Vol No : Joural of Facult of Egeerg & echolog JFE Pages 9- Uque Coo Fed Pot of Sequeces of Mags -Metrc Sace M. Ara * Noshee * Deartet of Matheatcs C Uverst Lahore Pasta. Eal: ara7@ahoo.co Deartet of Matheatcs Jah College for Woe Uverst of Peshawar Peshawar Pasta. Eal: oshha@gal.co Abstract. I ths aer we rove soe fed ot theores for sequeces of self ags usg A-te cotracto -etrc sace. we also show that these results eteds ad roves the corresodg results [ 4] ad others. AMS Matheatcs Subect Classfcato: 47H 54H5 Ke words: Fed ot -etrc sace A-te cotracto Setrc -etrc.. Itroducto he stud of Baach cotracto rcle have led to uber of geeralzatos ad odfcato of the rcle. It cocers certa ags of a colete etrc sace to tself. It states suffcet codto for the esteces ad uqueess of fed ots. he theore also gves a teratve rocess b whch we ca obta the aroato to the fed ots. Ma authers have geeralzed the well ow Baach cotracto rcle several dfferet fors we a see for eale [7 9 46 8]. I [6] Dhage troduced D-etrc sace as a geeralzato of etrc sace ad roved a results ths settg. But 5 Z. Mustafa ad B. Ss [] roved that these results are ot true toologcal structure ad hece the troduced -etrc sace as a geeralzed for of etrc sace. Sce the a authors cludg Z. Mustafa [] have bee studg fed ot results -etrc saces. I [] M. Ara A. A. Sddqu ad A. A. Zafar troduced a class of cotractos called A-cotractos ad roved soe fed ot theores for self as usg A-cotractos. hs geeral class of cotractos roerl cotas soe of the cotractos studed b R. Kaa [8] Bach [5] M. S. Kha [9] ad Rech [5] for detals see [ 7]. Further M Ara A. A. Sddqu ad A. A. Zafar have studed soe fed ot theores usg A-cotracto geeralzed etrc saces gs for detal see [] ad [4]. I ths aer we rove soe fed ot theores for sequeces of self ags usg A-te cotracto -etrc sace. Also we show that these results eteds ad roves the corresodg results [ 4] ad other corresodg results the curret lterature. 9
. Prelares I ths secto we gve soe basc deftos ad results o -etrc sace fro [] whch we requre the sequel. Defto. Let X be a oet set ad let : X X X R be a fucto satsfg the followg roertes. z f z. < for all X wth. z for all z X wth z 4. z z z...setr all three varables 5. z a a a z for all z a X rectagular equalt. he the fucto s called a geeralzed etrc or ore secfcall a -etrc o X ad the ar X s called a -etrc sace. Defto. A -etrc sace X s called setrc -etrc f for all X. Defto. Let X be a -etrc sace ad be a sequece of ots of X a ot X s sad to be the lt of the sequece f l ad oe ca sas that the sequece s -coverget to. hus f a -etrc sace X the for a > there est N N such that < for all N. Proosto.4 Let X be a -etrc sace the the followg are equvalet s -coverget to... as as 4. as. Defto.5 Let ever > there s N l as l. X be a -etrc sace a sequece N such that < l s called -Cauch f for for l N ; that s f Proosto.6 Let X be a -etrc sace the the followg are equvalet s -Cauch. 94
. for > there est N N such that < for all N. Defto.7 A -etrc sace X s sad to be -colete f ever -Cauch sequece X s -coverget X. Defto.8 Let ' ' a fucto the f s sad to be -cotuous at a ot X ad ' ' X be -etrc saces ad let : X X f be a X f ad ol f gve > ' there ests > such that X ; ad a < les f a f f <. A fucto f s -cotuous at X f ad ol f t s -cotuous at all a X.. Fed ot theores for sequeces of self ag Defto. [] Let A stads for the set of all fuctos : R. s -cotuous o the set R satsfg R of all trlets of oegatve reals wth resect to the Eucldea -etrc o R.. a b for soe wheever a a b b or a b a b or a b b a for all a br. Defto. A-te Cotracto A self a o a -etrc X s sad to be A-te cotracto of X f there ests A such that for all X. Net theore s the eteso of heore 6 of [] fro etrc sace setu to the -etrc sace setu. heore. Let A ad { } be a sequece of self ags o a colete -etrc sace X such that he { }... has a uque coo fed ot X. Proof: Defe a sequece { } X as where. Now. B usg Aga we get for soe [... 95
96 [. soe for Now usg we have. Slarl. 4 4 Cotug ths wa we get. As < as. Now b reeated use of the rectagular equalt of -etrc for ever teger > we ca wrte.... hs gves l whch les } { s a -Cauch sequece ad sce X s colete there est X such that as. Now for > we ca wrte. Let the. Whch gves. Now suose s aother fed ot of that s for soe X.. B usg we get
. Hece. Corollar.4 Let A ad { } be a sequece of self ags o a colete -etrc sace X such that a oe of the followg cotractve codto s satsfed. here est a uber a [ such that for all X a X he { }.. here est a uber h [ such that for all X h.. here est a uber h [ such that for all X ha{ }. 4. here est ubers a b c[ such that a b c < ad for all a b c. has a uque coo fed ot X. Proof: I [] t s show that above cotractos are A-te cotracto so b heore. we ca coclude that { } has a uque coo fed ot X. Net theore s aalogous to the heore of [] -etrc sace. heore.5 Let { } ad { S } be sequeces of self as o a colete setrc -etrc sace X satsfg S S S for all X for soe A ad for each N. he } ad S } have a uque coo fed ot. Proof: Defe a sequece { } X as ad S. Cosder S S Sce X s setrc -etrc Sace we ca wrte { {. 97
98 for soe [. Slarl we have for soe [. Whch gves. Proceedg the sae wa we get. I geeral we have for soe [. As < as. Now b reeated use of the rectagular equalt of -etrc for ever teger > we ca wrte.... hs gves l whch les } { s a -Cauch sequece ad sce X s colete there est X such that as. Now for each > cosder. g g g Let we get. hs les that. Slarl we ca show that S. hus } { ad } { S have coo fed ot. he uqueess ca be obta easl. hs coletes the roof. Corollar.6 Let A ad } { ad } { S be a sequeces of self ags o a colete -etrc sace X such that a oe of the followg cotractve codto s satsfed
X. here est a uber a [ such that for all X S S a S S.. here est a uber h [ such that for all X S S h S S.. here est a uber h [ such that for all X S S ha{ S S }. 4. here est ubers a b c[ such that a b c < ad for all S S a b c S S. { he { } ad S } has a uque coo fed ot X. Refereces: [] M. Ara ad Noshee: Soe Fed Pot heores of A-e Cotractos -Metrc Sace; Subtted. [] M. Ara A. A. Zafar ad A. A. Sddqu: A geeral class of cotractos: A-cotractos; Nov Sad J. Math. 88 5-. [] M. Ara A. A. Sddque ad A. A. Zafar: Coo fed ot theores for self as of a geeralzed etrc sace satsfg A-cotracto te codto It. Joural of Math. Aalss Vol. 5 757-76. [4] M. Ara ad A.A. Sddqu: A fed ot theore for A-cotracto o a class of geeralzed etrc saces; Korea J. Math. Sceces -5. [5] R. Bach: Su u roblea d S.Rech rguardabc la teora De ut fss; Boll. u. Math.Ital. 597-8. [6] B. C. Dhage: eeralzed etrc saces ad toologcal structure; I. Aalele sttfce ale uverstat AL. I. cuza d Las. Sere Noua. Mateatca 46-4. [7] A. K. Kalda: O a fed ot theore for Kaa te ags; Maths Jaoca. 5988 7-7. [8] R. Kaa: Soe results o fed ots; Bull Calcutta Math. soc. 6965 7-76. [9] M. S. Kha: O fed ot theore; Math. Jaoca. 978. [] Z. Lu: O fed ot theores for Kaa as; Puab U. J. Math. v 995-9. [] Z. Mustafa H. Obedat ad F. Awaedeh: Soe fed ot theore for ag o colete -etrc sace; Fed ot theor ad alcato artcle ID 8987 Vol.8 ages. [] Z. Mustafa ad B. Ss: Aroach to geeralzed etrc sace; Joural of olear ad cove aalss 76 89-97. [] Noshee: Stud of fed ot theores -etrc sace; M.Phl hess C Uverst. [4] F. U. Reha ad B. Ahad: Soe fed ot theores colete etrc sace; Math. Jaoca 6 99 9-4. [5] S. Rech: Kaa s fed ot theore; Boll. U. Math. Ital. 497 -. [6] B. E. Rhoades: Soe fed ot theores for ar of ags; Jaabha. 5985 99
5-56. [7] B. E. Rhoades: A coarso of varous deftos of cotractve ags; ras. Aer. Math. Soc 6977 57-9. [8] C. S. Wog: O Kaa ages; Proc. Aer. ath. Soc. 47975 5-.