Some Fixed Point Theorems in Generating Polish Space of Quasi Metric Family

Transcription

1 Global ad Stochastic Aalysis Special Issue: 25th Iteratioal Coferece of Forum for Iterdiscipliary Mathematics Some Fied Poit Theorems i Geeratig Polish Space of Quasi Metric Family Arju Kumar Mehra ad Maoj Kumar Shukla Ab s t r a c t: The geeratig space of quasi metric family is importat because of its ivolvemet with fuzzy ad probabilistic metric space. Mathematical Classificatio (2000) 49H10, 54H Itroductio ad Prelimiary The geeratig space of quasi 2-metric family is importat because of its ivolvemet with fuzzy ad probabilistic 2-metric space ad a miimizatio theorem [1], [3] is to obtai fied poit theorem. I 2008 V.B. Dhagat ad V.S. Thakur proved o cove miimizatio theorem for geeratig space of quasi 2-metric family. I this paper, we prove a miimizatio theorem for sequece of mappigs Ta for a Œ N ad further we prove fied poit theorem as a applicatio of miimizatio theorem with o commutig coditio Kow as weak compatible Defiitios Geeratig Polish Space of Quasi Metric Family. Let X be o empty set ad {d a : a Œ (0,1]} be a family of mappigs d a of (W X) (W X) ito R +, w Œ W be a selector. (X, d a : a Œ (0, 1]) is called geeratig Polish space of quasi metric family if it satisfies the followig coditios: 1. d a ((w, ), (w, y)) = 0 " a Œ (0, 1] = y 2. d a ((w, ), (w, y)) = d a ((w, y), (w, )) ", y Œ X, w Œ W ad a Œ (0, 1] 3. For ay a Œ (0, 1], there eists a umber m Œ (0, a] such that: d a ((w, ), (w, y)) = d m ((w, ), (w, z)) + d m ((w, z), (w, y)) ", y Œ X, w Œ W be a selector. 4. For ay, y Œ X, w Œ W, d a ((w, ), (w, y)) is o-icreasig ad left cotiuous i a Quasi Compatible. Let (X, d a : a Œ (0, 1]) be a geeratig Polish space of quasi metric family ad S ad T be mappigs from W X ito X. The mappig S ad T are said to be quasi compatible if d a (ST(w, ), TS(w, )) Æ 0 as Æ, a Œ (0, 1], w Œ W wheever {w, } be a sequece i W X such that lim S( w, ) = lim T( w, ) = p for some p Œ X. Keyword. Fied poit, Quasi 2-metric family, Geeratig polish Space.

2 110 Arju Kumar Mehra ad Maoj Kumar Shukla Compatible of Type (A). Let (X, d a : a Œ (0, 1]) be a geeratig Polish space of quasi metric family ad S ad T be mappigs from W X ito X. The mappig S ad T are said to be compatible of type (A) if: d a (TS(w, ), SS(w, )) = 0 ad d a (ST(w, ), TT(w, ))=0 wheever {w, }be a sequece i W X such that lim S( w, ) = lim T( w, ) = p for some p Œ X Implicit Relatio. Let f be the set of all real fuctios f: R + 4 Æ R such that: (F 1 ): F is cotiuous i each coordiate variable, (F 2 ): If either f(u, 0, u, v) 0 or f(u, 0, u + v, v) 0 for all u, v 0, the there eists a real costat 0 h 1 such that u hv. 2. Some Cocerig Results Lemma 2.1: Let (X, d a : a Œ (0, 1]) be a geeratig Polish space of quasi metric family ad S ad T be mappigs from W X ito X. Suppose that lim S( w, ) = lim T( w, ) = p for some p Œ X. The we have the followig 1. lim ST( w, ) = Tp if T is cotiuous ad 2. STp = TSp ad Sp = Tp if T is cotiuous Proof 2.1(1): Suppose that lim S( w, ) = lim T( w, ) = p for some p Œ X. Now, sice T is cotiuous, we have lim TS( w, ) = Tp By 1.1.1(3), we have d a (ST(w, ), Tp) = d m (ST(w, ), TS(w, )) + d m (TS(w, ), Tp); m Œ (0, a] Sice S ad T are quasi compatible, we have lim ST( w, ) = Tp Proof 2.1(2): Sice T is cotiuous, lim ST( w, ) = Tp Hece by uiqueess of limit, we have Sp = Tp Now agai d a (STp, TSp) = lim d ( (, ), (, ) ) a ST w TS w = 0 i.e. STp = TSp This completes the proof. Lemma 2.2: Let (X, d a : a Œ (0, 1]) be a geeratig Polish space of quasi metric family ad S ad T be mappigs from W X ito X. If S i ad T j are weakly compatible for ay a Œ (0, 1] ad for m Œ (0, a]. The, S i T j p = T j T j p = T j S i p = S i S i p

3 Some Fied Poit Theorems i Geeratig Polish Space of Proof: Suppose {w, } be a sequece i W X defied by = p as Æ ad Sp = Tp. The we have lim S( w, ) = lim T( w, ) = p Sice S ad T have weakly compatible), we have d a (STp, TTp) = lim ( ST( w, ), TT( w, )) = 0; a Œ( 0, 1] Hece, we have, STp = TTp Similarly, we have, TSp = SSp But, Tp = Sp It implies, TTp = TSp Therefore, STp = TTp = TSp = SSp 3. Mai Results Theorem 3.1: Let, (X, d a : a Œ (0, 1]) be a Geeratig Polish space of quasi metric family ad S, T & G are mappig from W X Æ X are cotiuous radom operator w.r.t. d. Suppose there are some a Œ (0, 1] such that for, y Œ X ad Œ W, we have the followig coditios 1. S(X) Õ G(X) ad T(X) Õ G(X) ( ) Ï S w,, T w, y, S w,, G w, y, 2. f Ì 0 ( G( w, ), T( w, y) ), ( G ( w, ), G ( w, y Ó )) ", y Œ X ad a Œ (0, 1], (3) G is cotiuous (4) The pairs {S, G} ad {T, G} are weakly compatible o X. The S, T ad G have commo fied poit. Proof: Let, 0 be ay arbitrary poit of X. Sice, S(X) Õ G(X) ad T(X) Õ G(X) ad SG(X) Õ GG(X) ad TG(X) Õ GG(X) So there eists 1 ad 2 i X such that GG(w, 1 ) = SG(w, 0 ) ad GG(w, 2 ) = TG(w, 1 ) I geeral GG(w, ) = SG(w, 2 ) ad GG(w, ) = TG(w, ) for, = 0, 1, 2, 3,... Let, d = d a (GG(w, ), GG(w, + 1 )) Also we kow d a (GG(w, 2 ), GG(w, )) Ï dm( GG( w, 2 ), GG( w, 2 + 1) ) Ì Ó + dm( GG( w, 2 + 1), GG( w, 2 + 2) ) ", y Œ X ad m Œ (0, a],

4 112 Arju Kumar Mehra ad Maoj Kumar Shukla Suppose 2, satisfy 3.1(2) the "a Œ (0, 1] ( ) Ï SG w,, TG w,, SG w,, GG w,, f Ì ( GG ( w, ) 2, TG ( w, 2 + )) 1, ( GG ( w, 2 ), GG ( w, 2 + ) Ó ) 1 ( ) Ï GG w,, GG w,, GG w,, GG w,, f Ì ( GG ( w, ), GG ( 2 w, 2 + )) 2, ( GG ( w, 2 ), GG ( w, 2 + ) Ó ) 1 ( ( 2 1) ( 2 2) ) ( ( + + 2) ( 2 + 1) ) ) Ï GG w,, GG w,, 0, È dm GG w,, GG w, ÎÍ f Ì + dm ( GG ( w, ), GG ( w, )), ( GG w,, GG w, Ó Thus by defiitio of implicit relatio we have d a (GG(w, ), GG(w, )) h{d a (GG(w, 2 ), GG(w, ))} d hd 2 d d 2 Similarly, d 2 hd Thus, {d 2 } be mootoe decreasig ad hece coverge to zero. Therefore, {GG(w, 2 )} is a Cauchy sequece ad coverge to Gp ad hece to poit X. Sice {SG(w, 2 )} ad {TG(w, 2 )} are subsequece of {GG(w, 2 )} ad so coverge to same poit p. Now by lemma 2.1 we obtai Similarly, Hece, Also, SGp = GSp ad Sp = Gp TGp = GTp ad Tp = Gp Sp = Tp = Gp Sp = p = Gp = Tp as Gp = p Hece, p is commo fied poit of S, T ad G. This completes the proof. Corollary 3.2: Let, (X, d a : a Œ (0, 1]) be a geeratig Polish space of quasi metric family ad S, T ad G be mappigs from W X ito X satisfyig 1. S(X) Õ G(X) ad T(X) Õ G(X) ( ) Ï S w,, T w, y, S w,, G w, y, 2. f Ì 0 ( G( w, ), T( w, y) ), ( G ( w, ), G ( w, y Ó )) ", y Œ X ad a Œ (0, 1], 3. G is cotiuous 4. The pairs {S, G} ad {T, G} are weakly compatible The S, T ad G have commo fied poit.

5 Some Fied Poit Theorems i Geeratig Polish Space of Proof: Similar to the proof of the theorem 3.1 usig the fact that Quasi compatible pair of maps is weakly compatible but coverse is ot always true. Refereces 1. D. Dowig ad W.A. Kirk, A geeralizatio of Caristi s theorem with applicatios to oliear mappig theory, Pacific j. Math. 69(1977), No. 2, V.B. Dhagat ad V.S. Thakur, No cove miimizatio i geeratig space, Tamkag Joural of Math, Taiwa Vol. 39, No. 3, , Autum J.S. Jug, Y.J. Cho ad J.K. Kim, Miimizatio theorems for fied poit theorems i fuzzy metric space ad applicatio, Fuzzy sets ad system 61(2), (1994), Ar j u Ku m a r Me h r a: Re s e a r c h Sc h o l o r, Ra i Du r g a v a t i Vi s h w a v i d y a l a y a, Ja b a l p u r, (MP), I d i a address: Marjumehra1111@gmail.com Ma o j Ku m a r Sh u k l a: De p a r t m e t o f Ma t h e m a t i c s, I s t i t u t e f o r E c e l l e c e i Hi g h e r Ed u c a t i o, Bhopal (MP) address: maojshukla2012@yahoo.com

6