Volume 2010, Article ID 189684,13pages doi:10.1155/2010/189684
Research Article
Some Fixed Point Theorems of Integral Type
Contraction in Cone Metric Spaces
Farshid Khojasteh,
1Zahra Goodarzi,
2and Abdolrahman Razani
2, 31Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran 14778, Iran 2Department of Mathematics, Faculty of Science, Imam Khomeini International University,
Qazvin, 34149-16818, Iran
3School of Mathematics, Institute for Research in Fundamental Sciences, P.O. Box 19395-5746,
Tehran, Iran
Correspondence should be addressed to Farshid Khojasteh,fr [email protected]
Received 1 October 2009; Accepted 11 February 2010
Academic Editor: Jerzy Jezierski
Copyrightq2010 Farshid Khojasteh et al. 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.
We define a new concept of integral with respect to a cone. Moreover, certain fixed point theorems in those spaces are proved. Finally, an extension of Meir-Keeler fixed point in cone metric space is proved.
1. Introduction
In 2007, Huang and Zhang in1introduced cone metric space by substituting an ordered Banach space for the real numbers and proved some fixed point theorems in this space. Many authors study this subject and many fixed point theorems are proved; see2–5. In this paper, the concept of integral in this space is introduced and a fixed point theorem is proved. In order to do this, we recall some definitions, examples, and lemmas from1,4as follows.
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, b≥0, andx, y∈P imply thataxby∈P, iiix∈Pand−x∈Pimply thatx0.
Given a coneP ⊂E,we define a partial ordering≤with respect toP byx ≤yif and only if
y−x ∈intP,where intP denotes the interior ofP.The coneP is called normal if there is a numberK > 0 such that 0≤ x ≤ yimpliesx ≤ Ky,for allx, y ∈ E.The least positive number satisfying above is called the normal constant1.
The conePis called regular if every increasing sequence which is bounded from above is convergent. That is, if{xn}n≥1is a sequence such that x1 ≤ x2 ≤ · · · ≤ yfor somey ∈ E,
then there isx∈Esuch that limn→ ∞xn−x 0. Equivalently, the conePis regular if and only if every decreasing sequence which is bounded from below is convergent1. Also every regular cone is normal4. In addition, there are some nonnormal cones.
Example 1.1. SupposeEC2
R0,1with the normff∞f∞and consider the cone
P {f ∈E:f ≥0}. For allK ≥1, setfx xandgx x2K.Then 0≤ g ≤f,f 2 and
g 2K1.SinceKf < g, Kis not normal constant ofP.Therefore,P is non-normal cone.
From now on, we suppose thatEis a real Banach space,Pis a cone inEwithint P /∅,
and≤is partial ordering with respect toP. LetX be a nonempty set. As it has been defined in1, a functiond : X ×X → E is called a cone metric onX if it satisfies the following conditions:
idx, y≥0 for allx, y∈Xanddx, y 0 if and only ifxy, iidx, y dy, x, for allx, y∈X,
iiidx, y≤dx, z dy, z, for allx, y, z∈X.
ThenX, dis called a cone metric space.
Example 1.2. SupposeEl1, P {{x
n}n∈N∈E:xn ≥0,for all n,X, ρis a metric space and d:X×X → Eis defined bydx, y {ρx, y/2n}
n∈N.ThenX, dis a cone metric space and
the normal constant ofPis equal to 1.
Definition 1.3. LetX, dbe a cone metric space. Let{xn}n∈Nbe a sequence inXandx∈X.If
for anyc∈Ewith 0c,there isn0 ∈Nsuch that for alln > n0, dxn, xc,then{xn}n∈N
is said to be convergent tox,andxis the limit of{xn}n∈N. We denote this by
lim
n→ ∞xnx or xn−→x n−→ ∞. 1.1
Definition 1.4. LetX, dbe a cone metric space and{xn}n∈Nbe a sequence inX.If for any c ∈Ewith 0 c, there isn0 ∈Nsuch that for allm, n > n0, dxn, xm c,then{xn}n∈Nis
called a Cauchy sequence inX.
Definition 1.5. LetX, dbe a cone metric space, if every Cauchy sequence is convergent inX,
thenXis called a complete cone metric space.
Definition 1.6. LetX, dbe a cone metric space. LetT be a self-map onX.If for all sequence
{xn}n∈NinX,
lim
n→ ∞xnx implies limn→ ∞Txn Tx, 1.2
The following lemmas are useful for us to prove the main result.
Lemma 1.7. LetX, dbe a cone metric space andP a normal cone with normal constant K.Let
{xn}n∈Nbe a sequence inX.Then{xn}n∈Nconverges toxif and only if
lim
n∈Ndxn, x 0. 1.3
Lemma 1.8. LetX, dbe a cone metric space andP a normal cone with normal constant K.Let
{xn}n∈Nbe a sequence inX. Then{xn}n∈Nis a Cauchy sequence if and only if
lim
m,n→ ∞dxm, xn 0. 1.4
Lemma 1.9. LetX, dbe a cone metric space and{xn}n∈Na sequence inX.If{xn}n∈Nis convergent, then it is a Cauchy sequence.
Lemma 1.10. LetX, dbe a cone metric space andPbe a normal cone with normal constantK. Let
{xn}and{yn}be two sequences inXandxn → x, yn → yn → ∞. Then
dxn, yn
−→dx, y n−→ ∞. 1.5
The following example is a cone metric space.
Example 1.11. LetER2, P {x, y∈Ex, y≥0},andX R.Suppose thatd:X×X → Eis defined bydx, y |x−y|, α|x−y|,whereα≥0 is a constant. ThenX, dis a cone metric space.
Theorem 1.12. LetX, dbe a complete cone metric space andPa normal cone with normal constant K.Suppose the mappingf:X → Xsatisfies the contractive condition
dfx, fy≤βdx, y 1.6
for allx, y∈X,whereβ∈0,1is a constant. Thenf has a unique fixed pointx0 ∈X.Also, for all
x∈X,the sequence{fnx}∞
n1converges tox0.
2. Certain Integral Type Contraction Mapping in Cone Metric Space
In 2002, Branciari in6introduced a general contractive condition of integral type as follows.
Theorem 2.1. LetX, dbe a complete metric space,α∈0,1, andf :X → Xis a mapping such that for allx, y∈X,
dfx,fy
0
φtdt≤α
dx,y
0
whereφ:0,∞ → 0,∞is nonnegative and Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of0,∞such that for each >0,0φtdt >0, thenfhas a unique fixed pointa∈X, such that for eachx∈X,limn→ ∞fnxa.
In this section we define a new concept of integral with respect to a cone and introduce the Branciari’s result in cone metric spaces.
Definition 2.2. Suppose thatPis a normal cone inE. Leta, b∈Eanda < b. We define
a, b:{x∈E:xtb 1−ta, for somet∈0,1},
a, b:{x∈E:xtb 1−ta, for somet∈0,1}.
2.2
Definition 2.3. The set{a x0, x1, . . . , xn b}is called a partition fora, bif and only if the sets{xi−1, xi}ni1are pairwise disjoint anda, b {
n
i1xi−1, xi} ∪ {b}.
Definition 2.4. For each partitionQofa, band each increasing functionφ :a, b → P,we define cone lower summation and cone upper summation as
LCon
n
φ, Q n−1
i0
φxixi−xi1,
UCon
n
φ, Qn−1
i0φxi1
xi−xi1,
2.3
respectively.
Definition 2.5. Suppose thatP is a normal cone inE.φ : a, b → P is called an integrable function ona, bwith respect to conePor to simplicity, Cone integrable function, if and only if for all partitionQofa, b
lim n→ ∞L
Con
n
φ, QSCon lim n→ ∞U
Con
n
φ, Q, 2.4
whereSConmust be unique.
We show the common valueSConby
b
a
φxdPx or to simplicity
b
a
φdp. 2.5
We denote the set of all cone integrable functionφ:a, b → PbyL1a, b, P.
Lemma 2.6. (1) Ifa, b ⊆ a, c, thenbafdp ≤
c
afdp,forf ∈ L1X, P.(2)
b
aαf βgdp αbafdpβ
b
agdp,forf, g∈ L1X, Pandα, β∈R.
Proof. 1Suppose thatPandRare partitions fora, bandb, c,respectively. That is,
LetQP∪R. Qis a partition fora, c.Therefore one can write
LConn f, P n−1
i0
fxixi−xi1 ≤
n−1
i0
fxixi−xi1
m−1
in
fxixi−xi1
LConn f, PLConn f, RLConn f, Q.
2.7
So
b
a fdp≤
c
a
fdp. 2.8
2SupposePis an partition fora, b, that is
P {ax0, x1, . . . , xn−1, xnb}. 2.9
Then
LConn f, P n−1
i0
αfxi βgxi
xi−xi1
α n−1
i0
fxixi−xi1β
n−1
i0
gxixi−xi1αLConn
f, PβLConn g, P. 2.10
Thus
b
a
αfβgdpα
b
a fdpβ
b
a
gdp. 2.11
Definition 2.7. The functionφ :P → Eis called subadditive cone integrable function if and only if for alla, b∈P
ab
0
φdP ≤
a
0
φdP
b
0
φdP. 2.12
Example 2.8. LetE X R,dx, y |x−y|,P 0,∞,andφt 1/t1for allt >0.
Then for alla, b∈P,
ab
0
dt
t1 lnab1,
a
0
dt
t1 lna1,
b
0
dt
Sinceab≥0,thenab1≤ab1ab a1b1. Therefore
lnab1≤lna1b1 lna1 lnb1. 2.14
This shows thatφis an example of subadditive cone integrable function.
Theorem 2.9. LetX, dbe a complete cone metric space and P a normal cone. Suppose thatφ :
P → Pis a nonvanishing map and a subadditive cone integrable on eacha, b⊂Psuch that for each 0,0φdp0. Iff :X → Xis a map such that, for allx, y∈X
dfx,fy
0
φdp≤α
dx,y
0
φdp, 2.15
for someα∈0,1,thenfhas a unique fixed point inX.
Proof. Letx1∈P.Choosexn1fxn.We have
dxn1,xn
0
φdp
dfxn,fxn−1
0
φdp
≤α
dxn,xn−1
0
φdp
.. .
≤αn−1
dx2,x1
0
φdp.
2.16
Sinceα∈0,1thus
lim n→ ∞
dxn1,xn
0
φdp0. 2.17
If limn→ ∞dxn1, xn/0 then limn→ ∞d0xn1,xnφdp/0 and this is a contradiction, so
lim
n→ ∞dxn1, xn 0. 2.18
We now show thatxnis a Cauchy sequence. Due to this, we show that
lim m,n→ ∞d
fxm, fxn
0. 2.19
By triangle inequality
dfxm,fxn
0
φdp≤
dfxn,fxn1dfxn1,fxn2···dfxm−1,fxm
0
and by sub-additivity ofφwe get
dfxm,fxn
0
φdp≤
dfxn,fxn1
0
φ dp· · ·
dfxm−1,fxm
0
φdp
≤ αnαn−1· · ·αm−1 dx2,x1
0
φ dp≤ αn 1−α
dx2,x1
0
φ dp−→0.
2.21
Thus
lim m,n→ ∞d
fxn, fxm
0. 2.22
This means that{xn}n∈Nis a Cauchy sequence and sinceXis a complete cone metric space,
thus{xn}n∈Nis convergent tox0∈X.Finally, since
dxn1,fx0
0
φ dp
dfxn,fx0
0
φdp≤α
dxn,x0
0
φ dp, 2.23
thus limn→ ∞dxn1, fx0 0.This means thatfx0 x0.Ifx0, y0 are two distinct fixed
points off,then
dx0,y0
0
φdp
dfx0,fy0
0
φdp≤α
dx0,y0
0
φdp 2.24
which is a contradiction. Thusfhas a unique fixed pointx0∈X.
Lemma 2.10. LetER2, P {x, y∈Ex, y≥0},andX R.Suppose thatd:X×X → Eis
defined bydx, y |x−y|, α|x−y|,whereα≥0is a constant. Suppose thatφ:0,0,a, b →
P is defined byφx, y φ1x, φ2y,whereφ1, φ2 : 0,∞ → 0,∞are two
Riemann-integrable functions. Then
a,b
0,0φdP
a2b2
1
a
a
0
φ1tdt,1
b
b
0
φ2tdt
Proof. LetQ {xi, yi}ni0 be a partition of set0,0,a, bsuch thatxi a/niandyi b/ni, thenby Definitions2.4and2.5
a,b
0,0φdP nlim→ ∞L
Con n
φ, Q lim n→ ∞
n−1
i0
φxi, yixi1, yi1
−xi, yi
lim n→ ∞
n−1
i0
φ1 a ni
, φ2
b ni
an,b
n
a2b2lim
n→ ∞
n−1
i0
1 n φ1 a ni
, φ2
b ni
a2b2
lim n→ ∞
1
n n−1
i0
φ1 a ni ,lim n→ ∞
1
n n−1
i0
φ2
b ni
a2b2
1
anlim→ ∞
a n
n−1
i0
φ1 a ni ,1 bnlim→ ∞
b n
n−1
i0
φ2
b ni
a2b2
1
a
a
0
φ1tdt,1
b
b
0
φ2tdt
.
2.26
Thus
a,b
0,0φdP
a2b2
1
a
a
0
φ1tdt,
1
b
b
0
φ2tdt
. 2.27
Example 2.11. LetX{1/n:n∈N} ∪ {0}, ER2andP {x, y∈E:x≥0, y≥0}.Suppose
dx, y |x−y|, α|x−y|,for some constantα >0.Firstly,X, dis a complete cone metric space. Secondly, iff :X → Xandφ:P → Eare defined by
fx
⎧ ⎨ ⎩
1
n1 if x 1
n, n∈N,
0 if x0,
φt, s
⎧ ⎨ ⎩
t1/t−21−lnt, s1/s−21−lns, t, s∈P\ {0,0},
0,0, t, s 0,0,
2.28
respectively, then
dfx,fy
0
φdP ≤ 1 2
dx,y
0
In order to obtain inequality2.29, setx1/nandy1/m,wherem > n.Hence
dfx, fy
m−n m1n1,
αm−n m1n1
,
dx, y
m−n mn ,
αm−n mn
.
2.30
Supposeφ1t φ2t t1/t−21−lntfor allt > 0 andφ10 φ20 0. Thusφt, s
φ1t, φ2s. ByLemma 2.10
dfx,fy
0
φdP
m−n/m1n1,αm−n/m1n1
0,0
φ1, φ2
dP
m−n
m1n1
1α2
m1n1
m−n
m−n/m1n1
0
φ1tdt,mαm1n−n1
αm−n/m1n1
0
φ2tdt
1α2
m−n/m1n1
0
φ1tdt,1
α
αm−n/m1n1
0
φ2tdt
.
2.31
Sinceτ0t1/t−21−lntdtτ1/τ,thus
m−n/m1n1
0
φ1tdt
m−n m1n1
m1n1/m−n
,
αm−n/m1n1
0
φ2tdt
αm−n
m1n1
m1n1/αm−n
.
2.32
It means that
dfx,fy
0
φdP
1α2
m−n m1n1
m1n1/m−n
,1 α
αm−n
m1n1
m1n1/αm−n
.
2.33
On the other side, Branciari in6shows that
m−n n1m1
n1m1/m−n ≤ 1
2
m−n nm
nm/m−n
for allm, n∈N. Therefore
m−n
m1n1
m1n1/m−n
,1 α
αm−n
m1n1
m1n1/αm−n
≤ 1
2
m−n
mn
mn/m−n
,1 α
αm−n
mn
mn/αm−n
.
2.35
Thus inequalities2.33and2.35imply that
dfx,fy
0
φdP ≤ 1 2
1α2
m−n
mn
mn/m−n ,1
α
αm−n
mn
mn/αm−n
1
2
dx,y
0
φdP, 2.36
or in other words
dfx,fy
0
φdP ≤ 1 2
dx,y
0
φdP. 2.37
Thus byTheorem 2.9,fhas a fixed point. But, on the other hand,
dfx, fy< dx, y, 2.38
and this means thatfdoes not satisfy inTheorem 1.12.
3. Extension of Meir-Keeler Contraction in Cone Metric Space
In 2006, Suzuki in7proved that the integral type contraction see6is a special case of Meir-Keeler contraction see 8. Haghi and Rezapour in 5 extended Meir-Keeler contraction in cone metric space as follows.
Theorem 3.1see5. LetX, dbe a complete regular cone metric space and f has the property (KMC) onX;that is, for all0/∈P, there existsδ0such that
dx, y< δ implies dfx, fy< 3.1
for allx, y∈X. Thenfhas a unique fixed point.
An extension ofTheorem 3.1is as follows.
Theorem 3.2. LetX, dbe a complete regular cone metric space andf a mapping onX. Suppose that there exists a functionθfromPinto itself satisfying the following:
B1θ0 0andθt0for allt0,
B3for all0/∈P,there existsδ0such that for allx, y∈X
θdx, y< δ impliesθdfx, fy< 3.2
B4for allx, y∈X
θxy≤θx θy. 3.3
Thenfhas a unique fixed point.
Proof. First, note thatθdfx, fy< θdx, yfor allx, y∈Xwithx /y.Sinceθ−1exists,
thusdfx, fy< dx, yfor allx, y∈Xwithx /y.Now Letx0∈X.Setxnfxn−1for
alln∈ N.If, there is a naturalm∈ Nsuch thatdxm1, xm 0,thenfxm xm and sof has a fixed point. Ifdxn1, xn/0 for alln ∈ N,thenθdxn1, xn < θdxn, xn−1.Hence,
according to regularity ofP,there existsα∈P such thatθdxn1, xn↓α.We claimα0.If α /0,then according toB3,there is 0dsuch thatθdfx, fy< αfor allx, y∈Xwith
θdx, y< αd.Chooser >0 such thatd/2 Nr0⊆Pand take the natural numberN such thatθdxn1, xn−α< r,for alln≥N.We obtain
d2 −θdxn1−xn−α− d
2
< r. 3.4
Thus
d
2 −θdxn1−xn−α∈
d
2 Nr0⊆P. 3.5
So,θdxn1, xn−αd.Sincefhas the propertyB3, θdxn2, xn1< αfor alln≥N.
This is a contradiction becauseα < θdxi1, xifor alli≥1.Thus
lim
n→ ∞θdxn1, xn 0. 3.6
Now, we show that{xn}∞n1 is a Cauchy sequence. If this is not, then there is a 0 csuch
that for all natural numberk,there aremk, nk > kso that the relationdxmk, xnk cdoes not hold. Sinceθ has continuous inverse thus there exists 0 c such that for all natural number k,there are mk, nk > k so that the relationθdxmk, xnk c does not hold. For each 0 e c,there exists 0 d such thatθdfx, fy < e, for allx, y ∈ X with
Also, takemM ≥ nM > Mso that the relationθdxmM, xnMcdoes not hold. ThenB4 yields
θdxnM−1, xnM1≤θdxnM−1, xnM θdxnM, xnM1
d
2
d
2
de.
3.7
Hence,θdxnM, xnM2e.Similarly,θdxnM, xnM3e.Thus
θdxnM, xmMec 3.8
which is a contradiction. Therefore{xn}∞n1is a Cauchy sequence. SinceX, dis a complete
cone metric space, there isu∈Xsuch thatxn → u. Sincedfx, fy< dx, y, for allx, y∈X withx /y, thus for each 0, there is a natural numberN > 0 such that for alln > N,
dxn, x .Sincedfxn, fu < dxn, uthusdfxn, fu for alln > N. It means that fxn → fu.In the other side,fxn xn1 → uand the limit point is unique in cone metric
spaces. Thusfhas at least one fixed point. Now, ifu, vare two distinct fixed points forf,then
du, v dfu, fv< du, v 3.9
which is a contradiction. Thereforefhas a unique fixed point.
Remark 3.3. 1Setθx x, thenTheorem 3.1is a direct result ofTheorem 3.2.
2Letφ:P → Pbe a nonvanishing map and a subadditive cone integrable on each
a, b⊂Psuch that for each0,0φdp0.Ifθx
x
0φdP,thenθsatisfies all conditions
ofTheorem 3.2. In other words,Theorem 2.9is a direct result ofTheorem 3.2.
Acknowledgment
The third author would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Teheran, Iran, for supporting this researchGrant no. 88470119.
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3 D. Ili´c and V. Rakoˇcevi´c, “Common fixed points for maps on cone metric space,”Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 876–882, 2008.
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