IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex Valued Geeralized Metric Spaces Ajay Sigh 1, Naweet Hooda 1, Departmet of Mathematics, Deebhu Chhotu Ram Uiversity of Sciece Techology, Murthal 131039, Haryaa, Idia mathajaysigh@gmailcom Abstract: I this paper, we prove some commo coupled fixed poit results usig ratioal type cotractive coditios i complex valued geeralized metric spaces Keywords: Commo coupled fixed poit, Weakly icreasig map, Complex valued geeralized metric space, Partially ordered set I Itroductio Ad Prelimiaries From last few years, the metric fixed poit theory has become a importat field of research i both pure applied scieces The first result i the directio was obtaied by Ra Reurigs [6], i this, the authors preseted some applicatios of their obtaied results of matrix equatios I [7,8], Nieto Lopez exteded the result of Ra Reurigs [6] for o decreasig mappigs applied their result to get a uique solutio for first order differetial equatio While Agarwal et al [9] O Rega Petrutel [10] studied some results for a geeralized cotractio i ordered metric spaces Baach s cotractio Priciple gives appropriate simple coditio to establish the existece uiqueess of a solutio of a operator equatio Fx x The there are may geeralizatios of the Baach s cotractio mappig priciple i the literature Bhaskar Lakshmikatham [11] itroduced the cocept of coupled fixed poit of a mappig F from X X ito X They established some coupled fixed poit results applied their results to the study of existece uiqueess of solutio for a periodic boudary value problem Lakshikatham Ciric [1] itroduced the cocept of coupled coicidece poit proved coupled coicidece coupled commo fixed poit results for a mappig F from X X ito X g from X ito X satisfyig oliear cotractio i ordered metric space The may results of coupled fixed poit are obtaied by may authors i may spaces refer as [6-9] Recetly, Azam et al [1] itroduced the cocept of complex coupled metric spaces proved some result The idea of complex valued metric spaces ca be exploited to defie complex valued ormed spaces complex valued Hilbert spaces, additioally, it offers umerous research activities i mathematical aalysis Let be the set of complex umbers let z1, z Defie a partial order as follows: z1 z if oly if Re( z1) Re( z) It follows that z1 z if oe of the followig is satisfied: (i) Re( z1) Re( z) (ii) Re( z1) Re( z) (iii) Re( z1) Re( z) (iv) Re( z1) Re( z) I particular, we will write z1 z if oe of (i), (ii) (iii) is satisfied we will write z1 z if ol y if (iii) is satisfied Note that 0 z z z z 1 1 z z, z z z z 1 3 1 3 Defiitio 1 ([1)] Let X be a oempty set Suppose that the mappig d : X X satisfies the followig: (i) 0 d( x, y) y X d( x, y) 0 if oly if x y; (ii) d( x, y) d( y, x) y X ; 69 Page
(iii) d( x, y) d( y, x) d( z, y) y X The d is called complex valued metric o X, ( Xd, ) is called a complex valued metric space Defiitio ([18]) Let X be a oempty set If a mappig d : X X satisfies: (iv) 0 d( x, y) y X d( x, y) 0 if oly if x y; (v) d( x, y) d( y, x) y X ; (vi) d( x, y) d( x, u) d( u, v) d( v, y) y X all distict u, v X each oe is differet from x y The d is called a complex valued geeralized metric o X ( Xd, ) is called a complex valued geeralized metric space The followig defiitio is due to Altu [15]: Defiitio 3 ([15]) Let ( X, ) be a partially ordered set A pair ( f, g ) of self maps of X is said to be weakly icreasig if fx gfx gx fgx for all x X If f g, the we have fx f x for all x X i case, we say that f is weakly icreasig map For coupled, we exted this as follows: Let ( X, ) be a partially ordered set Let f, g : X X X two mappig is said to be weakly icreasig if f ( x, y) g( f ( x, y), f ( y, x)); f ( y, x) g( f ( y, x), f ( x, y)) g( x, y) f ( g( x, y), g( y, x)); g( y, x) f ( g( y, x), g( x, y)) y X A poit x X is called iterior poit of a set A X wheever there exist 0 r such that B( x, r) { y X : d( x, y) r} A A poit x X is a limit poit of A wheever, for every 0 r B( x, r) ( A { x}) A is called ope wheever each elemet of A is a iterior poit of A Moreover, a subset B X is called closed wheever each limit poit of B belogs to B The family, F { B( x, r) : x X,0 r } is a sub basis for a Haudorff topology o X Let { x } be a sequece i X x X If for every c with 0 c there is 0 N such that, for all 0, d( x, x) c, the { x } is said to be coverget, { x } coverges to x, x is the limit poit of { x } We deote this by lim x x, or x x as If for every c with 0 c there is 0 N, d( x, xm ) c the { x } is called Cauchy sequece i ( Xd, ) If every Cauchy sequece is coverget i ( Xd, ), the ( Xd, ) is called a complete complex-valued geeralized space Lemma 1 Let ( Xd, ) be a complex-valued geeralized metric space, let { x } be a sequece i X The { x } coverges to x if oly if d( x, x) 0 as Lemma Let ( Xd, ) be a complex-valued geeralized metric space, { x } be a sequece i X The { x } is Cauchy sequece if oly if d( x, x m) 0 as Defiitio 4 ([11]) A elemet ( x, y) X X is called a coupled fixed poit of the mappig F : X X X if F( x, y) x F( y, x) y Defiitio 5 ([1]) (i) A coupled coicidece poit of the mappig F : X X X g : X X F( x, y) g( x) F( y, x) g( y) (ii) A commo coupled coicidece poit of the mappig F : X X X g : X X if F( x, y) g( x) x F( y, x) g( y) y 70 Page
I this paper, we exted the result of M Abbas et al [18] fixed poit to coupled fixed poit i complex valued geeralized metric space II Mai Results Theorem 1 Let ( X, ) be a partially ordered set such that there exist a complete complex valued geeralized metric d o X ( ST, ) a pair of weakly icreasig which defied as S, T : X X X Suppose that, for every comparable x, y, u, v X, we have either a1[ d( u, S( x, y)) d( x, T( u, v)) d( x, T( u, v)) d( u, S( x, y)) ] d( S( x, y), T( u, v)) d( x, T( u, v)) d( u, S( x, y)) d( x, T( u, v)) d( u, S( x, y)) a a3 d( x, S( x, y)) a4 d( u, T( u, v)) d( x, T( u, v)) d( u, S( x, y)) i case d( x, T( u, v)) d( u, S( x, y)) 0, a 0, i 1,,3,4 i 4 i i1 a 1, or d( S,( x, y), T( u, v)) 0 if d( x, T( u, v)) d( u, S( x, y)) 0 () If S or T is cotiuous or for ay odecreasig sequeces { x } { y } with x z y z, we ecessary have x z y z for all N, the S T have a commo coupled fixed poit Moreover, the set of commo coupled fixed poit of S T is totally ordered if oly if S T have oe oly oe commo coupled fixed poit Proof First of all, we show that if S or T has a coupled fixed poit, the it has a commo coupled fixed poit of S T Suppose ( xy, ) is a coupled fixed poit of S, that is S( x, y) x S( y, x) y The we shall show that ( xy, ) is a coupled fixed poit of T also Let if possible ( xy, ) is ot a coupled fixed poit of T The from (1), we get d( x, T( x, y)) d( S( x, y), T( x, y)) [ d( x, S( x, y)) d( x, T( x, y)) d( x, T( x, y)) d( x, S( x, y)) ] a1 d( x, T( x, y)) d( x, S( x, y)) d( x, T( x, y)) d( x, S( x, y)) a a3 d( x, S( x, y)) a4 d( x, T( x, y)) d( x, T( x, y)) d( x, S( x, y)) which implies that d( x, T( x, y)) a4 d( x, T( x, y)), which is a cotradictio as a4 1, so we get d( x, T( x, y)) 0, this implies that T( x, y) x Similarly, we ca show that T( y, x) y I the same way if we take ( xy, ) is coupled fixed poit of T that it is also S Now let ( x0, y 0) be a arbitrary poit of X X If S( x0, y0) x0 S( y0, x0 ) y0, the proof is fiished Let either S( x0, y0) x0 or S( y0, x0 ) y0 Defie the sequeces { x } { y } i X as follows: x S( x, y ) T( S( x, y ), S( y, x )) T( x, y ) x 1 0 0 0 0 0 0 1 1 y S( y, x ) T( S( y, x ), S( x, y )) T( y, x ) y 1 0 0 0 0 0 0 1 1 x T( x, y ) S( T( x, y ), S( y, x )) S( x, y ) x 1 1 1 1 1 1 3 y T( y1, x1 ) S( T( y1, x1 ), S( x1, y1)) S( y, x) y3 I similar fashio, we get x1 x x3 x x 1 y1 y y3 y y 1 Let d( x, x 1) 0 d( y, y 1) 0 for every N If ot, the x x 1 y y 1 for some For all those, x x1 S( x, y) y y1 S( y, x ), proof is fiished Now, let d( x, x 1) 0 d( y, y 1) 0 for 0,1,,3, As x, x 1 y, y 1 are comparable, so we have d( x, x ) d( S( x, y ), T( x, y )) 1 1 1 (1) 71 Page
d( x1, S( x, x )) d( x, T( x1, y1)) d( x, T( x1, y1)) d( x1, S( x, y)) 1 (, ( 1, 1)) ( 1, (, )) a d x T x y d x S x y [ d( x, T( x, y )) d( x, S( x, y ))] (, (, )) (, (, )) 1 1 1 a d x T x 1 y 1 d x 1 S x y a d( x, S( x, y )) a d( x, T( x, y )) 3 1 1 a4 1 3 4 1 1 1 a3 d( x, x1 ) a4 d( x1, x ) which implies that a d( x, x ) d( x, x ) for all 0 Hece d( x, x ) hd( x, x ) for all 0, 1 1 a3 where 0 h 1 Similarly, 1 a4 d( x, x ) hd( x, x ) for all 0 1 1 Hece for all 0, we have d( x1, x ) hd( x, x1) Cosequetly, 1 d( x1, x ) hd( x, x1) h d( x0, x1 ) for all 0 I similar way, d( y1, y) hd( y, y1) for all 0 0h 1 like above d( y, y1) hd( y1, y) for all 0 0h 1 We have d( y, y ) hd( y, y ) Cosequetly, 1 1 1 1 1 0 1 d( y, y ) h d( y, y ) h d( y, y ) Now for m, we have d( x, x ) d( x, x ) d( x, x ) ( x, x ) m 1 1 m1 m 1 m1 0 1 0 1 0 1 h d( x, x ) h d( x, x ) h d( x, x ) h d( x0, x1) 1 h h Therefore, d( x, xm) d( x0, x1) So d( x, ) 0 1 xm as m, gives that { x } h is a Cauchy sequece i X Sice X is complete So there exist a poit u i X such that { x } coverges to u Similarly, we ca easily show that the sequece { y } is a Cauchy sequece i X due to completeess of X, { y } coverges to a poit v i X If S or T is cotiuous, the it is clear that S( u, v) u T( u, v) S( v, u) v T( v, u) If either S or T is cotiuous, the by give assumptio x u y v for all N We claim that ( u, v ) is a coupled fixed poit of S Let if possible d( S( u, v), u) z 0 d( S( v, u), v) z 0 From (1), we have z d( S( u, v), u) d( u, x1 ) d( x1, x ) d( x, S( u, v)) d( u, x ) d( x, x ) d( S( u, v), T( x, y )) 1 1 1 1 7 Page
so d( x1, S( u, v)) d( u, T( x1, y1)) d( u, x ) d( x, x ) a d u T x y d x S u v d( u, T( x1, y1)) d( x1, S( u, v )) 1 1 1 (, ( 1, 1)) ( 1, (, )) d( u, T( x, y )) d( x, S( u, v)) (, (, )) (, (, )) 1 1 1 a d u T x 1 y 1 d x 1 S u v a d( u, S( u, v)) a d( x, T( x, y )) 3 4 1 1 1 d( x1, S( u, v)) d( u, x) d( u, x) d( x1, S( u, v)) d( u, x1) d( x1, x ) a 1 d ( u, x ) d ( x, S ( u, v )) d( u, x ) d( x, S( u, v)) (, ) (, (, )) 1 a d u x d x 1 S u v z d( u, x ) d( x, x ) a 1 1 1 1 a d( u, S( u, v)) a d( x, x )) 3 4 1 d( x1, S( u, v)) { d( u, x) } d( u, x ) { d( x1, S( u, v )) } 1 { d( u, x ) } { d( x1, S( u, v))} d( u, x ) d( x1, S( u, v)) a a3 d( u, S( u, v)) a4 d( x1, x)) d( u, x ) d( x, S( u, v)) which o takig limit as, we get z a3 z, a cotradictio, so S( u, v) u Similarly S( v, u) v Hece S( u, v) u T( u, v) S( v, u) v T( u, v) Now suppose that the set of commo coupled fixed poit of S T is totally ordered Now we shall show that the commo coupled fixed poit of S T is uique Let ( xy, ) ( uv, ) are two distict commo coupled fixed poit of S T From (1), we obtai [ d( u, S( x, y)) d( x, T( u, v)) d( x, T( u, v)) d( u, S( x, y)) ] d( S( x, y), T( u, v)) a1 d( x, T( u, v)) d( u, S( x, y)) d( x, T( u, v)) d( u, S( x, y)) a (, (, )) d ( u, S ( x, y 3d( x, S( x, y)) a4d( u, T( u, v)) )) a d x T u v [ d( u, x) d( x, u) d( x, u) d( u, x) ] d( x, u) d( u, x) a1 a d( x, u) d( u, x) d ( x, u ) d ( u, x ) a3d ( x, x) a4d( u, u) so 1 d( S( x, y), T( u, v)) a1 a d( S( x, y), T( u, v)) where 1 a1 a 1 Which is a cotradictio, so S( x, y) T( u, v) Similarly, S( y, x) T( v, u) Hece x u y v which proves the uiqueess Coversely, if S T have oly oe commo coupled fixed poit the the set of commo coupled fixed poit beig sigleto is totally ordered 73 Page
Corollary 1 Let ( X, ) be a partially ordered set such that there exist a complete complex valued geeralized metric d o X let T : X X X be weakly icreasig map Suppose that, for every comparable u, v, x, y X, either [ d( u, T( x, y)) d( x, T( u, v)) d( x, T( u, v)) d( u, T( x, y)) ] d( T( x, y), T( u, v)) a1 d( x, T( u, v)) d( u, T( x, y)) d( x, T( u, v)) d( u, T( x, y)) a3 d( x, T( x, y)) a4 d( u, T( u, v)) (3) (, (, )) (, (, )) a d x T u v d u T x y If d( x, T( u, v)) d( u, T( x, y)) 0, ai 0, i 1,,3,4 ai 1 or d( T( x, y), T( u, v)) 0 if d( x, T( u, v)) d( u, T( x, y)) 0 (4) If T is cotiuous or for a odecreasig sequeces { x } { y } with x z x z i X, we ecessary have x z x z for all N the T has a coupled fixed poit Moreover, the set of coupled fixed poit of T is totally ordered if oly if T has oe oly oe coupled fixed poit Proof Take S T i Theorem 1 Theorem Let ( X, ) be a partially ordered set such that there exist a complete complex valued geeralized metric d o X a pair ( ST, ) weakly icreasig which defied as S, T : X X X Suppose that, for every comparable x, y, u, v X, either [ d( x, S( x, y)) d( x, T( u, v)) d( u, T( u, v)) d( u, S( x, y))] d( S( x, y), T( u, v)) a d( x, T( u, v)) d( u, S( x, y)) 4 i1 d( x, T( u, v)) d( u, S( x, y)) b d ( x, S ( x, y )) d ( u, T ( u, v )) if d( x, T( u, v)) d( u, S( x, y)) 0 d( x, S( x, y)) d( u, T( u, v)) 0, with ab 1, or d( S( x, y), T( u, v)) 0 if d( x, T( u, v)) d( u, S( x, y)) 0 or d( x, S( x, y)) d( u, T( u, v)) 0 (6) If S or T is cotiuous of for a odecreasig sequeces { x } { y } with x z y z i X, we ecessarily have x z y z for all N The S T have a commo coupled fixed poit Proof First of all we shall show that if S or T has a coupled fixed poit, the it has a commo coupled fixed poit of S T Suppose that ( xy, ) is a coupled fixed poit of S The we shall show that ( xy, ) is coupled fixed poit of T also Let if possible ( xy, ) is ot a coupled fixed poit of T The from (5), we obtai T( y, x) d( x, T( x, y)) d( S( x, y), T( x, y)) [ d( x, S( x, y)) d( x, T( x, y)) d( x, T( x, y)) d( x, S( x, y))] a d( x, T( x, y)) d( x, S( x, y)) d( x, T( x, y)) d( x, S( x, y)) b d ( x, S ( x, y )) d ( x, T ( x, y )) [ d( x, x) d( x, T( x, y)) d( x, T( x, y)) d( x, x))] a d( x, T( x, y)) d( x, x) d( x, T( x, y)) d( x, x) b d ( x, x ) d ( x, T ( x, y )) 0 Hece d( x, T( x, y)) 0 this shows that T( x, y) y (5) x Similarly, we ca easily, show that Hece ( xy, ) is a commo coupled fixed poit of S T I the same way if we take ( xy, ) is a coupled fixed poit of T the it is also S Now let ( x0, y 0) be a arbitrary poit of X X If S( x0, y0) x0 S( y0, x0 ) y0, the proof is hold Let either S( x0, y0) x0 or S( y0, x0 ) y0 74 Page
Costruct the sequeces { x } { y } i X such that: x S( x, y ) T( S( x, y ), S( y, x )) T( x, y ) x 1 0 0 0 0 0 0 1 1 y S( y, x ) T( S( y, x ), S( x, y )) T( y, x ) y 1 0 0 0 0 0 0 1 1 x T( x, y ) S( T( x, y ), S( y, x )) S( x, y ) x 1 1 1 1 1 1 3 y T( y1, x1 ) S( T( y1, x1 ), S( x1, y1)) S( y, x) y3 I the same way, we obtai x1 x x3 x x 1 y1 y y3 y y 1 Let d( x, x 1) 0 d( y, y 1) 0, for every N If ot, the x x 1 y y 1 for some N For all those, x x1 S( x, y) y y1 S( y, y ), proof is hold Now, let d( x, x 1) 0 d( y, y 1) 0 for 0,1,,3, As x, x 1 y, y 1 are comparable, so we have d( x, x ) d( S( x, y ), T( x, y )) 1 1 1 d( x, S( x, y )) d( x, T( x1, y1)) d( x 1, T( x 1, y 1)) d( x 1, S( x, y)) a d( x, T( x, y )) d( x, S( x, y )) 1 1 1 d( x, T( x1, y1)) d( x1, S( x, y )) b d ( x, S ( x, y )) d ( x, T ( x, y )) 1 1 1 [ d( x, x1) d( x, x ) d( x1, x ) d( x1, x1)] a d( x, x ) d( x, x ) d( x, x ) d( x1, x1) b d ( x, x ) d ( x, x ) 1 1 1 1 d( x, x1) d( x, x) a d( x, x ) ad( x, ) x 1 Similarly, d( x, x1) ad( x1, x ), for all 0 Thus d( x1, x ) ad( x, x1), for all 0 Cosequetly, 1 d( x1, x ) ad( x, x1) a d( x0, x1 ), for all 0 I similar way, 1 d( y1, y) a d( y, y1) a d( y0, y1), for all 0 Now for m, we have d( x, x ) d( x, x ) d( x, x ) ( x, x ) m 1 1 m1 m 1 m1 0 1 0 1 0 1 a d( x, x ) a d( x, x ) a d( x, x ) a d( x0, x1) 1 a a So, d( x, xm) d( x0, x1) So d( x, ) 0 1 xm as m, It follows that { x } a is a Cauchy sequece i X Sice X is complete So there exist a poit u i X such that { x } coverges to u Similarly, we ca easily show that the sequece { y } is a Cauchy sequece i X due to completeess of X, { y } coverges to a poit v i X If S or T is cotiuous, the it is clear that S( u, v) u T( u, v) S( v, u) v T( v, u) If either S or T is cotiuous, the by give assumptio x u y v for all N We claim that ( u, v ) is a coupled fixed poit of S Let if possible d( S( u, v), u) z 0 d( S( v, u), v) z 0 75 Page
From (5), we have z d( S( u, v), u) d( u, x1 ) d( x1, x ) d( x, S( u, v)) d( u, x1 ) d( x1, x ) d( S( u, v), T( x1, y1)) d( u, x1 ) d( x1, x ) d( u, S( u, v)) d( u, T( x1, y1)) d( x1, T( x1, y1)) d( x1, S( u, v)) a d( u, T( x, y )) d( x, S( u, v)) 1 1 1 d( u, T( x1, y1)) d( x1, S( u, v)) b d ( u, S ( u, u )) d ( x 1, T ( x 1, y 1)) d( u, S( u, v)) d( u, x) d( x1, x ) d( x1, S( u, v)) d( u, x1) d( x1, x ) a d( u, x ) d( x, S( u, v)) d( u, x ) d( x1, S( u, v)) b d ( u, S ( u, v )) d ( x, x ) 1 1 Takig modulas lim, o both side, we have z 0, a cotradictio So S( u, v) u Similarly S( v, u) v Hece S( u, v) u T( u, v) S( v, u) v T( u, v) The S T have a commo coupled fixed poit Remark I Theorem, if i (5), we take b 0, the poit of S T is totally ordered if oly if S T have oe oly, oe commo coupled fixed poit Corollary Let ( X, ) be a partially ordered set such that there exist a complete complex valued geeralized metric d o X let T : X X X be weakly icreasig map Let for every comparable x, y, u, v X, either [ d( x, T( x, y)) d( x, T( u, v)) d( u, T( u, v)) d( u, T( x, y))] d( T( x, y), T( u, v)) a d( x, T( u, v)) d( u, T( x, y)) d( x, T( u, v)) d( u, T( x, y)) b d ( x, T ( x, y )) d ( u, T ( u, v )) if d( x, T( u, v)) d( u, T( x, y)) 0 d( x, T( x, y)) d( u, T( u, v)) 0 with ab 1, or d( T( x, y), T( u, v)) 0 if d( x, T( u, v)) d( u, T( x, y)) 0 or d( x, T( x, y)) d( u, T( u, v)) 0 (9) If T is cotiuous or for a odecreasig sequeces { x } { } ecessarily have x z y z for all N The S T have a commo coupled fixed poit Proof Put S T i Theorem y with x z y (8) z i X Refereces [1] A Azam, B Fisher M Kha, Commo fixed poit theorems i complex valued metric spaces, Numerical Fuctioal Aalysis Optimizatio, 3 (3) (011), 43-53 [] T Ghaa Bhaskar V Lakshmikatham, Fixed poit theorems i partially ordered metric spaces applicatios, Noliear Aalysis: Theory, Methods & Applicatios 65(1) (006), 1379-1393 [3] V Lakshmikatham L Ciric, Coupled fixed poit theorems for oliear cotractios i partially ordered metric spaces, Noliear Aalysis 70(1) (009), 4341-4349 [4] JX Fag, Commo fixed poit theorems of compatible weakly compatible maps i Meger spaces, Noliear Aalysis 71(5-6) (009), 1833-1843 [5] M Abbas, MA Kha S Radeovic, Commo coupled fixed poit theorems icoe metric spaces for w-compatible mappigs, Applied Mathematics Computatio 17 (1) (010), 195-0 76 Page
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