- Email: [email protected]
A parallel projection method for a system of nonlinear variational inequalities
Applied Mathematics and Computation 217 (2010) 1971–1975
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A parallel projection method for a system of nonlinear variational inequalities q Haijian Yang, Lijun Zhou, Qingguo Li * College of Mathematics and Econometrics, Hunan University Changsha, Hunan 410082, PR China
a r t i c l e
i n f o
Keywords: Parallel projection method System of nonlinear variational inequalities Relaxed g (c, r)-cocoercive mappings
a b s t r a c t In this paper, we introduce and consider a new system of nonlinear variational inequalities involving two different operators. Using the parallel projection technique, we suggest and analyze an iterative method for this system of variational inequalities. We establish a convergence result for the proposed method under certain conditions. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The study of various problems in partial differential equations with unilateral constraints, unilateral mechanics, and engineering science has led to characterization of solutions to these problems by the mathematical theory of variational inequalities in the case of convex energy functionals, and by hemivariational inequalities in the case of nonconvex energy functions. Motivated by the recent advances [3–6,10], Verma [15,17] introduced a new system of strongly monotonic variational inequalities, and studied the approximation solvability of this system based on an application of a projection method. This studied and modified the earlier work on a class of strongly monotonic variational inequalities on real Hilbert spaces. The main and basic idea in this technique is to establish the equivalence between the variational inequalities and the fixed point problems. This alternative equivalence has been used to develop several projection iterative methods for solving variational inequalities and related optimization problems. It is well known that the convergence of projection type methods requires that the underlying operator should be strongly monotone and Lipschitz continuous. These strict conditions rule out many applications of these methods. This fact motivates to develop other methods or modify the projection methods. Using the updating technique, Aslam Noor [11–13], Verma [14,16] and others have suggested some double projection methods, the convergence of which can be proved under mild conditions. The double projection/projection type methods contain several known as well as new projection methods as special cases, while some have been applied to problems arising, especially from complementarity problems, computational mathematics, convex quadratic programming, and other variational problems. However, these sequential iterative methods are only suitable for implementing on the traditional single-processor computer. To satisfy practical requirements of modern multiprocessor systems, efficient iterative methods having parallel characteristics need to be further developed for the system of variational inequalities. Inspired and motivated by research works in this field, we introduce and consider a system of variational inequalities involving two different nonlinear operators. Using the parallel projection technique, we suggest and analyze a parallel iterative method for solving this system. We also prove q The work is supported by the National Nature Science Foundation (No. 10771056) and the National High Technology Research and Development Program (No. 2006AA04A104), PR China. * Corresponding author. E-mail addresses: [email protected] (H. Yang), [email protected] (L. Zhou), [email protected] (Q. Li).
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.06.053
1972
H. Yang et al. / Applied Mathematics and Computation 217 (2010) 1971–1975
the convergence of the proposed iterative method under much weaker conditions. Our result represents a refinement and improvement of the recent result of Verma [15,17] and other known corresponding results in the recent literature. 2. Preliminaries In this section, we start with some notations, definitions and basic results that are useful for the proposed methods. Let H be a real Hilbert space whose inner product and norm are denoted by h , i and k k, respectively. Let K be a closed and convex set in H. Let T1, T2: K ? H be two nonlinear different operators. We consider a system of nonlinear variational inequalities involving two different operators (abbreviated as SNVI) as follows: determine elements x*, y* 2 K (and hence g(x*), g(y*) 2 K) such that
hqT 1 ðy Þ þ gðx Þ gðy Þ; gðxÞ gðx Þi P 0 for all gðxÞ 2 K; hgT 2 ðx Þ þ gðy Þ gðx Þ; gðxÞ gðy Þi P 0 for all gðxÞ 2 K;
ð2:1Þ ð2:2Þ
where g: K ? K is any mapping and q, g are positive constants. Clearly, if T1 = T2 = T and g = I, then problem (SNVI) reduces to the following system of variational inequalities (SVI) considered by Verma [15,17] of finding x*, y* 2 K such that
hqTðy Þ þ x y ; x x i P 0
for all x 2 K;
ð2:3Þ
hgTðx Þ þ y x ; x y i P 0 for all x 2 K;
ð2:4Þ
where q and g are positive constants. In the following, we suggest a parallel projection algorithm for solving the system of nonlinear variational inequalities (SNVI). Our results extend the corresponding results in [15,17]. First of all, we establish the equivalence between the system of variational inequalities and fixed point problems. For this purpose, we recall the following result. Lemma 2.1. Given elements z 2 H, u 2 K satisfy the inequality
hu z; v ui P 0 for all
v 2 K;
if and only if u 2 K satisfies the relation u = PKz, where PK is a projection from K onto H. Using Lemma 2.1, we can easily show that SNVI (2.1) and (2.2) is equivalent to the following projection formulas:
gðx Þ ¼ PK ½gðy Þ qT 1 ðy Þ;
ð2:5Þ
gðy Þ ¼ PK ½gðx Þ gT 2 ðx Þ:
ð2:6Þ
This alternative equivalence formulation enables us to suggest the following iterative method for solving the system of variational inequalities (SNVI). Algorithm 2.1. For arbitrarily chosen initial points x0, y0 2 K (and g(x0), g(y0) 2 K), sequences {xn} and {yn} are computed by
gðxnþ1 Þ ¼ ð1 an Þgðxn Þ þ an PK ½gðyn Þ qT 1 ðyn Þ;
for q > 0;
ð2:7Þ
gðynþ1 Þ ¼ ð1 bn Þgðyn Þ þ bn PK ½gðxn Þ gT 2 ðxn Þ;
for g > 0;
ð2:8Þ
where an 2 [0, 1] and bn 2 [0, 1] for all n P 0. One of the attractive features of Algorithm 2.1 is suitable for implementing on two different processor computers. Assume that xn and yn are given in Algorithm 2.1, in order to get the new iterative points, we can set one processor computer to compute xn+1 by using (2.7), and set the other processor computer to compute yn+1 by using (2.8). In other words, xn+1 and yn+1 are solved in parallel, and Algorithm 2.1 is the so-called parallel projection method. The sequential iterative methods introduced in [7,15,17] are only suitable for implementing on the traditional single-processor computer. That is, assume that xn and yn are given, in order to get the new iterative points, we need to solve xn+1 and yn+1 in sequence. It is obvious that, in order to satisfy practical requirements of modern multiprocessor systems, the parallel iterative methods are more attractive with respect to the sequence iterative methods. We refer the interested reader to the papers [1,2,8,9] and references therein for more examples and ideas of the parallel iterative methods. If T1 = T2 = T and g = I, then Algorithm 2.1 reduces to the following algorithm. Algorithm 2.2. For arbitrarily chosen initial points x0, y0 2 K, sequences {xn} and {yn} are computed by
xnþ1 ¼ ð1 an Þxn þ an PK ½yn qTðyn Þ for q > 0;
ð2:9Þ
ynþ1 ¼ ð1 bn Þyn þ bn PK ½xn gTðxn Þ for g > 0;
ð2:10Þ
where an 2 [0, 1] and bn 2 [0, 1] for all n P 0. Definition 2.2. A mapping g: K ? H is called to be a-expansive if for all x, y 2 H, there exists a constant a > 0, such that
kgðxÞ gðyÞk P akx yk:
1973
H. Yang et al. / Applied Mathematics and Computation 217 (2010) 1971–1975
Definition 2.3. A mapping T: K ? H is called to be (1) g r-strongly monotone if for all x, y 2 K, there exists a constant r > 0 such that
hTx Ty; gðxÞ gðyÞi P rkgðxÞ gðyÞk2 ; (2) g c-cocoercive if for all x, y 2 K, there exists a constant c > 0 such that
hTx Ty; gðxÞ gðyÞi P ðcÞkTx Tyk2 ; (3) relaxed g (c, r)-cocoercive if for all x, y 2 K, there exists a constant r > 0 and c > 0 such that
hTx Ty; gðxÞ gðyÞi P ðcÞkTx Tyk2 þ rkgðxÞ gðyÞk2 ; (4) g l-Lipschitz continuous if for all x, y 2 K, there exist constants l > 0 such that
kTx Tyk 6 lkgðxÞ gðyÞk: From Definition 2.3, clearly, the class of relaxed g (c, r)-cocoercive mappings is more general than the class of g rstrongly monotone mappings. For g = I, the identity operator, Definition 2.3 reduces to the standard definition of strongly monotonicity, cocoercivity, relaxed (c, r)-cocoercivity and Lipschitz continuation, respectively. Lemma 2.4 ([7,18]). Suppose fdn g1 n¼0 is a nonnegative sequence satisfying the following inequality:
dnþ1 6 ð1 kn Þdn þ r for all n P 0; with kn 2 ½0; 1;
P1
n¼0 kn
¼ 1, and rn = o(kn). Then limn?1 dn = 0.
3. Main results In this section, based on Algorithm 2.1, we present the approximation solvability of SNVI (2.1) and (2.2) involving relaxed g (c, r)-cocoercive and g l-Lipschitz continuous mappings in Hilbert settings. Theorem 3.1. Let K be a nonempty closed convex subset of a real Hilbert space H and let (x*, y*) be the solution of SNVI (2.1) and (2.2). Let Ti: K K ? H be relaxed g (ci, ri)-cocoercive and g li-Lipschitz continuous mappings for i = 1, 2, g: K ? K be an aexpansive mapping. Sequences {xn} and {yn} are generated by Algorithm 2.1 . In addition, the following assumptions hold: 1=2 (1) h1 ¼ 1 þ 2qc1 l21 2qr 1 þ q2 l21 such that 0 < h1 < 1. 2 2 2 1=2 (2) h2 ¼ 1 þ 2gc2 l2 2gr 2 þ g l2 such that 0 < h2 < 1. P P1 (3) 0 6 an, bn 6 1, (an h2bn) P 0 and (bn h1an) P 0 such that 1 n¼0 ðan h2 bn Þ ¼ 1; n¼0 ðbn h1 an Þ ¼ 1. Then sequences {xn} and {yn} converge to x* and y*, respectively. Proof. To prove the result, we first need to evaluate kg(xn+1) g(x*)k for all n P 0. From (2.5) and (2.7), and the nonexpansive property of the projection PK, we have
kgðxnþ1 Þ gðx Þk ¼ kð1 an Þgðxn Þ þ an PK ½gðyn Þ qT 1 ðyn Þ ð1 an Þgðx Þ an P K ½gðy Þ qT 1 ðy Þk 6 ð1 an Þkgðxn Þ gðx Þk þ an kPK ½gðyn Þ qT 1 ðyn Þ PK ½gðy Þ qT 1 ðy Þk 6 ð1 an Þkgðxn Þ gðx Þk þ an k½gðyn Þ qT 1 ðyn Þ ½gðy Þ qT 1 ðy Þk ¼ ð1 an Þkgðxn Þ gðx Þk þ an kgðyn Þ gðy Þ q½T 1 ðyn Þ T 1 ðy Þk:
ð3:1Þ
Since T1 is a relaxed g (c1,r1)-cocoercive and g l1-Lipschitz continuous mapping, we have
kgðyn Þ gðy Þ q½T 1 ðyn Þ T 1 ðy Þk2 ¼ kgðyn Þ gðy Þk2 2qhT 1 ðyn Þ T 1 ðy Þ; gðyn Þ gðy Þi þ q2 kT 1 ðyn Þ T 1 ðy Þk2 6 kgðyn Þ gðy Þk2 2q½c1 kT 1 ðyn Þ T 1 ðy Þk2 þ r1 kgðyn Þ gðy Þk2 þ q2 kT 1 ðyn Þ T 1 ðy Þk2 6 kgðyn Þ gðy Þk2 þ 2qc1 l21 kgðyn Þ gðy Þk2 2qr 1 kgðyn Þ gðy Þk2 þ q2 l21 kgðyn Þ gðy Þk2 ¼ 1 þ 2qc1 l21 2qr 1 þ q2 l21 kgðyn Þ gðy Þk2 :
Since h1 ¼ 1 þ 2qc1 l 2qr 1 þ q 2 1
2
l
2 1=2 1
ð3:2Þ
from assumption (1), it follows from (3.2) that
kgðyn Þ gðy Þ q½T 1 ðyn Þ T 1 ðy Þk 6 h1 kgðyn Þ gðy Þk: As a result, in light of (3.1) and (3.3), we have
ð3:3Þ
1974
H. Yang et al. / Applied Mathematics and Computation 217 (2010) 1971–1975
kgðxnþ1 Þ gðx Þk 6 ð1 an Þkgðxn Þ gðx Þk þ an h1 kgðyn Þ gðy Þk:
ð3:4Þ
Similarly, since T2 is a relaxed g (c2, r2)-cocoercive and g l2-Lipschitz continuous mapping, we obtain
kgðynþ1 Þ gðy Þk 6 ð1 bn Þkgðyn Þ gðy Þk þ bn h2 kgðxn Þ gðx Þk; 2 1=2 2
where h2 ¼ ½1 þ 2gc2 l 2gr2 þ g l It follows from (3.4) and (3.5) that 2 2
2
ð3:5Þ
from assumption (2).
kgðxnþ1 Þ gðx Þk þ kgðynþ1 Þ gðy Þk 6 ½1 ðan h2 bn Þkgðxn Þ gðx Þk þ ½1 ðbn h1 an Þkgðyn Þ gðy Þk 6 maxfx1 ; x2 gðkgðxn Þ gðx Þk þ kgðyn Þ gðy ÞkÞ;
ð3:6Þ
where x1 = 1 (an h2bn) and x2 = 1 (bn h1an). Now, define the norm k k* on H H by
kðu; v Þk ¼ kuk þ kv k;
for all ðu; v Þ 2 H H:
It observes that (H H, k k*) is a Banach space. Hence, (3.6) implies that
kðgðxnþ1 Þ; gðynþ1 ÞÞ ðgðx Þ; gðy ÞÞk 6 max x1 ; x2 gkðgðxn Þ; gðyn ÞÞ ðgðx Þ; gðy ÞÞk: Notice that 0 6 an, bn 6 1, (an h2bn) P 0 and (bn h1an) P 0 such that assumption (3), and hence from Lemma 2.4, we have
P1
n¼0 ðan
h2 bn Þ ¼ 1;
P1
n¼0 ðbn
h1 an Þ ¼ 1 from
lim jðgðxnþ1 Þ; gðynþ1 ÞÞ ðgðx Þ; gðy ÞÞk ¼ 0:
n!1
Therefore, we obtain
kgðxnþ1 Þ gðx Þk ! 0; as n ! 1; kgðynþ1 Þ gðy Þk ! 0; as n ! 1: Consequently, sequences {g(xn)} and {g(yn)} converge to {g(x*)} and {g(y*)}, respectively. Since g be a-expansive, sequences {xn} and {yn} converge to x* and y*, respectively. That completes the proof. h Remark 3.1. Theorem 3.1 extends the solvability of SVI (2.3) and (2.4) to the more general SNVI (2.1) and (2.2). The underlying operator Ti (i = 1, 2) in our paper needs to be relaxed (c, r)-cocoercive while the underlying operator T in [15,17] needs to be strongly monotone. Hence, Theorem 3.1 extends and improves the main results of Theorem 2.1 of [15] and Theorem 3.1 of [17]. Moreover, Algorithm 2.1 is suitable for implementing on two different processor computers, so the computation workload is much less than the projection method in [7,15,17] at each iteration step. If T1 = T2 = T and g = I, then the following theorem can be directly obtained from Theorem 3.1. Theorem 3.2. Let K be a nonempty closed convex subset of a real Hilbert space H and let (x*, y*) be the solution of SVI (2.3) and (2.4) . Let T: K ? H be (c, r)-cocoercive and l-Lipschitz continuous mappings. Sequences {xn} and {yn} are generated by Algorithm 2.2 . In addition, the following assumptions hold: (1) h1 = [1 + 2qcl2 2q r + q2l2]1/2 such that 0 < h1 < 1. (2) h2 = [1 + 2gcl2 2g r + g2l2]1/2 such that 0 < h2 < 1. P P1 (3) 0 6 an, bn 6 1, (an h2bn) P 0 and (bn h1an) P 0 such that 1 n¼0 ðan h2 bn Þ ¼ 1; n¼0 ðbn h1 an Þ ¼ 1. Then sequences {xn} and {yn} converge to x* and y*, respectively. Acknowledgements The authors are very much indebted to the editor and the referee for their constructive suggestions and helpful comments, which led to improvements of the original manuscript of this paper. References [1] G. Baudet, Asynchronous iterative methods for multiprocessors, J. Assoc. Comput. Mach. 25 (1978) 226–244. [2] D. Bertsekas, J. Tsitsiklis, Parallel and Distributed Computation, Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1989. [3] Y.P. Fang, N.J. Huang, Y.J. Cao, S.M. Kang, Stable iterative algorithms for a class of general nonlinear variational inequalities, Adv. Nonlinear Var. Inequal. 5 (2) (2002) 1–9. [4] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984. [5] B.S. He, A class of projection method and contraction methods for monotone variational inequalities, Appl. Math. Optim. 35 (1997) 69–75. [6] B.S. He, X.Z. He, Relations of merit functions methods and the projection-contraction methods for variational inequalities, Adv. Nonlinear Var. Inequal. 6 (2) (2003) 55–68.
H. Yang et al. / Applied Mathematics and Computation 217 (2010) 1971–1975
1975
[7] Z.Y. Huang, M. Aslam Noor, An explicit projection method for a system of nonlinear variational inequalities with different (c, r)-cocoercive mappings, Appl. Math. Comput. 190 (2007) 356–361. [8] K.-H. Hoffmann, J. Zou, Parallel algorithms of Schwarz variant for variational inequalities, Numer. Funct. Anal. Optim. 13 (1992) 449–462. [9] K.-H. Hoffmann, J. Zou, Parallel solution of variational inequality problems with nonlinear source terms, IMA J. Numer. Anal. 16 (1996) 31–45. [10] S. Karamardian, S. Schaible, Seven kinds of monotone maps, J. Optim. Theory Appl. 66 (1990) 37–46. [11] M. Aslam Noor, Some developments in general monotone mixed variational inequalities, Appl. Math. Comput. 152 (2004) 199–277. [12] M. Aslam Noor, K. Inayat Noor, Self-adaptive projection algorithms for general variational inequalities, Appl. Math. Comput. 151 (2004) 659–670. [13] M. Aslam Noor, Projection–proximal methods for general variational inequalities, J. Math. Anal. Appl. 316 (2006) 736–752. [14] R.U. Verma, A class of projection–contraction methods applied to monotone variational inequalities, Appl. Math. Lett. 13 (2000) 55–62. [15] R.U. Verma, Projection methods, algorithms and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41 (2001) 1025–1031. [16] R.U. Verma, Generalized class of partial relaxed monotonicity and its connections, Adv. Nonlinear Var. Inequal. 7 (2) (2004) 155–164. [17] R.U. Verma, General convergence analysis for two-step projection methods and applications to variational problems, Appl. Math. Lett. 18 (2005) 1286– 1292. [18] X.L. Weng, Fixed point iteration for local strictly pseudocontractive mappings, Proc. Am. Math. Soc. 113 (1991) 727–731.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A parallel projection method for a system of nonlinear variational inequalities q Haijian Yang, Lijun Zhou, Qingguo Li * College of Mathematics and Econometrics, Hunan University Changsha, Hunan 410082, PR China
a r t i c l e
i n f o
Keywords: Parallel projection method System of nonlinear variational inequalities Relaxed g (c, r)-cocoercive mappings
a b s t r a c t In this paper, we introduce and consider a new system of nonlinear variational inequalities involving two different operators. Using the parallel projection technique, we suggest and analyze an iterative method for this system of variational inequalities. We establish a convergence result for the proposed method under certain conditions. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The study of various problems in partial differential equations with unilateral constraints, unilateral mechanics, and engineering science has led to characterization of solutions to these problems by the mathematical theory of variational inequalities in the case of convex energy functionals, and by hemivariational inequalities in the case of nonconvex energy functions. Motivated by the recent advances [3–6,10], Verma [15,17] introduced a new system of strongly monotonic variational inequalities, and studied the approximation solvability of this system based on an application of a projection method. This studied and modified the earlier work on a class of strongly monotonic variational inequalities on real Hilbert spaces. The main and basic idea in this technique is to establish the equivalence between the variational inequalities and the fixed point problems. This alternative equivalence has been used to develop several projection iterative methods for solving variational inequalities and related optimization problems. It is well known that the convergence of projection type methods requires that the underlying operator should be strongly monotone and Lipschitz continuous. These strict conditions rule out many applications of these methods. This fact motivates to develop other methods or modify the projection methods. Using the updating technique, Aslam Noor [11–13], Verma [14,16] and others have suggested some double projection methods, the convergence of which can be proved under mild conditions. The double projection/projection type methods contain several known as well as new projection methods as special cases, while some have been applied to problems arising, especially from complementarity problems, computational mathematics, convex quadratic programming, and other variational problems. However, these sequential iterative methods are only suitable for implementing on the traditional single-processor computer. To satisfy practical requirements of modern multiprocessor systems, efficient iterative methods having parallel characteristics need to be further developed for the system of variational inequalities. Inspired and motivated by research works in this field, we introduce and consider a system of variational inequalities involving two different nonlinear operators. Using the parallel projection technique, we suggest and analyze a parallel iterative method for solving this system. We also prove q The work is supported by the National Nature Science Foundation (No. 10771056) and the National High Technology Research and Development Program (No. 2006AA04A104), PR China. * Corresponding author. E-mail addresses: [email protected] (H. Yang), [email protected] (L. Zhou), [email protected] (Q. Li).
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.06.053
1972
H. Yang et al. / Applied Mathematics and Computation 217 (2010) 1971–1975
the convergence of the proposed iterative method under much weaker conditions. Our result represents a refinement and improvement of the recent result of Verma [15,17] and other known corresponding results in the recent literature. 2. Preliminaries In this section, we start with some notations, definitions and basic results that are useful for the proposed methods. Let H be a real Hilbert space whose inner product and norm are denoted by h , i and k k, respectively. Let K be a closed and convex set in H. Let T1, T2: K ? H be two nonlinear different operators. We consider a system of nonlinear variational inequalities involving two different operators (abbreviated as SNVI) as follows: determine elements x*, y* 2 K (and hence g(x*), g(y*) 2 K) such that
hqT 1 ðy Þ þ gðx Þ gðy Þ; gðxÞ gðx Þi P 0 for all gðxÞ 2 K; hgT 2 ðx Þ þ gðy Þ gðx Þ; gðxÞ gðy Þi P 0 for all gðxÞ 2 K;
ð2:1Þ ð2:2Þ
where g: K ? K is any mapping and q, g are positive constants. Clearly, if T1 = T2 = T and g = I, then problem (SNVI) reduces to the following system of variational inequalities (SVI) considered by Verma [15,17] of finding x*, y* 2 K such that
hqTðy Þ þ x y ; x x i P 0
for all x 2 K;
ð2:3Þ
hgTðx Þ þ y x ; x y i P 0 for all x 2 K;
ð2:4Þ
where q and g are positive constants. In the following, we suggest a parallel projection algorithm for solving the system of nonlinear variational inequalities (SNVI). Our results extend the corresponding results in [15,17]. First of all, we establish the equivalence between the system of variational inequalities and fixed point problems. For this purpose, we recall the following result. Lemma 2.1. Given elements z 2 H, u 2 K satisfy the inequality
hu z; v ui P 0 for all
v 2 K;
if and only if u 2 K satisfies the relation u = PKz, where PK is a projection from K onto H. Using Lemma 2.1, we can easily show that SNVI (2.1) and (2.2) is equivalent to the following projection formulas:
gðx Þ ¼ PK ½gðy Þ qT 1 ðy Þ;
ð2:5Þ
gðy Þ ¼ PK ½gðx Þ gT 2 ðx Þ:
ð2:6Þ
This alternative equivalence formulation enables us to suggest the following iterative method for solving the system of variational inequalities (SNVI). Algorithm 2.1. For arbitrarily chosen initial points x0, y0 2 K (and g(x0), g(y0) 2 K), sequences {xn} and {yn} are computed by
gðxnþ1 Þ ¼ ð1 an Þgðxn Þ þ an PK ½gðyn Þ qT 1 ðyn Þ;
for q > 0;
ð2:7Þ
gðynþ1 Þ ¼ ð1 bn Þgðyn Þ þ bn PK ½gðxn Þ gT 2 ðxn Þ;
for g > 0;
ð2:8Þ
where an 2 [0, 1] and bn 2 [0, 1] for all n P 0. One of the attractive features of Algorithm 2.1 is suitable for implementing on two different processor computers. Assume that xn and yn are given in Algorithm 2.1, in order to get the new iterative points, we can set one processor computer to compute xn+1 by using (2.7), and set the other processor computer to compute yn+1 by using (2.8). In other words, xn+1 and yn+1 are solved in parallel, and Algorithm 2.1 is the so-called parallel projection method. The sequential iterative methods introduced in [7,15,17] are only suitable for implementing on the traditional single-processor computer. That is, assume that xn and yn are given, in order to get the new iterative points, we need to solve xn+1 and yn+1 in sequence. It is obvious that, in order to satisfy practical requirements of modern multiprocessor systems, the parallel iterative methods are more attractive with respect to the sequence iterative methods. We refer the interested reader to the papers [1,2,8,9] and references therein for more examples and ideas of the parallel iterative methods. If T1 = T2 = T and g = I, then Algorithm 2.1 reduces to the following algorithm. Algorithm 2.2. For arbitrarily chosen initial points x0, y0 2 K, sequences {xn} and {yn} are computed by
xnþ1 ¼ ð1 an Þxn þ an PK ½yn qTðyn Þ for q > 0;
ð2:9Þ
ynþ1 ¼ ð1 bn Þyn þ bn PK ½xn gTðxn Þ for g > 0;
ð2:10Þ
where an 2 [0, 1] and bn 2 [0, 1] for all n P 0. Definition 2.2. A mapping g: K ? H is called to be a-expansive if for all x, y 2 H, there exists a constant a > 0, such that
kgðxÞ gðyÞk P akx yk:
1973
H. Yang et al. / Applied Mathematics and Computation 217 (2010) 1971–1975
Definition 2.3. A mapping T: K ? H is called to be (1) g r-strongly monotone if for all x, y 2 K, there exists a constant r > 0 such that
hTx Ty; gðxÞ gðyÞi P rkgðxÞ gðyÞk2 ; (2) g c-cocoercive if for all x, y 2 K, there exists a constant c > 0 such that
hTx Ty; gðxÞ gðyÞi P ðcÞkTx Tyk2 ; (3) relaxed g (c, r)-cocoercive if for all x, y 2 K, there exists a constant r > 0 and c > 0 such that
hTx Ty; gðxÞ gðyÞi P ðcÞkTx Tyk2 þ rkgðxÞ gðyÞk2 ; (4) g l-Lipschitz continuous if for all x, y 2 K, there exist constants l > 0 such that
kTx Tyk 6 lkgðxÞ gðyÞk: From Definition 2.3, clearly, the class of relaxed g (c, r)-cocoercive mappings is more general than the class of g rstrongly monotone mappings. For g = I, the identity operator, Definition 2.3 reduces to the standard definition of strongly monotonicity, cocoercivity, relaxed (c, r)-cocoercivity and Lipschitz continuation, respectively. Lemma 2.4 ([7,18]). Suppose fdn g1 n¼0 is a nonnegative sequence satisfying the following inequality:
dnþ1 6 ð1 kn Þdn þ r for all n P 0; with kn 2 ½0; 1;
P1
n¼0 kn
¼ 1, and rn = o(kn). Then limn?1 dn = 0.
3. Main results In this section, based on Algorithm 2.1, we present the approximation solvability of SNVI (2.1) and (2.2) involving relaxed g (c, r)-cocoercive and g l-Lipschitz continuous mappings in Hilbert settings. Theorem 3.1. Let K be a nonempty closed convex subset of a real Hilbert space H and let (x*, y*) be the solution of SNVI (2.1) and (2.2). Let Ti: K K ? H be relaxed g (ci, ri)-cocoercive and g li-Lipschitz continuous mappings for i = 1, 2, g: K ? K be an aexpansive mapping. Sequences {xn} and {yn} are generated by Algorithm 2.1 . In addition, the following assumptions hold: 1=2 (1) h1 ¼ 1 þ 2qc1 l21 2qr 1 þ q2 l21 such that 0 < h1 < 1. 2 2 2 1=2 (2) h2 ¼ 1 þ 2gc2 l2 2gr 2 þ g l2 such that 0 < h2 < 1. P P1 (3) 0 6 an, bn 6 1, (an h2bn) P 0 and (bn h1an) P 0 such that 1 n¼0 ðan h2 bn Þ ¼ 1; n¼0 ðbn h1 an Þ ¼ 1. Then sequences {xn} and {yn} converge to x* and y*, respectively. Proof. To prove the result, we first need to evaluate kg(xn+1) g(x*)k for all n P 0. From (2.5) and (2.7), and the nonexpansive property of the projection PK, we have
kgðxnþ1 Þ gðx Þk ¼ kð1 an Þgðxn Þ þ an PK ½gðyn Þ qT 1 ðyn Þ ð1 an Þgðx Þ an P K ½gðy Þ qT 1 ðy Þk 6 ð1 an Þkgðxn Þ gðx Þk þ an kPK ½gðyn Þ qT 1 ðyn Þ PK ½gðy Þ qT 1 ðy Þk 6 ð1 an Þkgðxn Þ gðx Þk þ an k½gðyn Þ qT 1 ðyn Þ ½gðy Þ qT 1 ðy Þk ¼ ð1 an Þkgðxn Þ gðx Þk þ an kgðyn Þ gðy Þ q½T 1 ðyn Þ T 1 ðy Þk:
ð3:1Þ
Since T1 is a relaxed g (c1,r1)-cocoercive and g l1-Lipschitz continuous mapping, we have
kgðyn Þ gðy Þ q½T 1 ðyn Þ T 1 ðy Þk2 ¼ kgðyn Þ gðy Þk2 2qhT 1 ðyn Þ T 1 ðy Þ; gðyn Þ gðy Þi þ q2 kT 1 ðyn Þ T 1 ðy Þk2 6 kgðyn Þ gðy Þk2 2q½c1 kT 1 ðyn Þ T 1 ðy Þk2 þ r1 kgðyn Þ gðy Þk2 þ q2 kT 1 ðyn Þ T 1 ðy Þk2 6 kgðyn Þ gðy Þk2 þ 2qc1 l21 kgðyn Þ gðy Þk2 2qr 1 kgðyn Þ gðy Þk2 þ q2 l21 kgðyn Þ gðy Þk2 ¼ 1 þ 2qc1 l21 2qr 1 þ q2 l21 kgðyn Þ gðy Þk2 :
Since h1 ¼ 1 þ 2qc1 l 2qr 1 þ q 2 1
2
l
2 1=2 1
ð3:2Þ
from assumption (1), it follows from (3.2) that
kgðyn Þ gðy Þ q½T 1 ðyn Þ T 1 ðy Þk 6 h1 kgðyn Þ gðy Þk: As a result, in light of (3.1) and (3.3), we have
ð3:3Þ
1974
H. Yang et al. / Applied Mathematics and Computation 217 (2010) 1971–1975
kgðxnþ1 Þ gðx Þk 6 ð1 an Þkgðxn Þ gðx Þk þ an h1 kgðyn Þ gðy Þk:
ð3:4Þ
Similarly, since T2 is a relaxed g (c2, r2)-cocoercive and g l2-Lipschitz continuous mapping, we obtain
kgðynþ1 Þ gðy Þk 6 ð1 bn Þkgðyn Þ gðy Þk þ bn h2 kgðxn Þ gðx Þk; 2 1=2 2
where h2 ¼ ½1 þ 2gc2 l 2gr2 þ g l It follows from (3.4) and (3.5) that 2 2
2
ð3:5Þ
from assumption (2).
kgðxnþ1 Þ gðx Þk þ kgðynþ1 Þ gðy Þk 6 ½1 ðan h2 bn Þkgðxn Þ gðx Þk þ ½1 ðbn h1 an Þkgðyn Þ gðy Þk 6 maxfx1 ; x2 gðkgðxn Þ gðx Þk þ kgðyn Þ gðy ÞkÞ;
ð3:6Þ
where x1 = 1 (an h2bn) and x2 = 1 (bn h1an). Now, define the norm k k* on H H by
kðu; v Þk ¼ kuk þ kv k;
for all ðu; v Þ 2 H H:
It observes that (H H, k k*) is a Banach space. Hence, (3.6) implies that
kðgðxnþ1 Þ; gðynþ1 ÞÞ ðgðx Þ; gðy ÞÞk 6 max x1 ; x2 gkðgðxn Þ; gðyn ÞÞ ðgðx Þ; gðy ÞÞk: Notice that 0 6 an, bn 6 1, (an h2bn) P 0 and (bn h1an) P 0 such that assumption (3), and hence from Lemma 2.4, we have
P1
n¼0 ðan
h2 bn Þ ¼ 1;
P1
n¼0 ðbn
h1 an Þ ¼ 1 from
lim jðgðxnþ1 Þ; gðynþ1 ÞÞ ðgðx Þ; gðy ÞÞk ¼ 0:
n!1
Therefore, we obtain
kgðxnþ1 Þ gðx Þk ! 0; as n ! 1; kgðynþ1 Þ gðy Þk ! 0; as n ! 1: Consequently, sequences {g(xn)} and {g(yn)} converge to {g(x*)} and {g(y*)}, respectively. Since g be a-expansive, sequences {xn} and {yn} converge to x* and y*, respectively. That completes the proof. h Remark 3.1. Theorem 3.1 extends the solvability of SVI (2.3) and (2.4) to the more general SNVI (2.1) and (2.2). The underlying operator Ti (i = 1, 2) in our paper needs to be relaxed (c, r)-cocoercive while the underlying operator T in [15,17] needs to be strongly monotone. Hence, Theorem 3.1 extends and improves the main results of Theorem 2.1 of [15] and Theorem 3.1 of [17]. Moreover, Algorithm 2.1 is suitable for implementing on two different processor computers, so the computation workload is much less than the projection method in [7,15,17] at each iteration step. If T1 = T2 = T and g = I, then the following theorem can be directly obtained from Theorem 3.1. Theorem 3.2. Let K be a nonempty closed convex subset of a real Hilbert space H and let (x*, y*) be the solution of SVI (2.3) and (2.4) . Let T: K ? H be (c, r)-cocoercive and l-Lipschitz continuous mappings. Sequences {xn} and {yn} are generated by Algorithm 2.2 . In addition, the following assumptions hold: (1) h1 = [1 + 2qcl2 2q r + q2l2]1/2 such that 0 < h1 < 1. (2) h2 = [1 + 2gcl2 2g r + g2l2]1/2 such that 0 < h2 < 1. P P1 (3) 0 6 an, bn 6 1, (an h2bn) P 0 and (bn h1an) P 0 such that 1 n¼0 ðan h2 bn Þ ¼ 1; n¼0 ðbn h1 an Þ ¼ 1. Then sequences {xn} and {yn} converge to x* and y*, respectively. Acknowledgements The authors are very much indebted to the editor and the referee for their constructive suggestions and helpful comments, which led to improvements of the original manuscript of this paper. References [1] G. Baudet, Asynchronous iterative methods for multiprocessors, J. Assoc. Comput. Mach. 25 (1978) 226–244. [2] D. Bertsekas, J. Tsitsiklis, Parallel and Distributed Computation, Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1989. [3] Y.P. Fang, N.J. Huang, Y.J. Cao, S.M. Kang, Stable iterative algorithms for a class of general nonlinear variational inequalities, Adv. Nonlinear Var. Inequal. 5 (2) (2002) 1–9. [4] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984. [5] B.S. He, A class of projection method and contraction methods for monotone variational inequalities, Appl. Math. Optim. 35 (1997) 69–75. [6] B.S. He, X.Z. He, Relations of merit functions methods and the projection-contraction methods for variational inequalities, Adv. Nonlinear Var. Inequal. 6 (2) (2003) 55–68.
H. Yang et al. / Applied Mathematics and Computation 217 (2010) 1971–1975
1975
[7] Z.Y. Huang, M. Aslam Noor, An explicit projection method for a system of nonlinear variational inequalities with different (c, r)-cocoercive mappings, Appl. Math. Comput. 190 (2007) 356–361. [8] K.-H. Hoffmann, J. Zou, Parallel algorithms of Schwarz variant for variational inequalities, Numer. Funct. Anal. Optim. 13 (1992) 449–462. [9] K.-H. Hoffmann, J. Zou, Parallel solution of variational inequality problems with nonlinear source terms, IMA J. Numer. Anal. 16 (1996) 31–45. [10] S. Karamardian, S. Schaible, Seven kinds of monotone maps, J. Optim. Theory Appl. 66 (1990) 37–46. [11] M. Aslam Noor, Some developments in general monotone mixed variational inequalities, Appl. Math. Comput. 152 (2004) 199–277. [12] M. Aslam Noor, K. Inayat Noor, Self-adaptive projection algorithms for general variational inequalities, Appl. Math. Comput. 151 (2004) 659–670. [13] M. Aslam Noor, Projection–proximal methods for general variational inequalities, J. Math. Anal. Appl. 316 (2006) 736–752. [14] R.U. Verma, A class of projection–contraction methods applied to monotone variational inequalities, Appl. Math. Lett. 13 (2000) 55–62. [15] R.U. Verma, Projection methods, algorithms and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41 (2001) 1025–1031. [16] R.U. Verma, Generalized class of partial relaxed monotonicity and its connections, Adv. Nonlinear Var. Inequal. 7 (2) (2004) 155–164. [17] R.U. Verma, General convergence analysis for two-step projection methods and applications to variational problems, Appl. Math. Lett. 18 (2005) 1286– 1292. [18] X.L. Weng, Fixed point iteration for local strictly pseudocontractive mappings, Proc. Am. Math. Soc. 113 (1991) 727–731.