Solvability of Multivalued General Mixed Variational Inequalities

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1 Joural of Mathematical Aalysis ad Applicatios 261, doi: jmaa , available olie at http: o Solvability of Multivalued Geeral Mixed Variatioal Iequalities Muhammad Aslam Noor Mathematics, Etisalat College of Egieerig, P.O. Box 980, Sharjah, Uited Arab Emirates oormaslam@hotmail.com Submitted by William F. Ames Received Jauary 26, 2001 I this paper, we use the auxiliary priciple techique to suggest a ew class of predictor-corrector algorithms for solvig multivalued geeral mixed variatioal iequalities. The covergece of the proposed method oly requires the partially relaxed strog mootoicity of the operator, which is weaker tha co-coercivity. As special cases, we obtai a umber of kow ad ew results for solvig various classes of variatioal iequalities Academic Press Key Words: variatioal iequalities; auxiliary priciple; iterative methods; covergece. 1. INTRODUCTION Variatioal iequalities theory has emerged a iterestig ad fasciatig brach of applicable mathematics with a wide rage of applicatios i idustry, physical, regioal, social, pure, ad applied scieces. This field is dyamic ad is experiecig a explosive growth i both theory ad applicatios; as a cosequece, research techiques ad problems are draw from various fields. Variatioal iequalities have bee geeralized ad exteded i differet directios usig ovel ad iovative techiques. A importat ad useful geeralizatio of variatioal iequalities is called the multivalued geeral mixed variatioal iequality. For applicatios ad umerical methods, see 12, ad the refereces therei. There are several umerical methods for solvig variatioal iequalities ad related optimizatio problems. Amog the most efficiet umerical techiques are projectio ad its variat forms, Wieer Hopf equatios, auxiliary priciple, ad the pealty fuctio methods. It is well kow that the cover X 01 $35.00 Copyright 2001 by Academic Press All rights of reproductio i ay form reserved. 390

2 MIXED VARIATIONAL INEQUALITIES 391 gece aalysis of the projectio method requires that the uderlyig operator must be strogly mootoe ad Lipschitz cotiuous. These strict coditios rule out may applicatios of the projectio methods. These facts motivated us to modify the projectio methods usig the updatig techique of the solutio ad the Wieer Hopf equatios methods; see, for example, 17, 19, 21, 22, 29 ad the refereces therei for recet state-of-the-art techiques. It is well kow that all the projectio type methods caot be exteded ad geeralized to suggest ad aalyze iterative methods for solvig the mixed variatioal iequalities ivolvig the oliear terms. Secod the evaluatio of the projectio of the operator is very expasive. To overcome these drawbacks, we use the resolvet operator methods. I fact, if the oliear term ivolvig the mixed variatioal iequalities is proper, covex, ad lower-semicotiuous, the the mixed variatioal iequalities are equivalet to the fixed-poit problem ad the resolvet equatios. I this techique, the give operator is decomposed ito the sum of Ž maximal. mootoe operators, whose resolvets are easier to evaluate tha the resolvet of the give origial operator. I the cotext of the mixed variatioal iequalities, Noor has used the resolvet operator ad resolvet equatios techiques to develop various splittig ad predictor-corrector type methods for solvig mixed variatioal iequalities ad related optimizatio problems usig the updatig techique of the solutio. A useful feature of the forward backward splittig methods is that the resolvet operator ivolves the subdifferetial of the proper, covex, ad lower-semicotiuous fuctio ad the other part facilitates the problem decompositio oly. If the oliear term ivolvig the mixed variatioal iequalities is a idicator fuctio of a closed covex set i a space, the the resolvet operator is exactly the projectio operator from the space ito the covex set. Cosequetly, the resolvet equatios are equivalet to the Wieer Hopf equatios, which were itroduced by Shi 28 ad Robiso 27. For the recet applicatios of the Wieer Hopf equatios, see 13. I passig, we remark that the resolvet equatios play the same role i the mixed variatioal iequalities as the Wieer Hopf equatios i variatioal iequalities. To implemet these methods, oe has to evaluate the resolvet of the operator, which is itself a difficult problems. If the oliear term is odifferetiable, the oe caot use the resolvet type methods for solvig the mixed type variatioal iequalities. Furthermore, the updatig techique of the solutio caot be exteded to suggest two-step, three-step splittig, ad predictor-corrector type methods for multivalued Ž mixed. variatioal iequalities. These facts motivated us to cosider ad develop other methods. Oe of these techiques is called the auxiliary priciple, the origi of which ca be traced back to Lios ad Stampacchia 9. This techique deals with fidig the auxiliary variatioal iequality ad prov-

3 392 MUHAMMAD ASLAM NOOR ig that the solutio of the auxiliary problem is the solutio of the origial variatioal iequality by usig the fixed-poit techique. It tured out that this techique ca be used to fid the equivalet differetiable optimizatio problem, which eables us to costruct a gap Ž merit. fuctio. These gap Ž merit. fuctios have played a importat part i developig some efficiet iterative methods for solvig variatioal iequalities; see 5, 22, 24, 30. Glowiski et al. 6 used this techique to study the existece of a solutio of the mixed variatioal iequalities. Noor 11, 12, has used the auxiliary priciple techique to develop some iterative methods for solvig various classes of variatioal iequalities ad optimizatio problems. It has bee show that a substatial umber of umerical methods ca be obtaied as special cases from this techique; see 10, 11 16, 30 ad refereces therei. I this paper, we use the auxiliary priciple techique to suggest a class of three-step predictor-corrector iterative methods for multivalued geeral mixed variatioal iequalities. I particular, we show that oe ca obtai various forward backward splittig, modified resolvet, ad other methods as special cases from these methods. We also prove that the covergece of the suggested methods requires oly the partially relaxed strog mootoicity, which is a weaker coditio tha the co-coercivity. Cosequetly, our results represet a improvemet ad refiemet of the previously kow results. Our results ca be cosidered as a extesio of the results of Noor 11, 12, 14, 15 for solvig geeral mixed variatioal iequalities ad complemetarity problems. 2. PRELIMINARIES Let H be a real Hilbert space whose ier product ad orm are deoted by ², : ad, respectively. Let CŽ H. be a family of all oempty compact subset of H. Let T: H CŽ H. be a multivalued operator ad g: H H be a sigle-valued operator. Let K be a oempty closed covex set i H. Let : H R 4 be a fuctio. For a give sigle-valued operator NŽ,.: H H H, we cosider the problem of fidig u H, TŽ u. such that ² NŽ,., g Ž. gž u. : Ž g Ž.. Ž gž u.. 0, g Ž. H. Ž 2.1. The iequality of type Ž 2.1. is called the multi alued geeral mixed ariatioal iequality. It ca be show that a wide class of multivalued odd order ad osymmetric free, obstacle, movig, equilibrium, ad optimizatio problems arisig i pure ad applied scieces ca be studied via the multivalued variatioal iequalities Ž 2.1..

4 MIXED VARIATIONAL INEQUALITIES 393 If : H R 4 is a proper, covex, ad lower semicotiuous fuctio, the problem Ž 2.1. is equivalet to fidig u H, TŽ u. such that 0 NŽ,. Ž gž u.., Ž 2.2. where is the subdifferetial of ad is a maximal mootoe operator. Problem Ž 2.2. is also kow as fidig a zero of the sum of Ž maximal. mootoe mappigs. Problems of type Ž 2.2. has bee extesively studied recetly; see 5, We ote that if T: H H is a sigle-valued operator, the problem Ž 2.1. is equivalet to fidig u H such that ² NŽ u, u., g Ž. gž u. : g Ž. gž u. 0, g Ž. H, Ž 2.3. which is kow as the geeral mixed variatioal iequality. If Nu, Ž u. Tu, T: H H ad is the idicator fuctio of a closed covex set K i H, the problem Ž 2.3. is equivalet to fidig u H, gu K such that ² Tu, g Ž. gž u. : 0, g Ž. K, Ž 2.4. which is called the geeral ariatioal iequality itroduced ad studied by Noor 21 i It ca be show that a class of quasi variatioal iequalities ad ocovex programmig problems ca be studied by the geeral variatioal iequality approach; see Noor 15, 16, 25, 26. We remark that if g I, the idetity operator, the problem Ž 2.1. is equivalet to fidig u H, TŽ u. such that ² NŽ,., u: Ž. Ž u. 0, H, Ž 2.5. which are called the geeralized mixed variatioal iequalities. If is the idicator fuctio of a closed covex set K i H, the problem Ž 2.1. is equivalet to fidig u H, gu K, TŽ u. such that ² NŽ,., g Ž. gž u. : 0, g Ž. K, Ž 2.6. which is kow as the multivalued variatioal iequality, itroduced ad studied by Noor 25 recetly. I particular, for g I, the idetity operator, the problem is called the geeralized variatioal iequality problem itroduced ad is studied by Fag ad Peterso 4. If K * u H: ² u, : 0, K 4 is a polar coe of a covex coe K i H, the problem Ž 2.6. is equivalet to fidig u H such that gž u. K, NŽ,. K *, ad ² NŽ,., gž u. : 0, Ž 2.7.

5 394 MUHAMMAD ASLAM NOOR which is kow as the multivalued complemetarity problem. We ote that if gu u mu, where m is a poit-to-poit mappig, the problem Ž 2.7. is called the multivalued quasiž implicit. complemetarity problem. For Nu, Ž u. Tu, the problem is called the geeral oliear complemetarity problem; see the refereces for the formulatio ad umerical methods. It is clear that problems Ž 2.2. Ž 2.7. are special cases of the multivalued variatioal iequality Ž I brief, for a suitable ad appropriate choice of the operators T, g, NŽ,. ad the space H, oe ca obtai a wide class of variatioal iequalities ad complemetarity problems. This clearly shows that problem Ž 2.1. is quite geeral ad a uifyig oe. Furthermore, problem Ž 2.1. has may importat applicatios i various braches of pure ad applied scieces; see We also eed the followig well kow result ad cocepts. LEMMA 2.1. u, H, we ha e 2² u, : u 2 u 2 2. Ž 2.8. DEFINITION 2.1. u, u, z H, w TŽ u., w TŽ u , the operator NŽ,.: H H CŽ H. is said to be: Ž. i g-partially relaxed strogly mootoe, if there exists a costat 0 such that that ² NŽ w, w. NŽ w, w., gž z. gž u.: gž u. gž z. 2, ii iii g-co-coercive, if there exists a costat 0 such that ² NŽ w, w. NŽ w, w., gž u. gž u.: NŽ w, w. NŽ w, w. 2, M-Lipschitz cotiuous, if there exists a costat 0 such M TŽ u., TŽ u. u u, where MŽ,. is the Hausdorff metric o CŽ H.. We remark that if z u 1, the g-partially relaxed strog mootoicity is exactly g-mootoicity of the operator NŽ,.. For g I, the idetity operator ad Nu, Ž u. Tu, T: H H is a operator, Defiitio 2.1 reduces to the defiitio of partially relaxed strog mootoicity ad co-coercivity of the operator. Usig the techique of Noor 11, it ca be show that g-co-coercivity implies g-partially relaxed strog mootoicity, but ot coversely. Cosequetly, it follows that the cocept of g-partially relaxed strog mootoicity is weaker tha co-coercivity.

6 MIXED VARIATIONAL INEQUALITIES MAIN RESULTS I this sectio, we suggest ad aalyze a ew iterative method for solvig the problem Ž 2.1. by usig the auxiliary priciple techique. For a give u H, TŽ u. cosider the problem of fidig a uique w H, satisfyig the auxiliary variatioal iequality ² NŽ,. gž w. gž u., g Ž. gž w. : Ž g Ž.. Ž gž u.. 0, g Ž. H, Ž 3.1. where 0 is a costat. We ote that if w u, the clearly w is a solutio of the multivalued variatioal iequality Ž This observatio eables us to suggest the followig predictor-corrector method for solvig the multivalued mixed variatioal iequalities Ž For a give u0 H, compute the approximate soluby the iterative schemes ALGORITHM 3.1. tio u ad ² NŽ,. gž u. gž w., g Ž. gž u.: Ž g Ž.. Ž gž u.. 0, g Ž. H Ž 3.2. T w : Ž. MŽ TŽ w., TŽ w.. Ž 3.3. ² NŽ,. gž w. gž y., g Ž. gž w.: Ž g Ž.. Ž gž w.. 0, g Ž. H Ž 3.4. TŽ y.: M TŽ y., TŽ y. Ž 3.5. ² NŽ,. gž y. gž u., g Ž. gž y.: Ž g Ž.. Ž gž y.. 0, g Ž. H. Ž 3.6. T u : Ž. MŽ TŽ u., TŽ u.., 0,1,2,..., Ž 3.7. where 0, 0, ad 0 are costats. Note that if g I, the idetity operator, the Algorithm 3.1 reduces to the followig predictor-corrector method for solvig the mixed variatioal iequalities Ž 2.3..

7 396 MUHAMMAD ASLAM NOOR ALGORITHM 3.2. For a give u H, compute u by the iterative 0 schemes ² NŽ,. u w, u : Ž. Ž u. 0, TŽ w.: M TŽ w., TŽ w. H, ² NŽ,. w y, w : Ž. Ž w. 0, H TŽ y.: M TŽ y., TŽ y. ² NŽ,. y u, y : Ž. Ž y. 0, H. TŽ u.: M TŽ u., TŽ u., 0,1,2.... If is a proper, covex, ad lower-semicotiuous fuctio, Algorithm 3.1 ca be writte as ALGORITHM 3.3. For a give u0 H, compute u such that TŽ w., TŽ y., TŽ u. by the iterative schemes gž u. J gž w. NŽ,., gž w. J gž y. NŽ,., gž y. J gž u. NŽ,., 0,1,2,..., where J is the resolvet operator associated with the subdifferetial, which is a maximal mootoe operator; see 17, 18. Algorithm 3.3 is a three-step forward backward splittig method for solvig multivalued mixed variatioal iequalities Ž 2.1., which appears to be a ew oe. If T is a sigle-valued operator, the Algorithm 3.1 collapses to the followig predictor-corrector method for solvig geeral mixed variatioal iequalities Ž 2.2. ad appears to be a ew oe. ALGORITHM 3.4. For a give u0 H, compute u by the iterative schemes ² NŽ w, w. gž u. gž w., g Ž. gž u.: Ž g Ž.. Ž gž u.. 0, g Ž. H ² NŽ y, y. gž w. gž y., g Ž. gž w.: Ž g Ž.. Ž gž w.. 0, g Ž. H ² NŽ u, u. gž y. gž u., g Ž. gž y.: g Ž. gž y. 0, g Ž. H.

8 MIXED VARIATIONAL INEQUALITIES 397 We remark that Algorithm 3.4 ca be writte i the equivalet form as ALGORITHM 3.5. For a give u0 H, compute u by the iterative schemes gž y. J gž u. NŽ u, u. gž w. J gž y. NŽ y, y. gž u. J gž w. NŽ w, w., 0, 1,2.... If Nu, Ž u. Tu, T: H H is a sigle-valued operator, the Algorithm 3.1 becomes: ALGORITHM 3.6. For a give u0 H, compute the approximate solutio u by the iterative schemes ² TŽ w. gž u. gž w., g Ž. gž u.: Ž g Ž.. Ž gž u.. 0, g Ž. H ² TŽ y. gž w. gž y., g Ž. gž w.: Ž g Ž.. Ž gž y.. 0, g Ž. H ² Tu g y g u, g g y : Ž. Ž. Ž. Ž g Ž.. Ž gž y.., g Ž. H. If is the idicator fuctio of a closed covex set K i H, the Algorithm 3.6 reduces to the followig algorithm of Noor 12 for solvig the geeral variatioal iequalities Ž ALGORITHM 3.7. For a give u H, gu 0 0 K, compute the approximate solutio u by the iterative schemes ² TŽ w g u g w, g g u :. Ž. Ž. Ž. 0, g Ž. K ² TŽ y g w g y, g g w :. Ž. Ž. Ž. 0, g Ž. K ² Tu gž y. gž u., g Ž. gž y.: 0, g Ž. K. For the covergece aalysis of Algorithm 3.7, see Noor 12. For a suitable choice of the operators NŽ,., T ad the space H, oe ca obtai various ew ad kow methods for solvig variatioal iequalities ad complemetarity problems. For the covergece aalysis of Algorithm 3.1, we eed the followig result, which is proved by usig the techique of Noor 11.

9 398 MUHAMMAD ASLAM NOOR LEMMA 3.1. Let u H be the exact solutio of Ž 2.1. ad u be the approximate solutio obtaied from Algorithm 3.1. If the operator N Ž,. : H H CŽ H. is a g-partially relaxed strogly mootoe operator with costat 0, the Proof. gž u. gž u. 2 gž u. gž u. 2 Ž 1 2. gž u. gž u. 2. Ž 3.8. Let u H, TŽ u. be a solutio of Ž The ² NŽ,., g Ž. gž u. : g Ž. gž u. 0, g Ž. H Ž 3.9. ² NŽ,. g Ž. gž u. : g Ž. gž u. 0, g Ž. H Ž ² NŽ,., g Ž. gž u. : g Ž. g Ž. 0, where 0, 0, ad 0 are costats. Now takig u i Ž 3.9. ad u i Ž 3.2., we have g Ž. H, Ž ² NŽ,., gž u. gž u. : gž u. gž u. 0 Ž ad ² NŽ,. gž u. gž w., gž u. gž u.: gž u. gž u. 0. Ž Addig 3.12 ad 3.13, we have ² gž u. gž w., gž u. gž u.: ² NŽ,. NŽ,., gž u. gž u. : gž u. gž w. 2, Ž where we have used the fact that NŽ,. is g-partially relaxed strogly mootoe with costat 0. Settig u gu gu ad gu gw i Ž 2.6., we obtai ² gž u. gž w., gž u. gž u.: g u g w g u g u 2 gž u. gž w. 2. Ž

10 MIXED VARIATIONAL INEQUALITIES 399 Combiig 3.14 ad 3.15, we have gž u. gž u. 2 gž w. gž u. 2 Ž 1 2. gž u. gž w. 2. Ž Takig u i Ž 3.4. ad w i Ž 3.10., we have ad ² NŽ,., gž w. gž u. : gž w. gž u. 0 Ž ² NŽ,. gž w. gž y., gž u. gž w.: gž u. gž w. 0. Ž Addig 3.18 ad 3.17 ad rearragig the terms, we have ² gž w. gž y., gž u. gž w.: ² NŽ,. NŽ,., gž w. gž u. : gž y. gž w. 2, Ž sice NŽ,. is a g-partially relaxed strogly mootoe operator with costat 0. Now takig gw gž y. ad u gu gw i Ž 2.6., Ž ca be writte as gž u. gž w. 2 gž u. gž y. 2 Ž 1 2. gž y. gž w. 2 gž u. gž y. 2, for Ž Similarly, by takig u i Ž 3.5. ad u i Ž ad usig the g-partially relaxed strog mootoicity of the operator NŽ,., we have ² gž y. gž u., gž u. gž y.: gž y. gž u. 2. Ž Lettig y u, ad u u y with Ž 3.21., we have i Ž 2.6., ad combiig the resultat gž y. gž u. 2 gž u. gž u. 2 Ž 1 2. gž y. gž u. 2 Now gž u. gž u. 2, for 0 1. Ž gž u. gž w. 2 gž u. gž u. gž u. gž w. 2 gž u. gž u. 2 gž u. gž w ² gž u. gž u., gž u. gž w.:. Ž 3.23.

11 400 MUHAMMAD ASLAM NOOR Combiig 3.16, 3.20, 3.22, ad 3.23, we obtai gž u. gž u. 2 gž u. gž u. 2 Ž 1 2. gž u. gž u. 2, the required result 3.8. THEOREM 3.1. Let H be a fiite dimesioal space. Let g: H Hbe 1 i ertible ad 0. Let T: H CŽ H. 2 be a M-Lipschitz cotiuous operator. If u is the approximate solutio obtaied from Algorithm 3.1 ad u H is the exact solutio of Ž 2.1., the lim u u. 1 Ž. 2 Proof. Let u H be a solutio of 1. Sice 0, from 3.8, it follows that the sequece gu gu 4 is oicreasig ad cose- quetly u 4 is bouded. Furthermore, we have Ý Ž 2. Ž. Ž 0. Ž 2. 0 which implies that 1 2 g u g u g u g u, lim gž u. gž u. 0. Ž ˆ j Let u be the cluster poit of u 4ad let the subsequece u 4of the sequece u 4 coverge to u H. Replacig w ad y by u i Ž 3.2. ˆ j, Ž 3.4., ad Ž 3.6., takig the limit ad usig Ž 3.24., we have j ² NŽ ˆˆ,., g Ž. gž uˆ. : g Ž. gž uˆ. 0, g Ž. H, which implies that uˆ solves the multivalued mixed variatioal iequality Ž 2.1. ad gž u. gž u. 2 gž u. gž u. 2. Thus it follows from the above iequality that the sequece u 4 has exactly oe cluster poit uˆ ad Sice g is ivertible, thus lim gž u. gž u ˆ.. lim Ž u. u. ˆ It remais to show that T u. From 3.7 ad usig the M-Lipschitz cotiuity of T, we have M TŽ u., TŽ u. u u,

12 MIXED VARIATIONAL INEQUALITIES 401 which implies that as. Now cosider d, TŽ u. d, TŽ u. M TŽ u., TŽ u. u u 0 as, where dž, TŽ u.. if z : z TŽ u.4, ad 0 is the M-Lipschitz cotiuity costat. From the above iequality, it follows that dž, TŽ u.. 0. This implies that TŽ u., sice TŽ u. CŽ H.. This completes the proof. REFERENCES 1. W. F. Ames, Numerical Methods for Partial Differetial Equatios, 3rd ed., Academic Press, New York, E. Blum ad W. Oettli, From optimizatio ad variatioal iequalities to equilibrium problems, Math. Studet 63 Ž 1994., G. Cohe, Auxiliary problem priciple exteded to variatioal iequalities, J. Optim. Theory Appl. 59 Ž 1988., S. C. Fag ad E. L. Peterso, Geeralized variatioal iequalities, J. Optim. Theory Appl. 38 Ž 1982., F. Giaessi ad A. Maugeri, Variatioal Iequalities ad Network Equilibrium Problems, Pleum, New York, R. Glowiski, J. L. Lios, ad R. Tremolieres, Numerical Aalysis of Variatioal Iequalities, North-Hollad, Amsterdam, R. Glowiski ad P. Le Tallec, Augmeted Lagragias ad Operator Splittig Methods i Noliear Mechaics, SIAM, Philadelphia, D. Kiderleher ad G. Stampacchia, A Itroductio to Variatioal Iequalities ad Their Applicatios, Academic Press, New York, J. L. Lios ad G. Stampacchia, Variatioal iequalities, Comm. Pure Appl. Math. 20 Ž 1967., P. Marcotte ad J. H. Wu, O the covergece of projectio methods: Applicatios to decompositio of affie variatioal iequalities, J. Optim. Theory Appl. 85 Ž 1995., M. Aslam Noor, Some predictor-corrector algorithms for multivalued variatioal iequalities, J. Optim. Theory Appl. 108 Ž 2001., M. Aslam Noor, A predictor-corrector method for geeral variatioal iequalities, Appl. Math. Lett. 14 Ž 2001., M. Aslam Noor, Wieer Hopf equatios techique for variatioal iequalities, Korea J. Comput. Appl. Math. 7 Ž 2000., M. Aslam Noor, Some ew iterative methods for geeral mixed variatioal iequalities, Southeast Asia Bull. Math., i press. 15. M. Aslam Noor, A class of ew iterative methods for geeral variatioal iequalities, Math. Comput. Modellig 31 Ž 2000., M. Aslam Noor, New approximatio schemes for geeral variatioal iequalities, J. Math. Aal. Appl. 251 Ž 2000., M. Aslam Noor, Splittig methods for pseudomootoe geeral mixed variatioal iequalities, J. Global Optim. 18 Ž 2000.,

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