Research Article Some Extensions of Banach s Contraction Principle in Complete Cone Metric Spaces

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1 Hindwi Publishing Corportion Fixed Point Theory nd Applictions Volume 2008, Article ID , 11 pges doi: /2008/ Reserch Article Some Extensions of Bnch s Contrction Principle in Complete Cone Metric Spces P. Rj nd S. M. Vezpour Deprtment of Mthemtics nd Computer Sciences, Amirkbir University of Technology, P.O. Box 15914, Hfez Avenue, Tehrn, Irn Correspondence should be ddressed to S. M. Vezpour, vez@ut.c.ir Received 10 December 2007; Revised 2 June 2008; Accepted 23 June 2008 Recommended by Billy Rhodes In this pper we consider complete cone metric spces. We generlize some definitions such s c-nonexpnsive nd c, λ -uniformly loclly contrctive functions f-closure, c-isometric in cone metric spces, nd certin fixed point theorems will be proved in those spces. Among other results, we prove some interesting pplictions for the fixed point theorems in cone metric spces. Copyright q 2008 P. Rj nd S. M. Vezpour. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. 1. Introduction The study of fixed points of functions stisfying certin contrctive conditions hs been t the center of vigorous reserch ctivity, for exmple see 1 5 nd it hs wide rnge of pplictions in different res such s nonliner nd dptive control systems, prmeterize estimtion problems, frctl imge decoding, computing mgnetosttic fields in nonliner medium, nd convergence of recurrent networks, see Recently, Hung nd Zhng generlized the concept of metric spce, replcing the set of rel numbers by n ordered Bnch spce nd obtined some fixed point theorems for mpping stisfying different contrctive conditions 11. The study of fixed point theorems in such spces is followed by some other mthemticins, see The im of this pper is to generlize some definitions such s c-nonexpnsive nd c, λ -uniformly loclly contrctive functions in these spces nd by using these definitions, certin fixed point theorems will be proved. Let E be rel Bnch spce. A subset P of E is clled cone if nd only if the following hold: i P is closed, nonempty, nd P / {0}, ii, b R,,b 0, nd x, y P imply tht x by P, iii x P nd x P imply tht x 0.

2 2 Fixed Point Theory nd Applictions Given cone P E, we define prtil ordering with respect to P by x y if nd only if y x P. We will write x<yto indicte tht x y but x / y, while x y will stnd for y x int P, where int P denotes the interior of P. The cone P is clled norml if there is number K>0 such tht 0 x y implies x K y, for every x, y E. The lest positive number stisfying bove is clled the norml constnt of P. There re non-norml cones. Exmple 1.1. Let E C 2 R 0, 1 with the norm f f f, nd consider the cone P {f E : f 0}. For ech K 1, put f x x nd g x x 2K. Then, 0 g f, f 2, nd g 2K 1. Since K f < g, K is not norml constnt of P 16. In the following, we lwys suppose E is rel Bnch spce, P is cone in E with int P /,nd is prtil ordering with respect to P. Let X be nonempty set. As it hs been defined in 11, functiond : X X E is clled cone metric on X if it stisfies the following conditions: i d x, y 0, for every x, y X, ndd x, y 0ifnd only if x y, ii d x, y d y, x, for every x, y X, iii d x, y d x, z d y, z, for every x, y, z X. Then X, d is clled cone metric spce. Exmple 1.2. Let E l 1,P {{x n } n 1 E : x n 0, for ll n}, X, ρ metric spce, nd d : X X E defined by d x, y {ρ x, y /2 n } n 1. Then X, d is cone metric spce nd the norml constnt of P is equl to The sequence {x n } in X is clled to be convergent to x X if for every c E with 0 c, there is n 0 N such tht d x n,x c, for every n n 0, nd is clled Cuchy sequence if for every c E with 0 c, there is n 0 N such tht d x m,x n c, for every m, n n 0.A cone metric spce X, d is sid to be complete cone metric spce if every Cuchy sequence in X is convergent to point of X. A self-mpt on X is sid to be continuous if lim n x n x implies tht lim n T x n T x, for every sequence {x n } in X. The following lemms re useful for us to prove our min results. Lemm 1.3 see 11, Lemm 1. Let X, d be cone metric spce, P be norml cone with norml constnt K.Let{x n } be sequence in X. Then, {x n } converges to x if nd only if lim n d x n,x 0. Lemm 1.4 see 11, Lemm 3. Let X, d be cone metric spce, {x n } be sequence in X.If{x n } is convergent, then it is Cuchy sequence, too. Lemm 1.5 see 11, Lemm 4. Let X, d be cone metric spce, P be norml cone with norml constnt K. Let{x n } be sequence in X. Then, {x n } is Cuchy sequence if nd only if lim m,n d x m,x n 0. The following exmple is cone metric spce, see 11. Exmple 1.6. Let E R 2,P { x, y E x, y 0}, X R, ndd : X X E such tht d x, y x y,α x y, where α 0 is constnt. Then X, d is cone metric spce.

3 P. Rj nd S. M. Vezpour 3 2. Certin nonexpnsive mppings Definition 2.1. Let X, d be cone metric spce, where P is cone nd f : X X is function. Then f is sid to be c-nonexpnsive, for 0 c,if d f x,f y ) d x, y, 2.1 for every x, y X with d x, y c. If we hve d f x,f y ) <d x, y, 2.2 for every x, y X with x / y nd d x, y c, then f is clled c-contrctive. Definition 2.2. Let X, d be cone metric spce, where P is cone. A point y Y X is sid to belong to the f-closure of Y ndisdenotedbyy Y f,iff Y Y nd there re point x Y nd n incresing sequence {n i } N such tht lim i f n i x y. Definition 2.3. Let X, d be cone metric spce, where P is cone. A sequence {x i } X is sid to be c-isometric sequence if d x m,x n ) d xm k,x n k ), 2.3 for ll k, m N with d x m,x n <c.apointx X is sid to generte c-isometric sequence under the function f : X X, if{f n x } is c-isometric sequence. Theorem 2.4. Let X, d be cone metric spce, where P is norml cone with norml constnt K. If f : X X is c-nonexpnsive, for some 0 c, nd x X f, then there is n incresing sequence {m j } N such tht lim j f m j x x. Proof. Since x X f, there re y X nd sequence {n i } such tht lim i f n i y x. If f m y x, for some m N. Putm j n j m n j >m, is sequence s desired, then {m j } is sequence with desired property. Otherwise, for ɛ>0, fix δ, 0 <δ<ɛ. Choose c E with 0 c nd K c <δ. Then there is i i c such tht d x, f n i j y ) c 4, 2.4 for every j N {0}. Sobyc-nonexpnsivity of f nd putting j 0, we hve d f n i k n i x, f n i k y ) < c 4, 2.5 for every k N. Therefore, d f n i y, f n i k y ) d x, f n i y ) d x, f n i k y ) c 2, 2.6 for every k N. Hence, d x, f n i 1 n i x ) d x, f n i y ) d f n i y,f n i 1 y ) d f n i 1 y,f n i 1 n i x ) < c 4 c 2 c c, 4 2.7

4 4 Fixed Point Theory nd Applictions which implies d x, f n i 1 n i x ) K c <δ. 2.8 Put m 1 n i 1 n i nd suppose tht m 1 <m 2 < <m j 1 chosen such tht d x, f m i y ) 1 min d x, f m y ), m 1,...,m i 1 for i 2, 3,...,j 1. We put m j n l 1 n l, where l is chosen so s to stisfy d x, f l j y c/4, with δ replced by { min δ, 1 min d x, f m y ) } m 1,...,m i 1 It is esily seen tht the sequence {m j } tht is defined in the bove stisfies the requirements of the theorem. The proof is complete. Theorem 2.5. Let X, d be cone metric spce, where P is norml cone with norml constnt K. If f : X X is c-nonexpnsive function, then every x X f genertes c-isometric sequence. Proof. By contrdiction, suppose tht there re k, m, n N such tht d f m x,f n x <cnd p d f m x,f n x d f m k x,f n k x / 0. By the ssumption, p P nd 0 <p d f m x,f n x ) d f m l x,f n l x ), 2.11 for l k, l N. It mens tht p K d f m x,f n x ) d f m k x,f n k x ), 2.12 for l k, l N. Also by the ssumption nd Theorem 2.4, lim f n j f l x ) limf nj l x f l x j j Put δ p nd choose c E such tht 0 c nd c < 1/K 2 δ. ByLemm 1.4, there is i N such tht d f m n j x,f m x ) c 2, d f n n j x,f n x ) c 2, 2.14 for every j i. However, d f m x,f n x ) d f m x,f m n j x ) d f m n j x,f n n j x ) d f n n j x,f n x ) c d f m n j x,f n n j x ) c So It mens tht d f m x,f n x ) d f m n j x,f n n j x ) c. d f m x,f n x ) d f m n j x,f n n j x ) K c < 1 K δ, tht is contrdiction by 2.12, forn j complete. mx{n i,k}. Therefore, p 0 nd the proof is The following corollry implies immeditely. Corollry 2.6. Let X, d be cone metric spce, where P is norml cone with norml constnt K. If f : X X is nonexpnsive function nd x X f genertes n isometric sequence.

5 P. Rj nd S. M. Vezpour 5 3. Extended contrction principle We hve the following generlized form of Bnch s contrction for cone metric spces. Theorem 3.1 see 11, Theorem 1. Let X, d be complete cone metric spce, P be norml cone with norml constnt K. Suppose the mpping T : X X stisfies the contrctive condition d T x,t y ) βd x, y, 3.1 for every x, y X, whereβ 0, 1 is constnt. Then T hs unique fixed point in X, nd for ny x X, the sequence {T n x } converges to the fixed point. It is nturl to sk whether the mentioned theorem could be modified if 3.1 holds for just sufficiently close points. To be more specific, we introduce the following definitions. Definition 3.2. Let X, d be cone metric spce. A function f : X X is sid to be loclly contrctive, if for every x X there is c X with 0 c nd 0 λ<1 such tht d f p,f q ) λd p, q, 3.2 for every p, q {y X : d x, y c}. Afunctionf : X X is sid to be c, λ -uniformly loclly contrctive if it is loclly contrctive nd both c nd λ do not depend on x. It is esy to find cone metric spces which dmit loclly contrctive which re not globlly contrctive. Exmple 3.3. Let E R 2, X { x, y x cos t, y sin t; 0 t 3π 2 } R 2, 3.3 nd P { x, y E x, y 0}, ndd : X X E such tht d x, y x y,α x y, where α 0 is constnt. It is esily checked tht X, d is cone metric spce. Suppose tht f cos t, sin t cos t/2, sin t/2. It is not hrd to see tht f is loclly contrctive but not globlly contrctive. Note tht every loclly contrctive function is c-nonexpnsive for some c 0. Definition 3.4. A cone metric spce X, d is clled c-chinble, for 0 c, if for every, b X, there is finite set of points x 0,x 1,...,x n b, n depends on both nd b, such tht d x i 1,x i <c,fori, 1 i n. Exmple 3.5. It is esily seen tht the cone metric spce tht is defined in Exmple 1.6 is c- chinble. Theorem 3.6. Let X, d be complete c-chinble cone metric spce, P be norml cone with norml constnt K. Iff : X X is c, β -uniformly loclly contrctive, then there is unique point z X such tht f z z. Proof. Let x X be rbitrry. Consider the c-chin x x 0,x 1,...,x n f x. We hve d x, f x ) n d ) x i 1,x i <nc. i 1 3.4

6 6 Fixed Point Theory nd Applictions We hve d f x i 1 xi )) βd xi 1,x i ) <βc, 3.5 for every 1 i n, nd by induction d f m x i 1 m x i )) <βd f m 1 x i 1 m 1 x i )) < <β m c, 3.1 for every m N. Hence d f m x,f m 1 x ) n d f m ) x i 1,f m )) x i <β m nc, i for every m N. Now,form, p N with m<p, we hve It mens tht d f m x,f p x ) p 1 d f i x,f i 1 x ) <nc β m β p 1) <nc βm 1 β. 3.3 i m d f m x,f p x ) n c β m 1 β, 3.4 for m, p N with m < p. Since k 0, 1, then lim m,p d f m x,f p x 0. So lim m,p d f m x,f p x 0, nd by Lemm 1.5, {f m x } is Cuchy sequence. Since X is complete, then lim m f m x z, for some z X. From the continuity of f it follows tht f z z. To complete the proof it is enough to show tht z is the unique point with this property. To do this, suppose tht there is z X such tht f z z.letz x 0,x 1,...,x t z be c-chin. By 3.1,weobtin d f z,f z )) d f l z,f l z )) t d f l ) x i 1,f l )) x i <β l tc. i It mens tht d z, z ) d f z,f z )) β l t c. 3.6 Since β 0, 1, then d z, z 0ndz z. This completes the proof. Corollry 3.7. Let X, d be complete c-chinble cone metric spce, P be norml cone with norml constnt K.Iff is one to one, c, λ -uniformly loclly expnsive function of Y onto X,whereY X, then f hs unique fixed point. Proof. It is n immedite consequence of the fct tht for the inverse function ll ssumptions of the Theorem 3.6 re stisfied. In the following theorem we investigte kind of functions which re not necessrily contrctions but hve unique fixed point. First, we will prove the following lemm which will be used lter.

7 P. Rj nd S. M. Vezpour 7 Lemm 3.8. Let X, d be complete cone metric spce, P be norml cone with norml constnt K, f : X Xbe continuous function, nd β 0, 1 such tht for every x X, there is n n x N such tht d f n x x,f n x y ) βd x, y, 3.7 for every y X. Then for every x X, r x sup n d f n x x,x is finite. Proof. Let x X nd l x mx{ d f j x,x : j 1, 2,...,n x }.Ifn N nd n>n x, then there is s N {0} such tht sn x <n s 1 n x nd we hve d f n x,x ) d f n x f n n x x n x x ) d f n x x,x ) βd f n n x x,x ) d f n x x,x ) d f n x x,x ) β d f n n x x,f n x x ) d f n x x,x )) 3.8 d f n x x,x ) β βd f n 2n x x,x ) d f n x x,x )) d f n x x,x ) 1 β β 2 β s). It mens tht d f n x,x ) 1 K d f n x x,x ) 1 K l x β 1 β Hence r x is finite nd the proof is complete. Theorem 3.9. Let X, d be complete cone metric spce, P be norml cone with norml constnt K, β 0, 1, nd f : X X be continuous function such tht for every x X, there is n n x N such tht d f n x x,f n x y ) βd x, y, 3.10 for every y X. Thenf hs unique fixed point u X nd lim n f n x 0 u, for every x 0 X. Proof. Let x 0 X be rbitrry, nd m 0 n x 0. Define the sequence x 1 f m 0 x 0,x i 1 f m i x i, where m i n x i. We show tht {x n } is Cuchy sequence. We hve d ) x n 1,x n d f m n 1 f m n ) xn 1,f m n 1 )) xn 1 βd f m ) ) n x n 1,xn β n d f m ) ) n x 0,x0, for every n N. SobyLemm 3.8, d x n 1,x n Kβ n r x 0, for every n N. Now, suppose tht m, n N with m<n, we hve d ) n 1 x n,x m K d ) x i 1,x i K βn 1 β r ) x i m Since lim n β n / 1 β 0, then lim m,n d x n,x m 0, nd by Lemm 1.5, {x n } is Cuchy sequence. Completeness of X implies tht lim n x n u, for some u X. Now,we

8 8 Fixed Point Theory nd Applictions show tht f u u. By contrdiction, suppose tht f u / u. We clim tht there re c, d E such tht 0 c, 0 d nd B c u nd B d f u hve no intersection, where B e x {y X : d x, y e}, for every x X nd 0 e. If not, then suppose tht ɛ>0, nd choose c E with 0 c nd K c <ɛ. Then clerly, 0 c/2ndforz B c/2 u B c/2 f u, we hve d u, f u ) d u, z d z, f u ) c It mens tht d u, f u K c <ɛ. Since ɛ>0 is rbitrry, then d u, f u 0ndso f u u, contrdiction. Therefore, ssume tht c, d E with 0 c, 0 d re such tht B c u B d f u. Since f is continuous, then there is n 0 N such tht x n B c u nd f x n B d f u, for every n N nd n n 0. Then d f x n ),xn ) d f m n 1 f xn 1 m n 1 xn 1 )) βd f xn 1 ),xn 1 ) β n d f x 0 ),x0 ), 3.14 for every n N. It mens tht d f x n,x n Kβ n d f x 0,x 0, for every n N. So lim n d f x n,x n 0, contrdiction. Thus f u u. The uniqueness of the fixed point follows immeditely from the hypothesis. Now, suppose tht x 0 X is rbitrry. To show tht lim n f n x 0 u, set r 0 mx { d f m x 0 ) : m 0, 1,...,n u 1 } If n is sufficiently lrge, then n rn u q,forr>0nd0 q<n u, nd we hve d f n x 0 ) d f rn u q x 0 n u u ) βd f r 1 n u q x 0 )) β r d f q x 0 ) It mens tht d f n x 0 ) Kβ r d f q x 0 ) Kβ r r Therefore, lim n d f n x 0,u 0 nd hence lim n f n x 0 u. This completes the proof. Definition Let X be n ordered spce. A function ϕ : X X is sid to be comprison function if for every x, y X, x y, implies tht ϕ x ϕ y, ϕ x x, nd lim n ϕ n x 0, for every x X. Exmple Let E R 2,P { x, y E x, y 0}. It is esy to check tht ϕ : E E, with ϕ x, y x, y, for some 0, 1 is comprison function. Also if ϕ 1,ϕ 2 re two comprison functions over R, then ϕ x, y ϕ 1 x,ϕ 2 y is lso comprison function over E. Recll tht for cone metric spce X, d, where P is cone with norml constnt K, since for every x X, x x, nd therefore x K x, then K 1. Theorem Let X, d be complete cone metric spce, where P is norml cone with norml constnt K. Letf : X X be function such tht there exists comprison function ϕ : P P such tht d f x,f y ) ϕ d x, y ), 3.18 for every x, y X. Thenf hs unique fixed point.

9 P. Rj nd S. M. Vezpour 9 Proof. Let x 0 X be rbitrry. We hve d f n x 0 n 1 x 0 )) ϕ d f n 1 x 0 n x 0 ))) ϕ 2 d f n 2 x 0 n 1 x 0 ))) 3.19 ϕ n d x 0 )), for every n N. Since lim n ϕ n d x 0 0, for n rbitrry ɛ>0, we cn choose n N such tht d f n ) n 1 ) )) ɛ K ϕ c x 0 <, 3.20 K for every n n 0 nd c P with c < ɛ K, 1 ϕ c ϕ d f n ) x 2 0,f n 1 ))) x K For n n 0, we hve d f n ) n 2 )) x 0 d f n ) n 1 )) x 0 d f n 1 ) n 2 )) x So d f n ) n 2 )) x 0 K d f n ) n 1 )) x 0 K d f n 1 ) n 2 )) x 0 ɛ K ϕ c ) <K K 2 ϕ d f n ) n 1 ))) x 0 K ɛ. Now, for every n n 0, we hve d f n ) n 3 )) x 0 d f n ) n 1 )) x 0 d f n 1 ) n 3 )) x 0. Since K 1, then we hve d f n ) n 3 )) x 0 K d f n ) n 1 )) x 0 K d f n 1 ) n 3 )) x 0 ɛ K ϕ c ) <K K 2 ϕ d f n ) n 2 ))) x 0 K ɛ By induction, we hve d f n x 0,f n r x 0 < ɛ, for every r N nd n n 0. Hence by Lemm 1.5, we hve {f n x 0 } is Cuchy sequence in X, d. So lim n f n x 0 x, for some x X. Now, we will prove f x x. Since lim n f n x 0 x, for every c 0, there exists n c N such tht for every n n c, we hve d f n x 0,x <c. Therefore, d x,f x )) d x,f n 1 )) x 0 d f f n )) x )) d x,f n 1 )) x 0 ϕ d f n ) x 0,x )) 3.26 <d x,f n 1 x 0 )) d f n x 0 ),x ) < 2c, for every c 0. So f x x. For the uniqueness of the fixed point, suppose tht there exists y X such tht f y y. Hence d x,y ) d f n x n y )) ϕ n d y,x )) So d x,y ) K ϕ n d y,x )). Since lim n ϕ n d y,x 0, then x y nd the proof is complete. 3.28

10 10 Fixed Point Theory nd Applictions 4. Applictions Theorem 4.1. Consider the integrl eqution Suppose tht x t b k t, s, x s ) ds g t, t, b. i i k :, b, b R n R n nd g :, b R n ; ii k t, s, : R n R n is incresing for every t, s, b ; iii there exists continuous function p :, b, b R nd comprison function ϕ : R 2 R 2 such tht k t, s, u k t, s, v,α k t, s, u k t, s, v ) p t, s,αp t, s ) ϕ d u, v ), 4.1 for every t, s, b, u,v R n ; b iv sup t,b p t, s,αp t, s ds 1. Then the integrl eqution i hs unique solution x in C, b, R n. Proof. Let X C, b, R n,p { x, y : x, y 0} R 2,nddefined f, g f g,α f g, for every f, g X. Then it is esily seen tht X, d is cone metric spce. Define A : C, b, R n C, b, R n,by For every x, y X, we hve Ax t : b Ax t Ay t,α Ax t Ay t ) k t, s, x s ) ds g t, t, b. 4.2 b [ ) )] k t, s, x s k t, s, y s ds,α b [ k t, s, x s ) k t, s, y s )] ds ) b ) ) b k t, s, x s k t, s, y s ds, α ) ) k t, s, x s k t, s, y s ds b p t, s,αp t, s ) ϕ x s y s,α x s y s ) ds ϕ ) b ) x y,α x y p t, s,αp t, s ds ϕ x y,α x y ). 4.3 Hence d Ax, Ay ϕ d x, y, for every x, y X. The conclusion follows now from Theorem 3.12.

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