From condition (5) and Remark 1.1 (2), we have Therefore, from (3.16)–(3.19), we derive which implies where , .
Adding (3.14) and (3.21), we have
Define the norm on by it is easy to prove that is a Banach space.
From (3.22), by conditions (3.1), we have where . From condition (3.2), which implies , hence is a cauchy sequence, let .
By the Lipschitz continuities of and , and , we have Since is -Lipschitz, and , then we obtain From condition (5) and Remark 1.1 (2), we have Hence, as in (2.13), we obtain It is similar as above, we can obtain Therefore, is a solution of SGNMLVIP (1.1) and (1.2).

Example 3.2. Let be Lebesgue measurable and , then the dual space . For each , , let the norm , and the inner product .
For each , let the mappings , , , , be defined as for any , , , respectively, then we have(1) is Lipschitz continuous with constant and -strongly monotone in the first argument with constant ;(2) is Lipschitz continuous with constant and -strongly monotone in the second argument with constant ;(3)for each , is Lipschitz continuous with constant ;(4)for each , satisfy properties (i)–(iv) with constant ;(5)for each , is Lipschitz continuous with constant and -strongly monotone with constant .After simple calculations, conditions (3.1) and (3.2) imply that .

Remark 3.3. Example 3.2 shows that the constant which satisfies the conditions (3.1) and (3.2) can be obtained.

Acknowledgment

This work is partially supported by the National Natural Science Foundation of China Grant (10771050).

References

  1. G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Les Comptes Rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964. View at Zentralblatt MATH
  2. 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
  3. L. C. Zeng, S. Schaible, and J.-C. Yao, “Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities,” Journal of Optimation Theory and Applications, vol. 124, no. 3, pp. 725–738, 2005.
  4. Y. Yao, M. A. Noor, and Y.-C. Liou, “Iterative algorithms for general multi-valued variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 768272, 10 pages, 2012. View at Publisher · View at Google Scholar
  5. Y. Yao, M. A. Noor, and Y.-C. Liou, “Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequalities,” Fixed Point Theory and Applications, vol. 2011, p. 101, 2011. View at Publisher · View at Google Scholar
  6. M. A. Noor, “Mixed variational-like inequalities,” Communications on Applied Nonlinear Analysis, vol. 1, no. 4, pp. 63–75, 1994. View at Zentralblatt MATH
  7. M. A. Noor, “Iterative schemes for multivalued quasi variational inclusions,” Journal of Global Optimization, vol. 19, no. 2, pp. 141–150, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. X. H. Chen and Y. F. Liu, “A generalized nonlinear variational-like inequality in reflexive Banach spaces,” Journal of Nanjing University Mathematic Biquart, vol. 18, no. 1, pp. 96–103, 2001.
  9. J. Parida, M. Sahoo, and A. Kumar, “A variational-like inequality problem,” Bulletin of the Australian Mathematical Society, vol. 39, no. 2, pp. 225–231, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. R. Glowinski, J.-L. Lions, and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland Publishing, New York, NY, USA, 1981.
  11. J. Shen and L.-P. Pang, “A bundle-type auxiliary problem method for solving generalized variational-like inequalities,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2993–2998, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. J. Y. Park and J. U. Jeong, “Generalized nonlinear variational-like inequalities in reflexive Banach spaces,” Mathematical Methods in the Applied Sciences, vol. 33, no. 13, pp. 1637–1646, 2010. View at Zentralblatt MATH
  13. M. A. Noor, “Auxiliary principle for generalized mixed variational-like inequalities,” Journal of Mathematical Analysis and Applications, vol. 215, no. 1, pp. 75–85, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Iterative algorithm for finding approximate solutions of mixed quasi-variational-like inclusions,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 942–952, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. X. P. Ding, J.-C. Yao, and L.-C. Zeng, “Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces,” Computers & Mathematics with Applications, vol. 55, no. 4, pp. 669–679, 2008. View at Publisher · View at Google Scholar
  16. K. R. Kazmi and F. A. Khan, “Auxiliary problems and algorithm for a system of generalized variational-like inequality problems,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 789–796, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. X. P. Ding, “General algorithm for nonlinear variational-like inequalities in reflexive Banach spaces,” Indian Journal of Pure and Applied Mathematics, vol. 29, no. 2, pp. 109–120, 1998. View at Zentralblatt MATH
  18. C. E. Chidume, K. R. Kazmi, and H. Zegeye, “General auxiliary problem and algorithm for a generalized multi-valued variational-like inequality problem in reflexive Banach spaces,” Applicable Analysis. An International Journal, vol. 82, no. 12, pp. 1099–1109, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. 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
  20. S. S. Chang and S. W. Xiang, “Existence of solutions for a class of quasibilinear variational inequalities,” Journal of Systems Science and Mathematical Sciences, vol. 16, no. 2, pp. 136–140, 1996.