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Predictor–corrector methods for general mixed quasi variational inequalities
Applied Mathematics and Computation 157 (2004) 643–652 www.elsevier.com/locate/amc
Predictor–corrector methods for general mixed quasi variational inequalities Muhammad Aslam Noor a,*, Khalida Inayat Noor b, Kamel Al-Khaled b b
a Etisalat College of Engineering, Sharjah, United Arab Emirates Department of Mathematics and Computer Science, United Arab Emiratea University, P.O. Box 17551, Al Ain, United Arab Emirates
Abstract In this paper, we use the auxiliary principle technique to suggest a class of predictor– corrector methods for solving general mixed quasi variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain various known and new results for solving various classes of variational inequalities and related problems. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Variational inequalities; Auxiliary principle; Iterative methods; Convergence; Fixed points
1. Introduction Variational inequalities theory provides us with a simple, natural, unified, novel and general framework to study a wide class of unilateral, obstacle, free, moving and equilibrium problems arising in fluid flow through porous media, elasticity, transportation, finance, economics, and optimization, etc., see [1–25]. This field is dynamic and is experiencing an explosive growth in both theory
*
Corresponding author. E-mail addresses: [email protected] (M. Aslam Noor), [email protected] (K. Inayat Noor), [email protected] (K. Al-Khaled). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.060
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and applications; as a consequence, research techniques and problems are drawn from various fields. Variational inequalities have been generalized and extended in different directions using the novel and innovative techniques. An important and useful generalization of variational inequalities is called the mixed quasi variational inequality involving the nonlinear bifunction. Due to the presence of the nonlinear bifunction, projection method and its variant forms, Wiener–Hopf equations, descent methods cannot be extended and modified to suggest iterative methods for solving the mixed quasi variational inequalities. If the nonlinear term involving the general mixed variational inequalities is a proper, convex and lower-semicontinuous with respect to the first argument, then it has been shown [19] that the general mixed variational inequalities are equivalent to the fixed point and resolvent equations. These alternative formulations have been used to develop a number of iterative type methods for solving mixed variational inequalities. In this approach, one has to evaluate the resolvent operator, which is itself a difficult problem. Secondly, the convergence of these methods requires that the underlying operator must be strongly monotone and Lipschizt continuous. To overcome these drawbacks, the auxiliary principle technique has been developed, the origin of which can be traced back to Lions and Stampacchia [10]. Glowinski et al. [7] used this technique to study the existence of a solution of the mixed variational inequalities. In recent years, this technique has been used to suggest and analyze various iterative methods for solving various classes of variational inequalities. It has been shown that a substantial number of numerical methods can be obtained as special cases from this technique, see [16,17,20,21,25] and references therein. In this paper, we extend these ideas and techniques to suggest a class of predictor– corrector methods for solving general mixed quasi variational inequalities. The convergence of these methods requires only that the operator is partially relaxed strongly monotone, which is weaker than co-coercive. Our results can be viewed as a significant and novel application of the techniques developed in [20,21] to general mixed quasi variational inequalities (2.1). Since general mixed quasi variational ineqaualities include quasi variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.
2. Preliminaries 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 nonempty closed convex set in H . Let uð; Þ : H H ! R [ fþ1g be a continuous bifunction function. For given nonlinear operators T ; g : H ! H , consider the problem of finding u 2 H such that
M. Aslam Noor et al. / Appl. Math. Comput. 157 (2004) 643–652
hTu; gðvÞ gðuÞi þ uðgðvÞ; gðuÞÞ uðgðuÞ; gðuÞÞ P 0;
8gðvÞ 2 H :
645
ð2:1Þ
The inequality of type (2.1) is called the general mixed quasi variational inequality. It can be shown that a wide class of linear and nonlinear problems arising in pure and applied sciences can be studied via the general mixed variational inequalities (2.1), see [19]. We remark that if g I, the identity operator, then problem (2.1) is equivalent to finding u 2 H such that hTu; v ui þ uðv; uÞ uðu; uÞ P 0;
8v 2 H ;
ð2:2Þ
which is called the mixed quasi variational inequalities. For the applications, numerical methods and formulations, see [1,4,7–9,14–19] and the references therein. We note that if uðv; uÞ ¼ uðvÞ; 8u 2 H , then problem (2.1) is equivalent to finding u 2 H such that hTu; gðvÞ gðuÞi þ uðgðvÞÞ uðgðuÞÞ P 0;
8gðvÞ 2 H ;
ð2:3Þ
which is called the general mixed variational inequality. For the numerical methods and applications, see Noor [20,21]. We note that if u is the indicator function of a closed convex set K in H , that is, 0; if u 2 K; uðuÞ IK ðuÞ ¼ þ1; otherwise; then problem (2.1) is equivalent to finding u 2 H , gðuÞ 2 K such that hTu; gðvÞ gðuÞi P 0;
8gðvÞ 2 K:
ð2:4Þ
Inequality of the type (2.4) is known as the general variational inequality, which was introduced and studied by Noor [12] in 1988. It turned out that the odd-order and nonsymmetric free, unilateral, obstacle and equilibrium problems can be studied by the general variational inequality (2.4), see [13,16–20]. If K ¼ fu 2 H : hu; vi P 0; 8v 2 Kg is a polar cone of a convex cone K in H , then problem (2.4) is equivalent to finding u 2 H such that gðuÞ 2 K;
Tu 2 K
and
hTu; gðuÞi ¼ 0;
ð2:5Þ
which is known as the general complementarity problem. We note that if gðuÞ ¼ u mðuÞ, where m is a point-to-point mapping, then problem (2.5) is called the quasi (implicit) complementarity problem, see the references for the formulation and numerical methods. For g I, the identity operator, problem (2.4) collapses to: find u 2 K such that hTu; v ui P 0;
8v 2 K;
ð2:6Þ
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which is called the standard variational inequality, introduced and studied by Stampacchia [22] in 1964. For the recent state-of-the-art, see [1–25]. It is clear that problems (2.2)–(2.6) are special cases of the general mixed quasi variational inequality (2.1). In brief, for a suitable and appropriate choice of the operators T , g, uð; Þ and the space H , one can obtain a wide class of variational inequalities and complementarity problems. This clearly shows that problem (2.1) is quite general and unifying one. Furthermore, problem (2.1) has important applications in various branches of pure and applied sciences, see [1,4,7–11,19]. We also need the following concepts. Lemma 2.1. 8u; v 2 H , we have 2
2
2
2hu; vi ¼ ku þ vk kuk kvk : Proof. It is trivial.
ð2:7Þ
h
Definition 2.1. 8u; v; z 2 H , an operator T : H ! H is said to be: ii(i) g-relaxed strongly monotone, if there exists a constant c > 0 such that 2
hTu Tv; gðuÞ gðvÞi P ckgðuÞ gðvÞk : i(ii) g-partially relaxed strongly monotone, if there exists a constant a > 0 such that 2
hTu Tv; gðzÞ gðvÞi P akgðuÞ gðzÞk : (iii) g-co-coercive, if there exists a constant l > 0 such that hTu Tv; gðuÞ gðvÞi P lkTu Tvk2 : We remark that if z ¼ u, then g-partially relaxed strongly monotonicity is exactly g-monotonicity of the operator T . For g I, the identity operator, then Definition 2.1 reduces to the standard definition of relaxed strongly monotonicity [11], and co-coercivity of the operator. It can be shown that g-co-coercivity implies g-partially relaxed strongly monotonicity but the converse is not true. This shows that the concept of partially relaxed strongly monotonicity is weaker than co-coercivity. Definition 2.2. The bifunction uð; Þ : H H ! R [ f1g is said to be skewsymmetric, if uðu; uÞ uðu; vÞ uðv; uÞ þ uðv; vÞ P 0;
8u; v 2 H :
Note that if uð; Þ is a bilinear function then uð; Þ is nonnegative.
ð2:8Þ
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647
3. Main results In this section, we suggest and analyze a new iterative method for solving the problem (2.1) by using the auxiliary principle technique of Glowinski et al. [7] as developed by Noor [20,21]. For a given u 2 H such that gðuÞ 2 H , consider the problem of finding a unique w 2 H such gðwÞ 2 H satisfying the auxiliary variational inequality hqTu þ gðwÞ gðuÞ; gðvÞ gðwÞi þ quðgðvÞ; gðuÞÞ quðgðuÞ; gðuÞÞ P 0; 8v 2 H ;
ð3:1Þ
where q > 0 is a constant. We note that if w ¼ u, then clearly w is a solution of the general mixed variational inequality (2.1). This observation enables us to suggest the following iterative method for solving the general mixed variational inequalities (2.1). Algorithm 3.1. For a given u0 2 H , compute the approximate solution unþ1 by the iterative scheme hqTwn þ gðunþ1 Þ gðwn Þ; gðvÞ gðunþ1 Þi þ quðgðvÞ; gðunþ1 ÞÞ quðgðunþ1 Þ; gðunþ1 ÞÞ P 0;
8v 2 H ;
ð3:2Þ
hbTyn þ gðwn Þ gðyn Þ; gðvÞ gðwn Þi þ buðgðvÞ; gðwn ÞÞ buðgðwn Þ; gðwn ÞÞ P 0;
8v 2 H
ð3:3Þ
and hlTun þ gðyn Þ gðun Þ; gðvÞ gðyn Þi þ luðgðvÞ; gðyn ÞÞ luðgðyn Þ; gðyn ÞÞ P 0;
8v 2 H ;
ð3:4Þ
where q > 0, l > 0 and b > 0 are constants. Note that if g I, the identity operator, then Algorithm 3.1 reduces to: Algorithm 3.2. For a given u0 2 H , compute unþ1 by the iterative scheme hqTwn þ unþ1 wn ; v unþ1 i þ quðv; unþ1 Þ quðunþ1 ; unþ1 Þ P 0; hbTyn þ wn yn ; v wn i þ buðv; wn Þ buðwn ; wn Þ P 0;
8v 2 H ;
8v 2 H
and hlTun þ yn un ; v yn i þ luðv; yn Þ luðyn ; yn Þ P 0;
8v 2 H :
Algorithm 3.2 appears to be a new one for solving mixed variational inequalities (2.2).
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If uðv; uÞ is a proper, convex and lower-semicontinuous differentiable function with respect to the first argument, then Algorithm 3.1 collapses to: Algorithm 3.3. For a given u0 2 H , compute unþ1 by the iterative scheme gðunþ1 Þ ¼ Juðunþ1 Þ ½gðwn Þ qTwn ; gðwn Þ ¼ Juðwn Þ ½gðyn Þ bTyn ; gðyn Þ ¼ Juðyn Þ ½gðun Þ lTun ;
n ¼ 0; 1; 2 . . . ;
where JuðuÞ ðI þ qouð; uÞÞ1 is the resolvent operator associated with the subdifferential ouð; uÞ of the bifunction uð; Þ, see [19]. If the function uðv; uÞ ¼ uðvÞ, 8u 2 H , is the indicator function of a closed convex set K in H , then Algorithm 3.1 reduces to the following method for solving general variational inequalities (2.4). Algorithm 3.4. For a given u0 2 H such that gðu0 Þ 2 K, compute unþ1 by the iterative schemes hqTwn þ gðunþ1 Þ gðwn Þ; gðvÞ gðunþ1 Þi P 0; 8gðvÞ 2 K; hbTyn þ gðwn Þ gðyn Þ; gðvÞ gðwn Þi P 0; 8gðvÞ 2 K and hlTun þ gðyn Þ gðun Þ; gðvÞ gðyn Þi P 0;
8gðvÞ 2 K:
For the convergence analysis of Algorithm 3.4; see Noor [20]. For a suitable choice of the operators and the space H , one can obtain various new and known methods for solving variational inequalities and complementarity problems. For the convergence analysis of Algorithm 3.1, we need the following result. The analysis in the spirit of Noor [20,21]. 2 H be the exact solution of (2.1) and unþ1 be the approximate Lemma 3.1. Let u solution obtained from Algorithm 3.1. If the operator T : H ! H is a g-partially relaxed strongly monotone operator with constant a > 0, and the bifunction uð; Þ is skew-symmetric, then 2
2
2
kgðunþ1 Þ gð uÞk 6 kgðwn Þ gð uÞk ð1 2aqÞkgðunþ1 Þ gðwn Þk ; ð3:5Þ kgðwn Þ gð uÞk2 6 kgðyn Þ gð uÞk2 ð1 2abÞkgðwn Þ gðyn Þk2 ;
ð3:6Þ
uÞk2 6 kgðun Þ gð uÞk2 ð1 2alÞkgðyn Þ gðun Þk2 : kgðyn Þ gð
ð3:7Þ
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649
Proof. Let u 2 H be solution of (2.1). Then hqT u; gðvÞ gð uÞi þ quðgðvÞ; gð uÞÞ quðgðuÞ; gðuÞÞ P 0;
8v 2 H ; ð3:8Þ
hbT u; gðvÞ gð uÞi þ buðgðvÞ; gð uÞÞ buðgðuÞ; gðuÞÞ P 0;
8v 2 H ð3:9Þ
and hlT u; gðvÞ gð uÞi þ luðgðvÞ; gð uÞÞ luðgðuÞ; gðuÞÞ P 0;
8v 2 H ; ð3:10Þ
where q > 0, b > 0 and l > 0 are constants. Now taking v ¼ unþ1 in (3.8) and v ¼ u in (3.2), we have uÞi þ quðgðunþ1 Þ; gð uÞÞ quðgðuÞ; gðuÞÞ P 0 hqT u; gðunþ1 Þ gð
ð3:11Þ
and hqTwn þ gðunþ1 Þ gðwn Þ; gð uÞ gðunþ1 Þi þ quðgðuÞ; gðunþ1 ÞÞ quðgðunþ1 Þ; gðunþ1 ÞÞ P 0:
ð3:12Þ
Adding (3.11) and (3.12), we have hgðunþ1 Þ gðwn Þ; gð uÞ gðunþ1 Þi P qhTwn T u; gðunþ1 Þ gð uÞi þ uðgðuÞ; gðuÞÞ uðgðuÞ; gðunþ1 ÞÞ uðgðunþ1 Þ; gð uÞÞ þ uðgðunþ1 Þ; gðunþ1 ÞÞ P aqkgðunþ1 Þ gðwn Þk2 ;
ð3:13Þ
where we have used the fact that T is g-partially relaxed strongly monotone with constant a > 0 and the skew-symmetry of the bifunction uð; Þ. Setting u ¼ gð uÞ gðunþ1 Þ and v ¼ gðunþ1 Þ gðwn Þ in (2.7), we obtain hgðunþ1 Þ gðwn Þ; gð uÞ gðunþ1 Þi 1 2 2 2 uÞ gðwn Þk kgð uÞ gðunþ1 Þk kgðunþ1 Þ gðwn Þk g: ¼ fkgð 2 ð3:14Þ Combining (3.13) and (3.14), we have kgðunþ1 Þ gð uÞk2 6 kgðwn Þ gð uÞk2 ð1 2aqÞkgðunþ1 Þ gðwn Þk2 ; the required (3.5). Taking v ¼ u in (3.3) and v ¼ wn in (3.9), we have uÞi þ buðgðwn Þ; gð uÞÞ buðgðuÞ; gðuÞÞ P 0 hbT u; gðwn Þ gð
ð3:15Þ
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and uÞ gðwn Þi þ buðgðuÞ; gðwn ÞÞ hbTyn þ gðwn Þ gðyn Þ; gð buðgðwn Þ; gðwn ÞÞ P 0:
ð3:16Þ
Adding (3.15) and (3.16) and rearranging the terms, we have uÞ gðwn Þi hgðwn Þ gðyn Þ; gð ; gðwn Þ gð P bhTyn T u uÞi þ uðgðuÞ; gðuÞÞ uðgðuÞ; gðwn ÞÞ uðgðwn Þ; gð uÞÞ þ uðgðwn Þ; gðwn ÞÞ 2
P bakgðyn Þ gðwn Þk ;
ð3:17Þ
since T is a g-partially relaxed strongly monotone operator with constant a > 0 and uð; Þ is a skew-symmetry. Now taking v ¼ gðwn Þ gðyn Þ and u ¼ gðuÞ gðwn Þ in (2.7) and (3.17) can be written as 2
2
2
kgð uÞ gðwn Þk 6 kgð uÞ gðyn Þk ð1 2baÞkgðyn Þ gðwn Þk ; the required (3.6). Similarly, by taking v ¼ u in (3.4) and v ¼ unþ1 in (3.10) and using the gpartially relaxed strongly monotonicity of the operator T , and using the skewsymmetry of the bifunction uð; Þ, we have 2
hgðyn Þ gðun Þ; gð uÞ gðyn Þi P lakgðyn Þ gðun Þk :
ð3:18Þ
u yn in (2.7), and combining the resultant with Letting v ¼ yn un , and u ¼ (3.18), we have kgðyn Þ gð uÞk2 6 kgð uÞ gðun Þk2 ð1 2laÞkgðyn Þ gðun Þk2 ; the required (3.7).
h
Theorem 3.1. Let u 2 H be a solution of (2.1) and unþ1 be the approximate solution obtained from Algorithm 3.1. If H is a finite dimensional space, g is an injective map and 0 < q < 1=2a, 0 < b < 1=2a, 0 < l < 1=2a, then limn!1 un ¼ u. Proof. Let u 2 H be a solution of (2.1). Then, from (3.5)–(3.7), it follows that the sequences fkgðun Þ gð uÞkg, fkgðyn Þ gðuÞkg and fkgðwn Þ gðuÞkg are nonincreasing and consequently the sequences fwn g, fyn g and fun g are bounded and
M. Aslam Noor et al. / Appl. Math. Comput. 157 (2004) 643–652
651
1 X 2 2 ð1 2aqÞkgðunþ1 Þ gðun Þk 6 kgðw0 Þ gðuÞk ; n¼0 1 X 2 2 ð1 2abÞkgðwn Þ gðun Þk 6 kgðy0 Þ gðuÞk ; n¼0 1 X 2 2 ð1 2alÞkgðyn Þ gðun Þk 6 kgðu0 Þ gðuÞk ; n¼0
which implies that lim kgðunþ1 Þ gðwn Þk ¼ 0;
n!1
lim kgðwn Þ gðyn Þk ¼ 0;
n!1
lim kgðyn Þ gðun Þk ¼ 0:
n!1
Thus lim kgðunþ1 Þ gðun Þk ¼ lim kgðunþ1 Þ gðwn Þk
n!1
n!1
þ lim kgðwn Þ gðyn Þk þ lim kgðyn Þ gðun Þk ¼ 0: n!1
n!1
ð3:19Þ Let ^ u be the cluster point of fun g and the subsequence funj g of the sequence fun g converge to ^ u 2 H . Replacing wn and yn by unj in (3.2)–(3.4), taking the limit nj ! 1 and using (3.19), we have hT ^ u; gðvÞ gð^ uÞi þ uðgðvÞ; gð^ uÞÞ uðgð^uÞ; gð^uÞÞ P 0;
8v 2 H ;
which implies that ^ u solves the general mixed variational inequality (2.1) and 2
2
kgðunþ1 Þ gð uÞk 6 kgðun Þ gð uÞk : Thus it follows from the above inequality that the sequence fun g has exactly one cluster point ^ u and lim gðun Þ ¼ gð^ uÞ:
n!1
Since g is invertible, so lim ðun Þ ¼ ^ u;
n!1
the required result.
h
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References [1] C. Baiocchi, A. Capelo, Variational and Quasi-Variational Inequalities, J. Wiley and Sons, New York, London, 1984. [2] H. Brezis, Operateurs Maximaux Monotone et Semigroups de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973. [3] R.W. Cottle, F. Giannessi, J.L. Lions, Variational Inequalities and Complementarity Problems: Theory and Applications, J. Wiley and Sons, New York, 1980. [4] J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, UK, 1984. [5] F. Giannessi, A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, 1995. [6] F. Giannessi, A. Maugeri, P.M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic Publishers, Dordrecht, Holland, 2001. [7] R. Glowinski, J.L. Lions, R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. [8] N. Kikuchi, J.T. Oden, Contact Problems in Elasticity, SIAM Publishing Co., Philadelphia, PA, 1988. [9] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and their Applications, SIAM Publishing Co., Philadelphia, 2000. [10] J.L. Lions, G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math. 20 (1967) 493–512. [11] Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, 1995. [12] M. Aslam Noor, General variational inequalities, Appl. Math. Lett. 1 (1988) 119–121. [13] M. Aslam Noor, Wiener-Hopf equations and variational inequalities, J. Optim. Theory Appl. 79 (1993) 197–206. [14] M. Aslam Noor, On a class of variational inequalities, J. Math. Anal. Appl. 128 (1987) 138– 155. [15] M. Aslam Noor, Nonlinear variational inequalities in elastostatics, Int. J. Eng. Sci. 26 (1988) 1043–1053. [16] M. Aslam Noor, Some recent advances in variational inequalities, Part I, basic concepts, New Zealand J. Math. 26 (1997) 53–80. [17] M. Aslam Noor, Some recent advances in variational inequalities, Part II, other concepts, New Zealand J. Math. 26 (1997) 229–255. [18] M. Aslam Noor, Some algorithms for general monotone mixed variational inequalities, Math. Comput. Modell. 29 (7) (1999) 1–9. [19] M. Aslam Noor, Set-valued mixed quasi variational ineqaualities and implicit resolvent equations, Math. Comput. Modell. 29 (1999) 1–11. [20] M. Aslam Noor, Some predictor–corrector algorithms for multivalued variational inequalities, J. Optim. Theory Appl. 108 (2001) 659–670. [21] M. Aslam Noor, A class of new iterative methods for general mixed variational inequalities, Math. Comput. Modell. 31 (13) (2000) 11–19. [22] G. Stampacchia, Formes bilineaires coercivities sur les ensembles convexes, C.R. Acad. Sci. Paris 258 (1964) 4413–4416. [23] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Contr. Optim. 38 (2000) 431–446. [24] N.H. Xiu, J. Zhang, M. Aslam Noor, Tangent projection equations and general variational inequalities, J. Math. Anal. Appl. 258 (2001) 755–762. [25] D.L. Zhu, P. Marcotte, An extended descent framework for variational inequalities, J. Optim. Theory Appl. 80 (1994) 349–366.
Predictor–corrector methods for general mixed quasi variational inequalities Muhammad Aslam Noor a,*, Khalida Inayat Noor b, Kamel Al-Khaled b b
a Etisalat College of Engineering, Sharjah, United Arab Emirates Department of Mathematics and Computer Science, United Arab Emiratea University, P.O. Box 17551, Al Ain, United Arab Emirates
Abstract In this paper, we use the auxiliary principle technique to suggest a class of predictor– corrector methods for solving general mixed quasi variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain various known and new results for solving various classes of variational inequalities and related problems. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Variational inequalities; Auxiliary principle; Iterative methods; Convergence; Fixed points
1. Introduction Variational inequalities theory provides us with a simple, natural, unified, novel and general framework to study a wide class of unilateral, obstacle, free, moving and equilibrium problems arising in fluid flow through porous media, elasticity, transportation, finance, economics, and optimization, etc., see [1–25]. This field is dynamic and is experiencing an explosive growth in both theory
*
Corresponding author. E-mail addresses: [email protected] (M. Aslam Noor), [email protected] (K. Inayat Noor), [email protected] (K. Al-Khaled). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.060
644
M. Aslam Noor et al. / Appl. Math. Comput. 157 (2004) 643–652
and applications; as a consequence, research techniques and problems are drawn from various fields. Variational inequalities have been generalized and extended in different directions using the novel and innovative techniques. An important and useful generalization of variational inequalities is called the mixed quasi variational inequality involving the nonlinear bifunction. Due to the presence of the nonlinear bifunction, projection method and its variant forms, Wiener–Hopf equations, descent methods cannot be extended and modified to suggest iterative methods for solving the mixed quasi variational inequalities. If the nonlinear term involving the general mixed variational inequalities is a proper, convex and lower-semicontinuous with respect to the first argument, then it has been shown [19] that the general mixed variational inequalities are equivalent to the fixed point and resolvent equations. These alternative formulations have been used to develop a number of iterative type methods for solving mixed variational inequalities. In this approach, one has to evaluate the resolvent operator, which is itself a difficult problem. Secondly, the convergence of these methods requires that the underlying operator must be strongly monotone and Lipschizt continuous. To overcome these drawbacks, the auxiliary principle technique has been developed, the origin of which can be traced back to Lions and Stampacchia [10]. Glowinski et al. [7] used this technique to study the existence of a solution of the mixed variational inequalities. In recent years, this technique has been used to suggest and analyze various iterative methods for solving various classes of variational inequalities. It has been shown that a substantial number of numerical methods can be obtained as special cases from this technique, see [16,17,20,21,25] and references therein. In this paper, we extend these ideas and techniques to suggest a class of predictor– corrector methods for solving general mixed quasi variational inequalities. The convergence of these methods requires only that the operator is partially relaxed strongly monotone, which is weaker than co-coercive. Our results can be viewed as a significant and novel application of the techniques developed in [20,21] to general mixed quasi variational inequalities (2.1). Since general mixed quasi variational ineqaualities include quasi variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.
2. Preliminaries 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 nonempty closed convex set in H . Let uð; Þ : H H ! R [ fþ1g be a continuous bifunction function. For given nonlinear operators T ; g : H ! H , consider the problem of finding u 2 H such that
M. Aslam Noor et al. / Appl. Math. Comput. 157 (2004) 643–652
hTu; gðvÞ gðuÞi þ uðgðvÞ; gðuÞÞ uðgðuÞ; gðuÞÞ P 0;
8gðvÞ 2 H :
645
ð2:1Þ
The inequality of type (2.1) is called the general mixed quasi variational inequality. It can be shown that a wide class of linear and nonlinear problems arising in pure and applied sciences can be studied via the general mixed variational inequalities (2.1), see [19]. We remark that if g I, the identity operator, then problem (2.1) is equivalent to finding u 2 H such that hTu; v ui þ uðv; uÞ uðu; uÞ P 0;
8v 2 H ;
ð2:2Þ
which is called the mixed quasi variational inequalities. For the applications, numerical methods and formulations, see [1,4,7–9,14–19] and the references therein. We note that if uðv; uÞ ¼ uðvÞ; 8u 2 H , then problem (2.1) is equivalent to finding u 2 H such that hTu; gðvÞ gðuÞi þ uðgðvÞÞ uðgðuÞÞ P 0;
8gðvÞ 2 H ;
ð2:3Þ
which is called the general mixed variational inequality. For the numerical methods and applications, see Noor [20,21]. We note that if u is the indicator function of a closed convex set K in H , that is, 0; if u 2 K; uðuÞ IK ðuÞ ¼ þ1; otherwise; then problem (2.1) is equivalent to finding u 2 H , gðuÞ 2 K such that hTu; gðvÞ gðuÞi P 0;
8gðvÞ 2 K:
ð2:4Þ
Inequality of the type (2.4) is known as the general variational inequality, which was introduced and studied by Noor [12] in 1988. It turned out that the odd-order and nonsymmetric free, unilateral, obstacle and equilibrium problems can be studied by the general variational inequality (2.4), see [13,16–20]. If K ¼ fu 2 H : hu; vi P 0; 8v 2 Kg is a polar cone of a convex cone K in H , then problem (2.4) is equivalent to finding u 2 H such that gðuÞ 2 K;
Tu 2 K
and
hTu; gðuÞi ¼ 0;
ð2:5Þ
which is known as the general complementarity problem. We note that if gðuÞ ¼ u mðuÞ, where m is a point-to-point mapping, then problem (2.5) is called the quasi (implicit) complementarity problem, see the references for the formulation and numerical methods. For g I, the identity operator, problem (2.4) collapses to: find u 2 K such that hTu; v ui P 0;
8v 2 K;
ð2:6Þ
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which is called the standard variational inequality, introduced and studied by Stampacchia [22] in 1964. For the recent state-of-the-art, see [1–25]. It is clear that problems (2.2)–(2.6) are special cases of the general mixed quasi variational inequality (2.1). In brief, for a suitable and appropriate choice of the operators T , g, uð; Þ and the space H , one can obtain a wide class of variational inequalities and complementarity problems. This clearly shows that problem (2.1) is quite general and unifying one. Furthermore, problem (2.1) has important applications in various branches of pure and applied sciences, see [1,4,7–11,19]. We also need the following concepts. Lemma 2.1. 8u; v 2 H , we have 2
2
2
2hu; vi ¼ ku þ vk kuk kvk : Proof. It is trivial.
ð2:7Þ
h
Definition 2.1. 8u; v; z 2 H , an operator T : H ! H is said to be: ii(i) g-relaxed strongly monotone, if there exists a constant c > 0 such that 2
hTu Tv; gðuÞ gðvÞi P ckgðuÞ gðvÞk : i(ii) g-partially relaxed strongly monotone, if there exists a constant a > 0 such that 2
hTu Tv; gðzÞ gðvÞi P akgðuÞ gðzÞk : (iii) g-co-coercive, if there exists a constant l > 0 such that hTu Tv; gðuÞ gðvÞi P lkTu Tvk2 : We remark that if z ¼ u, then g-partially relaxed strongly monotonicity is exactly g-monotonicity of the operator T . For g I, the identity operator, then Definition 2.1 reduces to the standard definition of relaxed strongly monotonicity [11], and co-coercivity of the operator. It can be shown that g-co-coercivity implies g-partially relaxed strongly monotonicity but the converse is not true. This shows that the concept of partially relaxed strongly monotonicity is weaker than co-coercivity. Definition 2.2. The bifunction uð; Þ : H H ! R [ f1g is said to be skewsymmetric, if uðu; uÞ uðu; vÞ uðv; uÞ þ uðv; vÞ P 0;
8u; v 2 H :
Note that if uð; Þ is a bilinear function then uð; Þ is nonnegative.
ð2:8Þ
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647
3. Main results In this section, we suggest and analyze a new iterative method for solving the problem (2.1) by using the auxiliary principle technique of Glowinski et al. [7] as developed by Noor [20,21]. For a given u 2 H such that gðuÞ 2 H , consider the problem of finding a unique w 2 H such gðwÞ 2 H satisfying the auxiliary variational inequality hqTu þ gðwÞ gðuÞ; gðvÞ gðwÞi þ quðgðvÞ; gðuÞÞ quðgðuÞ; gðuÞÞ P 0; 8v 2 H ;
ð3:1Þ
where q > 0 is a constant. We note that if w ¼ u, then clearly w is a solution of the general mixed variational inequality (2.1). This observation enables us to suggest the following iterative method for solving the general mixed variational inequalities (2.1). Algorithm 3.1. For a given u0 2 H , compute the approximate solution unþ1 by the iterative scheme hqTwn þ gðunþ1 Þ gðwn Þ; gðvÞ gðunþ1 Þi þ quðgðvÞ; gðunþ1 ÞÞ quðgðunþ1 Þ; gðunþ1 ÞÞ P 0;
8v 2 H ;
ð3:2Þ
hbTyn þ gðwn Þ gðyn Þ; gðvÞ gðwn Þi þ buðgðvÞ; gðwn ÞÞ buðgðwn Þ; gðwn ÞÞ P 0;
8v 2 H
ð3:3Þ
and hlTun þ gðyn Þ gðun Þ; gðvÞ gðyn Þi þ luðgðvÞ; gðyn ÞÞ luðgðyn Þ; gðyn ÞÞ P 0;
8v 2 H ;
ð3:4Þ
where q > 0, l > 0 and b > 0 are constants. Note that if g I, the identity operator, then Algorithm 3.1 reduces to: Algorithm 3.2. For a given u0 2 H , compute unþ1 by the iterative scheme hqTwn þ unþ1 wn ; v unþ1 i þ quðv; unþ1 Þ quðunþ1 ; unþ1 Þ P 0; hbTyn þ wn yn ; v wn i þ buðv; wn Þ buðwn ; wn Þ P 0;
8v 2 H ;
8v 2 H
and hlTun þ yn un ; v yn i þ luðv; yn Þ luðyn ; yn Þ P 0;
8v 2 H :
Algorithm 3.2 appears to be a new one for solving mixed variational inequalities (2.2).
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If uðv; uÞ is a proper, convex and lower-semicontinuous differentiable function with respect to the first argument, then Algorithm 3.1 collapses to: Algorithm 3.3. For a given u0 2 H , compute unþ1 by the iterative scheme gðunþ1 Þ ¼ Juðunþ1 Þ ½gðwn Þ qTwn ; gðwn Þ ¼ Juðwn Þ ½gðyn Þ bTyn ; gðyn Þ ¼ Juðyn Þ ½gðun Þ lTun ;
n ¼ 0; 1; 2 . . . ;
where JuðuÞ ðI þ qouð; uÞÞ1 is the resolvent operator associated with the subdifferential ouð; uÞ of the bifunction uð; Þ, see [19]. If the function uðv; uÞ ¼ uðvÞ, 8u 2 H , is the indicator function of a closed convex set K in H , then Algorithm 3.1 reduces to the following method for solving general variational inequalities (2.4). Algorithm 3.4. For a given u0 2 H such that gðu0 Þ 2 K, compute unþ1 by the iterative schemes hqTwn þ gðunþ1 Þ gðwn Þ; gðvÞ gðunþ1 Þi P 0; 8gðvÞ 2 K; hbTyn þ gðwn Þ gðyn Þ; gðvÞ gðwn Þi P 0; 8gðvÞ 2 K and hlTun þ gðyn Þ gðun Þ; gðvÞ gðyn Þi P 0;
8gðvÞ 2 K:
For the convergence analysis of Algorithm 3.4; see Noor [20]. For a suitable choice of the operators and the space H , one can obtain various new and known methods for solving variational inequalities and complementarity problems. For the convergence analysis of Algorithm 3.1, we need the following result. The analysis in the spirit of Noor [20,21]. 2 H be the exact solution of (2.1) and unþ1 be the approximate Lemma 3.1. Let u solution obtained from Algorithm 3.1. If the operator T : H ! H is a g-partially relaxed strongly monotone operator with constant a > 0, and the bifunction uð; Þ is skew-symmetric, then 2
2
2
kgðunþ1 Þ gð uÞk 6 kgðwn Þ gð uÞk ð1 2aqÞkgðunþ1 Þ gðwn Þk ; ð3:5Þ kgðwn Þ gð uÞk2 6 kgðyn Þ gð uÞk2 ð1 2abÞkgðwn Þ gðyn Þk2 ;
ð3:6Þ
uÞk2 6 kgðun Þ gð uÞk2 ð1 2alÞkgðyn Þ gðun Þk2 : kgðyn Þ gð
ð3:7Þ
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649
Proof. Let u 2 H be solution of (2.1). Then hqT u; gðvÞ gð uÞi þ quðgðvÞ; gð uÞÞ quðgðuÞ; gðuÞÞ P 0;
8v 2 H ; ð3:8Þ
hbT u; gðvÞ gð uÞi þ buðgðvÞ; gð uÞÞ buðgðuÞ; gðuÞÞ P 0;
8v 2 H ð3:9Þ
and hlT u; gðvÞ gð uÞi þ luðgðvÞ; gð uÞÞ luðgðuÞ; gðuÞÞ P 0;
8v 2 H ; ð3:10Þ
where q > 0, b > 0 and l > 0 are constants. Now taking v ¼ unþ1 in (3.8) and v ¼ u in (3.2), we have uÞi þ quðgðunþ1 Þ; gð uÞÞ quðgðuÞ; gðuÞÞ P 0 hqT u; gðunþ1 Þ gð
ð3:11Þ
and hqTwn þ gðunþ1 Þ gðwn Þ; gð uÞ gðunþ1 Þi þ quðgðuÞ; gðunþ1 ÞÞ quðgðunþ1 Þ; gðunþ1 ÞÞ P 0:
ð3:12Þ
Adding (3.11) and (3.12), we have hgðunþ1 Þ gðwn Þ; gð uÞ gðunþ1 Þi P qhTwn T u; gðunþ1 Þ gð uÞi þ uðgðuÞ; gðuÞÞ uðgðuÞ; gðunþ1 ÞÞ uðgðunþ1 Þ; gð uÞÞ þ uðgðunþ1 Þ; gðunþ1 ÞÞ P aqkgðunþ1 Þ gðwn Þk2 ;
ð3:13Þ
where we have used the fact that T is g-partially relaxed strongly monotone with constant a > 0 and the skew-symmetry of the bifunction uð; Þ. Setting u ¼ gð uÞ gðunþ1 Þ and v ¼ gðunþ1 Þ gðwn Þ in (2.7), we obtain hgðunþ1 Þ gðwn Þ; gð uÞ gðunþ1 Þi 1 2 2 2 uÞ gðwn Þk kgð uÞ gðunþ1 Þk kgðunþ1 Þ gðwn Þk g: ¼ fkgð 2 ð3:14Þ Combining (3.13) and (3.14), we have kgðunþ1 Þ gð uÞk2 6 kgðwn Þ gð uÞk2 ð1 2aqÞkgðunþ1 Þ gðwn Þk2 ; the required (3.5). Taking v ¼ u in (3.3) and v ¼ wn in (3.9), we have uÞi þ buðgðwn Þ; gð uÞÞ buðgðuÞ; gðuÞÞ P 0 hbT u; gðwn Þ gð
ð3:15Þ
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and uÞ gðwn Þi þ buðgðuÞ; gðwn ÞÞ hbTyn þ gðwn Þ gðyn Þ; gð buðgðwn Þ; gðwn ÞÞ P 0:
ð3:16Þ
Adding (3.15) and (3.16) and rearranging the terms, we have uÞ gðwn Þi hgðwn Þ gðyn Þ; gð ; gðwn Þ gð P bhTyn T u uÞi þ uðgðuÞ; gðuÞÞ uðgðuÞ; gðwn ÞÞ uðgðwn Þ; gð uÞÞ þ uðgðwn Þ; gðwn ÞÞ 2
P bakgðyn Þ gðwn Þk ;
ð3:17Þ
since T is a g-partially relaxed strongly monotone operator with constant a > 0 and uð; Þ is a skew-symmetry. Now taking v ¼ gðwn Þ gðyn Þ and u ¼ gðuÞ gðwn Þ in (2.7) and (3.17) can be written as 2
2
2
kgð uÞ gðwn Þk 6 kgð uÞ gðyn Þk ð1 2baÞkgðyn Þ gðwn Þk ; the required (3.6). Similarly, by taking v ¼ u in (3.4) and v ¼ unþ1 in (3.10) and using the gpartially relaxed strongly monotonicity of the operator T , and using the skewsymmetry of the bifunction uð; Þ, we have 2
hgðyn Þ gðun Þ; gð uÞ gðyn Þi P lakgðyn Þ gðun Þk :
ð3:18Þ
u yn in (2.7), and combining the resultant with Letting v ¼ yn un , and u ¼ (3.18), we have kgðyn Þ gð uÞk2 6 kgð uÞ gðun Þk2 ð1 2laÞkgðyn Þ gðun Þk2 ; the required (3.7).
h
Theorem 3.1. Let u 2 H be a solution of (2.1) and unþ1 be the approximate solution obtained from Algorithm 3.1. If H is a finite dimensional space, g is an injective map and 0 < q < 1=2a, 0 < b < 1=2a, 0 < l < 1=2a, then limn!1 un ¼ u. Proof. Let u 2 H be a solution of (2.1). Then, from (3.5)–(3.7), it follows that the sequences fkgðun Þ gð uÞkg, fkgðyn Þ gðuÞkg and fkgðwn Þ gðuÞkg are nonincreasing and consequently the sequences fwn g, fyn g and fun g are bounded and
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651
1 X 2 2 ð1 2aqÞkgðunþ1 Þ gðun Þk 6 kgðw0 Þ gðuÞk ; n¼0 1 X 2 2 ð1 2abÞkgðwn Þ gðun Þk 6 kgðy0 Þ gðuÞk ; n¼0 1 X 2 2 ð1 2alÞkgðyn Þ gðun Þk 6 kgðu0 Þ gðuÞk ; n¼0
which implies that lim kgðunþ1 Þ gðwn Þk ¼ 0;
n!1
lim kgðwn Þ gðyn Þk ¼ 0;
n!1
lim kgðyn Þ gðun Þk ¼ 0:
n!1
Thus lim kgðunþ1 Þ gðun Þk ¼ lim kgðunþ1 Þ gðwn Þk
n!1
n!1
þ lim kgðwn Þ gðyn Þk þ lim kgðyn Þ gðun Þk ¼ 0: n!1
n!1
ð3:19Þ Let ^ u be the cluster point of fun g and the subsequence funj g of the sequence fun g converge to ^ u 2 H . Replacing wn and yn by unj in (3.2)–(3.4), taking the limit nj ! 1 and using (3.19), we have hT ^ u; gðvÞ gð^ uÞi þ uðgðvÞ; gð^ uÞÞ uðgð^uÞ; gð^uÞÞ P 0;
8v 2 H ;
which implies that ^ u solves the general mixed variational inequality (2.1) and 2
2
kgðunþ1 Þ gð uÞk 6 kgðun Þ gð uÞk : Thus it follows from the above inequality that the sequence fun g has exactly one cluster point ^ u and lim gðun Þ ¼ gð^ uÞ:
n!1
Since g is invertible, so lim ðun Þ ¼ ^ u;
n!1
the required result.
h
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