A Note on Convergence of a Sequence and its Applications to Geometry of Banach Spaces

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1 Advaces i Pure Mathematics -4 doi:46/apm9 Published Olie May ( A Note o Covergece of a Sequece ad its Applicatios to Geometry of Baach Spaces Abstract Hemat Kumar Pathak School of Studies i Mathematics Padit Ravishakar Shukla Uiversity Raipur Idia hkpathak5@gmailcom Received Jauary 9 ; revised April ; accepted April The purpose of this ote is to poit out several obscure places i the results of Ahmed ad Zeyada [J Math Aal Appl 74 () ] I der to rectify ad improve the results of Ahmed ad Zeyada we itroduce the cocepts of locally quasi-oepasive biased quasi-oepasive ad coditioally biased quasi-oepasive of a mappig wrt a sequece i metric spaces I the sequel we establish some theems o covergece of a sequece i complete metric spaces As cosequeces of our mai result we obtai some results of Ghosh ad Debath [J Math Aal Appl 7 (997) 96-] Kirk [A Uiv Mariae Curie-Sklodowska Sec A LI 5 (997) 67-78] ad Petryshy ad Williamso [J Math Aal Appl 4 (97) ] Some applicatios of our mai results to geometry of Baach spaces are also discussed Keywds: Locally Quasi-Noepasive Biased Quasi-Noepasive Coditioally Biased Quasi-Noepasive Drop Super Drop Itroductio I the last four decades of the last cetury there have bee a multitude of results o fied poits of oepasive ad quasi-oepasive mappigs i Baach spaces (eg [5-7 9-]) Our aim i this ote is to poit out several obscure places i the results of Ahmed ad Zeyada [J Math Aal Appl 74 () ] I der to rectify ad improve the results of Ahmed ad Zeyada we itroduce the cocepts of locally quasi-oepasive biased quasioepasive ad coditioally biased quasi-oepasive of a mappig wrt a sequece i metric spaces Let X be a metric space ad D a oempty subset F T be of X Let T be a mappig of D ito X ad let the set of all fied poits of T F a give D the sequece of iterate is determied by T T (I) Let X be a med space ad the sequece of iterates are defied by T T T I T (II) T T T I T[ I T] (III) The iteratio scheme (I) is called Teoplitz iteratio ad the iteratio scheme (II) was itroduced by Ma [] while the iteratio scheme (III) was itroduced by Ishikawa [9] The cocept of quasi-oepasive mappig was iitiated by Tricomi i 94 f real fuctios It was further studied by Diaz ad Metcalf [5] ad Dosto [67] f mappigs i Baach spaces Recetly this cocept was give by Kirk [] i metric spaces as follows: Defiitio The mappig T is said to be quasioepasive if f each D ad f every p FT dt p d p A mappig T is coditioally quasi-oepasive if it is quasi-oepasive wheever FT We ow itroduce the followig defiitio: Defiitio The mappig T is said to be locally quasi-oepasive at p FT if f each D dt p d p Obviously quasi-oepasive locally quasi-oep F T but the reverse implicatio pasive at each Copyright SciRes

2 4 may ot be true To this ed we observe the followig eample Eample Let X ad D be e- 4 dowed with the Euclidea metric d Defie the mappig T : D X by T f each D The we observe that FT f all D ad p F T we have that dt p d p ie T is locally quasi-oepasive at p FT However oe ca easily see that T is ot locally quasi- oepasive at p FT Ideed f all ad p FT we have d T p d p Hece we coclude that T is ot quasi-oepasive although it is locally quasi-oepasive at p FT The cocept of asymptotic regularity was fmally itroduced by Browder ad Petryshy [] f mappigs i Hilbert spaces Recetly it was defied by Kirk [] i metric spaces as follows: Defiitio The mappig T is said to be asymp- totically regular if lim d T T f each D Mai Results Let N deote the set of all positive itegers ad N Ahmed ad Zeyada [] itroduce-ed the followig: Defiitio The mappig T is said to be quasioepasive wrt a sequece if f all ad f each p FT d p d p The followig lemma was quoted by Ahmed ad Zeyada [] without proof Lemma A If T is quasi-oepasive the T is quasi-oepasive wrt a sequece T (respectively T T ) f each D Remark We otice that the above lemma is valid if T D f each ad a give D ( D is T-ivariet) So the crect versio of Lemma A should be read as follows: Lemma If T is quasi-oepasive ad f a give D ad each T D the T is quasi-oepasive wrt a sequece T (re spectively T T ) f each D Further they claimed that the reverse implicatio i Lemma A may ot be true i their Eample We agai otice that there are several obscure places i this eample We ow quote Eample of Ahmed ad Zeyada [] i the followig: 4 Eample A Let X ad D be e- 5 dowed with the Euclidea metric d We defie the mappig T : D X by T f each D F a give D we have 4 d T p d T p where T 4 DN ad FT ie T is quasi-oepasive wrt a sequece T 4 Furtherme the map T is quasi-oepa sive wrt a sequece T ad T They foud that T is either coditioally quasi-o- epasive quasi-oepasive f D ad 4 p FT d4 d4 ad D is ot closed Remark We otice that the followig claims made i Eample A were false: ) T : D X is a mappig I fact TD X 5 ) FT I fact FT ) T is quasi-oepasive wrt a sequece T 4 4) T is quasi-oepasive wrt a sequece T ad T However (i) ca be rectified by takig X as ay superset of 5 5 i Eve if this crectio is made we fid that the remaiig statemets ) - 4) will remai false Cosequetly the claim of Ahmed ad Zeyada [] that the reverse implicatio i Lemma may ot be true seems false We ow itroduce the followig defiitio Defiitio The mappig T is said to be locally p F T quasi-oepasive at wrt a sequece Copyright SciRes

3 if f all d p d p Obviously locally quasi-oepasiveess at p F T locally quasi-oepasiveess at p F T wrt a sequece We ow state the followig lemma without proof Lemma If T is quasi-oepasive wrt a sequece the T is locally quasi-oepasive at each p FT wrt the sequece The reverse implicatio i Lemma may ot be true as show i the followig eample: Eample Let X ad D be e- dowed with the Euclidea metric d Defie the map- pig T : D X by T f each D The we observe that FT F a give D 4 p F T we have that ad d T p d T p * where T D ie T is locally quasi- 4 oepasive at p FT wrt a sequece T However oe ca easily see that T is ot 4 locally quasi-oepasive at p FT wrt the sequece T Ideed we have 4 d T p d T p f all Cosequetly T is either quasi-oepasive quasi-oepasive wrt the sequece T 4 We ow itroduce the followig: Defiitio The mappig T : D X is said to be biased quasi-oepasive (bq) wrt a sequece X if f all ad at each p codf T where d p d p ** FT pft lim sup d p lim if d FT cod : A mappig T is coditioally biased quasi-oepasive (cbq) wrt a sequece if codft Remark We observe that the followig implicatios are obvious: (a) Coditioal biased quasi-oepasiveess wrt a sequece biased quasi-oepasiveess wrt a sequece but the reverse implicatio may ot be true (Ideed ay mappig T : D X f which codft is a biased quasi-oepasive wrt a sequece but ot coditioally biased quasioepasive wrt a sequece However uder certai coditios a biased quasi-oepasive map wrt a sequece may be a coditioal biased quasioepasive wrt a sequece (see Lemma 6 below) (b) If T is coditioally biased quasi-oepasive wrt a sequece ad codft F T the T is locally quasi-oepasive at each p F T wrt a sequece (c) If T is biased quasi-oepasive wrt a sequece ad codftö FT the T is locally quasi-oepasive at each p codf T wrt a sequece (d) Quasi-oepasivees locally quasi-oepasiveess at p FT locally quasi-oepasiveess at p FT wrt a sequece I Eample above we observe that p F T we have ) f lim sup d p limsup ) f p FT lim we have lim sup d p limsup lim lim if d F T lim if lim 5 Copyright SciRes

4 6 Here cod F T ad i view of (*) ad (**) it is evidet that T is coditioally biased quasi-o- epasive (cbq) wrt a sequece T 4 ad hece it is biased quasi-oepasive (bq) wrt a sequece T 4 We ow show i the followig eample that codft eed ot be a sigleto set Eample Let X ad D be edowed with the Euclidea metric d Defie the mappig T : D X by T f ad T f Clearly FT Cosider the sequece i X the we observe that p F T we have ad ) f d p lim sup lim sup lim ; ) f p FT we have lim sup d p lim sup lim ; d FT Thus we have cod F lim if lim T ad it is evidet that T is coditioally biased quasi oepasive (cbq) wrt the sequece i X ad hece it is biased quasi-oepasive (bq) wrt the sequece i X However iterested reader ca check that if we cosider the sequece such that the cod FT Further we observe that f p cod F T ad f all we have d p d p Thus T is coditioally biased quasi-oepasive (cbq) wrt the sequece i X O the other had if we cosider the sequece such that the cod FT ad T is coditioally biased quasi-oepasive (cbq) wrt the sequece i X Remark 4 Eample above also shows that cod FT is a closed set eve though T is discotiuous at p We eed the followig lemmas to prove our mai theem: Lemma Let T be locally quasioepasive at p FT wrt ad lim d F T The is a Cauchy sequece Proof Sice d F T lim the f ay give there eists such that f each d FT So there eists q FT such that f all d q Thus f ay m we have dm dm qd q q FT Hece is a Cauchy sequece Lemma 4 Let T be coditioally biased quasioepasive wrt ad lim if d F T The: ) coverges to a poit p i codft ad T is locally quasi-oepasive at p codft wrt ) is a Cauchy sequece Proof ) Sice T is coditioally biased quasioepasive wrt it follows that codft lim if d F T we have that As lim sup d p f some p codft have lim d p f some p codft coverges to a poit p i is locally quasi-oepasive at p codft ) From lim d p So we ; ie cod F T ad T wrt it follows that f ay give there eists N such that f each d p Thus f ay m we have dm dm qd q q FT Hece is a Cauchy sequece The followig lemma follows easily Lemma 5 Let T be biased quasi-oepasive wrt ad let coverges to a poit p i F T The: ) coverges to a poit p i codft ad T is coditioally biased quasi-oepasive wrt ; ) is a Cauchy sequece We ow state our mai theem i the preset paper Theem Let FT be a oempty closed set The lim d F T coverges to a po- ) if it p i FT ; ) coverges to a poit i F T if Copyright SciRes

5 7 lim d F T at p F Twrt ad X is complete Proof ) Sice is closed p FT mappig d FT the T is locally quasi-oepasive ad the is cotiuous (see [ p ]) lim d F T d lim F T d p F T ) From Lemma is a Cauchy sequece Sice X is complete the coverges to a poit F T is closed the say q i X Sice d F d F implies that q FT dp F lim T lim T T As cosequeces of Theem we have the followig: Collary Let FT a oempty closed set ad f a give D ad each T D The lim d T F T T coverges to if ) a poit p i FT ; ) T coverges to a poit i lim dt FT at p FT wrt F T if T is locally qusi-oepasive T ad X is complete Collary Let X be a med liear space FT a oempty closed set ad f a give D ad each T D T coverges to a poit p () If the sequece i FT the lim d T F T () If d T FT oepasive at p FT wrt T lim T is locally quasi- ad X is complete the T coverges to a poit p i FT Collary Let X be a med liear space FT a oempty closed set ad f a give D ad each T D The () lim d T FT if the sequece T coverges to a poit p i FT ; () T coverges to a poit p i FT if lim d T F T T is locally quasi-oepa- sive at p FT wrt T Note that the cotiuity of T implies that ad X is complete F T is closed but the coverse eed ot be true To effect this cosider the followig eample Eample Let X ad D be edowed with the Euclidea metric d Defie the map- ig T : D X by T if ad T if Obviously FT is a oempty closed but T is ot cotiuous at Remark 5 (a) I der to suppt the above fact Ahmed ad Zeyada [] stated wrogly i their Eample where X D T If X 4 ad T if 56 that T is ot cotiuous I fact we observe that i this eample T is cotiuous (b) From Lemma Eamples ad the cotiuity of T implies that FT is closed but the coverse may ot be true; the we have that Collaries ad are improvemet of Theem i [ p46] Theem i [ p 469] ad Theem i [8 p 98] respectively (c) Sice every quasi-oepasive map wrt a sequece is locally quasi-oepasive at each p F T wrt a sequece but the coverse may ot be true; we have that Theem Collaries ad are improvemet of crespodig Theem Collary ad of Ahmed ad Zeyada [] (d) By cosiderig the closedess of F T i lieu of the cotiuity of T ad T : D X istead of T : X X we have that our Collary improves Propositio of Kirk [ p 68] (e) The closedess coditio of D i Theem ad of Petryshy ad Williamso [ p ] ad Theem i [8 p 98] is superfluous (f) The coveity coditio of D i Theem of Petryshy ad Williamso [ p 469] is superfluous because the auth assumed i their theem that T D f each ad a give D i coditio Theem Let codft be a oempty closed set The coverges to a poit i codft if lim if d cod F T T is codioally biased quasi-oepasive wrt ad X is complete cod F T F T we have that Proof Sice if d cod F T lim implies limif d F T Now usig the techique of the proof of Theem the coclusio follows from Lemma The followig results follows easily from Lemma 5 Theem Let FT be a oempty closed set The coverges to a poit i codft if coverges to a poit p i FT T is biased quasi-oepasive wrt ad X is complete Copyright SciRes

6 8 let ) Theem 4 Let X be a complete metric space ad codf T be a oempty closed set Assume that ) T is biased quasi-oepasive wrt ; lim d is a Cauchy sequece; ) if the sequece y satisfies dy y the lim lim i f d y co d F T lim su pd y co dft The coverges to a poit i Proof Sice codft cod F T it follows from (i) that T is codioally biased quasi-oepasive wrt ad the sequece d cod FT is mootoically decreasig ad bouded from below by zero limif d cod F T eists The From ) ad ) we have that lim i f d co d F T pd co d The l d c F T lim su F T im od Therefe by The em the sequece coverges to a poit i codft As cosequeces of Theem 4 we obtai the followig: Collary 4 Let X be a complete metric space ad let cod F T be a oempty closed set Assume that ) T is biased quasi-oepasive wrt ; ) T is asymptoticc regular at D ( T is a Cauchy sequece ); ) if the sequece y satisfies lim dy y the lim i f d y co d F T lim su pd y co d F T The T coverges to a poit i cod F T Collary 5 Let X be a Baach space ad let cod F T be a oempty closed set Assume that ) T is biased quasi-oepasive wrt T ; D ( T ) T is asymptoticc regular at is a Cauchy sequece ); ) if the sequece y satisfies lim y T y the lim i f d y co d F T lim su pd y co d F T The T coverges to a poit i cod F T Collary 6 Let X be a Baach space ad let codft be a oempty closed set Assume that ) T is biased quasi-oepasive wrt T ; ) T is asymptoticc regular at D ( T is a Cauchy sequece ); ) if the sequece y satisfies lim y T y the lim i f d y co d F T lim su pd y co d F T The T cod F T coverges to a poit i Remark 6 From Lemmas ad Eamples ad Remark the cotiuity of T implies that F T is closed but the coverse may ot be true; we obtai that Collary 4 iclude Theem i [ p 464] ad Theem i [7 p 99] as special cases As aother cosequece of Theem we establish the followig theem: Theem 5 Let X be a complete metric space ad let codft be a oempty closed set Assume that ) T is biased quasi-oepasive wrt ; ) f every D codft there eists p codft such that d p d p; ) the sequece cotais a subsequece j covergig to D The coverges to a poit i codft Proof Sice codft it follows from (i) that T is codioally biased quasi-oepasive wrt ad the sequece d cod FT is mootoically decreasig ad bouded from below by zero FT The l co FT im d d d lim cod Copyright SciRes

7 r eists We ow apply Theem 4 It suffices to F T the show that r= If lim cod r = If cod F T the D cod FT Thus there eists p cod F T such that lim lim lim d p dlim p d p d p d p d p This is a cotradictio So codf T Collary 7 Let X be a complete metric space cod F T a oempty closed set ad f a give D ad each T D Assume that ) T is biased quasi-oepasive wrt T ; ) f every D codft there eists p codft such that d T p d T p ; ) the sequece j The T covergig to D T cotais a subsequece T coverges to a poit i cod F T Collary 8 Let X be a Baach space cod F T a oempty closed set ad f a give D ad each T D ) T is biased quasi-oepasive wrt T ) f every D codft there eists T p T p ; ) the sequece cod F T such that j covergig to D T The T j Assume that ; p T cotais a subsequece coverges to a poit i cod F T Collary 9 Let X be a Baach space cod F T a oempty closed set ad f a give p D ad each T D ) T is biased quasi-oepasive wrt T ) f every D codft codft such that T p T p ; ) the sequece Assume that ; there eists T cotais a subsequece j covergig to D The T coverges to a poit i T 9 cod F T Remark 7 From Lemmas ad Eamples ad Remark the cotiuity of T implies that FT is closed but the coverse may ot be true ; we obtai that Collary 7 is a improvemet of Theem i [ p 466] Applicatios to Geometry of Baach Spaces Throughout this sectio let R deote the set of real umbers Let K K z r be a closed ball i a Baach space X F a sequece Ú K covergig to X we defie lim D SD K where ad D cov K D cov D ad SD K is called a super drop Clearly f a costat sequece covergig to we have D D so that D k cov K ad is called a drop Thus the cocept of a drop is a special case of super drop It is also clear that if y D K the D y K D K ad if z the y Recall that a fuctio : X R is called a lower semicotiuous wheever X : a is closed f each a R Caristi [4] proved the followig: Theem A Let X d be complete ad : X R a lower semicotiuous fuctio with a fiite lower boud Let T : X X be ay fuctio such that d T T f each X The T has a fied poit We ow state ad prove some applicatios of our mai results i sectio to geometry of Baach Spaces Theem Let C be a closed subset of a Baach space X let z X C K K z r be a closed ball of radius r d z C R Let be a arbitrary elemet of C let be a sequece i C covergig to X ad let T : C X be ay cotiuous fuctio defied implicitly by T C SD K f each C i the sese that TC D f each The ad let Copyright SciRes

8 4 ) d F T lim if coverges to a poit p i FT ; ) coverges to a poit i FT if lim d F T T is locally quasi-oepasive at p F T wrt Proof Without loss of geerality we may assume that z Let R ad let X A SD K The it is clear that T maps X ito itself F give y X ad a sequece y covergig to y we shall estimate y Ty o X F give y X ad the crespodig sequece y there is a sequece b i X with Tytb ty t Now Ty t b t y we have t y b y T y so because y b R we fid that y Ty t R Thus y T y t y b r y Ty t y b r R r Defie d y y y X ad y r y the X is complete as a metric space ad R r : X R is a cotiuous fuctio So is a lower-semicotiuous fuctio Also the above iequality takes the fm dy T yy T y Proceedig to the limit as we obtai dy Ty y T y f each y X There- fe applyig the theem of Caristi we obtai that T has a fied poit p p f each C ie FT By cotiuity of T FT is closed Hece the coclusio follows from Theem Sice drop is a special case of super drop we have the followig: Collary Let C be a closed subset of a Baach space X let z X C K K z r be a closed ball of radius r d z C R Let be a arbitrary elemet of C ad let T : C X be ay (ot ecessarily cotiuous) fuctio defied implicitly by TC D K f each C The lim d F T if coverges to a poit () p i FT ; ad let () coverges to a poit i lim d F T at p F T wrt F T if T is locally quasi-oepasive We ow prove the followig result f biased quasioepasive mappig wrt a sequece Theem Let C be a closed subset of a Baach space X let z X C ad let K Kz r be a closed ball of radius r dz C R Let be a arbitrary elemet of C a sequece i C covergig to X ad let T : C X be ay co- tiuous fuctio defied implicitly by TC SD K f each C i the sese that T C D f each If coverges to a poit i FT T is biased quasi-oepasive wrt the coverges to a poit i codft Proof Usig Theem istead of Theem the coclusio follows o the lies of the proof techique of Theem As a cosequece of Theem we obtai the followig: Collary Let C be a closed subset of a Baach space X let z X C K K z r be a closed ball of radius r dz C R Let be a arbitrary elemet of C ad let T : C X be ay (ot ecessarily cotiuous) fuctio defied implicitly by T C D K f each C If coverges to a poit i FT T is biased quasi-oepasive wrt the coverges to a poit i codft Ope Questio To what etet ca the cotiuity hypothesis o T be muted i Theems ad? 4 Refereces ad let [] M A Ahmed ad F M Zeyad O Covergece of a Sequece i Complete Metric Spaces ad its Applicatios to Some Iterates of Quasi-Noepasive Mappigs Joural of Mathematical Aalysis ad Applicatios Vol 74 No pp doi:6/s-47x()4- [] J-P Aubi Applied Abstract Aalysis Wiley-Itersciece New Yk 977 [] F E Browder ad W V Petryshy The Solutio by Iteratio of Noliear Fuctioal Equatios i Baach Spaces Bulleti of the America Mathematical Society Vol pp doi:9/s [4] J Caristi Fied Poit Theems f Mappigs Satisfyig Iwardess Coditios Trasactio of the America Mathematical Society Vol pp 4-5 doi:9/s [5] J B Diaz ad F T Metcalf O the Set of Sequecial Limit Poits of Successive Approimatios Trasactios of the America Mathematical Society Vol 5 Copyright SciRes

9 4 969 pp [6] W G Dotso Jr O the Ma Iteratio Process Trasactio of the America Mathematical Society Vol pp 65-7 doi:9/s [7] W G Dotso Jr Fied Poits of Quasio-Epasive Mappigs Joural of the Australia Mathematical Society Vol 97 pp 67-7 [8] M K Ghosh ad L Debath Covergece of Ishikawa Iterates of Quasi-Noepasive Mappigs Joural of Mathematical Aalysis ad Applicatios Vol 7 No 997 pp 96- doi:6/jmaa [9] S Ishikawa Fied Poits by a New Iteratio Method Proceedigs of the America Mathematical Society Vol 44 No 974 pp 47-5 doi:9/s [] W A Kirk Remarks o Approimatioad Approimate Fied Poits i Metric Fied Poit They Aales Uiversitatis Mariae Curie-Skłodowska Sectio A Vol 5 No 997 pp [] W A Kirk Noepasive Mappigs Ad Asymptotic Regularity Ser A: They Methods Noliear Aalysis Vol 4 No -8 pp - [] W R Ma Mea Valued Methods I Iteratio Proceedigs of the America Mathematical Society Vol 4 No 95 pp 56-5 doi:9/s [] W V Petyshy ad T E Williamso Jr Strog ad Weak Covergece of The Sequece of Successive Approimatios f Quasi-Noepasive Mappigs Joural of Mathematical Aalysis ad Applicatios Vol 4 97 pp doi:6/-47x(7)987-5 Copyright SciRes