Common Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces
|
|
- Nathaniel Davis
- 5 years ago
- Views:
Transcription
1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn: x Volume 10, Issue 3 Ver II (May-Ju 014), PP 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 , 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 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
2 (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
3 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 y S( y, x ) T( S( y, x ), S( x, y )) T( y, x ) y x T( x, y ) S( T( x, y ), S( y, x )) S( x, y ) x 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) 71 Page
4 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 ))] (, (, )) (, (, )) 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 )) a 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 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, 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 m 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 )) Page
5 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, (, )) d( u, T( x, y )) d( x, S( u, v)) (, (, )) (, (, )) 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 )) 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 a d( u, S( u, v)) a d( x, x )) 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
6 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
7 Costruct the sequeces { x } { y } i X such that: x S( x, y ) T( S( x, y ), S( y, x )) T( x, y ) x y S( y, x ) T( S( y, x ), S( x, y )) T( y, x ) y x T( x, y ) S( T( x, y ), S( y, x )) S( x, y ) x 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 )) 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 )) d( x, T( x1, y1)) d( x1, S( x, y )) b d ( x, S ( x, y )) d ( x, T ( x, y )) [ 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 ) 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 m 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
8 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)) 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), [] T Ghaa Bhaskar V Lakshmikatham, Fixed poit theorems i partially ordered metric spaces applicatios, Noliear Aalysis: Theory, Methods & Applicatios 65(1) (006), [3] V Lakshmikatham L Ciric, Coupled fixed poit theorems for oliear cotractios i partially ordered metric spaces, Noliear Aalysis 70(1) (009), [4] JX Fag, Commo fixed poit theorems of compatible weakly compatible maps i Meger spaces, Noliear Aalysis 71(5-6) (009), [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), Page
9 [6] ACM Ra MCB Reurigs, A fixed poit theorem i partially ordered sets some applicatios to matrix equatios, Proc Am Math Soc 13, (004) doi:101090/s [7] JJ Nieto RR López, Cotractive mappig theorems i partially ordered sets applicatios to ordiary differetial equatios, Order, 3-39 (005) doi:101007/s [8] JJ Nieto RR López, Existece uiqueess of fixed poit i partially ordered sets applicatios to ordiary differetial equatios, Acta Math Siica Egl Ser 3(1), 05 1 (007) doi:101007/s [9] RP Agarwal, MA El-Gebeily D O Rega, Geeralized cotractios i partially ordered metric spaces, Appl Aal 87, 1-8 (008) doi:101080/ [10] D O Rega A Petrutel, Fixed poit theorems for geeralized cotractios i ordered metric spaces, J Math Aal Appl 341, (008) [11] G Bhaskar, V Lakshmikatham, Fixed poit theorems i partially ordered metric spaces applicatios, Noliear Aal (006) doi:101016/ja [1] V Lakshmikatham LJ Cirić, Coupled fixed poit theorems for oliear cotractios i partially ordered metric spaces, Noliear Aal 70, (009) doi:101016/ja [13] M Abbas, YJ Cho T Nazir, Commo fixed poit theorems for four mappigs i TVS-valued coe metric spaces, J Math Iequal 5, (011) [14] M Abbas, MA Kha S Radeović, Commo coupled fixed poit theorem i coe metric space for w-compatible mappigs, Appl Math Comput 17, (010) doi:101016/jamc [15] I Altu H Simsek, Some fixed poit theorems o ordered metric spaces applicatio, Fixed Poit Theory Appl, vol 010, Article ID 6149 [16] A Azam, B Fisher M Kha, Commo fixed poit theorems i complex valued metric spaces, Numer Fuct Aal Optim 3(3), (011) [17] F Rouzkard M Imbdad, Some commo fixed poit theorems o complex valued metric spaces, Comput Math Appl (01) doi:101016/jcamwa [18] M Abbas, VCRT Nazir S Radeović, Commo fixed poit of mappigs satisfyig ratioal iequalities i ordered complex valued geeralized metric spaces, Afr Mat (013), doi: /s z 77 Page
COMMON FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN COMPLEX VALUED b-metric SPACES
I S S N 3 4 7-9 J o u r a l o f A d v a c e s i M a t h e m a t i c s COMMON FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN COMPLEX VALUED b-metric SPACES Ail Kumar Dube, Madhubala Kasar, Ravi
More informationSome Common Fixed Point Theorems in Cone Rectangular Metric Space under T Kannan and T Reich Contractive Conditions
ISSN(Olie): 319-8753 ISSN (Prit): 347-671 Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio) Some Commo Fixed Poit Theorems i Coe Rectagular Metric
More informationA Fixed Point Result Using a Function of 5-Variables
Joural of Physical Scieces, Vol., 2007, 57-6 Fixed Poit Result Usig a Fuctio of 5-Variables P. N. Dutta ad Biayak S. Choudhury Departmet of Mathematics Begal Egieerig ad Sciece Uiversity, Shibpur P.O.:
More informationFixed Point Theorems for Expansive Mappings in G-metric Spaces
Turkish Joural of Aalysis ad Number Theory, 7, Vol. 5, No., 57-6 Available olie at http://pubs.sciepub.com/tjat/5//3 Sciece ad Educatio Publishig DOI:.69/tjat-5--3 Fixed Poit Theorems for Expasive Mappigs
More informationGeneralization of Contraction Principle on G-Metric Spaces
Global Joural of Pure ad Applied Mathematics. ISSN 0973-1768 Volume 14, Number 9 2018), pp. 1159-1165 Research Idia Publicatios http://www.ripublicatio.com Geeralizatio of Cotractio Priciple o G-Metric
More informationCOMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS
PK ISSN 0022-2941; CODEN JNSMAC Vol. 49, No.1 & 2 (April & October 2009) PP 33-47 COMMON FIXED POINT THEOREMS IN FUZZY METRIC SPACES FOR SEMI-COMPATIBLE MAPPINGS *M. A. KHAN, *SUMITRA AND ** R. CHUGH *Departmet
More informationA Common Fixed Point Theorem Using Compatible Mappings of Type (A-1)
Aals of Pure ad Applied Mathematics Vol. 4, No., 07, 55-6 ISSN: 79-087X (P), 79-0888(olie) Published o 7 September 07 www.researchmathsci.org DOI: http://dx.doi.org/0.457/apam.v4a8 Aals of A Commo Fixed
More informationII. EXPANSION MAPPINGS WITH FIXED POINTS
Geeralizatio Of Selfmaps Ad Cotractio Mappig Priciple I D-Metric Space. U.P. DOLHARE Asso. Prof. ad Head,Departmet of Mathematics,D.S.M. College Jitur -431509,Dist. Parbhai (M.S.) Idia ABSTRACT Large umber
More informationA Common Fixed Point Theorem in Intuitionistic Fuzzy. Metric Space by Using Sub-Compatible Maps
It. J. Cotemp. Math. Scieces, Vol. 5, 2010, o. 55, 2699-2707 A Commo Fixed Poit Theorem i Ituitioistic Fuzzy Metric Space by Usig Sub-Compatible Maps Saurabh Maro*, H. Bouharjera** ad Shivdeep Sigh***
More informationUnique Common Fixed Point Theorem for Three Pairs of Weakly Compatible Mappings Satisfying Generalized Contractive Condition of Integral Type
Iteratioal Refereed Joural of Egieerig ad Sciece (IRJES ISSN (Olie 239-83X (Prit 239-82 Volume 2 Issue 4(April 23 PP.22-28 Uique Commo Fixed Poit Theorem for Three Pairs of Weakly Compatible Mappigs Satisfyig
More informationOn Weak and Strong Convergence Theorems for a Finite Family of Nonself I-asymptotically Nonexpansive Mappings
Mathematica Moravica Vol. 19-2 2015, 49 64 O Weak ad Strog Covergece Theorems for a Fiite Family of Noself I-asymptotically Noexpasive Mappigs Birol Güdüz ad Sezgi Akbulut Abstract. We prove the weak ad
More informationCOMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPS IN PARTIAL METRIC SPACES
Iteratioal Joural of Egieerig Cotemporary Mathematics ad Scieces Vol. No. 1 (Jauary-Jue 016) ISSN: 50-3099 COMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPS IN PARTIAL METRIC SPACES N. CHANDRA M. C. ARYA
More informationOn the Variations of Some Well Known Fixed Point Theorem in Metric Spaces
Turkish Joural of Aalysis ad Number Theory, 205, Vol 3, No 2, 70-74 Available olie at http://pubssciepubcom/tjat/3/2/7 Sciece ad Educatio Publishig DOI:0269/tjat-3-2-7 O the Variatios of Some Well Kow
More informationA COMMON FIXED POINT THEOREM IN FUZZY METRIC SPACE USING SEMI-COMPATIBLE MAPPINGS
Volume 2 No. 8 August 2014 Joural of Global Research i Mathematical Archives RESEARCH PAPER Available olie at http://www.jgrma.ifo A COMMON FIXED POINT THEOREM IN FUZZY METRIC SPACE USING SEMI-COMPATIBLE
More informationCOMMON FIXED POINT THEOREM FOR FINITE NUMBER OF WEAKLY COMPATIBLE MAPPINGS IN QUASI-GAUGE SPACE
IJRRAS 19 (3) Jue 2014 www.arpapress.com/volumes/vol19issue3/ijrras_19_3_05.pdf COMMON FIXED POINT THEOREM FOR FINITE NUMBER OF WEAKLY COMPATIBLE MAPPINGS IN QUASI-GAUGE SPACE Arihat Jai 1, V. K. Gupta
More informationProperties of Fuzzy Length on Fuzzy Set
Ope Access Library Joural 206, Volume 3, e3068 ISSN Olie: 2333-972 ISSN Prit: 2333-9705 Properties of Fuzzy Legth o Fuzzy Set Jehad R Kider, Jaafar Imra Mousa Departmet of Mathematics ad Computer Applicatios,
More informationConvergence of Random SP Iterative Scheme
Applied Mathematical Scieces, Vol. 7, 2013, o. 46, 2283-2293 HIKARI Ltd, www.m-hikari.com Covergece of Radom SP Iterative Scheme 1 Reu Chugh, 2 Satish Narwal ad 3 Vivek Kumar 1,2,3 Departmet of Mathematics,
More informationCommon Fixed Points for Multivalued Mappings
Advaces i Applied Mathematical Bioscieces. ISSN 48-9983 Volume 5, Number (04), pp. 9-5 Iteratioal Research Publicatio House http://www.irphouse.com Commo Fixed Poits for Multivalued Mappigs Lata Vyas*
More informationCOMMON FIXED POINT THEOREM USING CONTROL FUNCTION AND PROPERTY (CLR G ) IN FUZZY METRIC SPACES
Iteratioal Joural of Physics ad Mathematical Scieces ISSN: 2277-2111 (Olie) A Ope Access, Olie Iteratioal Joural Available at http://wwwcibtechorg/jpmshtm 2014 Vol 4 (2) April-Jue, pp 68-73/Asati et al
More informationOn common fixed point theorems for weakly compatible mappings in Menger space
Available olie at www.pelagiaresearchlibrary.com Advaces i Applied Sciece Research, 2016, 7(5): 46-53 ISSN: 0976-8610 CODEN (USA): AASRFC O commo fixed poit theorems for weakly compatible mappigs i Meger
More informationCommon Fixed Point Theorems for Four Weakly Compatible Self- Mappings in Fuzzy Metric Space Using (JCLR) Property
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 3 Ver. I (May - Ju. 05), PP 4-50 www.iosrjourals.org Commo Fixed Poit Theorems for Four Weakly Compatible Self- Mappigs
More informationINTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010
Fixed Poits theorem i Fuzzy Metric Space for weakly Compatible Maps satisfyig Itegral type Iequality Maish Kumar Mishra 1, Priyaka Sharma 2, Ojha D.B 3 1 Research Scholar, Departmet of Mathematics, Sighaia
More informationSome Fixed Point Theorems in Generating Polish Space of Quasi Metric Family
Global ad Stochastic Aalysis Special Issue: 25th Iteratioal Coferece of Forum for Iterdiscipliary Mathematics Some Fied Poit Theorems i Geeratig Polish Space of Quasi Metric Family Arju Kumar Mehra ad
More informationGeneralized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces
Turkish Joural of Aalysis a Number Theory, 08, Vol. 6, No., 43-48 Available olie at http://pubs.sciepub.com/tjat/6// Sciece a Eucatio Publishig DOI:0.69/tjat-6-- Geeralize Dyamic Process for Geeralize
More informationBETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios
More informationA FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE. Abdolrahman Razani (Received September 2004)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 35 (2006), 109 114 A FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE Abdolrahma Razai (Received September 2004) Abstract. I this article, a fixed
More informationCOMMON FIXED POINT THEOREMS VIA w-distance
Bulleti of Mathematical Aalysis ad Applicatios ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3, Pages 182-189 COMMON FIXED POINT THEOREMS VIA w-distance (COMMUNICATED BY DENNY H. LEUNG) SUSHANTA
More informationOn n-collinear elements and Riesz theorem
Available olie at www.tjsa.com J. Noliear Sci. Appl. 9 (206), 3066 3073 Research Article O -colliear elemets ad Riesz theorem Wasfi Shataawi a, Mihai Postolache b, a Departmet of Mathematics, Hashemite
More informationResearch Article Convergence Theorems for Finite Family of Multivalued Maps in Uniformly Convex Banach Spaces
Iteratioal Scholarly Research Network ISRN Mathematical Aalysis Volume 2011, Article ID 576108, 13 pages doi:10.5402/2011/576108 Research Article Covergece Theorems for Fiite Family of Multivalued Maps
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationJournal of Applied Research and Technology ISSN: Centro de Ciencias Aplicadas y Desarrollo Tecnológico.
Joural of Applied Research ad Techology ISSN: 665-64 jart@aleph.cistrum.uam.mx Cetro de Ciecias Aplicadas y Desarrollo Tecológico México Shahi, Priya; Kaur, Jatiderdeep; Bhatia, S. S. Commo Fixed Poits
More informationInternational Journal of Mathematical Archive-7(6), 2016, Available online through ISSN
Iteratioal Joural of Mathematical Archive-7(6, 06, 04-0 Available olie through www.ijma.ifo ISSN 9 5046 COMMON FIED POINT THEOREM FOR FOUR WEAKLY COMPATIBLE SELFMAPS OF A COMPLETE G METRIC SPACE J. NIRANJAN
More informationResearch Article Approximate Riesz Algebra-Valued Derivations
Abstract ad Applied Aalysis Volume 2012, Article ID 240258, 5 pages doi:10.1155/2012/240258 Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of
More informationSOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS
ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz
More informationOscillation and Property B for Third Order Difference Equations with Advanced Arguments
Iter atioal Joural of Pure ad Applied Mathematics Volume 3 No. 0 207, 352 360 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu ijpam.eu Oscillatio ad Property B for Third
More informationA General Iterative Scheme for Variational Inequality Problems and Fixed Point Problems
A Geeral Iterative Scheme for Variatioal Iequality Problems ad Fixed Poit Problems Wicha Khogtham Abstract We itroduce a geeral iterative scheme for fidig a commo of the set solutios of variatioal iequality
More informationOn a fixed point theorems for multivalued maps in b-metric space. Department of Mathematics, College of Science, University of Basrah,Iraq
Basrah Joural of Sciece (A) Vol.33(),6-36, 05 O a fixed poit theorems for multivalued maps i -metric space AMAL M. HASHM DUAA L.BAQAIR Departmet of Mathematics, College of Sciece, Uiversity of Basrah,Iraq
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationExistence of viscosity solutions with asymptotic behavior of exterior problems for Hessian equations
Available olie at www.tjsa.com J. Noliear Sci. Appl. 9 (2016, 342 349 Research Article Existece of viscosity solutios with asymptotic behavior of exterior problems for Hessia equatios Xiayu Meg, Yogqiag
More informationON THE FUZZY METRIC SPACES
The Joural of Mathematics ad Computer Sciece Available olie at http://www.tjmcs.com The Joural of Mathematics ad Computer Sciece Vol.2 No.3 2) 475-482 ON THE FUZZY METRIC SPACES Received: July 2, Revised:
More informationStrong Convergence Theorems According. to a New Iterative Scheme with Errors for. Mapping Nonself I-Asymptotically. Quasi-Nonexpansive Types
It. Joural of Math. Aalysis, Vol. 4, 00, o. 5, 37-45 Strog Covergece Theorems Accordig to a New Iterative Scheme with Errors for Mappig Noself I-Asymptotically Quasi-Noexpasive Types Narogrit Puturog Mathematics
More informationBoundaries and the James theorem
Boudaries ad the James theorem L. Vesely 1. Itroductio The followig theorem is importat ad well kow. All spaces cosidered here are real ormed or Baach spaces. Give a ormed space X, we deote by B X ad S
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationIterative Method For Approximating a Common Fixed Point of Infinite Family of Strictly Pseudo Contractive Mappings in Real Hilbert Spaces
Iteratioal Joural of Computatioal ad Applied Mathematics. ISSN 89-4966 Volume 2, Number 2 (207), pp. 293-303 Research Idia Publicatios http://www.ripublicatio.com Iterative Method For Approimatig a Commo
More informationResearch Article A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property
Discrete Dyamics i Nature ad Society Volume 2011, Article ID 360583, 6 pages doi:10.1155/2011/360583 Research Article A Note o Ergodicity of Systems with the Asymptotic Average Shadowig Property Risog
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationVECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS
Dedicated to Professor Philippe G. Ciarlet o his 70th birthday VECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS ROMULUS CRISTESCU The rst sectio of this paper deals with the properties
More informationCoupled coincidence point results for probabilistic φ-contractions
PURE MATHEMATICS RESEARCH ARTICLE Coupled coicidece poit results for probabilistic φ-cotractios Biayak S Choudhury 1, Pradyut Das 1 * P Saha 1 Received: 18 February 2016 Accepted: 29 May 2016 First Published:
More informationA Characterization of Compact Operators by Orthogonality
Australia Joural of Basic ad Applied Scieces, 5(6): 253-257, 211 ISSN 1991-8178 A Characterizatio of Compact Operators by Orthogoality Abdorreza Paahi, Mohamad Reza Farmai ad Azam Noorafa Zaai Departmet
More informationIntroduction to Optimization Techniques. How to Solve Equations
Itroductio to Optimizatio Techiques How to Solve Equatios Iterative Methods of Optimizatio Iterative methods of optimizatio Solutio of the oliear equatios resultig form a optimizatio problem is usually
More information1 Introduction. 1.1 Notation and Terminology
1 Itroductio You have already leared some cocepts of calculus such as limit of a sequece, limit, cotiuity, derivative, ad itegral of a fuctio etc. Real Aalysis studies them more rigorously usig a laguage
More informationMA541 : Real Analysis. Tutorial and Practice Problems - 1 Hints and Solutions
MA54 : Real Aalysis Tutorial ad Practice Problems - Hits ad Solutios. Suppose that S is a oempty subset of real umbers that is bouded (i.e. bouded above as well as below). Prove that if S sup S. What ca
More informationSome Approximate Fixed Point Theorems
It. Joural of Math. Aalysis, Vol. 3, 009, o. 5, 03-0 Some Approximate Fixed Poit Theorems Bhagwati Prasad, Bai Sigh ad Ritu Sahi Departmet of Mathematics Jaypee Istitute of Iformatio Techology Uiversity
More informationOn Summability Factors for N, p n k
Advaces i Dyamical Systems ad Applicatios. ISSN 0973-532 Volume Number 2006, pp. 79 89 c Research Idia Publicatios http://www.ripublicatio.com/adsa.htm O Summability Factors for N, p B.E. Rhoades Departmet
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More information2 Banach spaces and Hilbert spaces
2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationMath 140A Elementary Analysis Homework Questions 3-1
Math 0A Elemetary Aalysis Homework Questios -.9 Limits Theorems for Sequeces Suppose that lim x =, lim y = 7 ad that all y are o-zero. Detarime the followig limits: (a) lim(x + y ) (b) lim y x y Let s
More informationCouncil for Innovative Research
ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, 34210 Eseler, Istabul egul34@gmail.com mehmetalbayrak12@gmail.com I this
More informationKeywords- Fixed point, Complete metric space, semi-compatibility and weak compatibility mappings.
[FRTSSDS- Jue 8] ISSN 48 84 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
More informationA Note On L 1 -Convergence of the Sine and Cosine Trigonometric Series with Semi-Convex Coefficients
It. J. Ope Problems Comput. Sci. Math., Vol., No., Jue 009 A Note O L 1 -Covergece of the Sie ad Cosie Trigoometric Series with Semi-Covex Coefficiets Xhevat Z. Krasiqi Faculty of Educatio, Uiversity of
More informationCommon Fixed Point Theorem for Expansive Maps in. Menger Spaces through Compatibility
Iteratioal Mathematical Forum 5 00 o 63 347-358 Commo Fixed Poit Theorem for Expasive Maps i Meger Spaces through Compatibility R K Gujetiya V K Gupta M S Chauha 3 ad Omprakash Sikhwal 4 Departmet of Mathematics
More informationCOMMON FIXED POINT THEOREM OF FOUR MAPPINGS IN CONE METRIC SPACE
Joural of Mathematical Scieces: Avaces a Applicatios Volume 5 Number 00 Pages 87-96 COMMON FIXED POINT THEOREM OF FOUR MAPPINGS IN CONE METRIC SPACE XIAN ZHANG School of Scieces Jimei Uiversit Xiame 60
More informationThe value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng
The value of Baach limits o a certai sequece of all ratioal umbers i the iterval 0, Bao Qi Feg Departmet of Mathematical Scieces, Ket State Uiversity, Tuscarawas, 330 Uiversity Dr. NE, New Philadelphia,
More informationON BI-SHADOWING OF SUBCLASSES OF ALMOST CONTRACTIVE TYPE MAPPINGS
Vol. 9, No., pp. 3449-3453, Jue 015 Olie ISSN: 190-3853; Prit ISSN: 1715-9997 Available olie at www.cjpas.et ON BI-SHADOWING OF SUBCLASSES OF ALMOST CONTRACTIVE TYPE MAPPINGS Awar A. Al-Badareh Departmet
More informationIntroduction to Optimization Techniques
Itroductio to Optimizatio Techiques Basic Cocepts of Aalysis - Real Aalysis, Fuctioal Aalysis 1 Basic Cocepts of Aalysis Liear Vector Spaces Defiitio: A vector space X is a set of elemets called vectors
More information1. Introduction. g(x) = a 2 + a k cos kx (1.1) g(x) = lim. S n (x).
Georgia Mathematical Joural Volume 11 (2004, Number 1, 99 104 INTEGRABILITY AND L 1 -CONVERGENCE OF MODIFIED SINE SUMS KULWINDER KAUR, S. S. BHATIA, AND BABU RAM Abstract. New modified sie sums are itroduced
More informationHomework 4. x n x X = f(x n x) +
Homework 4 1. Let X ad Y be ormed spaces, T B(X, Y ) ad {x } a sequece i X. If x x weakly, show that T x T x weakly. Solutio: We eed to show that g(t x) g(t x) g Y. It suffices to do this whe g Y = 1.
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationEquivalent Banach Operator Ideal Norms 1
It. Joural of Math. Aalysis, Vol. 6, 2012, o. 1, 19-27 Equivalet Baach Operator Ideal Norms 1 Musudi Sammy Chuka Uiversity College P.O. Box 109-60400, Keya sammusudi@yahoo.com Shem Aywa Maside Muliro Uiversity
More informationReal Analysis Fall 2004 Take Home Test 1 SOLUTIONS. < ε. Hence lim
Real Aalysis Fall 004 Take Home Test SOLUTIONS. Use the defiitio of a limit to show that (a) lim si = 0 (b) Proof. Let ε > 0 be give. Defie N >, where N is a positive iteger. The for ε > N, si 0 < si
More informationExistence Theorem for Abstract Measure Delay Integro-Differential Equations
Applied ad Computatioal Mathematics 5; 44: 5-3 Published olie Jue 4, 5 http://www.sciecepublishiggroup.com/j/acm doi:.648/j.acm.544. IN: 38-565 Prit; IN: 38-563 Olie istece Theorem for Abstract Measure
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationIJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor
IJITE Vol Issue-, (November 4) ISSN: 3-776 ATTRACTIVITY OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION Guagfeg Liu School of Zhagjiagag Jiagsu Uiversit of Sciece ad Techolog, Zhagjiagag, Jiagsu 56,PR
More informationA NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p
A NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p SEICK KIM Abstract. Assume that Ω is a bouded domai i R with 2. We study positive solutios to the problem, u = u p i Ω, u(x) as x Ω, where p > 1. Such solutios
More informationPositive solutions of singular (k,n-k) conjugate eigenvalue problem
Joural of Applied Mathematics & Bioiformatics, vol.5, o., 5, 85-97 ISSN: 79-66 (prit), 79-699 (olie) Sciepress Ltd, 5 Positive solutios of sigular (k,-k) cojugate eigevalue problem Shujie Tia ad Wei Gao
More informationBrief Review of Functions of Several Variables
Brief Review of Fuctios of Several Variables Differetiatio Differetiatio Recall, a fuctio f : R R is differetiable at x R if ( ) ( ) lim f x f x 0 exists df ( x) Whe this limit exists we call it or f(
More informationThe average-shadowing property and topological ergodicity
Joural of Computatioal ad Applied Mathematics 206 (2007) 796 800 www.elsevier.com/locate/cam The average-shadowig property ad topological ergodicity Rogbao Gu School of Fiace, Najig Uiversity of Fiace
More informationSome vector-valued statistical convergent sequence spaces
Malaya J. Mat. 32)205) 6 67 Some vector-valued statistical coverget sequece spaces Kuldip Raj a, ad Suruchi Padoh b a School of Mathematics, Shri Mata Vaisho Devi Uiversity, Katra-82320, J&K, Idia. b School
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationarxiv: v2 [math.fa] 21 Feb 2018
arxiv:1802.02726v2 [math.fa] 21 Feb 2018 SOME COUNTEREXAMPLES ON RECENT ALGORITHMS CONSTRUCTED BY THE INVERSE STRONGLY MONOTONE AND THE RELAXED (u, v)-cocoercive MAPPINGS E. SOORI Abstract. I this short
More informationFUNDAMENTALS OF REAL ANALYSIS by
FUNDAMENTALS OF REAL ANALYSIS by Doğa Çömez Backgroud: All of Math 450/1 material. Namely: basic set theory, relatios ad PMI, structure of N, Z, Q ad R, basic properties of (cotiuous ad differetiable)
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationMcGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected problems
McGill Uiversity Math 354: Hoors Aalysis 3 Fall 212 Assigmet 3 Solutios to selected problems Problem 1. Lipschitz fuctios. Let Lip K be the set of all fuctios cotiuous fuctios o [, 1] satisfyig a Lipschitz
More informationWeighted Approximation by Videnskii and Lupas Operators
Weighted Approximatio by Videsii ad Lupas Operators Aif Barbaros Dime İstabul Uiversity Departmet of Egieerig Sciece April 5, 013 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig
More informationDANIELL AND RIEMANN INTEGRABILITY
DANIELL AND RIEMANN INTEGRABILITY ILEANA BUCUR We itroduce the otio of Riema itegrable fuctio with respect to a Daiell itegral ad prove the approximatio theorem of such fuctios by a mootoe sequece of Jorda
More informationUniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations
Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 3 Issue Versio 0 Year 03 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationAPPROXIMATE FUNCTIONAL INEQUALITIES BY ADDITIVE MAPPINGS
Joural of Mathematical Iequalities Volume 6, Number 3 0, 46 47 doi:0.753/jmi-06-43 APPROXIMATE FUNCTIONAL INEQUALITIES BY ADDITIVE MAPPINGS HARK-MAHN KIM, JURI LEE AND EUNYOUNG SON Commuicated by J. Pečarić
More informationReal Numbers R ) - LUB(B) may or may not belong to B. (Ex; B= { y: y = 1 x, - Note that A B LUB( A) LUB( B)
Real Numbers The least upper boud - Let B be ay subset of R B is bouded above if there is a k R such that x k for all x B - A real umber, k R is a uique least upper boud of B, ie k = LUB(B), if () k is
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationOn Strictly Point T -asymmetric Continua
Volume 35, 2010 Pages 91 96 http://topology.aubur.edu/tp/ O Strictly Poit T -asymmetric Cotiua by Leobardo Ferádez Electroically published o Jue 19, 2009 Topology Proceedigs Web: http://topology.aubur.edu/tp/
More informationOn Orlicz N-frames. 1 Introduction. Renu Chugh 1,, Shashank Goel 2
Joural of Advaced Research i Pure Mathematics Olie ISSN: 1943-2380 Vol. 3, Issue. 1, 2010, pp. 104-110 doi: 10.5373/jarpm.473.061810 O Orlicz N-frames Reu Chugh 1,, Shashak Goel 2 1 Departmet of Mathematics,
More informationStatistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function
Applied Mathematics, 0,, 398-40 doi:0.436/am.0.4048 Published Olie April 0 (http://www.scirp.org/oural/am) Statistically Coverget Double Sequece Spaces i -Normed Spaces Defied by Orlic Fuctio Abstract
More informationThe Australian Journal of Mathematical Analysis and Applications
The Australia Joural of Mathematical Aalysis ad Applicatios Volume 6, Issue 1, Article 10, pp. 1-10, 2009 DIFFERENTIABILITY OF DISTANCE FUNCTIONS IN p-normed SPACES M.S. MOSLEHIAN, A. NIKNAM AND S. SHADKAM
More informationCOMMON FIXED POINTS OF COMPATIBLE MAPPINGS
Iterat. J. Math. & Math. Sci. VOL. 13 NO. (1990) 61-66 61 COMMON FIXED POINTS OF COMPATIBLE MAPPINGS S.M. KANG ad Y.J. CHO Departmet of Mathematics Gyeogsag Natioal Uiversity Jiju 660-70 Korea G. JUNGCK
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More information