On the Variations of Some Well Known Fixed Point Theorem in Metric Spaces

Transcription

1 Turkish Joural of Aalysis ad Number Theory, 205, Vol 3, No 2, Available olie at Sciece ad Educatio Publishig DOI:0269/tjat O the Variatios of Some Well Kow Fixed Poit Theorem i Metric Spaces Krisha Patel *, G M Deheri Departmet of Mathematics, Sardar Patel Uiversity, Aad, Gujarat, Idia *Correspodig author: krishappatel0@gmailcom Received April 20, 205; Revised May 25, 205; Accepted Jue 28, 205 Abstract Results dealig with a Fixed Poit for a map eed ot be cotiuous o a metric space, which improves a famous classical result, has bee preseted here, wherei the covergece aspect is duly addressed This paper aims to preset some fixed poit theorems o metric spaces It ca be easily observed that these are sigificat improvemet of some of the well-kow classical result dealig with the geeralizatio of Baach fixed poit theorem Further, here the rate of covergece aspects is duly take care of Keywords: fixed poit, metric space, sequetial coverget, subsequetial coverget, uiqueess Cite This Article: Krisha Patel, ad G M Deheri, O the Variatios of Some Well Kow Fixed Poit Theorem i Metric Spaces Turkish Joural of Aalysis ad Number Theory, vol 3, o 2 (205): doi: 0269/tjat Itroductio The fixed poit theorem most frequetly cited i literature is Baach cotractio priciple, which asserts that if ( X, d) is a complete metric space ad T : X X is a cotractive mappig ( T is cotractive if there exist λ [ 0,) such that for all xy, X, d ( Tx, Ty) λd ( x, y) ) the T uique fixed poit I 968, Kaa [3] established a fixed poit T for mappig satisfyig: d Tx Ty d x Tx d y Ty for λ [ 0, 2) ad for all, xy X Kaa s paper [3] dealig with the geeralizatio of Baach fixed poit theorem was followed by a spate of papers cotaiig a variety of cotractive defiitios i metric spaces Rhodes [6] cosidered 250 types of cotractive defiitios ad aalyzed the iterrelatio amog them Jugck ad Rhoades [8] itroduced the cocept of weakly compatible maps for extedig some well kow fixed poit theorems to the settig of set valued o cotiuous fuctios Here the fixed poit theorems were proved for set valued fuctios without appealig to the cotiuity I 2008 Azam ad Arshad [] exteded Kaa s theorem for the geeralized metric space itroduced by Braciari [2] by replacig triagular iequality by rectagular oe i the cotext of fixed poit theorem I 200 Moradi ad Beiravad [7] ad Moardi ad Omid [5] itroduced ew classes of cotractive fuctios as followig ad established the Baach cotractio priciple Defiitio d ) be a metric space A mappig T : X X is said to be sequetially coverget if we have, for every sequece{ y }, if { Ty } is coverget the { y } is also coverget T is said to be subsequetially coverget if Ty is we have, for every sequece { } y, if { } coverget the { y } coverget subsequece I 20, Moradi ad Alimohammadi [4] exteded the Kaa s theorem ad the theorem due to Azam ad Arshad [] as followig: 2 Theorem d ) be a complete metric space ad T, S : X X be mappigs such that T is cotiuous, oe- to- oe ad subsequetially coverget If λ 0, 2 ad for all xy, X d TSx TSy d Tx TSx d Ty TSy the S uique fixed poit Also if T is sequetially coverget the for every x0 X the sequece of iterates S x coverges to this fixed poit I the preset paper, sufficiet coditios were obtaied for the existece of the uique fixed poit of Kaa s type mappig o complete metric spaces depedig o aother fuctio Of course, a variatio of this aspect has bee discussed by Patel ad Deheri [9] i 203 where

2 Turkish Joural of Aalysis ad Number Theory 7 commo fixed poit theorems were proved i the light of aother fuctio Of course, the uderlyig spaces were Baach spaces Ideed, It has bee deemed proper to provide some geeralizatios ad variatios of the mai results preseted i Moradi ad Alimohammadi [5] 2 Mai Results The mai result of the paper is cotaied i 2 Theorem d ) be a complete metric space ad T, S : X X be mappigs such that T is cotiuous, oe- to- oe ad subsequetially coverget If λ 0, 2 ad for all xy, X d TSx TSy d Tx TSy d Ty TSx () the S uique fixed poit Also if T is sequetially coverget the for every x0 X the sequece of iterates { S x 0} coverges to this fixed poit Proof Let x0 X be a arbitrary poit i X We defie the iterative sequece { x } by x+ = Sx (equivaletly, x = S x0 ), =,2,3 Usig equatio () oe gets d Tx, Tx+ = d( TSx, TSx) λ[ d( Tx, TSx) + d( Tx, TSx )] leadig to d ( Tx, Tx+ ) d( Tx, Tx) Usig iductio ad equatio (3), oe fids that λ d( Tx, Tx+ ) d Tx0, Tx λ (2) (3) (4) By (4), for every m, such that m> oe obtais d( Txm, Tx) d( Txm, Txm ) + d( Txm, Txm 2) d( Tx+, Tx) m m 2 λ λ λ λ λ + d Tx0 + λ λ λ λ λ d ( Tx0, Tx) λ (5) Lettig m, i equatio (5) oe cocludes that { Tx } is a Cauchy sequece, ad sice X is a complete lim Tx = v (6) Sice T is a subsequetially coverget, { x } coverget subsequece So there exists u X ad { xk } such that lim x k = u Sice T is cotiuous ad lim x k = u, lim Tx k = Tu v By equatio (6) oe gets that Tu =, which results i k k k + k + + d( TS x 0, Tu) λ[ d( Tu, TSx ) + d( Tx, TSu)] k k k λ + d( Txk +, Tu) + d( Tx0, Tx) = λd( Tu, TSxk ) + λd( Txk, TSu) ( k) λ + d( Txk +, Tu) + d( Tx0, Tx) Lettig k i equatio (7) oe gets λ (7) This implies d( TSu, Tu ) = 0 Sice T is oe-to-oe Su = u ad S fixed poit Uiqueess For the uiqueess of the fixed poit let us assume that there are u ad u 2 i X, ( u u2) such that Su = u ad Su2 = u2 The by equatio () oe get d TSu TSu d Tu TSu d Tu TSu implies (, ) 2 λd ( Tu, Tu ) d Tu Tu2 2 Sice T is oe-to-oe ad λ [ 0, 2) oe fids u = u 2 Also, if T is sequetially coverget, by replacig { } with { k } we coclude that lim x = u ad this shows that { x } coverges to the fixed poit of S To uderstad the importace of this result oe ca cast a glace at the followig: 22 Example Let X = {0},,, edowed with the Euclidea metric Defie S : X X by S (0) = 0 ad

3 72 Turkish Joural of Aalysis ad Number Theory S = for all 4 Obviously the coditio i + Kaa s theorem is ot true for every λ > 0 So we caot use Kaa s theorem By defiig T : X X by T (0) = 0 ad T = for all 4 oe have, for m, ( m > ), TS TS = m + m+ m < ( + ) ( + ) ( + ) ( + ) m m+ ( + ) m ( m+ ) = T TS + T TS 3 m m Therefore, by theorem 2 S uique fixed poit u = 0 Now, a little variatio i the iequality of above result leads to: 23 Theorem d ) be a complete metric space ad T, S : X X be mappigs such that T is cotiuous, oe- to- oe ad subsequetially coverget If λ 0, 3 ad for all xy, X + (, ) d TSx TSy d Tx TSx d Ty TSy d Tx Ty (8) the S uique fixed poit Also if T is sequetially coverget the for every x0 X the sequece of iterates S x coverges to this fixed poit Proof Let x0 X be a arbitrary poit i X We defie the iterative sequece { } x by x + = Sx (equivaletly, x = S x0 ), =,2,3 Usig equatio (8) oe gets d Tx, Tx+ = d( TSx, TSx) λ[ d( Tx, TSx ) + d( Tx, TSx) + d( Tx, Tx)] leadig to d ( Tx, Tx+ ) d( Tx, Tx) Usig iductio ad equatio (0), oe fids that d( Tx, Tx+ ) d Tx0, Tx (9) (0) () By equatio (), for every m, such that m > oe obtais d( Txm, Tx) d( Txm, Txm ) + d( Txm, Txm 2) d( Tx+, Tx) m m 2 λ λ λ λ λ + d Tx0 + λ λ λ λ λ d Tx0 λ (2) Lettig m, i equatio (2) oe cocludes that { Tx } is a Cauchy sequece, ad sice X is a complete lim Tx = v (3) Sice T is a subsequetially coverget, { x } coverget subsequece So there exists u X ad { xk } such that lim x k = u Sice T is cotiuous ad lim x k = u, lim Tx k = Tu By equatio (3) oe gets that Tu = v, which leads to hece, k k k + k ( + + d TS x, Tu) 0 d( Tu, TSu) λ d( Txk, TSxk ) d( Tu, Tx ) + + k k + d( Tx0, Tx ) + d( Txk +, Tu) k λd( Tu, TSu) + λ d( Tx0, Tx) k + λd( Tu, Txk ) + d( Tx0, Tx) k + λ d( Tx0, Tx) λ λ + d( Txk +, Tu) λ + d( Tu, Txk ) Lettig k i equatio (4) oe gets Sice T is oe-to-oe Su d( TSu, Tu ) = 0 (4) = u ad S fixed poit

4 Turkish Joural of Aalysis ad Number Theory 73 Uiqueess For the uiqueess of the fixed poit let us assume that there are u ad u 2 i X, ( u u ) such that Su = u 2 ad Su2 = u2 The by equatio (8) oe obtais (, 2) λ d ( Tu, TSu ) + d ( Tu, TSu ) + d ( Tu, Tu ) d TSu TSu which implies that λd ( Tu Tu ) d Tu, Tu2, 2 Sice T is oe-to-oe ad λ [ 0, 3) oe fids u = u 2 Also if T is sequetially coverget, by replacig { } with { k } we coclude that lim x = u ad this shows that { x } coverges to the fixed poit of S The followig result marks the ed of the discussio: 24 Theorem d ) be a complete metric space ad T, S : X X be mappigs such that T is cotiuous, oe- to- oe ad subsequetially coverget If λ 0, 3 ad for all xy, X (, ) d ( Ty, TSx) d ( Tx, Ty) d Tx TSy d ( TSx, TSy) λ + + (5) the S uique fixed poit Also if T is sequetially coverget the for every x0 X the sequece of iterates S x coverges to this fixed poit Proof Let x 0 be a arbitrary poit i X We defie the x by x+ = Sx (equivaletly, iterative sequece { } x S x0 = ) =,2,3, Usig equatio (5) oe gets d Tx + = d( TSx, TSx) λ + + (6) leadig to [ d( Tx, TSx) d( Tx, TSx ) d( Tx, Tx) ] [ d( Tx, Tx ) d( Tx, Tx ) d( Tx, Tx )] λ d ( Tx, Tx+ ) d( Tx, Tx) Usig iductio ad equatio (7), oe fids that d( Tx, Tx+ ) d Tx0, Tx (7) (8) By equatio (8), for every m, such that m > oe obtais d( Txm, Tx) d( Txm, Txm ) + d( Txm, Txm 2) d( Tx+, Tx) m m 2 λ λ λ λ λ + d Tx0 + λ λ λ λ λ d Tx0 λ (9) Lettig m, i equatio (9) oe cocludes that { Tx } is a Cauchy sequece, ad sice X is a complete lim Tx = v (20) Sice T is a subsequetially coverget, { x } coverget subsequece So there exists u X ad { xk } such that lim x k = u Sice T is cotiuous ad lim x k = u, lim Tx k = Tu By equatio (20) oe gets that Tu = v, which results i k k k + k d( TS + + x, Tu) 0 λ d( Tu, TSxk ) + d( Txk, TSu) k + d( Tu, Txk ) + d( Tx0, Tx) + d( Txk +, Tu) Lettig k i equatio (2) oe gets λ (2) This implies d( TSu, Tu ) = 0 Sice T is oe-to-oe Su = u ad S fixed poit Uiqueess For the uiqueess of the fixed poit let us assume that there are u ad u 2 i X, ( u u ) such that Su = u 2 ad Su2 = u2 The by equatio (5) oe obtais which implies that (, 2) d ( Tu, Tu ) d Tu TSu d ( TSu, TSy) λ + d Tu2, TSu + 2 (, ) 3 λ (, ) d Tu Tu2 d Tu2 Tu

5 74 Turkish Joural of Aalysis ad Number Theory Sice T is oe-to-oe ad λ [ 0, 3) oe fids u = u 2 Also, if T is sequetially coverget, by replacig { } with { k } we coclude that lim x = u ad this shows that { x } coverges to the fixed poit of S 3 Coclusio As ca be see the results preseted here ot oly are far more geeralized versio, but also improve some of the well kow classical results, addressig covergece itervals Ackowledgemet The correspodig author ackowledges the fudig agecy CSIR Refereces [] Azam, A ad Arshad, M, Kaa fixed poit theorem o geeralized metric spaces, The J Noliear Sci Appl, (), 45-48, Jul2008 [2] Braciari, A, A fixed poit theorem of Baach- Caccippoli type o a class of geeralized metric spaces, Publ Math Debrece, 57 (-2), 3-37, 2000 [3] Kaa, R, Some results o fixed poits, Bull Calcutta Math Soc, 60, 7-76, 968 [4] Moradi, S ad Alimohammadi, D, New extesios of Kaa fixed-poit theorem o complete metric ad geeralized metric spaces, It Joural of math Aalysis, 5(47), , 20 [5] Moradi, S ad Omid, M, A fixed poit theorem for itegral type iequality depedig o aother fuctio, It J Math Aal, 4, , 200 [6] Rhoades, B E, A Compariso of Various Defiitios of Cotractive Mappigs, Amer Math Soc, 226, ,977 [7] Moradi, S ad Beiravad, A, Fixed Poit of TF-cotractive Sigle-valued Mappigs, Iraia Joural of Mathematical scieces ad iformatics, 5, 25-32, 200 [8] Jugck, G ad Rhoades, B E, Fixed poits for set valued fuctios without cotiuity, Idia J pure appl Math, 29(3), , March 998 [9] Patel, K ad Deheri, G, Extesio of some commo fixed poit theorems, Iteratioal Joural of Applied Physics ad Mathematics, 3(5), , Sept203