Applied Mathematics 04 5 5-34 Published Olie Jauary 04 (http://www.scirp.org/joural/am http://d.doi.org/0.436/am.04.5004 ommo Fied Poit Theorems or Totally Quasi-G-Asymptotically Noepasive Semigroups with the Geeralized -Projectio hujie Wag Yuaheg Wag Departmet o Mathematics Zhejiag Normal Uiversity Jihua hia Email: chujwag@sia.c wagyuahegmath@63.com Received July 7 03; revised August 7 03; accepted August 4 03 opyright 04 hujie Wag Yua-Heg Wag. This is a ope access article distributed uder the reative ommos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited. I accordace o the reative ommos Attributio Licese all opyrights 04 are reserved or SIRP ad the ower o the itellectual property hujie Wag Yuaheg Wag. All opyright 04 are guarded by law ad by SIRP as a guardia. ABSTRAT I this paper we itroduce some ew classes o the totally quasi-g-asymptotically oepasive mappigs ad the totally quasi-g-asymptotically oepasive semigroups. The with the geeralized -projectio operator we prove some strog covergece theorems o a ew modiied Halper type hybrid iterative algorithm or the totally quasi-g-asymptotically oepasive semigroups i Baach space. The results preseted i this paper eted ad improve some correspodig oes by may others. KEYWORDS Totally Quasi-G-Asymptotically Noepasive Semigroup; Geeralized -Projectio Operator; Modiied Halper Type Hybrid Iterative Algorithm; Strog overgece Theorem. Itroductio I this paper we deote by ad the set o real umber ad the set o ature umber respectively. Let E be a real Baach space with its dual E ad be a oempty closed ad cove subset o E. The mappig J : E E is the ormalized duality mappig deied by { } J = E : = = E Recall that a mappig T : is said to be oepasive [] i or each y T Ty y. A mappig T : is said to be totally asymptotic ally oepasive i there eists oegative real sequeces { µ } ad { ν } with µ 0 ν 0 as ( ad a strictly icreasig cotiuous uctio ϕ : ϕ 0 = 0 such that or each y with 0. T T y y νϕ y µ We use φ : E E to deote the Lyapuov uctio deied by Obviously we have orrespodig author. φ y = Jy y y E.
6. J. WANG Y. H. WANG ( y φ ( y ( y. Recetly hag et al. [3-5] ad Li [6] itroduced the uiormly totally quasi-φ -asymptotically oepasive mappigs ad studied the strog covergece o some iterative methods or the mappigs i Baach space. T is said to be uiormly totally quasi-φ -asymptotically Deiitio. [] A coutable amily o mappig { i } oepasive i F ( T = i ad there eist oegative sequeces { µ } { } (as ad a strictly icreasig cotiuous uctio ψ : with i 0 ad each p F( T = i ( pt ( p ( ( p ν with µ 0 ν 0 ψ 0 = 0 such that or each φ φ µψ φ ν. ( i More recetly Wag et al. [7] studied the strog covergece or a coutable amily o total quasi-φ - asymptotically oepasive mappigs by usig the hybrid algorithm i -uiormly cove ad uiormly smooth real Baach spaces. Qua et al. [8] itroduced total quasi-φ -asymptotically oepasive semigroup cotaiig may kids o geeralized oepasive mappigs as its special cases ad used the modiied Halper-Ma iteratio algorithm to prove strog covergece theorems i Baach spaces. We use F ( T to deote the commo ied poit set o the semigroup T i.e. F( T = t 0 F( T( t. Deiitio. [8] Oe-parameter amily T : = { T( t : t 0} is said to be a quasi-φ -asymptotically oepasive semigroup i F ( T ad the ollowig coditios are satisied: (a T( 0 = or each ; (b For each T( s t = T( s T( t ts ; (c For each the mappig t T( t is cotiuous; (d For each p F( T there eists a sequeces { k} [ with k as such that ( φ Oe-parameter amily : = { T( t : t 0} pasive semigroup i F ( T (e I F ( T there eist sequeces { µ } { } φ pt t k p. ( T is said to be a totally quasi-φ -asymptotically oe- the coditios (a-(c ad the ollowig coditio are satisied: ν with µ ν 0 as ad a strictly icreasig cotiuous uctio ψ : with ψ (0 = 0 such that ( ( φ pt t φ p µψ φ p ν (3 or all p F T. O the other had Wu et al. [9] itroduced the geeralized -projectio which eteds the geeralized projectio ad always eists i a real releive Baach space. Li et al. [0] proved some properties o the geeralized -projectio operator ad studied the strog covergece theorems or the relatively oepasive mappigs. I 03 by usig the geeralized -projectio operator Seawa et al. [] itroduced the modiied Ma type hybrid projectio algorithm or a coutable amily o totally quasi-φ -asymptotically oepasive mappigs i a uiormly smooth ad strictly cove Baach space with Kadec-Klee property. Motivated by the above researches i this paper we itroduce a ew class o the totally quasi-g-asymptotically oepasive mappigs which cotais the class o the totally quasi-φ -asymptotically oepasive mappigs ad we eted rom a coutable amily o mappigs to the totally quasi-g-asymptotically oepasive semigroup. The we modiy the Halper type hybrid projectio algorithm by usig the geeralized -projectio operator or uiormly total quasi-g-asymptotically oepasive semigroup ad prove some strog covergece theorems uder some suitable coditios. The results preseted i this paper eted ad improve some correspodig oes by may others such as [780].. Prelimiaries This sectio cotais some deiitios ad lemmas which will be used i the proos o our mai results i the
. J. WANG Y. H. WANG 7 et sectio. Throughout this paper we assume that E be a real Baach space with its dual space E. A Baach space y E is said to be strictly cove i < or all y E with = y = ad y. E is said to be uiormly cove i y 0 y i E with = y = = or ay two sequeces { } { } lim y ty ad lim =. A Baach space E is said to be smooth i limt 0 eists or each t y E with = y =. E is said to be uiormly smooth i the limit is attaited uiormly or each = y =. It is well kow that the ormalized dual mappig J : E E holds the properties: ( I E is a smooth Baach space the J is sigle-valued ad semi-cotiuous; ( I E is uiormly smooth Baach space the J is uiormly orm-to-orm cotiuous operator o each bouded subset o E. A Baach space E is said to have Kadec-Klee property i or ay sequece { } E satisies E ad the. As we all kow i E is uiormly cove the E has the Kadec-Klee property. G: E deied by Now we give a uctioal { } G ( ξη ξ ξη η ρ ( ξ = (4 where ξ E : is proper cove ad lower semi-cotiuous. From the deiitio o G ad it is easy to see the ollowig properties: ( G ( ξη is cove ad cotiuous with respect to η whe ξ is ied; ( G ( ξη is cove ad lower semi-cotiuous with respect to ξ whe η is ied. Deiitio. [9] : E is said to be a geeralized -projectio operator i or ay η E η ρ is a positive real umber ad { } { u G( u ig( } η = : η = η. (5 Lemma. [9] Let E be a real releive Baach space with its dual E be a oempty closed ad cove subset o E. The y is a oempty closed ad cove subset o or all y E. Moreover i E is strictly cove the is a sigle-valued mappig. Recall that i E is a smooth Baach space the the ormalized dual mappig J is sigle-valued i.e. there eists uique η E such that η = J or each E. The (4 is equivalet to ( ξ ξ ρ ( ξ G J = J J. (6 Ad i a smooth Baach space the deiitio o the geeralized -projectio operator trasorms ito: Deiitio.3 [0] Let E be a real smooth Baach space ad be a oempty closed ad cove subset o E. The mappig : E is called geeralized -projectio operator i or all E { i ( ξ } ξ = u : G u J = G J. (7 Now we give the deiitio o the totally quasi- G -asymptotically oepasive mappig ad the totally quasi- G -asymptotically oepasive semigroup. Deiitio.4 A mappig T : is said to be a quasi-g-asymptotically oepasive i F( T k with k (as such that ad there eists a sequece { } [ ] or ay A mappig T : G p JT k G p J 0 (8 ad p F( T. is said to be a totally quasi-g-asymptotically oepasive i F( T µ { δ } with 0 eist sequeces { } ad there µ δ as ad a strictly icreasig cotiuous uctio
8. J. WANG Y. H. WANG τ : with τ ( 0 = 0 such that ad ( G pjt G p µτ G pj δ (9 or all p F T. Remark.5 It is easy to see that a quasi-φ -asymptotically oepasive mappig is a quasi-g-asymptotically oepasive mappig with ( p = 0 or all p F( T. A totally quasi-φ -asymptotically oepasive mappig is a totally quasi-g-asymptotically oepasive mappig with δ = µψ ( ( p. Thereore our totally quasi-g-asymptotically oepasive mappigs here are more widely tha the totally quasi-φ - asymptotically oepasive mappigs which cotai may kids o geeralized oepasive mappigs as their special cases. Deiitio.6 Oe-parameter amily T : = { T( t : t 0} is said to be a quasi-g-asymptotically oepasive semigroup o i the coditios (a-(c i Deiitio. ad the ollowig coditio are satisied: k with k as such that ( There eists a sequece { } [ ] G( p T ( t kg( p J (0 holds or all y. Oe-parameter amily T : = { T( t : t 0} is said to be a totally quasi-g-asymptotically oepasive semigroup o i the above coditios (a-(c i Deiitio. ad the ollowig coditio are satisied: (g i F ( T ad there eist sequeces { µ } { δ } with µ ν 0 as ad a strictly icreasig cotiuous uctio τ : τ 0 = 0 such that or all p F T with ( ( ( ( ad G p JT t G p J µτ G p J δ ( holds or each. Remark.7 It is easy to see that a quasi-φ -asymptotically oepasive semigroup is a quasi-g-asymptotically oepasive semigroup with ( p = 0 or all p F( T. A totally quasi- φ -asymptotically oepasive semigroup is a totally quasi-g-asymptotically oepasive semigroup with δ = µψ ( ( p. Whe we use tm ( m istead o t i Deiitio.6 ad deote T( t m by T m T : = { Tm : } m= the a quasi-g-asymptotically oepasive semigroup becomes a coutable amily o total quasi-g-asymptotically oepasive mappigs which cotais a coutable amily o total quasi-φ -asymptotically oepasive mappigs (see [347] as it s special case. So our totally quasi-g-asymptotically oepasive semigroup here is the most widely amily o the oepasive mappigs so ar. The ollowig Lemmas are ecessary or provig the mai results i this paper. Lemma.8 [] Let E be a uiormly cove ad smooth Baach space ad { } { y } be two sequeces o E. I φ ( y 0 ad either { } or { y } is bouded the y 0. Lemma.9 [3] I E is a strictly cove releive ad smooth Baach space the or y E φ ( y = 0 i ad oly i = y. Lemma.0 [4] Let E be a real Baach space ad : E { } be a lower semicotiuous cove uctioal. The there eists E ad α such that α ( or each E. Lemma. [0] Let E be a real releive ad smooth Baach space ad be a oempty closed ad cove subset o E. Let E z. The φ y z G z J G y J y. (3 Lemma. Let E be a uiormly smooth ad strictly cove Baach space be a oempty closed ad cove subset o E. Let T : be a totally quasi-g-asymptotically oepasive mappig deied by (9. I µ δ 0 F T o T is closed ad cove subset o. = = the the ied poit set
F T with p p sice T is a quasi-g-asymptotically oepasive mappig we have Proo Let { p } be a sequece i Sice µ = δ = 0 it is equivalet to that So. J. WANG Y. H. WANG 9 ( ( ( ( as we prove that p F( T G p JTp G p Jp µτ G p Jp δ. ρ ρ p p JTp JTp p p p Jp Jp p. φ ( p Tp φ( p p 0.. I act By lemma.8 we have that p F( T which implies that F( T is closed. Net we prove that cove i.e. or ay y FT λ ( 0 we prove that z = λ ( λ y F( T. I act ( = ρ G z JT z z z JT z JT z z = z λ JT z λ y JT z JT z ρ z = z λg Jz λ G y JT z λ λ y. ( ( λ G ( y JT z G ( Jz ( G ( Jz ( G ( y Jz ( G ( y Jz λg Jz λ µτ δ λ µτ δ ( = λ Jz Jz ρ µτ G Jz δ ( y y Jz Jz ( y ( G ( y Jz λ ρ µτ δ ( ( G ( Jz ( G ( y Jz = λ λ y z ρ z δ λµτ λ µτ. Submittig (5 ito (4 we have ( z T z z z JT z JT z ( G ( Jz ( G ( y Jz φ = λµτ λ µτ δ. This implies that T z z ad the proo o Lemma.. 3. Mai Results T z TT z z =. Hece we have z Tz = i.e. z F( T F T is (4 (5. This completes Theorem 3. Let E be a uiormly cove ad uiormly smooth Baach space ad be a oempty closed ad cove subset o E. Let : E be a cove ad lower semicotiuous uctio with it ( D( such that ( >0 or all 0 0 T = T t : t 0 be a closed ad totally ad =. Let { } quasi-g-asymptotically oepasive semigroup deied by Deiitio.6. Assume that T( t is uiormly asymptotically regular or all t 0 ad F( T = t 0 F( T( t. Let the sequece { } be deied by Echose arbitrarily; = yt = J αj ( α JT ( t = { z : supg ( z Jyt αg ( z J ( α G ( z J ξ} t 0 = (6
30. J. WANG Y. H. WANG where ξ = µ sup p F( T τ ( G ( p J δ ad the sequece { } ( 0 the { } coverges strogly to α. I lim α = 0 ad µ = δ = 0 F ( T. Proo We divide the proo ito ive steps. Step. Firstly we prove that F ( T ad are closed ad cove subsets i. Sice T( t is a totally quasi-g-asymptotically oepasive mappig it ollows the Lemma. that F( T( t is a closed ad cove subset o. So F( T = t 0 F( T( t is closed ad cove subset o. Agai by the assumptio = is closed ad cove. Suppose that is the closed ad cove subset o or. I view o the deiitio o G we have that { : sup ( α ( ( α ( ξ } t 0 { : ( α ( ( α ( ξ } = z G z Jy G z J G z J t = z G z Jy G z J G z J t 0 t 0 t { ξ } = z : z J Jy y. t t This shows that is closed ad cove or all. F T. Step. Net we prove that. Sice { } I act FT =. Suppose that FT or some T = T t : t 0 is a totally quasi-g-asymptotically oepasive semigroup or each p FT we have ( t = ( α ( α G p y G p J JT t = p p α J α JT t α J α JT t ρ p α α α p p J p JT t J ( α JT ( t ρ ( p G ( p J ( α G p JT ( t G ( p J ( G ( p J ( ( G ( p J ( ( α G ( p J ξ = α α α α µτ δ G p J where supt 0 ( G ( p J ξ = µ τ δ. This shows that p which implies that F( T or all. Step 3. We prove that { } is bouded ad { G( } is coverget. Sice : E is a cove ad lower semicotiuous uctio by virtue o Lemma.0 we have that there eists E ad α such that ( α or each E. The or each E we have that ( = ρ G J J ρ J ρα ρ = J ρα (7 Agai sice ρ J ρα ( = J ρ J ρ ρα. = ad F( T rom Lemma. we have G ( J G ( p J p F( T. Hece rom (7 we have or ay
. J. WANG Y. H. WANG 3 ( G p J G J J ρ J ρ ρα. Thereore { } ad { ( } Lemma. we have that G J are bouded. As = ad = by usig φ G J G J 0. This implies that { G ( J } is bouded ad odecreasig. Hece the limit lim G( Step 4. Net we prove that F( T. By the deiitio o or ay positive iteger m we have m = m Lemma. we have that ( G ( J G ( J φ <= 0 m m m eists.. Agai rom as m. It ollows rom Lemma.8 that lim m m 0 is a auchy sequece i. Sice is a oempty closed ad cove subset o Baach space E we ca assume that. Thereore we have =. Hece { } limξ = lim µ sup τ G ( p J µ = 0. p FT Sice ad α 0 it ollows rom the deiitio o that we have ( ( ( G Jy α G J α G J ξ. sup t t 0 (8 That is Sice ρ ( Jy ρ ( Jy y t t α t ( J ( α ρ ξ. ( α α( Jyt ξ. t t Jy y J ( y ( ( φ αφ α φ ξ. (9 t u ad α 0 rom (8 (9 we ca get ( y limφ t = 0. The by Lemma.8 we have limy t = (0. As J is uiormly cotiuous o each bouded subset o E we have t 0 we have ( α ( α ( α J 0= lim Jy J = lim α J JT t J t lim JT t Ju α J J = lim JT t J. J. The rom (0 or ay
3. J. WANG Y. H. WANG Sice ( α lim = we have that lim uiormly or all t 0. Sice J is uiormly cotiuous we obtai that = 0 JT t J 0 lim T t = ( uiormly or all t 0. T t is asymptotically regular or all t 0 rom ( we have Sice lim T t = T t T t T t = 0 lim The T ( t = T( t T ( t as. By virtue o the closedess o T( t ad as we ca obtai that T( t = which implies F( T( t or all t 0. Hece F( T = F( T( t t 0. Step 5. Fially we prove that Sice Assume that F = T. F T E is closed ad cove by Lemma. we kow that F ϖ = T. Sice ϖ F. As we kow ( all have As F T ad F T t T is sigle-valued. = we have G ( J G ϖ J or G y J is cove ad lower semicotiuous with respect to y whe is ied. So we limsup ( ϖ G J limig J G J G. T rom the deiitio o F ( T we ca obtai that F = ϖ = T ad as. This completes the proo o Theorem 3.. Just as i Remark.7 we use tm ( m istead o t i Deiitio.6 ad deote T( t m by T m T : = { Tm : } m= becomes a coutable amily o total quasi-g-asymptotically oepasive mappigs. The we get the ollowig corollary. orollary 3. Let E be a uiormly cove ad uiormly smooth Baach space ad be a oempty closed ad cove subset o E. Let T : = { Tm : } m= be a coutable amily o closed ad totally quasi-gasymptotically oepasive mappigs. Let : E be a cove ad lower semicotiuous uctio with it ( D( such that ( >0 or all ad ( 0 = 0. Assume that T m is uiormly asympto- tically regular or all m F = F T deied by ad ( T. Let the sequece { } m m Echose arbitrarily; = ym = J αj ( α JTm = z : supg ( z Jym αg ( z J ( α G ( z J ξ m = c where ξ = µ sup p F( T τ ( G ( p J δ ad { } ( 0 α. I lim α = 0 ad µ δ 0 ( = = the { } coverges strogly to F ( T. I orollary 3. whe ( 0 or all T = { Tm : } m= be a coutable amily o closed ad totally quasi-φ -asymptotically oepasive mappigs. The we ca get the ollowig theorem.
. J. WANG Y. H. WANG 33 orollary 3.3 Let E be a uiormly cove ad uiormly smooth Baach space ad be a oempty closed ad cove subset o E. Let T = { Tm : } m= be a coutable amily o closed ad totally quasi-φ - asymptotically oepasive mappigs. Assume that T m is uiormly asymptotically regular or all m F = F T deied by ad ( T. Let the sequece { } m m E chose arbitrarily; = ym = J αj ( α JTm = z : supφ( z Jym αφ ( z J ( α φ( z J ξ m = Πc where ξ = µ sup p F( T τ ( φ( p J δ ad { } ( 0 α. I lim α = 0 ad 0 coverges strogly to F ( T. Remark 3.4 The results i this paper improve ad eted may recet correspodig mai results o other authors (see or eample [347805-9] i the ollowig ways: (a we itroduce a ew class o totally quasi-g-asymptotically oepasive mappigs which cotais the classes o the totally quasi- φ -asymptotically oepasive mappigs ad may o-epasive mappigs; (b we eted rom a coutable amily o mappigs to the totally quasi-g-asymptotically oepasive semigroup; (c we modiy the Halper type hybrid projectio algorithm by usig the geeralized -projectio operator or uiormly total quasi-g-asymptotically oepasive semigroup. For eample orollary 3. eteds the mai result o Seawa et al. [] rom the modiied Ma type iterative algorithm to modiied Halper iterative by the geeralized -projectio method. orollary 3.3 is the mai result o hag et al.[3]. otributios (3 δ = the { } All authors cotributed equally ad sigiicatly i this research work. All authors read ad approved the ial mauscript. Ackowledgemets The authors would like to thak editors ad reerees or may useul commets ad suggestios or the improvemet o the article. This study was supported by the Natioal Natural Sciece Foudatios o hia (Grat No. 7330 ad the Natural Sciece Foudatios o Zhejiag Provice o hia (Grat No. Y6070. REFERENES [] Ya. I. Alber. E. hidume ad J. L. Li Stochastic Approimatio Method or Fied Poit Problems Applied Mathematics Vol. 0 No. 3 0 pp. 3-3. [] L. J. he ad J. H. Huag Strog overgece o a Iterative Method or Geeralized Mied Equilibrium Problems ad Fied Poit Problems Applied Mathematics Vol. 0 No. 0 pp. 3-0. [3] S. S. hag L. H. W. Joseph. K. ha ad W. B. Zhag A Modiied Halper-Type Iteratio Algorithm or Totally Quasi- ϕ-asymptotically Noepasive Mappigs with Applicatios Applied Mathematics ad omputatio Vol. 8 No. 0 pp. 6489-6497. http://d.doi.org/0.06/j.amc.0..09 [4] S. S. hag L. H. W. Joseph. K. ha ad L. Yag Approimatio Theorems or Total Quasi-ϕ-Asymptotically Noepasive Mappigs with Applicatio Applied Mathematics ad omputatio Vol. 8 No. 6 0 pp. 9-93. http://d.doi.org/0.06/j.amc.0.08.036 [5] S. S. Zhag L. Wag ad Y. H. Zhao Multi-Valued Totally Quai-Phi-Asymptotically Noepasive Semigrops ad Strog overgece Theorems i Baach Spaces Acta Mathematica Scietia Vol. 33B No. 03 pp. 589-599. http://d.doi.org/0.06/s05-960(3600-3 [6] Y. Li Fied Poit o a outable Family o Uiormly Totally Quasi-Phi-Asymptotically Noepasive Multi-Valued Mappigs i Releive Baach Spaces with Applicatios Applied Mathematics Vol. 03 No. 4 03 pp. 6-. [7] X. R. Wag S. S. hag L. Wag Y. K. Tag ad Y. G. Xu Strog overgece Theorem or Noliear Operator Equa-
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