Fixed Points Theorems In Three Metric Spaces

Transcription

1 Maish Kumar Mishra et al / () INERNIONL JOURNL OF DVNCED ENGINEERING SCIENC ND ECHNOLOGI Vol No 1, Issue No, Fixed Poits heorems I hree Metric Saces ( FPMS) Maish Kumar Mishra Deo Brat Ojha mkm781@rediffmailcom, deobratojha@rediffmailcom Deartmet of Mathematics RKGIstitute of echology Ghaziabad,UP,INDI satisfyig itegral tye iequality d ( RSx, RSy ) c max{d ( x, RSy ), d ( x, RSx), ( y, x), ( Sy, Sx)} (RSy, Sz ) c max{ ( y, Rz ), ( x, RSy), ( z, Sy), d ( Rz, RSy)} ( SRz, Sx) c max{ ( z, Sx), ( z, SRz), d ( x, Rz), (x, Rz)} bstract We obtaied fixed oit theorem o three metric saces x i, y i Y z i Z, c < 1 he RS has Mathematics Subject Classificatio: 54H5 a uique fixed oit u i, RS has a uique fixed oit v i Y SR has a uique fixed oit w i Z Further u = v, Keywords- hree metric sace, fixed oit, itegral tye iequality Sv = w Rw = u he ext theorem was roved i [3] heorem 13 : Let (, d ), (Y, ) ( Z, ) be comlete I metric saces suose is a cotiuous maig of INRODUCION he followig fixed oit theorem was roved by Fisher [1] ito Y, S is a cotiuous maig of Y ito Z R is a cotiuous maig of Z ito satisfyig the iequalities: comlete metric saces If S is a cotiuous maig of ito Z, R is a cotiuous d ( RSx, RSx ') c max{d ( x, x '), d ( x, RSx), d ( x ', RSx '), (x, x '), Sx, Sx ' } heorem 11: Let (, d ) ( Z, ) be maig of Z ito satisfyig the (RSy, RSy ') c max{ ( y, y '), ( y, RSy), ( y ', RSy '), ( Sy, Sy '), d RSy, RSy ' } (SRz, SRz ') c max{ ( z, z '), ( z, SRz), ( z ', SRz '), d Rz, Rz ' (Rz, Rz ')} iequalities: d ( RSx, RSx ') c max{d ( x, x '), d ( x, RSx), d ( x ' RSx '), d ( Sx, Sx ')} (SRz, SRz ') c max{ ( z, z '), ( z, SRz), ( z ' SRz '), ( Rz, Rz ')} x, x i, y, y i Y z, z i Z c <1 he RS has a uique fixed oit u i, RS x, x' i, z, z' i Z, c < 1, the has a uique fixed oit v i Y SR has a uique RS has a uique fixed oit u i RS has a fixed oit w i Z Further, u = v, Sv = w Rw = u I uique fixed oit w i Z Further Su = w Rw = recetly sari, Sharma[6] geerate Related Fixed Poits u he ext theorem was roved i [] heorems o hree Metric Saces Now We obtaied related heorem 1: Let (, d ), (Y, ) ( Z, ) be comlete metric saces suose is a cotiuous maig of ito Y, S is a cotiuous maig of Y ito Z R is a fixed oit theorem o three metric saces satisfyig itegral tye iequality Let (,d ) be [,1], f : a comlete metric a maig such that each cotiuous maig of Z ito satisfyig the iequalities: ISSN: 1 htt://wwwijaestiserorg ll rights Reserved sace, Page 13

2 Maish Kumar Mishra et al / () INERNIONL JOURNL OF DVNCED ENGINEERING SCIENC ND ECHNOLOGI Vol No 1, Issue No, d ( fx, fy ) x, y, (t )dt d ( x, y ) (t )dt, Where d1 ( SRy, SRx ) d (Sz,SRy ) d3 summable, is a lebesgue itegrable maig which is oegative such that,, ( Rx, RSz ) max M ( y, z ) max M 3 ( z, x ) b Y RS Proof Let coditio by the followig sequeces Let (, d ) be a metric sace let Ojha (1) [5] a multi-valued ma resectively, suose that f F are occasioally weakly ad ( fx, fy ) d P 1 ( fx, Fx ), ad ( fx, fy ) d P 1 ( fy, Fy ), P 1 P 1 ad ( fx, Fx ) d ( fy, Fy ), cd ( fx, Fy ) d ( fy, Fx ) x, y i, is a iteger a c 1 the f F have uique commo fixed oit x, y i c 1 the f, F is a iteger a all (t )dt N We will saces maigs satisfyig the followig iequalities: x x y y M1 ( x, y ) Similarly, if z z 1, x, y i (1) (4), we obtai: { d1 ( x, SRy ), d1 ( x, SRx ), d ( y, x )} ( x, x ), d1 ( x, x 1 ), d {, d1 ( x, x 1 ), d ( y, y 1 )} ( y, y 1 )} d1 ( x, x 1 ) d1 ( SRy, SRx ) max{, d1 ( x, x 1 ), d ( y, y 1 )} ( y, y 1 ) d1 ( x, x 1 ) F d sice, if F mi M 1 ( x, y ) max M1 ( x, y ) { d1 max M1 ( x, y ) d1 ( x, x 1 ) the by the iequality d1 ( x, x 1 ) 1 htt://wwwijaestiserorg ll rights Reserved If the latter equality imlies that it follows x 1 x sice 1 ISSN: we z are Cauchy sequeces are three that Otherwise, if x a, y 1 b z 1 c First, we rove that the sequeces MIN RULS : Y, R :Y Z, S : Z assume x x 1 be three metric Further, y 1 y, z 1 z some, the ut the agai have uique commo fixed oit i (, d1 ), (Y, d ) ( Z, d3 ) c Z be a arbitrary oit We defie three y y 1, the z z 1 Now, we will give rove our theorem as follows: heorem 1 Let SR the could II 6 x, y z with, Y, Z resectively as x x 1 i has a uique fixed oit SRx 1 SRx, ie x x 1 max 5 1, x x 1, y y 1 z z 1 commutative (OWC) satisfy the iequality J P ( Fx, Fy ) x f :, F : CB( ) be a sigle 4 x SR x, y x 1, z Ry follows: 1 max{d ( x, y ), d ( x, fx ), d ( y, fy ), [ d ( x, fy ) d ( y, fx )] F mi M 3 ( z, x ) has a uique fixed oit Rhoades(3)[4], exteded this result by relacig the above d ( fx, fy ) a, SR a b, Rb c Sc a, z such that each x, lim f x z F mi M ( y, z ) x, y Y, z Z has a uique fixed oit he f has a uique commo fixed each ; R R Page 131

3 Maish Kumar Mishra et al / () INERNIONL JOURNL OF DVNCED ENGINEERING SCIENC ND ECHNOLOGI Vol No 1, Issue No, d1 ( x, x 1 ) d ( y, y 1 ) y y, z z 1 M ( y, z 1 ) x x { d ( y, Sz 1 ), d ( y, SRy ), d 3 ( z 1, Ry )} d ( y, y 1 ) { d ( y, y ), d ( y, y 1 ), d 3 ( z 1, z )} {, d ( y, y 1 ), d3 ( z 1, z )} d (Sz 1,SRy ) d1 ( SRb, x 1 ) max{, d ( y, y 1 ), d3 ( z 1, z )} d3 ( z 1, z ) F max M1 ( x, b ) M1 ( x, b ) max{d ( y, y 1 ), d3 ( z 1, z )} d ( y, y 1 ) the by the d1 ( SRb, a ) d ( y, y 1 ) ( y, y 1 ) d3 ( z 1, z ) { d3 ( z, z 1 ), d1 {, d3 ( z, z 1 ), d1 ( x 1, x )} ( x 1, Sz )} d3 ( Rx 1, RSz ) max{, d3 ( z, z 1 ), d1 ( x 1, x )} d3 ( z 1, z ) 1 we obtai: with d1 ( x, x 1 ) 9 Usig (7), (8) (9) we get d1 ( x, x 1 ) d ( y, y 1 ) 3 d1 ( x, x 1 ) F mi M1 ( a, y ) d3 ( z 1, z ) 4 d1 ( x 4, x 3 ) 3 k d1 ( x, x1 ) 3 k d1 ( x1, x ) { d1 ( a, x ), d1 ( a, SRa ), d ( y,a )} d1 ( a, SRa ) max{d1 ( a, a ), d1 ( a, SRa ), d ( b,a )} max{d1 ( a, SRa ), d ( b,a )} from which it follows or d1 ( a, SRa ) or d1 ( a, SRa ) d1 ( a, SRa ) SRa a d ( b,a ) which ca be also writte i the followig m d1 ( a, Sc ) d ( b,a ) 11 sice R a = c, z z, y b above we k obtai: x, y z Lettig ted to ifiity we get k 1 Sice 1, the sequeces { d1 ( a, SRy ), d1 ( a, SRa ), d ( y,a )} F Relacig M 1 ( a, y ) F mi M 3 ( z, x 1 ) max M 3 ( z, x 1 ) ( x 1, Sz ) d1 ( SRy, SRa ) x a, y y we get d1 SRb a max M1 ( a, y ) ( z, z ), d3 d1 ( x, SRa ) 8 { d 3 ( z, Rx 1 ), d 3 ( z, RSz ), d1 ( x 1, Sz )} ( z, z 1 ) Usig (4), if we take d1 ( a, SRb ) I the same way it ca be show that Sc = b R a = c x x 1 z z i (3) (6) we obtai: M 3 ( z, x 1 ) from which it follows d hus Lettig ted to ifiity i the iequality (1) by the fact that F {d1 ( x, SRb ), d1 ( x, x 1 ), d ( b, y 1 )} it follows y y 1 sice 1 { d1 ( x, SRb ), d1 ( x, SRx ), d ( b,x )} ( y, y 1 ) 1 iequality F mi M1 ( x, b ) d3 is cotiuous i we get sice, if d1 ( SRb, SRx ) F mi M ( y, z 1 ) max M ( y, z 1 ) y b i the iequality (4) we obtai d i () (5), we obtai: lim x a, lim y b Y, lim z c Z have 7 are Cauchy d ( b,a ) d3 ( c, Rb ) i the iequality (5) i the same way as 1 I the same way, we obtai: sequeces Sice (, d1 ), (Y, d ) ( Z, d3 ) are comlete metric saces we ISSN: 1 htt://wwwijaestiserorg ll rights Reserved Page 13

4 Maish Kumar Mishra et al / () INERNIONL JOURNL OF DVNCED ENGINEERING SCIENC ND ECHNOLOGI Vol No 1, Issue No, d3 ( c, Rb ) d1 ( a, Sc ) 13 d3 ( Ra ', c ) d3 ( Ra ', c ) d1 ( a, Sc ) 3 d1 ( a, Sc ) hus, agai d1 ( a, SRa ) Sc a d1 ( a, a ') 16 or By (11), (1), (13) it follows: d3 ( Ra ', c ) 17 By (17) it follows that Ra ' c sice 1 d1 ( a, Sc ) By (14), (15) (16) we obtai d1 ( a, a ') or SRa a d (a ',b ) d1 ( a, a ') d3 ( Ra ', c ) 3 d1 ( a, a ') 3 d1 ( a, a ') 1, it follows So, we roved that a is a fixed oit of SR by I the same way it ca be show that b is a fixed oit of SR c that is a fixed oit of R S hus we roved that a is the uique fixed oit of SR Further, we showed that a = b, Rb = c, Sc = a I the same way, it ca be show that b is the uique fixed oit of is aother fixed oit of SR, x a, y a ', we get differet from a Usig (4), if we take d1 ( a, a ') d1 ( SRa ', SRa ) max M1 ( a,a ') F mi M1 ( a,a ') { d1 ( a, SRa '), d1 ( a, SRa ), d (a ',a )} { d1 ( a, a '), d1 ( a, a ), d (a ', b )} m which it follows d1 ( a, a ') Z Y, d1 d ), the maig (a ', b ) i the iequality (5) we obtai: d (Sc,SRa ') max M1 (a ', c ) F mi M (a ', c ) { d (a ',Sc ), d (a ',SRa '), d3 ( c, Ra ')} { d (a ', b ), d (a ',a '), d3 ( c, Ra ')} he d ( b,a ') d3 ( c, Ra ') z c, y a ' d3 ( Ra ',c ) max M 3 ( c, a ') 15 i the iequality (6) we obtai: d3 ( Ra ', RSc ) F mi M ( c, a ') M 3 ( c, a ') {d3 ( c, Ra '), d ( c, RSc ), d1 ( a ', Sc )} { d3 ( c, Ra '),, d1 ( a ', a )} d1 ( Sy, Sx ) aly the R iequalities (4)(5) (6) as the idetity maig M1 ( x, y ) d (Sy,Sy ) { d1 ( x, Sy ), d1 ( x, Sx ), d ( y, x )} max M ( y, y ) F mi M ( y, y ) he iequality (6) takes the m: d (x,sy ) max M 3 ( y, x ) F mi M 3 ( y, x ) M3 ( y, x ) { d ( y, x ), d ( y, Sy ), d1 ( x, Sy )} x, y Y I sequel, is deoted with M 3 ( y, x) M ( y, x) hus, the followig theorem (heorem 1[1]) is obtaied: (, d1 ), (Y, d ) be : Y, S :Y which holds always sice the left h is zero two metric saces d1 ( Sy, Sx ) M1 ( x, y ) { d1 ( x, Sy ), d1 ( x, Sx ), d ( y, x )} ISSN: 1 htt://wwwijaestiserorg ll rights Reserved are two maigs satisfyig the followig iequalities: he Let We he iequality (5) takes the m: M (a ', c ) Proof Z Y, d3 d, z y as the idetity maig of Y Ry y, y Y, the we obtai heorem 1[1] 14 z c, y a ' d ( b,a ') R he iequality (4) takes the m: d the same with the metric sace (Y, d ), (that is SR c is the uique fixed oit of R S ( Z, d3 ) a' a Corollary If we cosider i heorem 1 the metric sace M1 ( a,a ') a ' Let assume ow that Page 133

5 Maish Kumar Mishra et al / () INERNIONL JOURNL OF DVNCED ENGINEERING SCIENC ND ECHNOLOGI Vol No 1, Issue No, d ( x, Sy) max M ( y, x) F mi M ( y, x) ( ) ( ) () M ( y, x) t dt t dt t dt { d ( y, x), d ( y, Sy ), d1 ( x, Sy)} ( t) dt ( t) dt, x, y Y 1, the S has a uique fixed oit a S has a uique fixed oit b Y a = b Sb = a Further, It is clear that i the heorem 1[1] the fuctios F 1 F ca be relaced by F such that F t max F1 t, F t c1, c ca be relaced by c max c1, c Corollary 3 For 1 Ft t R, by heorem 1 we obtai a theorem which exteds the result of heorem 1([]) to three metric saces Corollary 4 For Ft t R, we obtai a geeralizatio of heorem [3], exteded to three metric saces REFERENC [1] S CNesi, fixed oit theorem i two metric saces, Bull Math Soc Sci Math Roumaie (NS) 44, No 9 (1), [] B Fisher, Fixed oits o two metric saces, Glasik Mat16, No 36(1981), [3] V Poa, Fixed oits o two comlete metric saces, Zb Rad Prirod-Mat Fak (NS) Ser mat 1, No 1 (1991), [4] R K Jai, H K Sahu, B Fisher,Related fixed oit theorems three metric saces, Novi Sad J Math 6, No1 (1996), [5] Luljeta Kikia, Fixed Poits heorems o hree Metric Saces, It Joural of Math alysis, Vol 3, 9, o 13, [6] Deo Brat Ojha, Maish Kumar Mishra Udayaa Katoch, Commo Fixed Poit heorem Satisfyig Itegral ye Occasioally Weakly Comatible Mas, Research Joural of lied Scieces, Egieerig echology (3): 39-44, 1 [7] Rhoades, BE,wo fixed oit theorem maig satisfyig a geeral cotractiv coditio of itegral tye It J Math Sci, 3: ISSN: 1 htt://wwwijaestiserorg ll rights Reserved Page 134