Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2010 (2010), Article ID 472760, 9 pages
http://dx.doi.org/10.1155/2010/472760
Research Article

Some Existence Theorems for Nonconvex Variational Inequalities Problems

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 9 November 2010; Accepted 18 December 2010

Academic Editor: Roman Dwilewicz

Copyright © 2010 Narin Petrot. 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

By using nonsmooth analysis knowledge, we provide the conditions for existence solutions of the variational inequalities problems in nonconvex setting. We also show that the strongly monotonic assumption of the mapping may not need for the existence of solutions. Consequently, the results presented in this paper can be viewed as an improvement and refinement of some known results from the literature.

1. Introduction

Variational inequalities theory, which was introduced by Stampacchia [1], provides us with a simple, natural, general, and unified framework to study a wide class of problems arising in pure and applied sciences. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solutions to important classes of problems. On the other hand, it also enables us to develop highly efficient and powerful new numerical methods for solving, for example, obstacle, unilateral, free, moving, and complex equilibrium problems.

It should be pointed out that almost all the results regarding the existence and iterative schemes for solving variational inequalities and related optimizations problems are being considered in the convexity setting; see [25] for examples. Moreover, all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex. Notice that the convexity assumption, made by researchers, has been used for guaranteeing the well definedness of the proposed iterative algorithm which depends on the projection mapping. In fact, the convexity assumption may not require for the well definedness of the projection mapping because it may be well defined, even in the nonconvex case (e.g., when the considered set is a closed subset of a finite dimensional space or a compact subset of a Hilbert space, etc.).

The main aim of this paper is intending to consider the conditions for the existence solutions of some variational inequalities problems in nonconvex setting. We will make use of some recent nonsmooth analysis techniques to overcome the difficulties that arise from the nonconvexity. Also, it is worth mentioning that we have considered when the mapping may not satisfy the strongly monotonic assumption. In this sense, our result represents an improvement and refinement of the known results.

2. Preliminaries

Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed subset of . We denote by the usual distance function to the subset ; that is, . Let us recall the following well-known definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis.

Definition 2.1. Let be a point not lying in . A point is called a closest point or a projection of onto if . The set of all such closest points is denoted by ; that is,

Definition 2.2. Let be a subset of . The proximal normal cone to at is given by

The following characterization of can be found in [6].

Lemma 2.3. Let be a closed subset of a Hilbert space . Then,

Clarke et al. [7] and Poliquin et al. [8] have introduced and studied a new class of nonconvex sets, which are called uniformly prox-regular sets. This class of uniformly prox-regular sets has played an important part in many nonconvex applications such as optimization, dynamic systems, and differential inclusions.

Definition 2.4. For a given , a subset of is said to be uniformly prox-regular with respect to if, for all and for all , one has We make the convention for .

It is well known that a closed subset of a Hilbert space is convex if and only if it is proximally smooth of radius . Thus, in view of Definition 2.4, for the case of , the uniform -prox-regularity is equivalent to the convexity of . Moreover, it is clear that the class of uniformly prox-regular sets is sufficiently large to include the class -convex sets, submanifolds (possibly with boundary) of , the images under a diffeomorphism of convex sets, and many other nonconvex sets; see [6, 8].

Now, let us state the following facts, which summarize some important consequences of the uniform prox-regularity. The proof of this result can be found in [7, 8].

Lemma 2.5. Let be a nonempty closed subset of and set . If is uniformly -uniformly prox-regular, then the following hold: (1)for all , , (2)for all , is Lipschitz continuous with constant on , (3)the proximal normal cone is closed as a set-valued mapping.

In this paper, we are interested in the following classes of nonlinear mappings.

Definition 2.6. A mapping is said to be (a)-strongly monotone if there exists a constant such that (b)-Lipschitz if there exist a constants such that

3. System of Nonconvex Variational Inequalities Involving Nonmonotone Mapping

Let be a real Hilbert space, and let be a nonempty closed subset of . In this section, we will consider the following problem: find such that where and are fixed positive real numbers, is a closed subset of , and is a mapping.

The iterative algorithm for finding a solution of the problem (3.1) was considered by Moudafi [9], when is -uniformly prox-regular and is a strongly monotone mapping. He also remarked that two-step models (3.1) for nonlinear variational inequalities are relatively more challenging than the usual variational inequalities since it can be applied to problems arising, especially from complementarity problems, convex quadratic programming, and other variational problems. In this section, we will generalize such result by considering the conditions for existence solution of problem (3.1) when is not necessary strongly monotone. To do so, we will use the following algorithm as an important tool.

Algorithm 3.1. Let be an -uniformly prox-regular subset of . Assume that is a nonlinear mapping. Letting be an arbitrary point in , we consider the following two-step projection method: where , are positive reals number, which were appeared in problem (3.1).

Remark 3.2. The projection algorithm above has been introduced in the convex case, and its convergence was proved see [10]. Observe that (3.2) is well defined provided the projection on is not empty. Our adaptation of the projection algorithm will be based on Lemma 2.5.

Now we will prove the existence theorems of problem (3.1), when is a closed uniformly -prox-regular. Moreover, from now on, the number will be understood as a finite positive real number (if not specified otherwise). This is because, as we know, if , then such a set is nothing but the closed convex set.

We start with an important remark.

Remark 3.3. Let be a uniformly -prox-regular closed subset of . Let be such that is a -Lipschitz continuous, -strongly monotone mapping and is a -Lipschitz continuous mapping. If , then for each we have where .

It is worth to point out that, in Remark 3.3, we have to assume that . Thus, from now on, without loss of generality we will always assume that .

Theorem 3.4. Let be a uniformly -prox-regular closed subset of a Hilbert space , and let be a nonlinear mapping. Let be such that is a -Lipschitz continuous and -strongly monotone mapping, is a -Lipschitz continuous mapping. If and the following conditions are satisfied: (a), where ;(b) there exists such that where , , and .
Then the problem (3.1) has a solution. Moreover, the sequence which is generated by (3.2) strongly converges to a solution of the problem (3.1).

Proof. Firstly, by condition (b), we can easily check that and belong to the set , for all . Thus, from Lemma 2.5 (1), we know that (3.2) is well defined. Consequently, from (3.2) and Lemma 2.5 (2), we have Since the mapping is -strongly monotone and -Lipschitz continuous, we obtain On the other hand, since is -Lipschitz continuous, we have Thus, by (3.5), (3.6), and (3.7), we obtain Similarly, we have Combining (3.8) and (3.9), we get where and . Moreover, by (3.4), we know that and are elements of the interval . Thus, from (3.10), it follows that for all , where . Hence, for any , it follows that Since , it follows that as , and this implies that is a Cauchy sequence. Consequently, from (3.9), we also have that is a Cauchy sequence in . Thus, by Lemma 2.5 (3), the closedness property of implies that there exists such that as .
We claim that is a solution of the problem (3.1). Indeed, by the definition of the proximal normal cone, from (3.2), we have By letting , using the closedness property of the proximal cone together with the continuity of , we have This completes the proof.

Immediately, by setting , we have the following result.

Theorem 3.5. Let be a uniformly -prox-regular closed subset of a Hilbert space . Let be a -Lipschitz continuous and -strongly monotone mapping. If the following conditions are satisfied: (a), where ; (b)there exists such that where and .
Then the problem (3.1) has a solution. Moreover, the sequence which is generated by (3.2) strongly converges to a solution of the problem (3.1).

In view of proving Theorem 3.4, we can obtain the following result, which contains a recent result presented by Moudafi [9] as a special case.

Theorem 3.6. Let be a uniformly -prox-regular closed subset of a Hilbert space , and let be a mapping. Let be such that is a -Lipschitz continuous and -strongly monotone mapping, is a -Lipschitz continuous mapping. If and there exists such that for all , where , and the sequence was generated by (3.2), then the sequence strongly converges to a solution of the problem (3.1).

Remark 3.7. (i) An inspection of Theorem 3.6 shows that the sequences and are bounded.
(ii) By setting , we see that Theorem 3.6 reduces to a result presented by Moudafi [9].

Remark 3.8. If is a convex set, by the definition of the proximal normal cone, we can reformulate (3.1) as follows: find such that The problem (3.17) was introduced and studied by Verma [10], when is a strong monotone mapping. Hence, Theorem 3.4 extends and improves the results presented by Verma [10]. For further recent results related to the problem (3.17), see also [2, 3, 5, 1113].

4. Further Results

By using the techniques as in Theorem 3.4, we can also obtain an existence theorem of the following problem: find such that The problem of type (4.1) was studied by Noor [14] but in a finite dimension Hilbert space setting. In this section, we intend to consider the problem (4.1) in an infinite dimension Hilbert space. To do this, the following remark is useful.

Remark 4.1. Let be a -strongly monotone and -Lipschitz continuous mapping. Then, the function which is defined by is a continuous decreasing function on its domain.

We now close this section by proving an existence theorem to the problem (4.1) in a nonconvex infinite dimensional setting.

Theorem 4.2. Let be a uniformly -prox-regular closed subset of a Hilbert space , and let be a -strongly monotone and -Lipschitz continuous mapping. If , then the problem (4.1) has a solution.

Proof. Firstly, by using an elementary calculation, we know that the function which is defined by is a continuous increasing function on . Moreover, we see that the net which is defined by converges to 1 as . Using these observations, together with the fact that as , we can find such that . It is worth to notice that, from the choice of , we have .
Now, we choose a fixed positive real number such that Next, let us start with an element and use an induction process to obtain a sequence satisfying Note that, because of the choice of , we can easily check that for all . Following the proof of Theorem 3.4, we know that is a Cauchy sequence in . If as , the closedness property of the proximal cone together with the continuity of , from (4.5), we see that is a solution of the problem (4.1). This completes the proof.

Remark 4.3. Theorems 3.4, 3.5, and 4.2 not only give the conditions for the existence solution of the problems (3.1) and (4.1), respectively, but also provide the algorithm to find such solutions for any initial vector .

Acknowledgments

This study was supported by the Commission on Higher Education and the Thailand Research Fund (Project no. MRG5380247).

References

  1. G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964. View at Google Scholar
  2. Y. J. Cho and X. Qin, “Systems of generalized nonlinear variational inequalities and its projection methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4443–4451, 2008. View at Publisher · View at Google Scholar
  3. Y. J. Cho, S. M. Kang, and X. Qin, “On systems of generalized nonlinear variational inequalities in Banach spaces,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 214–220, 2008. View at Publisher · View at Google Scholar
  4. N. Petrot, “Existence and algorithm of solutions for general set-valued Noor variational inequalities with relaxed (μ,ν)-cocoercive operators in Hilbert spaces,” Journal of Applied Mathematics and Computing, vol. 32, no. 2, pp. 393–404, 2010. View at Publisher · View at Google Scholar
  5. S. Suantai and N. Petrot, “Existence and stability of iterative algorithms for the system of nonlinear quasi-mixed equilibrium problems,” Applied Mathematics Letters, vol. 24, no. 3, pp. 308–313, 2011. View at Publisher · View at Google Scholar
  6. F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1998.
  7. F. H. Clarke, R. J. Stern, and P. R. Wolenski, “Proximal smoothness and the lower-C2 property,” Journal of Convex Analysis, vol. 2, no. 1-2, pp. 117–144, 1995. View at Google Scholar
  8. R. A. Poliquin, R. T. Rockafellar, and L. Thibault, “Local differentiability of distance functions,” Transactions of the American Mathematical Society, vol. 352, no. 11, pp. 5231–5249, 2000. View at Publisher · View at Google Scholar
  9. A. Moudafi, “Projection methods for a system of nonconvex variational inequalities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp. 517–520, 2009. View at Publisher · View at Google Scholar
  10. R. U. Verma, “General convergence analysis for two-step projection methods and applications to variational problems,” Applied Mathematics Letters, vol. 18, no. 11, pp. 1286–1292, 2005. View at Publisher · View at Google Scholar
  11. R. P. Agarwal and R. U. Verma, “General system of A(η)-maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 238–251, 2010. View at Publisher · View at Google Scholar
  12. M. A. Noor, “On a system of general mixed variational inequalities,” Optimization Letters, vol. 3, no. 3, pp. 437–451, 2009. View at Publisher · View at Google Scholar
  13. N. Petrot, “A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems,” Applied Mathematics Letters, vol. 23, no. 4, pp. 440–445, 2010. View at Publisher · View at Google Scholar
  14. M. A. Noor, “Iterative schemes for nonconvex variational inequalities,” Journal of Optimization Theory and Applications, vol. 121, no. 2, pp. 385–395, 2004. View at Publisher · View at Google Scholar