Joura of Mathematca Sceces: Advaces ad Appcatos Voume 4 umber 2 2 Pages 33-34 COVERGECE OF HE PROJECO YPE SHKAWA ERAO PROCESS WH ERRORS FOR A FE FAMY OF OSEF -ASYMPOCAY QUAS-OEXPASVE MAPPGS HUA QU ad S-SHEG YAO Departmet of Mathematcs Kumg Uverst Yua 653 P. R. Cha e-ma: quh79@ahoo.c Abstract ths paper we cosder the strog covergece of the proecto tpe shawa terato process to a commo fed pot of a fte fam of osef - asmptotca quas-oepasve mappgs. Our resuts of ths paper mprove ad eted the correspodg resuts of emr ad Gu [] emr [] ad hawa [2].. troducto hroughout ths paper et C be a oempt subset of a rea ormed ear space X ad deote the set of a fed pots of a mappg b 2 Mathematcs Subect Cassfcato: 47H 47H76. Kewords ad phrases: osef - asmptotca quas-oepasve mappg commo fed pot proecto tpe shawa terato process. hs wor was supported b the Scetfc Research Foudato of Kumg Uverst. Receved ovember 6 29 2 Scetfc Advaces Pubshers
34 HUA QU ad S-SHEG YAO F ( ) the -th terate ( ( ) ) of b ad E where E deotes the mappg E : C C defed b E respectve. et be a sef-mappg of C. s sad to be asmptotca oepasve f there ests a rea sequece { λ } [ ) wth m λ such that ( λ ) C. A mappg s caed uform -pschtza f there ests a rea umber > such that for ever K ad each. t was proved [2] that f X s uform cove ad f C s bouded cosed ad cove subset of X the ever asmptotca oepasve mappg has a fed pot. s caed -asmptotca quas-oepasve o C f there ests v wth m v such that u p sequece { } [ ) ( v ) u p for a u C p F ( ) F ( ) ad 2. Remar.. From above deftos t s eas to see f F ( ) s oempt a asmptotca oepasve mappg must be - asmptotca quas-oepasve. t s obvous that a asmptotca oepasve mappg s aso uform -pschtza wth sup { v : }. However the coverses of these cams are ot true geera. the past few decades ma resuts o fed pots o asmptotca oepasve quas-oepasve ad asmptotca quas-oepasve mappgs Baach space ad metrc spaces are obtaed (see e.g. [7 9]). Recet Rhoades ad emr [5] studed the covergece theorems for -oepasve mappgs emr ad Gu [] studed the covergece theorems for -asmptotca quasoepasve mappg Hbert space. Ver recet emr [] studed the covergece theorems of mpct terato process for a fte fam of -asmptotca oepasve mappgs. most papers [ 4 9] whch cocer the terato methods the shawa terato scheme as foows: for a gve C
COVERGECE OF HE PROJECO YPE SHKAWA 35 as a b b (.) where { a } { b } { a } ad { b } are rea sequeces [ ) wth a b a b are bouded sequeces C. O oe had S have bee assumed to map C to tsef (.) ad the covet of C esures that the sequece { } gve b (.) s we defed. f however C s a proper subset of the rea Baach space X ad maps C to X the the sequece gve b (.) ma ot be we defed. Oe method that has bee used to overcome ths the case of sge mappg s to geeraze the terato scheme b troducg a retracto P : X C the recurso formua (.). For osef oepasve mappgs some authors (see e.g. [8 4]) have studed the strog ad wea covergece theorems Hbert space or uform cove Baach spaces. As a mportat geerazato of the cass of asmptotca oepasve sef-mappgs Chdume [] 23 geerazed oepasve asmptotca oepasve uform -pschtza to Defto.. et C be a oempt subset of a rea ormed space X. et P : X C be a oepasve retracto of X oto C. A osef mappg : C X s caed asmptotca f there ests a sequece { } [ ) wth as such that for ever ( P ) ( P ) for ever C. s sad to be uform -pschtza f there ests a costat > such that ( P ) ( P ) for ever C. Ad f et : C X the mappg s sad to be Γ- pschtza f there ests Γ such that ( P ) ( P ) Γ ( P ) ( P ) for ever C.
36 HUA QU ad S-SHEG YAO 26 Wag [3] geerazed the wor to prove strog ad wea covergece theorems for a par of osef asmptotca oepasve mappgs. O the other had 99 Schu [7] troduced a modfed Ma terato process to appromate fed pots of asmptotca oepasve sef-mappgs Hbert space as foow: ( a ) a. (.2) Sce the Schu s terato process has bee wde used to appromate fed pots of asmptotca oepasve sef-mappgs Hbert space or Baach spaces [4 6]. 29 hawa [2] geerazed ther wor to prove strog ad wea covergece theorems of proecto tpe shawa terato for a par of osef asmptotca oepasve mappgs. Motvated b above wors ths paper we cosder the foowg proecto tpe shawa terato process wth errors (.3) to appromatg commo fed pots for a fte fam of osef - asmptotca quas-oepasve mappgs ad obta the strog covergece theorems for such mappgs uform cove Baach spaces. Defto.2. et : X C { } s osef - asmptotca quas-oepasve mappgs s osef asmptotca oepasve. he a teratve scheme s the sequeces of mappgs { } defed b for gve C P( a ( )( ( )) ( ) P b cu ) ( ( )( ( )) ( ) P a P b c v ) (.3) where { a } { b } { c } { a } { b } ad { c } are rea sequeces [ δ δ] for some δ ( ) wth a b c a b c ( ( ) ) ( ) ( ) { } ad { u } { } are bouded sequeces C. v
COVERGECE OF HE PROJECO YPE SHKAWA 37 We restate the foowg deftos ad emmas whch pa a mportat roes our proofs. Defto.3. et X be a Baach space C be a oempt subset of X. et : C C. he s sad to be () demcosed at f wheever { } C such that C ad the. (2) sem-compact f for a bouded sequece { } C such that as there ests a subsequece { } of { } such that { } coverges strog to some K. (3) compete cotuous f the sequece { } C coverges wea to mpes that { } coverges strog to. emma. [2]. et { α } { β } { γ } ad { µ } be four oegatve rea sequeces satsfg α ( γ ) ( µ ) α β for a. f µ < γ < ad β < the m α ests. emma.2 [7]. et E be a rea uform cove Baach space ad p t q < for a postve teger. Aso suppose { } ad { } are two sequeces of E such that m sup r m sup r ad m sup t ( t ) r hod for some r the. m emma.3 []. et X be a rea uform cove Baach space C be a oempt cosed subset of X ad et : C X be osef asmptotca oepasve mappg wth a sequece { } [ ) as. he E s demcosed at zero. 2. Ma Resuts ad emma 2.. et X be a uform cove Baach space K be a oempt cosed cove subset of X { : { 2 }} : K X be uform Γ -pschtza - asmptotca quas-oepasve o-
38 HUA QU ad S-SHEG YAO sef-mappgs wth sequeces { } [ ) v such that v < ad : { } : C X be uform -pschtza asmptotca oepasve osef-mappgs wth { } [ ) / u such that u < ad F F ( ) F ( ). Suppose that for a gve K the sequece { } s geerated b (.3) where c < c <. f F / the m m 2. Proof. Sce C s bouded there ests M > such that u M ad M > such that v M for a. For a F ( ) F ( ). p F / p ( )( ( )) ( ) a P b cu p a ( )( ( )) ( ) P p ( a ) p c u ( a v ) p c M. (2.) p a ( )( ( )) ( ) P b c v p a ( )( ( )) ( ) P p ( a ) p c v ( a ) p a ( u ) ( v ) p c M [ a ( u v u v )] p c M. (2.2) rasposg ad smpfg above equat ad otcg that [ δ δ]. We have a p ( av )[ a ( u v uv ) p c M ] cm ( γ )( µ ) p β (2.3) where γ a v µ a ( u v u v ) β c M c M.
COVERGECE OF HE PROJECO YPE SHKAWA 39 Sce u < < v for a { 2 } ad c < < c for a thus γ < µ < ad β <. B emma. m p ests for each p F. et m p d >. Sce ( )( ( )) ( ) P p c ( u ) ( v ) p c M. B (2.2) we have m sup ( ) ( P ( )) ( ) p c ( u ) d. Ad p c ( u ) p cm whch mpes m sup p c ( u ) d. m p d meas that [ ( )( ( )) ( m ) a P ( u )] ( a )[ p c ( u )] d. p c B emma.2 we have m ( )( ( )) ( ) P. (2.4) Usg (2.) p ( av ) p cm. We have d m p m f p. t foows from (2.2) m sup p m p d that m p d. hs mpes that m p m a ( ( )( P ( )) ( ) p c ( v )) ( a ) ( p c ( v )) d. Sce ( )( P ( )) ( ) p c ( v ) ( u )( v ) p c M ad p c ( v ) ( p) c M we have m sup ( )( ( )) ( ) P p c ( v p) d ad m sup p c ( v p) d. B emma 2.2 we have m ( )( ( )) ( ) P. (2.5) a ( )( ( )) ( ) P b c v
HUA QU ad S-SHEG YAO 3 ( )( ( )) ( ) ( ). as p v c P a (2.6) Aso ( )( ( )) ( ) P ( )( ( )) ( ) ( )( ( )) ( ) P P ( )( ( )) ( ) P ( ) ( )( ( )) ( ). P hus t foows from (2.4) ad (2.6) that ( )( ( )) ( ). m P (2.7) addto ( )( ( )) ( ) u c b P a ( )( ( )) ( ) u c P a ( ) c b ( )( ( )) ( ) M c P a b (2.4) ad (2.6) we have m as we as for a { } 2. m (2.8) otce that for each ( ) ( ) mod > ad ( ) ( ) ( ) hece ( ) ( ) ( ) ( ) ( ) ( ) ( ) that s ( ) ( ) ad ( ) ( ). From (2.5) (2.7) ad (2.8)
COVERGECE OF HE PROJECO YPE SHKAWA 3 ( )( ( )) ( ) ( )( ( )) ( ) P P ( )( ( )) ( ) ( )( ( )) ( ) P P Γ ( )( ( )) ( ) P ( )( ( )) ( ) P ( )( ( )) ( ) P ( )( ( )) ( ) P Γ ( ) ( ). Γ hs mpes that. m ow for a { }. 2 Γ ( ) ( ). Γ So m for a { }. 2 Cosequet we have. m (2.9) ( )( ( )) ( ) ( )( ( )) ( ) P P ( )( ( )) ( ) ( )( ( )) ( ) P P 2 ( )( ( )) ( ) ( )( ( )) ( ) P P 2 2 ( )( ( )) ( ) P ( )( ( )) ( ) P 2 ( ) ( ).
32 HUA QU ad S-SHEG YAO hs mpes that m. (2.) Ad ( ). ag m o both sdes the above equat the we get m for a { 2 }. Cosequet we have he proof s competed. m. (2.) heorem 2.2. et X be a uform cove Baach space ad C { } be same as emma 2.. f oe of et m s a semcompact mappg ad F / the { } coverges strog to a commo fed pot of { } ad { }. Proof. Sce m s sem-compact mappg { } s bouded ad m the there ests a subsequece { } of { } m such that { } coverges to. t foows from emma.3 ad F ( m ). addto sce s a uform Γ- pschtza mappg s a uform -pschtza mappg m ad m. So ad. hs mpes that F ( ) F ( ). Sce the subsequece { } of { } such that { } coverges strog to ad m ests the { } coverges strog to the commo fed pot F. he proof s competed.
COVERGECE OF HE PROJECO YPE SHKAWA 33 heorem 2.3. et X be a uform cove Baach space ad C { } be same as emma 2.. f oe of et m s compete cotuous mappg ad F / the { } coverges strog to a commo fed pot of { } ad { }. Proof. B emma 2. { } s bouded. Sce m m the { } ad { } are bouded. Sce m s compete cotuous that ests subsequece { m } of { m } such that { m } p as. hus we have m m. Hece b the cotut of m ad emma.3 we have m p ad p F ( m ). Further for a { 2 } p m m p p m m p. hus m p ad m p. hs mpes that { } { } coverges strog to p. Sce s a uform - pschtza mappg s uform Γ- pschtza so s cotuous. So p p p. Hece p F. B emma 2. m p ests. hus m p. he proof s competed. Refereces [] C. E. Chdume E. U. Ofoedu ad H. Zegee Strog ad wea covergece theorems for asmptotca oepasve mappgs J. Math. Aa. App. 28 (23) 364-374. [2] K. Goebe ad W. A. Kr A fed pot theorem for asmptotca oepasve mappgs Proc. Amer. Math. Soc. 35 (972) 7-74. [3] F. GU ad J. u A ew composte mpct terato process for a fte fam of o-epasve mappgs Baach spaces Fed Pot heor App. (26) - Artce D 82738. [4] M. O. Ose ad A. Udomee Demcosedess prcpe ad covergece theorems for strct pseudocotractve mappgs of Browder-Petrsh tpe J. Math. Aa. App. 256 (2) 43-445.
34 HUA QU ad S-SHEG YAO [5] B. E. Rhoades Fed pot teratos for certa oear mappgs J. Math. Aa. App. 67 (979) 274-276. [6] B. E. Rhoades ad S. emr Covergece theorems for -oepasve mappgs t. J. Math. Math. Sc. (26) -4. [7] J. Schu teratve costructo of fed pots of asmptotca oepasve mappgs J. Math. Aa. App. 58 (99) 47-43. [8]. Shahzad Appromatg fed pots of o-sef oepasve mappgs Baach spaces oear Aa. 6 (25) 3-39. [9] K. K. a ad H. K. Xu Appromatg fed pots of oepasve mappgs b shawa terato process J. Math. Aa. App. 78 (993) 3-38. [] S. emr ad O. Gu Covergece theorem for -asmptotca oepasve mappg Hbert space J. Math. Aa. App. 329 (27) 759-765. [] S. emr O the covergece theorems of mpct terato process for a fte fam of -asmptotca oepasve mappgs J. Comput. App. Math. 225 (29) 398-45. [2] S. hawa Commo fed pots of ew teratos for two asmptotca oepasve osef-mappgs a Baach space J. Comput. App. Math. 224 (29) 688-695. [3]. Wag Strog ad wea covergece theorems for commo fed pots of osef asmptotca oepasve mappgs J. Math. Aa. App. 323 (26) 55-557. [4] H. K. Xu ad X. M. Y Strog covergece theorems for oepasve osefmappgs oear Aa. 2(24) (995) 223-228. [5] H. K. Xu ad R. G. Or A mpct terato process for oepasve mappgs umber. Fuct. Aa. Optm. 22(5-6) (2) 767-773. g