Volume 2008, Article ID 768294,11pages doi:10.1155/2008/768294
Research Article
Some Extensions of Banach’s Contraction Principle in Complete Cone Metric Spaces
P. Raja and S. M. Vaezpour
Department of Mathematics and Computer Sciences, Amirkabir University of Technology, P.O. Box 15914, Hafez Avenue, Tehran, Iran
Correspondence should be addressed to S. M. Vaezpour,[email protected] Received 10 December 2007; Revised 2 June 2008; Accepted 23 June 2008 Recommended by Billy Rhoades
In this paper we consider complete cone metric spaces. We generalize some definitions such as c-nonexpansive andc, λ-uniformly locally contractive functionsf-closure,c-isometric in cone metric spaces, and certain fixed point theorems will be proved in those spaces. Among other results, we prove some interesting applications for the fixed point theorems in cone metric spaces.
Copyrightq2008 P. Raja and S. M. Vaezpour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The study of fixed points of functions satisfying certain contractive conditions has been at the center of vigorous research activity, for example see 1–5 and it has a wide range of applications in different areas such as nonlinear and adaptive control systems, parameterize estimation problems, fractal image decoding, computing magnetostatic fields in a nonlinear medium, and convergence of recurrent networks, see 6–10. Recently, Huang and Zhang generalized the concept of a metric space, replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mapping satisfying different contractive conditions 11. The study of fixed point theorems in such spaces is followed by some other mathematicians, see 12–15. The aim of this paper is to generalize some definitions such asc-nonexpansive andc, λ-uniformly locally contractive functions in these spaces and by using these definitions, certain fixed point theorems will be proved.
LetEbe a real Banach space. A subsetPofEis called a cone if and only if the following hold:
iPis closed, nonempty, andP /{0},
iia, b∈R, a, b0,andx, y∈Pimply thataxby∈P, iiix∈Pand−x∈P imply thatx0.
Given a coneP ⊂E, we define a partial orderingwith respect toP byxyif and only ify−x∈P. We will writex < yto indicate thatxybutx /y, whilexywill stand fory−x∈intP, where intPdenotes the interior ofP.
The coneP is called normal if there is a numberK >0 such that 0 x yimplies
||x|| K||y||, for everyx, y ∈ E. The least positive number satisfying above is called the normal constant ofP.
There are non-normal cones.
Example 1.1. LetE CR20,1with the norm||f|| ||f||∞||f||∞, and consider the cone P {f ∈E:f 0}. For eachK1, putfx xandgx x2K. Then, 0gf, ||f||2, and||g||2K1. SinceK||f||<||g||,Kis not normal constant ofP16.
In the following, we always supposeEis a real Banach space,P is a cone inEwith intP /∅, andis partial ordering with respect toP.
LetX be a nonempty set. As it has been defined in11, a functiond :X×X → Eis called a cone metric onXif it satisfies the following conditions:
idx, y0, for everyx, y∈X, anddx, y 0 if and only ifxy, iidx, y dy, x, for everyx, y∈X,
iiidx, ydx, z dy, z, for everyx, y, z∈X.
ThenX, dis called a cone metric space.
Example 1.2. LetE l1, P {{xn}n1 ∈ E : xn 0, for all n}, X, ρa metric space, and d:X×X →Edefined bydx, y {ρx, y/2n}n1. ThenX, dis a cone metric space and the normal constant ofPis equal to 116.
The sequence{xn}inXis called to be convergent tox∈Xif for everyc∈Ewith 0c, there isn0 ∈ Nsuch thatdxn, x c, for everyn n0, and is called a Cauchy sequence if for everyc ∈Ewith 0 c, there isn0 ∈Nsuch thatdxm, xn c, for everym, nn0. A cone metric spaceX, dis said to be a complete cone metric space if every Cauchy sequence inXis convergent to a point ofX. A self-mapTonXis said to be continuous if limn→∞xn x implies that limn→∞Txn Tx, for every sequence{xn}inX. The following lemmas are useful for us to prove our main results.
Lemma 1.3see11, Lemma 1. LetX, dbe a cone metric space,Pbe a normal cone with normal constantK. Let{xn}be a sequence inX. Then,{xn}converges toxif and only if limn→∞dxn, x 0.
Lemma 1.4see11, Lemma 3. LetX, dbe a cone metric space,{xn}be a sequence inX. If{xn} is convergent, then it is a Cauchy sequence, too.
Lemma 1.5 see11, Lemma 4. LetX, d be a cone metric space, P be a normal cone with normal constant K. Let{xn} be a sequence inX. Then,{xn} is a Cauchy sequence if and only if limm,n→∞dxm, xn 0.
The following example is a cone metric space, see11.
Example 1.6. LetE R2, P {x, y ∈E |x, y 0}, X R, andd: X×X → Esuch that dx, y |x−y|, α|x−y|, whereα0 is a constant. ThenX, dis a cone metric space.
2. Certain nonexpansive mappings
Definition 2.1. LetX, dbe a cone metric space, wherePis a cone andf :X→Xis a function.
Thenfis said to bec-nonexpansive, for 0c, if d
fx, fy
dx, y, 2.1
for everyx, y∈Xwithdx, yc. If we have d
fx, fy
< dx, y, 2.2
for everyx, y∈Xwithx /yanddx, yc, thenfis calledc-contractive.
Definition 2.2. LetX, dbe a cone metric space, wherePis a cone. A pointy∈Y ⊆Xis said to belong to thef-closure ofY and is denoted byy ∈Yf, iffY⊆ Y and there are a point x∈Y and an increasing sequence{ni} ⊆Nsuch that limi→∞fnix y.
Definition 2.3. LetX, dbe a cone metric space, whereP is a cone. A sequence{xi} ⊆ X is said to be ac-isometric sequence if
d xm, xn
d
xmk, xnk
, 2.3
for allk, m∈Nwithdxm, xn < c. A pointx∈Xis said to generate ac-isometric sequence under the functionf :X →X, if{fnx}is ac-isometric sequence.
Theorem 2.4. LetX, dbe a cone metric space, wherePis a normal cone with normal constantK.
Iff :X → Xisc-nonexpansive, for some 0c, andx∈Xf, then there is an increasing sequence {mj} ⊆Nsuch that limj→∞fmjx x.
Proof. Since x ∈ Xf, there are y ∈ X and a sequence{ni} such that limi→∞fniy x. If fmy x, for somem∈N. Putmj nj−mnj > m, is a sequence as desired, then{mj}is a sequence with desired property. Otherwise, for >0, fixδ, 0 < δ < . Choosec ∈Ewith 0candK||c||< δ. Then there isiicsuch that
d
x, fnijy c
4, 2.4
for everyj ∈N∪ {0}. So byc-nonexpansivity offand puttingj 0, we have d
fnik−nix, fniky
< c
4, 2.5
for everyk∈N. Therefore, d
fniy, fniky
d
x, fniy d
x, fniky c
2, 2.6
for everyk∈N. Hence, d
x, fni1−nix
d
x, fniy d
fniy, fni1y d
fni1y, fni1−nix
< c 4 c
2 c 4 c,
2.7
which implies
d
x, fni1−nixKc< δ. 2.8
Putm1 ni1−niand suppose thatm1< m2<· · ·< mj−1chosen such that d
x, fmiy 1 2 min
m1,...,mi−1
d
x, fmy, 2.9
fori2,3, . . . , j−1. We putmjnl1−nl, wherelis chosen so as to satisfydx, fljyc/4, withδreplaced by
min
δ,1 2 min
m1,...,mi−1
d
x, fmy
. 2.10
It is easily seen that the sequence{mj}that is defined in the above satisfies the requirements of the theorem. The proof is complete.
Theorem 2.5. LetX, dbe a cone metric space, wherePis a normal cone with normal constantK.
Iff:X→Xis ac-nonexpansive function, then everyx∈Xf generates ac-isometric sequence.
Proof. By contradiction, suppose that there arek, m, n∈Nsuch thatdfmx, fnx< cand pdfmx, fnx−dfmkx, fnkx/0. By the assumption,p∈Pand
0< pd
fmx, fnx
−d
fmlx, fnlx
, 2.11
forlk, l∈N. It means that pKd
fmx, fnx
−d
fmkx, fnkx, 2.12
forlk, l∈N. Also by the assumption andTheorem 2.4,
j→∞limfnj flx
lim
j→∞fnjlx flx. 2.13
Putδ pand choosec ∈ Esuch that 0 candc < 1/K2δ. ByLemma 1.4, there is i∈Nsuch that
d
fmnjx, fmx c
2, d
fnnjx, fnx c
2, 2.14
for everyj i. However, d
fmx, fnx
d
fmx, fmnjx d
fmnjx, fnnjx d
fnnjx, fnx
cd
fmnjx, fnnjx c
2. 2.15
So
d
fmx, fnx
−d
fmnjx, fnnjx
c. 2.16
It means that
d
fmx, fnx
−d
fmnjx, fnnjxKc< 1
Kδ, 2.17
that is a contradiction by 2.12, for nj max{ni, k}. Therefore, p 0 and the proof is complete.
The following corollary implies immediately.
Corollary 2.6. LetX, dbe a cone metric space, whereP is a normal cone with normal constantK.
Iff:X→Xis a nonexpansive function andx∈Xf generates an isometric sequence.
3. Extended contraction principle
We have the following generalized form of Banach’s contraction for cone metric spaces.
Theorem 3.1see11, Theorem 1. LetX, dbe a complete cone metric space,Pbe a normal cone with normal constantK. Suppose the mappingT :X→Xsatisfies the contractive condition
d
Tx, Ty
βdx, y, 3.1
for everyx, y∈X, whereβ∈0,1is a constant. ThenT has a unique fixed point inX, and for any x∈X, the sequence{Tnx}converges to the fixed point.
It is natural to ask whether the mentioned theorem could be modified if3.1holds for just sufficiently close points. To be more specific, we introduce the following definitions.
Definition 3.2. LetX, dbe a cone metric space. A functionf : X → X is said to be locally contractive, if for everyx∈Xthere isc∈Xwith 0cand 0λ <1 such that
d
fp, fq
λdp, q, 3.2
for everyp, q ∈ {y ∈X :dx, y c}. A functionf :X → Xis said to bec, λ-uniformly locally contractive if it is locally contractive and bothcandλdo not depend onx.
It is easy to find cone metric spaces which admit locally contractive which are not globally contractive.
Example 3.3. LetER2, X
x, y|xcost, ysint; 0t 3π 2
⊆R2, 3.3
andP {x, y ∈ E | x, y 0}, andd : X×X → Esuch thatdx, y |x−y|, α|x−y|, whereα0 is a constant. It is easily checked thatX, dis a cone metric space. Suppose that fcost,sint cost/2,sint/2. It is not hard to see thatf is locally contractive but not globally contractive.
Note that every locally contractive function isc-nonexpansive for somec0.
Definition 3.4. A cone metric spaceX, dis calledc-chainable, for 0c, if for everya, b∈X, there is a finite set of pointsa x0, x1, . . . , xn b, ndepends on bothaand b, such that dxi−1, xi< c, fori, 1in.
Example 3.5. It is easily seen that the cone metric space that is defined inExample 1.6 isc- chainable.
Theorem 3.6. LetX, dbe a completec-chainable cone metric space,Pbe a normal cone with normal constantK. Iff :X→X isc, β-uniformly locally contractive, then there is a unique pointz∈X such thatfz z.
Proof. Letx∈Xbe arbitrary. Consider thec-chainxx0, x1, . . . , xnfx. We have d
x, fx
n
i1
d xi−1, xi
< nc. 3.4
We have
d f
xi−1 , f
xi
βd
xi−1, xi
< βc, 3.5
for every 1in, and by induction d
fm xi−1
, fm xi
< βd fm−1
xi−1 , fm−1
xi
<· · ·< βmc, 3.1
for everym∈N. Hence
d
fmx, fm1x
n
i1
d fm
xi−1 , fm
xi
< βmnc, 3.2
for everym∈N. Now, form, p∈Nwithm < p, we have
d
fmx, fpx p−1
im
d
fix, fi1x
< nc
βm· · ·βp−1
< nc βm
1−β. 3.3 It means that
d
fmx, fpxnc βm
1−β, 3.4
for m, p ∈ N with m < p. Since k ∈ 0,1, then limm,p→∞dfmx, fpx 0. So limm,p→∞dfmx, fpx 0, and by Lemma 1.5, {fmx} is a Cauchy sequence. Since X is complete, then limm→∞fmx z, for somez∈X. From the continuity off it follows that fz z. To complete the proof it is enough to show that zis the unique point with this property. To do this, suppose that there isz∈Xsuch thatfz z. Letzx0, x1, . . . , xtz be ac-chain. By3.1, we obtain
d
fz, f z
d
flz, fl z
t
i1
d fl
xi−1 , fl
xi
< βltc. 3.5
It means that
d
z, zd
fz, f
zβltc. 3.6
Sinceβ∈0,1, thendz, z0 andzz. This completes the proof.
Corollary 3.7. LetX, dbe a completec-chainable cone metric space,Pbe a normal cone with normal constantK. Iffis a one to one,c, λ-uniformly locally expansive function ofYontoX, whereY ⊆X, thenfhas a unique fixed point.
Proof. It is an immediate consequence of the fact that for the inverse function all assumptions of theTheorem 3.6are satisfied.
In the following theorem we investigate a kind of functions which are not necessarily contractions but have a unique fixed point. First, we will prove the following lemma which will be used later.
Lemma 3.8. LetX, dbe a complete cone metric space,Pbe a normal cone with normal constantK, f :X →Xbe a continuous function, andβ∈0,1such that for everyx∈X, there is annx∈N such that
d
fnxx, fnxy
βdx, y, 3.7
for everyy∈X. Then for everyx∈X, rx supndfnxx, xis finite.
Proof. Letx∈Xandlx max{dfjx, x:j1,2, . . . , nx}. Ifn∈Nandn > nx, then there iss∈N∪ {0}such thatsnx< ns1nxand we have
d
fnx, x
d
fnx
fn−nxx
, fnxx d
fnxx, x
βd
fn−nxx, x d
fnxx, x
d
fnxx, x β
d
fn−nxx, fnxx d
fnxx, x
d
fnxx, x β
βd
fn−2nxx, x d
fnxx, x · · ·d
fnxx, x
1ββ2· · ·βs .
3.8
It means that
d
fnx, xK 1 1−βd
fnxx, xK 1
1−βlx. 3.9
Hencerxis finite and the proof is complete.
Theorem 3.9. LetX, dbe a complete cone metric space,P be a normal cone with normal constant K, β ∈ 0,1, andf : X → X be a continuous function such that for everyx ∈ X, there is an nx∈Nsuch that
d
fnxx, fnxy
βdx, y, 3.10
for everyy∈X. Thenfhas a unique fixed pointu∈Xand limn→∞fnx0 u, for everyx0∈X.
Proof. Letx0 ∈ X be arbitrary, andm0 nx0. Define the sequence x1 fm0x0, xi1 fmixi, whereminxi. We show that{xn}is a Cauchy sequence. We have
d
xn1, xn d
fmn−1 fmn
xn−1 , fmn−1
xn−1
βd
fmn xn−1
, xn−1 · · ·βnd
fmn x0
, x0 ,
3.11
for everyn∈N. So byLemma 3.8,dxn1, xnKβnrx0, for everyn∈N. Now, suppose thatm, n∈Nwithm < n, we have
d
xn, xmK
n−1
im
d
xi1, xiK βn 1−βr
x0
. 3.12
Since limn→∞βn/1−β 0, then limm,n→∞dxn, xm 0, and by Lemma 1.5,{xn}is a Cauchy sequence. Completeness ofXimplies that limn→∞xn u, for someu∈X. Now, we
show thatfu u. By contradiction, suppose thatfu/u. We claim that there arec, d∈E such that 0c, 0dandBcuandBdfuhave no intersection, whereBex {y∈X : dx, y e}, for everyx ∈X and 0 e. If not, then suppose that > 0, and choosec∈ E with 0candKc< . Then clearly, 0c/2 and forz∈Bc/2u∩Bc/2fu, we have
d
u, fu
du, z d
z, fu
c. 3.13
It means thatdu, fu ≤Kc < . Since >0 is arbitrary, thendu, fu 0 and so fu u, a contradiction. Therefore, assume thatc, d ∈ Ewith 0 c, 0 dare such that Bcu∩Bdfu ∅. Sincef is continuous, then there isn0 ∈Nsuch thatxn ∈ Bcuand fxn∈Bdfu, for everyn∈Nandnn0. Then
d f
xn , xn
d fmn−1
f xn−1
, fmn−1 xn−1
≤βd f
xn−1 , xn−1
· · ·βnd f
x0 , x0
, 3.14 for every n ∈ N. It means that dfxn, xn Kβndfx0, x0, for every n ∈ N. So limn→∞dfxn, xn 0, a contradiction. Thusfu u. The uniqueness of the fixed point follows immediately from the hypothesis.
Now, suppose thatx0∈Xis arbitrary. To show that limn→∞fnx0 u, set r0maxd
fm x0
, u:m0,1, . . . , nu−1 . 3.15 Ifnis sufficiently large, thennrnu q, forr >0 and 0q < nu, and we have
d fn
x0 , u
d
frnuq x0
, fnuu
βd
fr−1nuq
x0 , u
≤ · · ·βrd fq
x0 , u
. 3.16 It means that
d fn
x0
, uKβrd fq
x0
, uKβrr0. 3.17 Therefore, limn→∞dfnx0, u0 and hence limn→∞fnx0 u. This completes the proof.
Definition 3.10. LetXbe an ordered space. A functionϕ:X →X is said to be a comparison function if for every x, y ∈ X, x y, implies that ϕx ϕy, ϕx x, and limn→∞||ϕnx||0, for everyx∈X.
Example 3.11. LetE R2, P {x, y ∈ E | x, y 0}. It is easy to check thatϕ : E → E, withϕx, y ax, ay, for somea∈ 0,1is a comparison function. Also ifϕ1, ϕ2are two comparison functions overR, then ϕx, y ϕ1x, ϕ2y is also a comparison function overE.
Recall that for a cone metric spaceX, d, whereP is a cone with normal constantK, since for everyx∈X, xx, and thereforexKx, thenK1.
Theorem 3.12. LetX, dbe a complete cone metric space, whereP is a normal cone with normal constantK. Letf : X →X be a function such that there exists a comparison functionϕ :P → P such that
d
fx, fy
ϕ
dx, y
, 3.18
for everyx, y∈X. Thenfhas a unique fixed point.
Proof. Letx0 ∈Xbe arbitrary. We have d
fn x0
, fn1 x0
ϕ d
fn−1 x0
, fn x0
ϕ2
dfn−2 x0
, fn−1 x0
· · ·ϕn
d x0, f
x0
,
3.19
for everyn∈ N. Since limn→∞ϕndx0, fx0 0, for an arbitrary > 0, we can choose n∈Nsuch that
d fn
x0 , fn1
x0<
−Kϕc
K , 3.20
for everyn≥n0andc∈P with c<
K2, 1
Kϕcϕ d
fn x0
, fn1
x0. 3.21
Forn≥n0, we have d
fn x0
, fn2 x0
d fn
x0
, fn1 x0
d fn1
x0
, fn2 x0
. 3.22 So d
fn x0
, fn2
x0Kd fn
x0
, fn1
x0Kd fn1
x0
, fn2 x0
< K
−Kϕc K
K2ϕ d
fn x0
, fn1 x0
.
3.23
Now, for everynn0, we have d
fnx0
, fn3 x0
d
fn x0
, fn1 x0
d fn1
x0 , fn3
x0
. 3.24
SinceK≥1, then we have d
fn x0
, fn3
x0Kd fn
x0
, fn1
x0Kd fn1
x0
, fn3 x0
< K
−Kϕc K
K2ϕ d
fn x0
, fn2 x0
.
3.25
By induction, we have dfnx0, fnrx0 < , for everyr ∈ Nand n n0. Hence by Lemma 1.5, we have{fnx0}is a Cauchy sequence inX, d. So limn→∞fnx0 x∗, for some x∗∈X. Now, we will provefx∗ x∗. Since limn→∞fnx0 x∗, for everyc0, there exists nc∈Nsuch that for everynnc, we havedfnx0, x∗< c. Therefore,
d x∗, f
x∗
d
x∗, fn1 x0
d f
fn x0
, f x∗
d
x∗, fn1 x0
ϕ d
fn x0
, x∗
< d
x∗, fn1 x0
d fn
x0
, x∗
<2c,
3.26
for everyc0. Sofx∗ x∗. For the uniqueness of the fixed point, suppose that there exists y∗∈Xsuch thatfy∗ y∗. Hence
d x∗, y∗
d fn
x∗ , fn
y∗ ϕn
d
y∗, x∗
. 3.27
So d
x∗, y∗Kϕn d
y∗, x∗. 3.28
Since limn→∞ϕndy∗, x∗0, thenx∗y∗and the proof is complete.
4. Applications
Theorem 4.1. Consider the integral equation xt
b
a
k
t, s, xs
dsgt, t∈a, b. i
Suppose that
ik:a, b×a, b×Rn→Rnandg:a, b→Rn; iikt, s,·:Rn→Rnis increasing for everyt, s∈a, b;
iiithere exists a continuous function p : a, b×a, b → R and a comparison function ϕ:R2→R2such that
kt, s, u−kt, s, v, αkt, s, u−kt, s, v
pt, s, αpt, s ϕ
du, v
, 4.1
for everyt, s∈a, b, u, v∈Rn; ivsupt∈a,bb
apt, s, αpt, sds1.
Then the integral equationihas a unique solutionx∗inCa, b,Rn.
Proof. LetXCa, b,Rn, P {x, y:x, y0} ⊆R2, and definedf, g f−g∞, αf− g∞, for every f, g ∈ X. Then it is easily seen that X, d is a cone metric space. Define A:Ca, b,Rn→Ca, b,Rn, by
Axt:
b
a
k
t, s, xs
dsgt, t∈a, b. 4.2
For everyx, y∈X, we have
Axt−Ayt, αAxt−Ayt
b
a
k
t, s, xs
−k
t, s, ys ds
, α b
a
k
t, s, xs
−k
t, s, ys ds
b
a
k
t, s, xs
−k
t, s, ysds, b
a
αkt, s, xs−k
t, s, ysds
b
a
pt, s, αpt, s
ϕxs−ys, αxs−ysds
ϕ
x−y∞, αx−y∞b
a
pt, s, αpt, s ds ϕ
x−y∞, αx−y∞ .
4.3
HencedAx, Ayϕdx, y, for everyx, y ∈X. The conclusion follows now from Theorem 3.12.
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