A Review article on some generalizations of Banach s contraction principle

Transcription

1 A Revew artcle on some generalzatons of Banach s contracton prncple Sujata Goyal Assstant professor, Dept. Of Mathematcs, A.S. College, Khanna, Punjab, Inda Abstract: Let (X,d) be a metrc space. The well known Banach s Contracton Prncple states that f T: X X s a contracton on X(.e.d (Tx,Ty) c d (x,y) for some 0 c< and for all x,y n X ) and (X,d) s complete then T has a fxed pont n X (.e. Tx = x for some x n X) In ths paper, a number of extensons of Banach contracton prncple have been dscussed. Keywords: Contracton, Multvalued contractons, quas contracton, - generalzed contracton, f-contracton and generalzed f- contracton, -weak contracton, - dstance, weakly unform strct contracton Introducton: S.Nadler has extended the result to multvalued contractons: Let (X,d) be a complete metrc space (a)cb(x)={c : C s closed and bounded subset of X} (b)n(ɛ,c)={xx:d (x,c)< ɛ for some cc} (c)h(a,b)=nf{ ɛ: A N(ɛ,B) or B N(ɛ,A)} where A, BCB(X) The functon H s a metrc on CB(X) and s called Hausdorff metrc. Let (X,d ) and (Y,d ) be two metrc spaces. Defnton : A functon T: X CB(Y) s called multvalued contracton mappng of X nto Y ff H(Tx,Ty) cd (x,y) for some 0<c< and for all x,y n X A pont x s sad to be fxed pont of T f x Tx Therom : Let (X,d) be a complete metrc space. If T : X CB(X) s a multvalued contracton mappng then T has a fxed pont (.e. there exsts xx such that x Tx). Above theorem generalzes the Banach s Contracton Prncple.( the map J: X CB(X) defned by J(x)={x} s an sometry. T: X X be a contracton then J T: X CB(X) s a multvalued contracton,thus there exsts xx such that xj T(x).e. x J(T(x)) whch mples x{tx}.thus x=tx 06, IRJET Impact Factor value:. ISO 900:008 Certfed Journal Page 6

2 Lj.B.Crc and S.B. Presc extended the result as follows: Theorem : Let (X,d) be a complete metrc space, k a postve nteger and T: X K X a map satsfyng the condton: d(t(x,x,... x k ), T(x,x 3,... x k )) c max{d(x,x ) : k} where c[0,) s a constant and x,x,.. x k n X are arbtrary then there exsts a pont x n X such that T(x, x,... x) = x If n addton, on dagonal X K, d(t(u, u,...u),t(v,v,...v)) < d(u,v) holds for all u,v n X wth u v then x s the unque pont n X wth T(x,x...x) = x For k= the above theorem gves Banach s contracton prncple. James Merryfeld and James D. Sten gave the proof of generalzed banach contracton conjecture : Theorem : Let (X,d) be a complete metrc space.t : X X be a map and let 0 c<.let J be a postve nteger.assume for each par x, y n X Mn{d(T K x,t K y): k J } c d(x,y) then T has a unque fxed pont. For J =, the above theorem gves Banach s contracton prncple. Lj.B.Crc has extended the result to quas contractons: Defnton : A mappng T : X X s sad to be quas-contracton f there exsts a number 0 c< such that d(tx,ty) c max{d(x.y), d(x,ty), d(y,tx), d(x,tx), d(y,ty)} for all x, y n X For xx, Let O(x, )={x, Tx, T x,....} Defnton : A space X sad to be T-orbtally complete f every Cauchy sequence n O(x, ) for some xx s convergent n X. Theorem : Let T : X X be a quas - contracton and X s T orbtally complete.then T has a unque fxed pont n X. Above theorem s clearly generalzaton of Banach s Contracton Prncple as every contracton s quas contracton. M.S. Khan ntroduced alterng dstance functon to generalze the result: Defnton : A functon :[0, ) [0, ) s called an alterng dstance functon f the followng condtons are satsfed: (a) (0)=0 (b) s contnuous and monotoncally non-decreasng. Theorem : Let (X,d) be a complete metrc space.let be an alterng dstance functon and lett: X X be a map whch satsfes the followng nequalty: (d(tx,ty)) c d(x,y) for all x,y n X and for some 0 c< then T has a unque fxed pont. Clearly for (t)=t, above theorem gves Banach s Contracton Prncple. Lj.B. Crc ntroduced generalzed contracton to extend the result: 06, IRJET Impact Factor value:. ISO 900:008 Certfed Journal Page 6

3 Defnton : A mappng T: X X s sad to be a generalzed contracton f for every x,y n X, there exst non negatve numbers q(x,y), r(x,y), s(x,y), t(x,y) such that sup {q(x,y)+r(x,y)+s(x,y)+t(x,y)}= < and d(tx,ty) q(x,y) x, yx d(x,y)+r(x,y) d(x,tx)+s(x,y) d(y,ty)+t(x,y) (d(x.ty)+d(y,tx)) holds for all x,y n X Theorem : Let T be - generalzed contracton of T orbtally complete metrc space X nto tself then T has a unque fxed pont n X. If we take r(x,y)=s(x,y)=t(x,y)=0 and q(x,y)=c for all x,y where 0<c< then above theorem gves Banach contracton prncple. Mlan R.Tascovc Theorem : Let T : X X be a map and X be T orbtally complete.if T satsfes the followng condton: There exst real numbers, for every x,y n X such that : > and and d(tx,ty)+ d(x,tx)+ 3 d(y,ty)+ mn{d(x,ty),d(y,tx)} d(x,y) then T has a fxed pont. Above theorem generalzes banach s contracton prncple as every contracton satsfes the above condton. Mlan R.Tascovc also defned f-contracton and generalzed f-contracton : Defnton : A mappng f: R k R s semhomogenous f f( x, x,... x k ) f(x,x,... x k ): >0 A mappng T : X X s sad to be f- contracton f for every x,y n X there exst non negatve real numbers such that d(tx,ty) f( (x,y)d(x,y), (x,y)d(x,tx), 3 (x,y)d(y,ty), (x,y)d(x.ty), (x,y)d(y,tx)) (x,y) =,,... where sup{f(,, 3,, ): x, yx}= <and the exstng mappng f:(r 0 ) R 0 s ncreasng and semhomogenous. Theorem : Let T be a f-contracton on a metrc space X and let X be T orbtally complete then T has a unque fxed pont. Every contracton mappng satsfes the above condton for f(s,t,u,v,w)= s and = c, = 3 = = =0 where 0<c<,thus above theorem generalzes Banach s contracton prncple. Defnton : A mappng f: R K R K s semhomogenous of order ff f( x, x,... x k ) f(x,x,... x k ) Where belongs to [, ) Defnton : A mappng T : X X s sad to be generalzed f- contracton f for every x,y n X d(tx,ty) f(d(x,y), d(x,tx), d(y,ty), d(x.ty), d(y,tx)) where f : (R 0 ) R 0 s ncreasng,semhomogenous of order and wth the propertes f(t,t,...t) < t 06, IRJET Impact Factor value:. ISO 900:008 Certfed Journal Page 63

4 lm sup f(y,y,... y)<t for all t(0, ] yt0 Theorem : Let T be a genertalzed f- contracton on a metrc space X and let X be T orbtally complete then T has a unque fxed pont. Every contracton mappng satsfes the above condton for f(s,t,u,v,w)= c s, where 0<c<,thus above theorem generalzes Banach s contracton prncple. A.Mer and Emmett Keeler gves an - condton to generalze the result: Theorem : Let (X,d) be a complete metrc space. T : X X be a map satsfyng the followng weakly unform strct contracton : Gven >0 there exsts >0 such that d(x,y) <+ d(tx, Ty) < Above theorem generalzes the Banach s contracton prncple snce every contractve mappng s unformly contnuous and every unformly contnuous map satsfes the above condton. Ch Song Wong : Theorem : Let S and T be self mappngs of a complete metrc space X. Suppose there exst functons XX nto [0, ) such that,=,,... from (a) r = sup { ( x, y) : x,y X}< (b) = 3, = (c) For any dstnct pont x, y n X d(sx,ty) (x,y) d(x,y)+ (x,y) d(x,ty)+ 3 (x,y) d(y,sx)+ (x,y) d(x,sx)+ (x,y) d(y,ty ) then S or T has a fxed pont.if both S and T has fxed ponts then each of S and T has a unque fxed pont and these two fxed ponts concde every contracton mappng T satsfes the above condton wth S=T, (x,y)=c ; 0<c<, (x,y) = 3 (x,y) = (x,y) = (x,y) =0, thus above theorem generalzes the Banach s contracton prncple. Theorem : T be a self mappng of a complete metrc space (X,d).Suppose that there exst functons, =,,3,, of (0, ) nto [0, ) such that Each s upper semcontnuous from the rght such that (t)+ (t)+ 3 (t)+ (t)+ (t) < t, t > 0 and for any dstnt ponts x,y n X, 06, IRJET Impact Factor value:. ISO 900:008 Certfed Journal Page 6

5 d(tx,ty) a d(x,tx)+a d(y,ty)+a 3 d(x,ty)+a d(y,tx)+a d(x,y) where a = (d(x,y))/d(x,y) Then T has a unque fxed pont. Every contracton mappng satsfes the above condton for (t) = c t where 0<c< and (t)= (t)= 3 (t)= (t) =0 for all t. Thus above theorem generalzes Banach s contracton prncple. B.E. Rhoades made use of weak contracton to generalze the result: Defnton : let (X,d) be a metrc space.t: X X be a map.t s sad to be weak contracton f d(tx,ty) d(x,y)- (d(x,y)) where :[0, ) [0, ) s contnuous and non decreasng functon wth (t)=0 ff t=0 Theorem : If (X,d) s complete metrc space and T s a -weak contracton on X then T has a unque fxed pont. Takng (t)=(-c)t,where 0<c<,the above theorem gves Banach s contracton prncple. P.N. Dutta and B.S. Chaudhury: Theorem : Let (X,d) be a complete metrc space and let T: X X be a map satsfyng (d(tx,ty)) (d(x,y))- (d(x,y)) for all x,y n X where, : [0, ) [0, ) are both contnuous and monotonc decreasng wth (t)= (t)=0 ff t=0.then T has a unque fxed pont. Takng (t)=t and (t)=(-c)t where 0<c<,above theorem gves Banach s contracton prncple. Mark Voorneveld : Defnton : Let X be a metrc space wth metrc d. We defne an - dstance on X to be a functon : XX [0, ) such that (a) satsfes the trangle nequalty (b) (x.) : X [0, ) s lower sem contnuous for every x X.e. f y m y then (x,y) lm nf m (x,y m ) (c) for every > 0, there exsts a > 0 such that for every x,y,z n X f (z,x) and (z,y) then (x,y) Defnton : Let (X,d) be a metrc space and an - dstance on X. Let F( ) denote the famly of functons on X X satsfyng the followng condtons: (a) For each (x,y) n XX, (x,y) depends only on the - dstance (x,y), ths allows us to wrte ( (x,y)) nstead of (x,y) (b) 0 (d) < for every d > 0 06, IRJET Impact Factor value:. ISO 900:008 Certfed Journal Page 6

6 (c) (d) s ncreasng functon of d. Theorem : Let (X,d) be a complete metrc space, an - dstance on X and T :X X a map. If there exsts F( ) such that (Tx,Ty) (x,y) (x,y) then T has a unque fxed pont x n X. Above theorem gves banach s contracton prncple for = d and a constant n [0,). Maher Berzg : Defnton : let, : [0, ) R be two functons. The par of functons (, ) s sad to be a par of shftng dstance functon f the followng condtons hold: a) For u,v [0, ) f (u) (v) then u v b) For {u n },{v n } [0, ) wth lm u n = lm v n = w f (u n ) (v n ) for all n 0 then w = 0 n n Theorem : Let (X,d) be a complete metrc space. T: X X be a mappng.suppose there exsts a par of shftng dstance functons (, ) such that (d(tx,ty)) (d(x,y)) for all x, y n X then T has a unque fxed pont n X. Above theorem generalzes banach s contracton prncple snce for a contracton mappng, above condtons are satsfed wth (x) = x and (x) = c x where 0 < c < References : Nadler Jr, Sam B. "Mult-valued contracton mappngs." Pacfc J. Math 30. (969): Ćrć LB, Prešć SB. On Prešć type generalzaton of the Banach contracton mappng prncple. Acta Mathematca Unverstats Comenanae. New Seres. 007;76(): Merryfeld J, Sten JD. A generalzaton of the Banach contracton prncple. Journal of Mathematcal Analyss and Applcatons. 00 ;73():-0.. Ćrć LB. A generalzaton of Banach s contracton prncple. Proceedngs of the Amercan Mathematcal Socety. 97;(): Khan MS, Swaleh M, Sessa S. Fxed pont theorems by alterng dstances between the ponts. Bulletn of the Australan Mathematcal Socety. 98 ;30(0): Crc LB. Generalzed contractons and fxed-pont theorems. Publ. Inst. Math.(Beograd)(NS). 97;(6): Taskovc MR. Some results n the fxed pont theory. Publ. Inst. Math. 976;0(3): Taskovc M. A generalzaton of Banach s contracton prncple. Publ. Inst. Math.(Beograd). 978;3(37): Mer A, Keeler E. A theorem on contracton mappngs. Journal of Mathematcal Analyss and Applcatons. 969;8(): Wong CS. Fxed pont theorems for generalzed nonexpansve mappngs. Journal of the Australan Mathematcal Socety. 97 ;8(03): , IRJET Impact Factor value:. ISO 900:008 Certfed Journal Page 66

7 . Rhoades BE. Some theorems on weakly contractve maps. Nonlnear Analyss: Theory, Methods & Applcatons. 00 ;7(): Dutta PN, Choudhury BS. A generalsaton of contracton prncple n metrc spaces. Fxed Pont Theory and Applcatons. 008 ;008(). 3. Voorneveld M. A Generalzaton of the Banach Contracton Prncple. Faculty of Economcs and Busness Admnstraton, Tlburg Unversty; Berzg M. Generalzaton of the Banach contracton prncple. arxv preprnt arxv: , IRJET Impact Factor value:. ISO 900:008 Certfed Journal Page 67