Research Article Finite-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities

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1 Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2008, Article ID , 13 pages doi: /2008/ Research Article Fiite-Step Relaxed Hybrid Steepest-Descet Methods for Variatioal Iequalities Ye-Cherg Li Departmet of Occupatioal Safety ad Health, Geeral Educatio Ceter, Chia Medical Uiversity, Taichug 404, Taiwa Correspodece should be addressed to Ye-Cherg Li, Received 22 August 2007; Revised 2 Jauary 2008; Accepted 13 March 2008 Recommeded by Jog Kim The classical variatioal iequality problem with a Lipschitzia ad strogly mootoe operator o a oempty closed covex subset i a real Hilbert space was studied. A ew fiite-step relaxed hybrid steepest-descet method for this class of variatioal iequalities was itroduced. Strog covergece of this method was established uder suitable assumptios imposed o the algorithm parameters. Copyright q 2008 Ye-Cherg Li. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. 1. Itroductio Let H be a real Hilbert space with ier product, ad orm.letc be a oempty closed covex subset of H, ad let F : C H be a operator. The classical variatioal iequality problem: fid u C such that VI F, C F u ),v u 0, v C, 1.1 was iitially studied by Kiderlehrer ad Stampacchia. It is also kow that the VI F, C is equivalet to the fixed-poit equatio u P C u μf u )), 1.2 where P C is the earest poit projectio from H oto C, thatis,p C x argmi y C x y for each x H ad where μ>0 is a arbitrarily fixed costat. If F is strogly mootoe ad Lipschitzia o C ad μ > 0 is small eough, the the mappig determied by the right-had side of this equatio is a cotractio. Hece the Baach cotractio priciple guaratees that the Picard iterates coverge i orm to the uique solutio of the VI F, C. Such a method is called the projectio method. However, Zeg ad Yao 2 poit out that the fixed-poit

2 2 Joural of Iequalities ad Applicatios equatio ivolves the projectio P C which may ot be easy to compute due to the complexity of the covex set C. To reduce the complexity problem probably caused by the projectio P C,a class of hybrid steepest-descet methods for solvig VI F, C has bee itroduced ad studied recetly by may authors see, e.g., 3, 4. Zeg ad Yao 2 have established the method of two-step relaxed hybrid steepest-descet for variatioal iequalities. A atural arisig problem is whether there exists a geeral relaxed hybrid steepest-descet algorithm that is more tha two steps for fidig approximate solutios of VI F, C or ot. Motivated ad ispired by the recet research work i this directio, we itroduce the followig fiite step relaxed hybrid steepest-descet algorithm for fidig approximate solutios of VI F, C ad aim to uify the covergece results of this kid of methods. Algorithm 1.1. Let {α k } 0, 1, {λ k } 0, 1, fork 1, 2,...,m, ad take fixed umbers 0, 2η/κ 2, k 1, 2,...,m. Startig with arbitrarily chose iitial poits u 0 H, compute the sequeces {u k } such that u 1 u 3 α u α 2 u α 3 u 1 α )[ 2 Tu λ 1 t F T )], 1 α 2 )[ 3 Tu λ 2 1 t 2 F Tu 3 )], 1 α 3 )[ 4 Tu λ 3 1 t 3 F Tu 4 )], 1.3 u m α m u. 1 α m )[ Tu λ m 1 t m F Tu )]. We will prove a strog covergece result for Algorithm 1.1 uder suitable restrictios imposed o the parameters. 2. Prelimiaries The followig lemmas will be used for provig the mai result of the paper i ext sectio. Lemma 2.1 see 5. Let {s } be a sequece of oegative real umbers satisfyig the iequality where {α }, {τ }, ad {γ } satisfy the followig coditios: s 1 1 α ) s α τ γ, 0, 2.1 i {α } 0, 1, 0 α, or equivaletly, 0 1 α 0; ii lim sup τ 0; iii {γ } 0,, 0 γ <. The lim s 0. Lemma 2.2 see 6. Demiclosedess priciple: assume that T is a oexpasive self-mappig o a oempty closed covex subset C of a Hilbert space H. If T has a fixed poit, the I T is demiclosed; that is, wheever {x } is a sequece i C weakly covergig to some x C ad the sequece { I T x } strogly coverges to some y H, it follows that I T x y. Here I is the idetity operator of H.

3 Ye-Cherg Li 3 The followig lemma is a immediate cosequece of a ier product. Lemma 2.3. I a real Hilbert space H, there holds the iequality x y 2 x 2 2 y, x y, x, y H. 2.2 Lemma 2.4. Let C be a oempty closed covex subset of H. For ay x, y H ad z C, the followig statemets hold: i P C x x, z P C x 0; ii P C x P C y 2 x y 2 P C x x y P C y Covergece theorem Let H be a real Hilbert space ad let C be a oempty closed covex subset of H. Let F : C H be a operator such that for some costats κ, η > 0,F is κ-lipschitzia ad η-strogly mootoe o C;thatis,F satisfies the coditios Fx Fy κ x y, x, y C, Fx Fy, x y η x y 2, x, y C, 3.1 respectively. Sice F is η-strogly mootoe, the variatioal iequality problem VI F, C has a uique solutio u C see, e.g., 7. Assume that T : H H is a oexpasive mappig with the fixed poits set Fix T C. Note that obviously Fix P C C. For ay give umbers λ 0, 1 ad μ 0, 2η/κ 2,we defie the mappig T λ μ : H H by T λ μx : Tx λμf Tx, x H. Lemma 3.1 see 3. Let T λ μ be a cotractio provided that 0 <λ<1 ad 0 <μ<2η/κ 2. Ideed, where τ 1 1 μ 2η μκ 2 0, 1. T λ μx T λ μy 1 λτ x y, x, y H, 3.2 We ow state ad prove the mai result of this paper. Theorem 3.2. Let H be a real Hilbert space ad let C be a oempty closed covex subset of H. Let F : C H be a operator such that for some costats κ, η > 0, Fis κ-lipschitzia ad η- strogly mootoe o C. Assume that T : H H is a oexpasive mappig with the fixed poits set Fix T C, the real sequeces {α k }, {λ k },fork 1, 2,...,m,iAlgorithm 1.1 satisfy the followig coditios: i 1 α k α k 1 <, fork 1, 2,...,m; ii lim α 0 ad lim α k 1, for k 2, 3,...,m;

4 4 Joural of Iequalities ad Applicatios iii lim λ iv λ 0, lim λ /λ 1 1, 1 λ ; max{λ k : k 2, 3,...,m}, for all 1. The the sequeces {u k } geerated by Algorithm 1.1 coverge strogly to u which is the uique solutio of the VI F, C. Proof. Sice F is η-strogly mootoe, by 7,theVI F, C has the uique solutio u C.Next we divide the rest of the proof ito several steps. Step 1. Let {u k } is bouded for each k 1, 2,...,m. Ideed, let us deote that Tt λu Tu λtf Tu,thewehave u 1 u α α u u α u α u 1 α λ 1 u 1 α u 1 α ) T λ u 1 )[ T λ 1 u T λ 1 u T λ 1u u ] u 1 α )[ 1 λ 1 τ ) u λ 1 t F u ) ], 3.3 where τ 1 1 2η κ 2 0, 1. Moreover, we also have u α 2 u α 2 u α 2 u α 2 u 1 α 2 λ 2 1 t 2 u 1 α 2 u 3 u )[ T λ 2 1 t 2 u 3 T λ 2 1 t 2 u T λ 2 1u u ] t 2 u 2 )[ 2 1 α 1 λ 1 τ 2 ) 3 u u 2 λ ) 2 λ u 2 ) 1 α u 3 u 1 α 2 1 t 2 F u ) ] 1 t 2 F u ), 3.4 where τ t 2 2η t 2 κ 2 0, 1, adfork 2, 3,...,m 1, u k u α k u α k α k α k u u u 1 α k λ k 1 u 1 α k u k )[ T λ k u 1 u k T λ k 1 u T λ k 1u u ] u 1 α k )[ k 1 λ 1 τ k ) u k u λ k ) k λ u 1 α k ) u k u 1 α k 1 t k F u ) ] 1 t k F u ), 3.5

5 Ye-Cherg Li 5 where τ k 1 1 2η κ 2 0, 1, ad u m u α m u α m α m u u u 1 α m λ m 1u t m u u 1 α m )[ T λ m 1u t m T λ m 1u T λ m 1u u ] t m t m u 1 α m )[ m 1 λ 1 τ m ) u u λ m 1 t m F u ) ] u λ m 1 t m F u ), where τ m 1 1 t m 2η t m κ 2 0, 1. Thus we obtai m 1 u u α m 1 u u m 1 ) 1 α u m α m 1 u u u m 1 λ 1 t m 1 F u ) u m 1 )[ 1 α u u m λ 1 t m F u ) ] λ m 1 ) { m } max λ t m t m 1 ) F u ), u 1 α m 1 1,λ m t m 1 F u ) ] u k u u u 1 α k ) { j } ) m max λ t j F u ), k j m 1 j k 3.7 for k 2, 3,...,m 1. I particular, u u u 1 α 2 ) { j } ) m max λ t j F u ). 2 j m Hece, substitutig 3.8 i 3.3 ad by coditio iv,weobtai u 1 u α u u 1 α )[ 1 λ 1 τ ) 2 u λ 1 t F u ) ] α u α u u ) [ 1 α 1 λ 1 τ ) u u m max 2 j m {λ j 1 } u 1 α ) [ 1 λ 1 τ ) u 1 t j F u ) λ u { j max λ 1 j m 1 1 t F ] u ) } m t j F ] u ). j By iductio, it is easy to see that u u M, 0, 3.10

6 6 Joural of Iequalities ad Applicatios where M max{ u 0 u, m j 1 t j /τ F u }. Ideed, for 0, from 3.9 we obtai u 1 u α 0 u 0 u 1 α ) [ 0 1 λ 1 τ ) u 0 u max 1 j m λ j 1 m ) t j F ] u ) j 1 α 0 M 1 α )[ 0 1 λ 1 τ ) M λ 1 τ M ] M Suppose that u u 1 u α u M, for 1. We wat to claim that u 1 u M. Ideed, u u 1 α ) [ 1 λ 1 τ ) u u λ 1 m t j F ] u ) j 1 α M 1 α )[ 1 λ 1 τ ) M λ 1 τ M ] M Therefore, we have u u M, for all 0, ad u m u M λ m 1 τ M τ M, for all 0. I this case, from 3.8, it follows that u k u { j M max λ 1} τ M 1 τ ) M, 0, k 2, 3,...,m k j m Step 2. Let u 1 Tu 0,. Ideed by Step 1, {u k } is bouded for 1 k m ad so are {Tu k } ad {F Tu k } for 1 k m. Thus from the coditios that lim α 0, lim α k 1, for k 2, 3,...,mad lim λ 0, we have, for k 2,...,m, u k u α k u 1 α k ) k Tu λ k 1 t k F Tu k )) u 1 α k ) u 1 α k ) k Tu λ k 1 t k F 3.14 Tu k )) ad so u 1 α k 1 Tu α u ) u k ) 1 α Tu k 1 α ) 2 Tu λ 1 t F T k λ 1 t k k F Tu )) Tu ) ) 0 α u Tu ) ) 2 1 α Tu Tu ) 1 λ 1 t F T )) α u Tu ) 1 α T Tu ) 1 α λ 1 t 2 ) F Tu α u Tu u λ 1 t F T ) 0 as. 3.15

7 Ye-Cherg Li 7 Step 3. Let u u m u m 1 1 u 0, as. Ideed, we observe that m α u α m 1 u 1 1 α m ) λ T m 1 α m u u α m α m 1 1 u t m u 1 1 α m 1 λ m u t m 1 1 α m ) T λ m 1u t m T λ m 1u t m 1 1 α m ) λ T m 1u t m 1 1 α m ) λ 1 T m u t m 1 α m u u 1 α m α m 1 u m ) m 1 1 α 1 λ 1 τ m ) u u 1 α m α m 1 Tu 1 1 α m ) m λ 1 1 α m ) m 1 λ t m F Tu ) α m ) m λ 1 τ m ) u u 1 1 α m ) m λ 1 1 α m ) m 1 λ t m ) F Tu 1 α m α m 1 u 1 Tu ) 1 u α m u 1 1 α m ) m λ 1 1 α m ) m 1 λ t m F Tu ) 1 α m u Tu ), ad, for 2 k m 1, k u u k 1 k α u α k 1 u 1 1 α k α k u u α k α k 1 1 λ k 1 u k u 1 1 α k ) λ 1 T k 1 α k ) T λ k u k 1 1 u k T λ k 1 u k 1 1 α k ) λ T k 1u k 1 1 α k ) λ 1 T k u k 1 α k u u 1 α k α k 1 u k ) k 1 1 α 1 λ 1 τ k ) k u u k 1 α k α k k 1 Tu 1 1 α k ) k λ 1 1 α k ) k 1 λ F Tu k ) 1 α k u u α k α k u Tu k ) α k ) k 1 λ 1 τ k ) u k u k 1 1 α k ) k λ 1 1 α k ) k 1 λ F Tu k ) 1,

8 8 Joural of Iequalities ad Applicatios m u u m u u α m α m u u m Tu ) 1 1 α m ) m λ 1 1 α m ) m 1 λ t m F Tu 1), u m 1 u u 1 α m ) m λ 1 2 u α m 1 ) m 1 λ 1 α m 1 α m... 1 α m α m 1 α m 1 u Tu m ) 1 1 ) m λ t m FTu ) m 1 λ t m 1 m FTu 1 α m u Tu ), u u m 1 1 α k ) k λ 1 k α k ) k λ t k k FTu m 1 k α α k u Tu k ) m α α m u Tu ). k 2 1 Hece it follows from the above iequalities that u 1 u α u α 1 u 1 1 α λ α ) λ 1 T 1 α u u α α u ) 1 α T λ T λ α α α u λ α 1 u α α 1 1 λ 1 u 1 1 α ) 1 λ 1 τ ) 1 α 2 1 Tu 1 1 α ) λ 1 1 α ) 1 λ F T ) 1 α u u α α k u Tu k ) α ) 1 λ 1 τ ) α ) λ 1 1 α ) 1 λ F T 1). 3.20

9 Ye-Cherg Li 9 Let us substitute 3.19 ito 3.20,thewehave u 1 u α 1 α ) 1 λ 1 τ )) u u 1 m 1 α k k 1 α k u Tu k ) m 1 1 α k ) k λ 1 1 α k ) k 1 λ k 1 k FTu 1 1 α m ) m λ 1 1 α m ) m 1 λ t m FTu 1 α m α m 1 u 1 Tu ) α ) λ 1 τ ) u u 1 1 α ) λ 1 τ ν δ, 3.21 where m 1 δ k α α k u Tu k ) m α α m u Tu ), k m ) m ν 1 α 1 α λ 1 τ λ 1 1 α m ) m 1 λ t m FTu 1 m 1 1 α k ) k λ 1 1 α k ) k 1 λ ) t k k FTu 1. k We put ξ sup { } { u : 0 sup Tu k } : 0, k 1, 2,...,m sup { FTu k : 0, k 1, 2,...,m }, M ) m u F u ) ξ. k The δ 2M m k 1 α k α k 1 0, as, ad m ν t )M k k 1 1 α 1 ) λ 1 τ m 1 α k ) k λ 1 1 α k ) k 1 λ ) 1 α λ 1 ) 1 α ) 1 λ. k

10 10 Joural of Iequalities ad Applicatios From ii iv,weobtaiν 0as. Furthermore, from i, 1 δ <. By Lemma 2.1, we deduce that u 0as. Step 4. Let u 1 u Tu 0as.FromSteps2 ad 3,wehave u Tu u 1 u u 1 Tu as. Step 5. Let lim sup F u,tu k of {Tu } such that u 0, for k 2, 3,...,m.Let{Tu i } be a subsequece lim sup F u ),Tu u lim F u ),Tu i u i Without loss of geerality, we assume that Tu i ũ weakly for some ũ H. ByStep 4, we derive u i ũ weakly. But by Lemma 2.2 ad Step 4, wehaveũ Fix T C. Sice u is the uique solutio of the VI F, C, we obtai lim sup F u ),Tu u F u ), ũ u From the proof of Step 2, Tu k Tu u 0, as, 3.28 for k 2, 3,...,m.The lim sup F u ),Tu k u [ lim sup F u ),Tu k F u ),Tu k lim sup lim sup 0, Tu F u ),Tu u ] Tu F u ),Tu u lim sup F u ),Tu u 3.29 for k 2, 3,...,m.

11 Ye-Cherg Li 11 Step 6. Let u u i orm ad so does {u k } for k 2, 3,...,m. Ideed usig Lemma 2.3 ad 3.7 we get u 1 u 2 α u α u α u α α u u u ) 1 α u 2 1 α u 2 1 α u 2 1 α λ 1 ) T λ 1 ) T λ 1 )[ T λ 1 u ) 2 u 2 T λ 1 T λ 1 u ) T λ 1u u ) 2 u 2 2 T λ 1 u 2 1 α ) 1 λ 1 τ ) 2 u 2 2 λ 1 F u ),T λ 1 t F T ) u α u u 2 [ 1 α ) 1 λ 1 τ ) 2 u u 1 α α ) λ 1 t F u ),T λ 1 t F T ) u α 1 α ) 1 λ 1 τ )) u u α 1 α ) 2 ) 1 α 1 λ ) 1 α 2 1 τ ) 2 max 2 j m {λ j 1 } m ) 2 1 λ 1 τ ) 2 2 λ 1 F u ),T u λ 1 t F T ) 1 1 α ) λ 1 α 1 τ ) u ) 1 α 2 u α ) 2 1 λ 1 τ ) 2 2λ 1 t F u ),T u λ 1 t F T ) 1 1 α ) λ 1 τ ) u u u,t λ 1 ) m max u ] { j } ) m F λ t j u ) 1 t j ) F u ) u ) 2 ) m 2 max 2 j m {λ j 1 } t j F u ) 2 ) 2 ) 1 α 1 λ 1 τ ) 2 max 2 j m ) 2 ) m 2 max 2 j m {λ j 1 } t j M 2 u 2 1 α ) λ 1 τ u { j } ) m λ t j 1 M 2 ] 2

12 12 Joural of Iequalities ad Applicatios [ 2t F u ),T u λ 1 t F T ) τ 1 α ) 2 1 α 2 ) 1 λ 1 τ ) 2 max2 j m {λ j 1 } m t j ) M 2 τ λ 1 2 ) 2 1 α 1 λ 1 τ ) 2 { j }) 2 max2 j m λ m 1 t j ) 2 M 2 ] τ λ α ) λ 1 τ ) u u 2 1 α ) λ 1 τ [ 2t F u ),T u λ 1 t F T ) τ 1 α ) 2 2 ) 1 α τ 1 λ 1 ) ) 2 ) 2 m 2 1 α τ 1 λ 1 τ ) 2 max 2 j m {λ j 1 } t j M ]. 2 From ii, iii, adstep 5, wehavelim α lim α k by Lemma 2.4, we coclude that lim sup lim λ k 1, for k 2,...,m, lim sup F u,t 2t F u ),T lim sup u λ 1 t F T τ 1 α 2 ) 2 1 α 1 λ 1 τ ) 2 max2 j m {λ j 1 }) m 2 τ 1 α τ ) F u ),T Cosequetly from Lemma 2.1, weobtai u u 0, for k 2, 3,...,m,that u k ) u lim sup 1 τ ) 2 ) m t j M , for k 1, 2,...,m ad } is bouded; u 0, ad {F T ) 2 1 α 2 1 λ 1 τ ) 2 m t j ) M 2 τ t j ) 2 M 2 ) 2 ) 2 λ 1 τ 1 α ) F u ), F T ) 3.31 u 0 ad hece it follows from u k u 0, for k 2, 3,...,m. Ackowledgmet This research was partially supported by Grat o. NSC M from the Natioal Sciece Coucil of Taiwa. Refereces D. Kiderlehrer ad G. Stampacchia, A Itroductio to Variatioal Iequalities ad Their Applicatios, vol. 88 of Pure ad Applied Mathematics, Academic Press, New York, NY, USA, 1980.

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