The Penalty Method for a New System of Generalized Variational Inequalities
We consider a new system of generalized variational inequalities (SGVI). Using the penalty methods, we prove the existence of solution of SGVI in Hilbert spaces. Our results extend and improve some known results.
Throughout this work, let be real Hilbert space with a norm and inner product . Let be a nonempty closed and convex subset of . Given nonlinear mappings and , , we consider the following problem:
which is called the system of generalized variational inequality problem (SGVI), where and are constants. An element is called a solution of the problem (1.1) if
Special cases of problem (1.1) are as follows.
(1) If , , and , , then problem (1.1) reduces to the variational inequality
Problem (1.3) was introduced by Browder [1, 2] and studied by many authors (e.g., see [3–7]).
(2) If , , , and , then problem (1.1) reduces to the system of variational inequality problem
Problem (1.4) was introduced and studied by Verma .
(3) If , , and , then problem (1.1) becomes the following system of nonlinear variational inequalities
which was considered by Chang et al. .
Remark 1.1. For a suitable choice of and , the problem (1.1) includes many kinds of known systems of variational inequalities as special case (see [4–10] and the references therein). In this work, by using the penalty method, we study the existence of solutions for SGVI.
In the sequel, we give some definitions and lemmas. In what follows, and stand for strong and weak convergence, respectively.
Definition 2.1 (see [11, pages 96–105]). Let be a mapping.(i)The mapping is said to be pseudo-monotone if is closed convex set and its restrictions to finite-dimensional subspaces are demicontinuous, and for every sequence , in , the inequality implies that (ii)The mapping is said to have the generalized pseudo-monotone property if for any sequence with in and in such that we have and as .(iii)The mapping is said to be monotone if for any , the inequality holds.(iv)The monotone mapping is said to be maximal if the inequality for all (graph of implies .(v)The mapping is said to be coercive if there exists a continuous increasing function with as such that for
Definition 2.2. Let be a mapping.(i)The mapping is said to be continuous if there exist constants such that (ii)The mapping is said to be strongly monotone in the first argument if there exists such that for each fixed , we have (iii)The mapping is said to be strongly monotone in the second argument if there exists such that for each fixed , we have (iv)The mapping is said to be coercive in the first (second) argument if there exists a continuous increasing function with as such that for each fixed , we have
Definition 2.3 (see ). (i) A single-valued bounded demicontinuous monotone operator with the property that is said to be the penalty operator of .
(ii) is demicontinuous if implies that .
(iii) is hemicontinuous if , , , implies that , where stands for weak convergence in .
Lemma 2.4 (see [11, page 267]). Let be the projection on , then the mapping is a penalty operator of .
Lemma 2.5 (see [11, page 98]). (i) Any maximal monotone mapping with is a pseudo-monotone one.
(ii) Any pseudo-monotone mapping has the generalized pseudo-monotone property.
(iii) The monotone mapping is maximal if and only if ,.
3. Main Results
We consider first the problem of finding such that
which is called the system of nonlinear variational equation, where , and , is the penalty operator of (see Lemma 2.4). We will prove the existence of solutions for problem (3.1).
Lemma 3.1. Let be real Hilbert space and be a nonempty closed convex subset of . Let mapping be continuous and strongly monotone in the first argument and mapping be continuous and strongly monotone in the second argument. If then for each , problem (3.1) admits a solution .
Proof. Since is continuous monotone operator, is hemicontinuous monotone operator. By Corollary in , is maximal monotone. So, problem (3.1) is equivalent to the following problem
where , , and is the identity mapping on .
For any given , we can compute the sequences and by Picard iterative schemes: Since is nonexpansive  and is continuous and strongly monotone in the first argument, then by (3.4), we have Similarly, we get from (3.5) By (3.6) and (3.7), we have where . By (3.2), we know that , and (3.8) implies that and are both Cauchy sequences. Thus, there exist , such that (as . By continuity of , , and and algorithm (3.4) and (3.5), we know that satisfies the following relation: Therefore, is a solution of problem (3.1). This completes the proof of Lemma 3.1.
Remark 3.2. If is Lipschitz continuous, then is bounded, that is, map bounded set to bounded set.
Now, we give our main results.
Theorem 3.3. Let be a real Hilbert space and be a nonempty closed convex subset with . Let be the same as in Lemma 3.1. If the mapping is coercive with respect to the first argument and the mapping is coercive with respect to the second argument, then for any , there exists one solution of the SGVI (1.1).
Proof. Let . Then, by Lemma 3.1, the problem (3.1) has a solution .
By the monotonicity of , we have since is coercive with respect to the first argument. Hence remains bounded as . Similarly, we have Hence remains also bounded as . The boundedness of and and the fact that implies that and for some constant . By reflexivity of , we can choose sequences and , , such that , and as and . Using the fact that and as and taking the limit in the monotonicity relation, we get that Set with and let be arbitrarily chosen in . Then and by the hemicontinuity of , we get that for all . Hence , that is, . Similarly, we have , that is, .
By (3.1) and monotonicity of , we have Therefore that is, By Lemma 2.5(ii) and Definition 2.1, we deduce that By (3.1), (3.17), and monotonicity of , we have Hence This completes the proof of Theorem 3.3.
If , , , and , then we have the following theorem.
Theorem 3.4. Let be a real Hilbert space and be a nonempty closed convex subset with . Let be strongly monotone, continuous, and coercive operator. If then for each , there exists at least one solution of problem (1.3).
The authors wish to thank the reviewers for many valuable suggestions to improve this paper, and we also would like to thank the editors' kind work. Project supported by National Natural Science Foundations of China (10561007) and The Natural Science Foundations of Jiangxi Province (0411036, 2007GQS2063).
F. E. Browder, “Nonlinear monotone operators and convex sets in Banach spaces,” Bulletin of the American Mathematical Society, vol. 71, pp. 780–785, 1965.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F. E. Browder, “Existence and approximation of solutions of nonlinear variational inequalities,” Proceedings of the National Academy of Sciences of the United States of America, vol. 56, pp. 1080–1086, 1966.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L.-W. Liu and L.-H. Wu, “The penalty method for generalized multivalued nonlinear variational inequalities in Banach spaces,” Applied Mathematics Letters, vol. 20, no. 8, pp. 908–912, 2007.View at: Publisher Site | Google Scholar | MathSciNet
Ya. 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, A. Kartsators, Ed., vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.View at: Google Scholar | Zentralblatt MATH | MathSciNet
Ya. I. Alber and S. Guerre-Delabriere, “Problems of fixed point theory in Hilbert and Banach spaces,” Functional Differential Equations, vol. 2, pp. 5–10, 1994.View at: Google Scholar | Zentralblatt MATH | MathSciNet
Ya. I. Alber, “The penalty method for variational inequalities with nonsmooth unbounded operators in Banach space,” Numerical Functional Analysis and Optimization, vol. 16, no. 9-10, pp. 1111–1125, 1995.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. S. Chang, “The Mann and Ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings,” Computers & Mathematics with Applications, vol. 37, no. 9, pp. 17–24, 1999.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. U. Verma, “Nonlinear variational inequalities on convex subsets of Banach spaces,” Applied Mathematics Letters, vol. 10, no. 4, pp. 25–27, 1997.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. S. Chang, H. W. Joseph Lee, and C. K. Chan, “Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces,” Applied Mathematics Letters, vol. 20, no. 3, pp. 329–334, 2007.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. P. Agarwal, N.-J. Huang, and M.-Y. Tan, “Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions,” Applied Mathematics Letters, vol. 17, no. 3, pp. 345–352, 2004.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Martinus Nijhoff, The Hague, The Netherlands, 1978.View at: MathSciNet