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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 949141, 15 pages
Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems
1Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
Received 21 September 2011; Revised 16 November 2011; Accepted 17 November 2011
Academic Editor: Khalida Inayat Noor
Copyright © 2012 Yeong-Cheng Liou 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.
We consider a general variational inequality and fixed point problem, which is to find a point with the property that (GVF): and where is the solution set of some variational inequality is the fixed points set of nonexpansive mapping , and is a nonlinear operator. Assume the solution set of (GVF) is nonempty. For solving (GVF), we suggest the following method , . It is shown that the sequence converges strongly to which is the unique solution of the variational inequality , for all .
Let and be two nonlinear mappings. We concern the following generalized variational inequality of finding such that The solution set of (1.1) is denoted by . It has been shown that a large class of unrelated odd-order and nonsymmetric obstacle, unilateral, contact, free, moving, and equilibrium problems arising in regional, physical, mathematical, engineering, and applied sciences can be studied in the unified and general framework of the general variational inequalities (1.1), see [1–16] and the references therein. Noor  has introduced a new type of variational inequality involving two nonlinear operators, which is called the general variational inequality. It is worth mentioning that this general variational inequality is remarkably different from the so-called general variational inequality which was introduced by Noor  in 1988. Noor  proved that the general variational inequalities are equivalent to nonlinear projection equations and the Wiener-Hopf equations by using the projection technique. Using this equivalent formulation, Noor  suggested and analyzed some iterative algorithms for solving the special general variational inequalities and further proved that these algorithms have strong convergence.
For , where is the identity operator, problem (1.1) is equivalent to finding such that which is known as the classical variational inequality introduced and studied by Stampacchia  in 1964. This field has been extensively studied due to a wide range of applications in industry, finance, economics, social, pure and applied sciences. For related works, please see [20–35]. Our main purposes in the present paper is devoted to study this topic.
Motivated and inspired by the works in this field, in this paper, we consider a general variational inequality and fixed point problem, which is to find a point with the property that where is the fixed points set of nonexpansive mapping . Assume the solution set of (GVF) is nonempty. For solving (GVF), we suggest the following method It is shown that the sequence converges strongly to which is the unique solution of the following variational inequality Our results contain some interesting results as special cases.
Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of . Recall that a mapping is said to be nonexpansive if for all . We denote by the set of fixed points of . A mapping is said to be -Lipschitz continuous, if there exists a constant such that for all . A mapping is said to be -inverse strongly -monotone if and only if for some and for all . A mapping is said to be strongly monotone if there exists a constant such that for all .
Let be a mapping of into . The effective domain of is denoted by , that is, . A multivalued mapping is said to be a monotone operator on if and only if for all , and . A monotone operator on is said to be maximal if and only if its graph is not strictly contained in the graph of any other monotone operator on . Let be a maximal monotone operator on and let .
It is well known that, for any , there exists a unique such that We denote by , where is called the metric projection of onto . The metric projection of onto has the following basic properties:(i) for all ;(ii) for every ;(iii) for all , .
It is easy to see that the following is true:
We use the following notation:(i) stands for the weak convergence of to ;(ii) stands for the strong convergence of to .
We need the following lemmas for the next section.
Lemma 2.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonlinear mapping and let the mapping be -inverse strongly -monotone. Then, for any , one has
Proof. Consider the following: If , we have
Lemma 2.2 (see ). Let be a closed convex subset of a Hilbert space . Let be a nonexpansive mapping. Then is a closed convex subset of and the mapping is demiclosed at 0, that is, whenever is such that and , then .
Lemma 2.3 (see ). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all and . Then, .
Lemma 2.4 (see ). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that(1);(2) or . Then .
3. Main Results
In this section, we will prove our main results.
Theorem 3.1. Let be a nonempty closed and convex subset of a real Hilbert space . Let be an -Lipschitz continuous mapping, be a weakly continuous and -strongly monotone mapping such that . Let be an -inverse strongly -monotone mapping and let be a nonexpansive mapping. Suppose that . Let and . For given , let be a sequence generated by where satisfies and . Then the sequence generated by (3.1) converges strongly to which is the unique solution of the following variational inequality:
Proof. First, we show the solution set of variational inequality (3.2) is singleton. Assume also solves (3.2). Then, we have
It follows that
Since is -strongly monotone, we have
In particular, . By (3.4), we deduce
which implies that because of by the assumption. Therefore, the solution of variational inequality (3.2) is unique.
Pick up any . It is obvious that and . Set . From (2.6), we know for any . Hence, we have From (3.6), (3.8), and Lemma 2.1, we get It follows from (3.1) that This indicates by induction that Hence, is bounded. By (3.6), we have . This implies that is bounded. Consequently, , and are all bounded.
Note that we can rewrite (3.1) as for all . Next, we will use Lemma 2.3 to prove that . In fact, we firstly have It follows that Since and the sequences , and are bounded, we have By Lemma 2.3, we obtain Hence, This together with (3.6) imply that By the convexity of the norm and (3.9), we have From Lemma 2.1, we derive Thus, So, Since and , we obtain Set for all . By using the property of projection, we get It follows that From (3.18) and (3.24), we have Then, we obtain Since , and , we have Next, we prove where is the unique solution of (3.2). We take a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to some point . Without loss of generality, we may assume that . This implies that due to the weak continuity of . Now, we show . First, we note that from (3.15) and (3.27) that . Hence, . By the demiclosedness principle of the nonexpansive mapping (see Lemma 2.2), we deduce . Next, we only need to prove . Set By , we know that is maximal -monotone. Let . Since and , we have From , we get It follows that Then, Since and , we deduce that by taking in (3.33). Thus, by the maximal -monotonicity of . Hence, . Therefore, . From (3.28), we obtain We take in (3.23) to get It follows that Therefore, where and . From condition , we have . By (3.34), we have . We can therefore apply Lemma 2.4 to conclude that and . This completes the proof.
Corollary 3.2. Let be a nonempty closed and convex subset of a real Hilbert space . Let be an -contraction. Let be an -inverse strongly monotone mapping and let be a nonexpansive mapping. Suppose that . Let and . For given , let be a sequence generated by where satisfies and . Then the sequence generated by (3.38) converges strongly to which is the unique solution of the following variational inequality:
The authors are very grateful to the referees for their comments and suggestions which improved the presentation of this paper. The first author was partially supported by the Program TH-1-3, Optimization Lean Cycle, of Subprojects TH-1 of Spindle Plan Four in Excellence Teaching and Learning Plan of Cheng Shiu University and was supported in part by NSC 100-2221-E-230-012. The second author was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105.
- M. Fukushima, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,” Mathematical Programming, vol. 53, no. 1, pp. 99–110, 1992.
- F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, New York, NY, USA, Plenum Press, 1995.
- F. Giannessi, A. Maugeri, and P. M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic, Dodrecht, The Netherlands, 2001.
- R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, Berlin, Germany, 1984.
- R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, vol. 8, North-Holland, Amsterdam, The Netherlands, 1981.
- R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, vol. 9, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1989.
- D. Han and H. K. Lo, “Two new self-adaptive projection methods for variational inequality problems,” Computers & Mathematics with Applications, vol. 43, no. 12, pp. 1529–1537, 2002.
- P. T. Harker and J.-S. Pang, “Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications,” Mathematical Programming, vol. 48, no. 2, pp. 161–220, 1990.
- Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011.
- B. He, “Inexact implicit methods for monotone general variational inequalities,” Mathematical Programming, vol. 86, no. 1, pp. 199–217, 1999.
- J.-S. Pang and J. C. Yao, “On a generalization of a normal map and equation,” SIAM Journal on Control and Optimization, vol. 33, no. 1, pp. 168–184, 1995.
- J. C. Yao, “Variational inequalities with generalized monotone operators,” Mathematics of Operations Research, vol. 19, no. 3, pp. 691–705, 1994.
- Y. Yao and S. Maruster, “Strong convergence of an iterative algorithm for variational inequalities in Banach spaces,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 325–329, 2011.
- Y. Yao, Y. -C. Liou, S. M. Kang, and Y. Yu, “Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces,” Nonlinear Analysis, vol. 74, no. 17, pp. 6024–6034, 2011.
- F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem,” Fixed Point Theory and Applications, Article ID 383740, 19 pages, 2010.
- M. Aslam Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004.
- M. A. Noor, “Differentiable non-convex functions and general variational inequalities,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 623–630, 2008.
- M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp. 119–122, 1988.
- G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964.
- M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000.
- F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,” Journal of Inequalities and Applications, Article ID 208692, 13 pages, 2009.
- Y. Yao, R. Chen, and H.-K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis, vol. 72, no. 7-8, pp. 3447–3456, 2010.
- M. A. Noor, “Some algorithms for general monotone mixed variational inequalities,” Mathematical and Computer Modelling, vol. 29, no. 7, pp. 1–9, 1999.
- M. A. Noor and K. I. Noor, “Self-adaptive projection algorithms for general variational inequalities,” Applied Mathematics and Computation, vol. 151, no. 3, pp. 659–670, 2004.
- M. A. Noor, K. I. Noor, and T. M. Rassias, “Some aspects of variational inequalities,” Journal of Computational and Applied Mathematics, vol. 47, no. 3, pp. 285–312, 1993.
- 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.
- Y. Yao, R. Chen, and Y. C. Liou, “A unified implicit algorithm for solving the triplehierarchical constrained optimization problem,” Mathematical and Computer Modelling.
- Y. Yao, M. Aslam Noor, and Y. C. Liou, “Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities,” Abstract and Applied Analysis, p. 9, 2012.
- Y. Yao, Y. C. Liou, C. L. Li, and H. T. Lin, “Extended extra-gradient methods for generalized variational inequalities,” Journal of Applied Mathematics, p. 14, 2012.
- Y. Yao, M. A. Noor, Y. C. Liou, and S. M. Kang, “Iterative algorithms for general multi-valued variational inequalities,” Abstract and Applied Analysis, p. 10, 2012.
- Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters.
- Y. Yao, Y.-C. Liou, and S. M. Kang, “Algorithms construction for variational inequalities,” Fixed Point Theory and Applications, Article ID 794203, 12 pages, 2011.
- V. Colao, G. L. Acedo, and G. Marino, “An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 2708–2715, 2009.
- 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.
- M. Aslam Noor and Z. Huang, “Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 504–510, 2007.
- 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.
- T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, no. 1, pp. 103–123, 2005.
- H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
- L. J. Zhang, J. M. Chen, and Z. B. Hou, “Viscosity approximation methods for nonexpansive mappings and generalized variational inequalities,” Acta Mathematica Sinica, vol. 53, no. 4, pp. 691–698, 2010.