Keywords- Fixed point, Complete metric space, semi-compatibility and weak compatibility mappings.

Transcription

1 [FRTSSDS- Jue 8] ISSN DOI:.58/eoo.989 Impact Factor- 5.7 GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES COMPATIBLE MAPPING AND COMMON FIXED POINT FOR FIVE MAPPINGS O. P. Gupta Shri Yogira Sagar Istitute of Techology & Sciece Ratlam ABSTRACT I this paper it is prove that the existece of uique commo fixe poit theorem ivolvig for five mappigs with semi-compatibility weak compatibility a commutativity o Metric space. This result improves a geeralies some kow result of Ima a Kha [7] by usig fuctioal expressios. Subject Classificatio. Primary 54H5 Secoary 47H Keywors- Fixe poit Complete metric space semi-compatibility a weak compatibility mappigs. I. INTRODUCTION The stuy of commo fixe poit of mappig satisfyig ifferet cotractio coitio has bee a very active fiel of research activity a may be extee to the abstract spaces. Fisher[45] geeralies affixe poit theorem of Jugck[6]. Hicks a Kubicek [] prove the Ma iteratio process i Hilbert space. Pahare a Waghmoe [9] prove a commo fixe poit theorem i Hilbert space. Sriivas.V [] prove a commo fixe poit theorem o compatible mappigs of type (p). Shrivastava [] a prove compatible mappig a commo fixe poit theorem. Gupta [] Commo fixe poit theorem for compatible mappigs of type (A-) i complete fuy metric space. Sessa [] itrouce the otio of weak commutativity which asserts that a pair of self mappig (AB) o a metric space (X ) is sai to be weakly commutig if ( BAx) (Bx Ax) for all x i X. Motivate by Sessa [] The otio of compatible mappig was itrouce by Jugck [7] which asserts that a pair self mappig (AB) of a metric space (X ) is sai to be compatible if lim ( BAx ) = whever lim Ax = lim Bx = t X. A weakly commutig pair is compatible but ot coversely as emostrate i Jugck [7]. Lohai a Bashah [8] prove some commo fixe poit theorem for four compatible mappigs o Metric space Ima [] prove a uique commo fixe poit theorem o five mappigs. Defiitio. Let S a T be mappigs from a metric space (X) ito itself. The mappigs S a T are sai to be compatible if lim STx TSx x is a sequece i X such that lim Sx limtx t for some t X. wheever Defiitio. Let S a T be mappigs from a metric space (X) ito itself. The mappigs S a T are sai to be weakly compatible if they commute at their coiciece poit that is STx=TSx wheever Sx=Tx xx. Defiitio. Let S a T be mappigs from a metric space (X) ito itself. The mappigs S a T are sai to be semi-compatible if lim STx Tx x is a sequece i X such that lim Sx limtx t t X. wheever for some Note that compatible mappigs are weakly compatible but weakly compatible mappigs are ot ecessarily compatible a clearly the pair (ST) is semi-compatible the they are weakly compatible. I this paper we prove a commo fixe poit theorem ivolvig five mappigs which geeralies earlier result ue to Ima a Kha [] by improvig cotractio coitio besies optimally chose suitable semi compatible weak compatible a commutig coitio o Complete Metric space by usig a ratioal iequality. 5 (C)Global Joural Of Egieerig Sciece A Researches

2 [FRTSSDS- Jue 8] ISSN DOI:.58/eoo.989 Impact Factor- 5.7 (C)Global Joural Of Egieerig Sciece A Researches 5 Theorem. Let A B S T a P be self mappigs of complete metric space (X) satisfyig the AB(X) P(X) ST(X) P(X) a X P X ST X AB a STy STy STy () for each x y X a + < either if (a){ab P} are semi-compatible P or AB is cotiuous a (STP) are weakly compatible or (b){st P} are semi-compatible P or ST is cotiuous a (AB P) are weakly compatible. The AB ST a P have a uique commo fixe poit. Furthermore if the pairs (AB)(AP)(BP)(ST)(SP)a (TP) are commutig mappig the ABST a P have a uique commo fixe poit. Proof. Let x be a arbitrary poit i X sice AB(X) P(X) we ca fi a poit x i X such that =. Also sice ST(X) P(X) we ca choose a poit x with STx = Ix usig this argumet repeately oe ca costruct a sequece { } such that = = + + =STx + = + for =. ( + + )=( + STx + ) STx STx where k. Thus for every we have k where k () which shows that { } is a Cauchy sequece i the Metric space (X) a so has a limit poit i X. Hece the sequece = + a STx + = + which are subsequeces also coverge to the poit. Let us ow assume that P is cotiuous so that the sequeces {P x } a {P }coverges to P a also i view of semi-compatibility of {ABP} {AB } coverges to P. Now put x = a y = x + i equatio () we have

3 [FRTSSDS- Jue 8] ISSN DOI:.58/eoo.989 Impact Factor- 5.7 AB STx AB P x STx P x AB STx P x P x 5 lettig we have P P P P P P P so that P Now put x= a y=x + i equatio () AB P STx AB STx P (C)Global Joural Of Egieerig Sciece A Researches P P lettig we have AB AB AB STx P P AB AB AB so that AB. Sice AB(X) P(X) there always exists a poit such that P = so that ST = ST(P ). Now put x = x a y = i equatio () ST' ST' P' P' P' ST' P' lettig we have ST' ST' ST' ST' so that ST'. Hece ST = = P which shows that is the coiciece poit of ST a P. Now usig the weak compatibility of (ST P) we have ST = ST (P ) = P(ST ) = P which shows that is also a coiciece poit of the pair (STP). Now put x = a y = i equatio () AB P ST P AB ST P P AB P ST P P P

4 [FRTSSDS- Jue 8] ISSN DOI:.58/eoo.989 Impact Factor- 5.7 ST ST ST ST so that ST. Hece = ST = P which shows that is commo fixe poit of AB ST a P. Now suppose that AB is cotiuous so that the sequece {AB x } a {AB } coverges AB.Sice (ABP) is semi-compatible it follows that {P } also coverges to AB. Thus put x = a y = x + i equatio () we have AB x P STx AB x STx P AB x STx P P lettig we have AB AB AB AB AB so that AB. AB AB AB Let there exist i X such that AB = = P. The put x = a y = i equatio () AB x P AB x ST' P lettig we have AB ST' ST' P' P' AB x P' ST' P P P' AB AB ST' AB - ST' so that AB ST' AB AB ST '. This gives ST = = P Thus is a coiciece poit of (STP) sice the pair (STP) is weakly compatible oe has ST =ST (P ) = P which show that ST = P. Put x = x a y = i equatio () we have ST P ST P P ST P 5 (C)Global Joural Of Egieerig Sciece A Researches

5 [FRTSSDS- Jue 8] ISSN DOI:.58/eoo.989 Impact Factor- 5.7 lettig we have ST ST ST ST ST which implies so that ST = =P. The poit therefore is i rage of ST a sice ST(X) P(X) there exists a poit i X such that P =. Thus put x = a y = i equatio () AB' ' P' ' ST P AB' ' ST P' ' P AB' ' P ST P' ' P" P AB' ' AB' ' AB' ' AB" which implies AB" Also sice (ABP) are semi-compatible are hece weakly commutig we obtai AB = P = Thus we have prove that is a commo fixe poit of AB ST a P. If mappigs ST or P is cotiuous istea of AB or P the the proof that is a commo fixe poit of ABST a P is similar. Let v be aother fixe poit of P AB a ST the v =Pv =ABv =STv AB P STv Pv AB STv AB Pv P Pv v v v v v v v v v 54 STv P P Pv which implies = v. Fially we ow show that is also a commo fixe poit of the family F={ABSTP}. Whe the pairs (AB)(AP)(BP)(ST)(SP)a (TP) are commutig pairs. For this evet we write A=A(AB) = A(BA) = AB(A) A = A(P) =AP()=PA() =P(A) B = B(AB) = BA (B) = AB (B) B = B(P)= BP()= PB()= P (B) S=S(ST) = S(TS) = ST(S) S = S(P) =SP()=PS() =P(S) T = T(ST) = TS (T) = ST (T) T = T(P)= TP()= PT()= P (T) (C)Global Joural Of Egieerig Sciece A Researches

6 [FRTSSDS- Jue 8] ISSN DOI:.58/eoo.989 Impact Factor- 5.7 which shows that A a B are commo fixe poit of (ABP) yielig thereby A =B =P = AB. where as S a T are commo fixe poit of (STP) it also shows that S = = T = P =ST. Now we ee to show that A = S (B = T) also remais a commo fixe poit of both the pairs (ABP) a (STP). For this (A S) = (A(BA)S (TS) ) = (AB(A) ST (S)) ABA PA ST S P S PA PS ABA PS STS PA PA PS Implies that (- )(AS) so that A = S. Similarly it ca be show that B=T Thus is the uique commo fixe poit of AB S T a P. Example. Let A B ST a P be self mappig of Hilbert space H. Let X= [] be a close subset of H. We efie mappig 4 Ax x Bx x Sx x Tx x a x. 4 9 Clearly AB X PX a ST X P X a 5 AB X STX PX 5 so that AB X STX PX. 5 Also the pair (AB P) (ST P) (AB) (ST) (AP) (BP) (SP) a (TP) are commutig a semi-compatible or weak compatible. For all xy i X (x>y) with 9 a we have x x y 5 x y 5 x y Usig y y we get 5 x y x y 5 5 which verifies the cotractio coitio (). y x y y x x 5 Clearly is uique commo fixe poit of A B S T a P. y. 55 (C)Global Joural Of Egieerig Sciece A Researches

7 [FRTSSDS- Jue 8] ISSN DOI:.58/eoo.989 Impact Factor- 5.7 REFERENCES [] Hicks T.L. a Kybicek J.P. O the Ma iteratio process i Hilbert spaces. J. Math. Aal. Appl. 59 (977) [] ImaM. Five mappigs with a commo fixe poit. Bull.Cal. Math.Soc.96(999) [] ImaM. a KhaQ.H. A commo fixe poit theorem for six mappig satisfyig a ratioal iequality. 44() () [4] Fisher B. Mappigs with a commo fixe poit.math.sem. Notes. 7(979) 8. [5] Fisher B. A aeum to Mappigs with a commo fixe poit. Math.Sem. Notes. 8(98) 5. [6] Jugck G. Commutig mappig a fixe poit.amer. Math. Mothly. 8 (976) 6. [7] Jugck G. Compatible mappig o commo fixe poit Iterat J. Math. a Math. Sci. 9 (4) (986) [8] Lohai P.C. a Bashah V.H. Compatible mappigs a commo fixe poit for four mappigs Bull. Cal. Math. Soc. 9 (998) -8. [9] Pahare D.M. a Waghmae B.B. A commo fixe poit theorem i Hilbert space. Acta. Ciecia. Iica. vol. XXIIM. (997) 7-. [] Sessa S. O a weak commutativity coitio of mappigs i fixe poit cosieratios. Publ. Irt. Math. (Beogra). (98)49-5.S [] Sriivas.V a Naga Raju.V. Commo Fixe Poit Theorem o Compatible Mappigs of Type (P) Ge. Math. Notes Vol. No. April 4 pp [] Shrivastava. R Jai. N a Qureshi.K. Compatible Mappig a Commo Fixe Poit Theorem. IOSR Joural of Mathematics e-issn: p-ISSN: 9-765X Volume 7 Issue (May. - Ju. ) PP [] Gupta.V Bashah.V.H a Malviya. P. Library Commo fixe poit theorem for compatible mappigs of type (A-) i complete fuy metric space. Pelagia Research Library Avaces i Applie Sciece Research 6 ISSN: CODEN (USA): 7():-7 56 (C)Global Joural Of Egieerig Sciece A Researches