Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 589725, 15 pages
doi:10.1155/2011/589725
Research Article
Common Fixed Point Theorems for Four Mappings
on Cone Metric Type Space
Aleksandar S. Cvetkovi´c,
1
Marija P. Stani´c,
2
Sladjana Dimitrijevi´c,
2
and Suzana Simi´c
2
1
Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade,
Kraljice Marije 16, 11120 Belgrade, Serbia
2
Department of Mathematics and Informatics, Faculty of S cience, University of Kragujevac,
Radoja Domanovi
´
ca 12, 34000 Kragujevac, Serbia
Correspondence should be addressed to Aleksandar S. Cvetkovi
´
c, [email protected]
Received 9 December 2010; Revised 26 January 2011; Accepted 3 February 2011
Academic Editor: Fabio Zanolin
Copyright q 2011 Aleksandar S. Cvetkovi
´
c et al. This is an open access article distributed under
/
{θ};
ii a, b ∈
, a, b ≥ 0, and x, y ∈ P imply ax by ∈ P;
iii P ∩ −P{θ}.
For a given cone P, a partial ordering with respect to P is introduced in the following
way: x y if and only if y − x ∈ P.Onewritesx ≺ y to indicate that x y,butx
/
y.If
y − x ∈ int P,onewritesx y.
If int P
/
∅, the cone P is called solid.
InthesequelwealwayssupposethatE is a real Banach space, P is a solid cone in E,
and is partial ordering with respect to P.
Analogously with definition of metric type space, given in 16, we consider cone
metric type space.
Definition 2.1. Let X be a nonempty set and
E a real Banach space with cone P.Avector-
valued function d : X × X → E is said to be a cone metric type function on X with constant
K ≥ 1 if the following conditions are satisfied:
d
1
θ dx, y for all x, y ∈ X and dx, yθ if and only if x y;
d
2
dx, ydy, x for all x, y ∈ X;
d
3
dx, y Kdx, zdz, y for all x, y, z ∈ X.
m
c for all
n, m > n
0
,then{x
n
} is called a Cauchy sequence in X.
If every Cauchy sequence is convergent in X,thenX is called a complete CMTS.
Fixed Point Theory and Applications 3
Example 2.4. Let B {e
i
| i 1, ,n}be orthonormal basis of
n
with inner product ·, ·.Let
p>0, and define
X
p
x
| x :
0, 1
−→
n
,
. 2.2
We show that P
B
is a solid cone. Let y
k
∈ P
B
, k ∈ , with property lim
k →∞
y
k
y. Since scalar
product is continuous, we get lim
k →∞
y
k
,e
i
lim
k →∞
y
k
,e
i
y, e
i
, i 1, ,n. Clearly,
it must be y, e
i
follows that z, e
i
0, i 1, ,n,and,sinceB is complete, we get z 0. Let us choose
z
n
i1
e
i
. It is obvious that z ∈ int P
B
, since if not, for every ε>0thereexistsy/∈ P
B
such
that |1 − y, e
i
|≤
n
i1
|1 −y, e
i
|
2
1/2
z − y <ε. If we choose ε 1/4, we conclude that
it must be y, e
i
> 1 −1/4 > 0, hence y ∈ P
,e
i
p
dt, f, g ∈ X
p
. 2.3
Then it is obvious that X
p
,d is CMTS with K 2
p−1
.Letf, g, h be functions such that
f, e
1
1, g,e
1
−2, h, e
1
0, and f, e
i
g, e
i
h, e
i
0, i 2, ,n,withp 2 give
df, g9e
1
, df, he
→ θ in E and let θ c. Then there exists positive integer n
0
such that x
n
c for
each n>n
0
.
Definition 2.6 see 17.LetF, G : X → X be mappings of a set X.Ify Fx Gx for some
x ∈ X,thenx is called a coincidence point of F and G,andy is called a point of coincidence
of F and G.
4 Fixed Point Theory and Applications
Definition 2.7 see 17.LetF and G be self-mappings of set X and CF, G{x ∈ X : Fx
Gx}.Thepair{F, G} is called weakly compatible if mappings F and G commute at all their
coincidence points, that is, if FGx GFx for all x ∈CF, G.
Lemma 2.8 see 5. Let F and G be weakly compatible self-mappings of a set X.IfF and G have a
unique point of coincidence y Fx Gx,theny is the unique common fixed point of F and G.
3. Main Results
Theorem 3.1. Let X, d be a CMTS with constant 1 ≤ K ≤ 2 and P a solid cone. Suppose that
self-mappings F, G, S, T : X → X are such that SX ⊂ GX, TX ⊂ FX and that for some constant
λ ∈ 0, 1/K for all x, y ∈ X there exists
u
x, y
∈
Kd
Fx,Gy
x, y
, 3.2
holds. If one of SX, TX, FX,orGX is complete subspace of X,then{S, F} and {T, G} have a unique
point of coincidence in X.Moreover,if{S, F } and {T, G} are weakly compatible pairs, then F, G, S,
and T have a unique common fixed point.
Proof. Let us choose x
0
∈ X arbitrary. Since SX ⊂ GX,thereexistsx
1
∈ X such that Gx
1
Sx
0
z
0
.SinceTX ⊂ FX,thereexistsx
2
∈ X such that Fx
2
Tx
1
z
1
.Wecontinueinthis
manner. In general, x
2n1
∈ X is chosen such that Gx
,n≥ 1, 3.3
where α max{λ, λK/2 − λK}, which will lead us to the conclusion that {z
n
} is a Cauchy
sequence, since α ∈ 0, 1it is easy to see that 0 <λK/2 − λK < 1.Toprovethis,itis
necessary to consider the cases of an odd integer n and of an even n.
For n 2 1, ∈
0
,wehavedz
21
,z
22
dSx
22
,Tx
21
,andfrom3.2 there
exists
u
x
22
,x
21
∈
Kd
Fx
21
,Sx
22
2
Kd
z
21
,z
2
,Kd
z
21
,z
22
,
Kd
z
2
,z
22
22
, which, because of property p
3
, implies
dz
21
,z
22
θ;
iii dz
21
,z
22
λ/2dz
2
,z
22
,thatis,byusingd
3
,
d
z
21
,z
22
λK
2
,wehave
d
z
2
,z
21
d
Sx
2
,Tx
21
λ
K
u
x
2
,x
21
, 3.6
where
u
x
Fx
2
,T
21
d
Gx
21
,Sx
2
2
Kd
z
2−1
,z
2
,Kd
z
2
,z
21
21
, which implies dz
2
,z
21
θ;
iii dz
2
,z
21
λ/2dz
2−1
,z
21
λK/2dz
2−1
,z
2
λK/2dz
2
,z
21
,
which implies dz
2
,z
21
λK/2 − λKdz
2
,z
Kd
z
n
,z
n1
K
2
d
z
n1
,z
n2
··· K
m−n−1
d
z
m−1
,z
m
Kα
n
Now, by p
4
and p
1
, it follows that for every c ∈ int P there exists positive integer n
0
such
that dz
n
,z
m
c for every m>n>n
0
,so{z
n
} is a Cauchy sequence.
Let us suppose that SX is complete subspace of X. Completeness of SX implies
existence of z ∈ SX such that lim
n →∞
z
2n
lim
n →∞
Sx
2n
z. Then, we have
lim
n →∞
Gx
2n1
Sx
2n
,z
λu
x
2n
,y
Kd
z
2n
,z
, 3.11
where
u
x
2n
,y
∈
Kd
Fx
2n
z
2n−1
,z
,Kd
z
2n−1
,z
2n
,Kd
z, Ty
,K
d
z
2n−1
,Ty
d
z, z
2n
2
K
1 −Kλ
·
1 −Kλ
K
· c c, as n −→ ∞; 3.13
iv dTy,z Kλ/2dz
2n−1
,Tydz, z
2n
Kdz
2n
,z,thatis,becauseofd
3
,
d
Ty,z
Kλ
2
Kd
z
2n−1
,z
2
λ
2
d
z
2n−1
,z
Kλ
2
K
d
z
2n
,z
K
2
λ
2 −K
2
λ
2 −K
λ/2 > 0.
Fixed Point Theory and Applications 7
Therefore, dTy,z c for each c ∈ int P .So,byp
2
we have dTy,zθ,thatis,
Ty Gy z, y is a coincidence point, and z is a point of coincidence of T and G.
Since TX ⊂ FX,thereexistsv ∈ X such that z Fv.LetusprovethatSv z.From
d
3
and 3.2,wehave
d
Sv, z
Kd
Sv, Tx
2n1
Kd
Tx
2n1
,z
λu
v, x
2n1
,K
d
Fv,Tx
2n1
d
Gx
2n1
,Sv
2
Kd
z, z
2n
,Kd
z, Sv
,Kd
z
2n
,z
2n1
Kdz
2n1
,z;
iv dSv, z Kλ/2dz, z
2n1
dz
2n
,Sv Kdz
2n1
,z.
By the same arguments as above, we conclude that dSv, zθ,thatis,Sv Fv z.
So, z is a point of coincidence of S and F, too.
Now we prove that z is unique point of coincidence of pairs {S, F}and {T, G}. Suppose
that there exists z
∗
which is also a point of coincidence of these four mappings, that is, Fv
∗
Gy
∗
Sv
∗
Ty
∗
z
∗
.From3.2,
d
,Kd
Fv,Sv
,d
Gy
∗
,Ty
∗
,K
d
Fv,Ty
∗
d
Gy
∗
,Sv
2
{
Kd
≤
x − y
2
y − z
2
2
x − y
|
x − z
|
≤ 2
x − y
2
M
2
dFx,Gy, that is, there exists λ M
2
< 1/2 1/K such
that 3.2 is satisfied.
According to Theorem 3 .1, {S, F} and {T, G} have a unique point of coincidence in X,
that is, there exists unique z ∈ X and there exist x, y ∈ X such that z Sx Fx Ty Gy.It
is easy to see that x −b/a, y −d/c,andz 0.
If {S, F} is weakly compatible pair, we have SFx FSx, which implies Mb b,that
is, b 0. Similarly, if {T, G} is weakly compatible pair, we have TGy GTy, which implies
Md d,thatis,d 0. Then x y 0, and z 0 is the unique common fixed point of these
four mappings.
The following two theorems can be proved in the same way as
Theorem 3.1,sowe
omit the proofs.
Theorem 3.3. Let X, d be a CMTS with constant K ≥ 2 and P a solid cone. Suppose that self-
mappings F, G, S, T : X → X are such that SX ⊂ GX, TX ⊂ FX and that for some constant
λ ∈ 0, 2/K
2
for all x, y ∈ X there exists
u
x, y
∈
Kd
u
x, y
, 3.23
holds. If one of SX, TX, FX,orGX is complete subspace of X,then{S, F} and {T, G} have a unique
point of coincidence in X.Moreover,if{S, F } and {T, G} are weakly compatible pairs, then F, G, S,
and T have a unique common fixed point.
Fixed Point Theory and Applications 9
Theorem 3.4. Let X, d be a CMTS with constant K ≥ 1 and P a solid cone. Suppose that self-
mappings F, G, S, T : X → X are such that SX ⊂ GX, TX ⊂ FX and that for some constant
λ ∈ 0, 1/K for all x, y ∈ X there exists
u
x, y
∈
Kd
Fx,Gy
,Kd
Fx,Sx
,Kd
Gy, Ty
If we choose T S and G F,fromTheorems3.1, 3.3,and3.4 we get the following
resultsfortwomappingsonCMTS.
Corollary 3.5. Let X, d be a CMTS with constant 1 ≤ K ≤ 2 and P a solid cone. Suppose that
self-mappings
F, S : X → X are such that SX ⊂ FX and that for some constant λ ∈ 0, 1/K for all
x, y ∈ X there exists
u
x, y
∈
Kd
Fx,Fy
,Kd
Fx,Sx
,Kd
Fy,Sy
,K
d
Fx,Sy
d
∈
Kd
Fx,Fy
,Kd
Fx,Sx
,Kd
Fy,Sy
,K
d
Fx,Sy
d
Fy,Sx
2
, 3.28
such that the following inequality
d
Sx, Sy
Fy,Sy
,
d
Fx,Sy
d
Fy,Sx
2
, 3.30
such that the following inequality
d
Sx, Sy
λ
K
u
x, y
, 3.31
holds. If FX or SX is complete subspace of X,thenF and S have a unique point of coincidence in X.
Moreover, if {F, S} is a weakly compatible pair, then F and S have a unique common fixed point.
< 1, 3.32
such that for all x, y ∈ X inequality
d
Sx, Ty
a
1
d
Fx,Gy
a
2
d
Fx,Sx
a
3
d
Gy, Ty
a
4
d
Fx,Ty
,n≥ 1, 3.34
where
α max
a
1
a
3
a
5
K
1 −a
2
− a
5
K
,
a
1
a
2
a
4
K
1 −a
3
− a
4
K
Fx
22
,Gx
21
a
2
d
Fx
22
,Sx
22
a
3
d
Gx
21
,Tx
21
a
4
d
Fx
22
,Tx
a
2
d
z
21
,z
22
a
3
d
z
2
,z
21
a
4
d
z
21
,z
21
a
5
d
5
d
z
2
,z
22
a
1
a
3
d
z
2
,z
21
a
2
d
z
21
,z
22
z
2
,z
21
a
2
a
5
K
d
z
21
,z
22
.
3.37
Therefore,
d
z
21
,z
22
2
,Tx
21
,andfrom3.33
we have
d
Sx
2
,Tx
21
a
1
d
Fx
2
,Gx
21
a
2
d
Fx
2
,Sx
2
d
z
2
,z
21
a
1
d
z
2−1
,z
2
a
2
d
z
2−1
,z
2
a
3
d
z
z
2−1
,z
2
a
3
d
z
2
,z
21
a
4
d
z
2−1
,z
21
a
1
a
2
z
2
,z
21
a
1
a
2
a
4
K
d
z
2−1
,z
2
a
3
a
4
K
K
d
z
2−1
,z
2
, 3.41
and inequality 3.34 holds in this case, too.
BythesameargumentsasinTheorem 3.1 we conclude that {z
n
} is a Cauchy sequence.
Let us suppose that SX is complete subspace of X. Completeness of SX implies
existence of z ∈ SX such that lim
n →∞
z
2n
lim
n →∞
Sx
2n
z. Then, we have
lim
n →∞
Gx
2n1
lim
n →∞
Sx
a
1
Kd
Fx
2n
,Gy
a
2
Kd
Fx
2n
,Sx
2n
a
3
Kd
Gy, Ty
a
4
Kd
Fx
2n
2n
a
3
Kd
z, Ty
a
4
Kd
z
2n−1
,Ty
a
5
Kd
z, z
2n
Kd
z
2n
,z
a
,z
a
4
K
2
d
z, Ty
a
5
Kd
z, z
2n
Kd
z
2n
,z
.
3.43
The sequence {z
n
} converges to z,soforeachc ∈ int P there exists n
0
∈ such that for every
2n
a
4
K
2
d
z
2n−1
,z
a
5
Kd
z, z
2n
Kd
z
2n
,z
a
1
K
1 −a
3
K − a
4
K
2
a
2
K
·
c
5
a
4
K
2
1 −a
3
K − a
4
K
2
·
1 −a
3
K − a
4
K
2
K
1 −a
3
K − a
4
K
2
·
1 −a
3
K − a
4
K
2
K
·
c
5
c,
3.44
Fixed Point Theory and Applications 13
because of 3.32.Now,byp
2
it follows that dTy,zθ,thatis,Ty z.So,wehave
Ty Gy z,thatis,y is a coincidence point, and z is a point of coincidence of m appings T
and G.
Since TX ⊂ FX,thereexistsv ∈ X such that z Fv.LetusprovethatSv z, too.
From d
3
3
Kd
Gx
2n1
,Tx
2n1
a
4
Kd
Fv,Tx
2n1
a
5
Kd
Gx
2n1
,Sv
Kd
Tx
2n1
,z
a
Kd
z
2n
,Sv
Kd
Tx
2n1
,z
a
1
Kd
z, z
2n
a
2
Kd
z, Sv
a
3
Kd
z
Tx
2n1
,z
,
3.45
and by the same arguments as above, we conclude that dSv, zθ,thatis,Sv Fv z.
Thus, z is a point of coincidence of mappings S and F, too.
Suppose that there exists z
∗
which is also a point of coincidence of these four
mappings, that is, Fv
∗
Gy
∗
Sv
∗
Ty
∗
z
∗
.From3.33 we have
d
z, z
∗
d
∗
a
4
Kd
Gy
∗
,Sv
a
1
Kd
z, z
∗
a
2
Kd
z, z
a
3
Kd
z
∗
,z
,
3.46
and because of p
3
it follows that z z
∗
. Therefore, z is the unique point of coincidence of
pairs {S, F } and {T, G}, and we have z Sv Fv Gy Ty.If{S, F} and {T, G} are weakly
compatible pairs, then z is the unique common fixed point of S, F, T,andG,byLemma 2.8.
The proofs for the cases in which FX, GX,orTX is complete are similar.
Theorem 3.8 is a generalization of 13,Theorem2.8. Choosing K 1from
Theorem 3.8 we get the following corollary.
Corollary 3.9. Let X, d be cone metric space and P a solid cone. Suppose that self-mappings
F, G, S, T : X → X are such that SX ⊂ GX, TX ⊂ FX and that there exist nonnegative constants
a
i
, i 1, ,5, satisfying a
1
a
2
a
3
2max{a
4
,a
5
} < 1, such that for all x, y ∈ X inequality
d
Gy, Sx
,
3.47
14 Fixed Point Theory and Applications
holds. If one of SX, TX, FX,orGX is complete subspace of X,then{S, F} and {T, G} have a unique
point of coincidence in X.Moreover,if{S, F } and {T, G} are weakly compatible pairs, then F, G, S,
and T have a unique common fixed point.
If we choose T S and G F,fromTheorem 3 .8, we get the following result for two
mappings on CMTS.
Corollary 3.10. Let X, d be a CMTS with constant K ≥ 1 and P a solid cone. Suppose that self-
mappings F, S : X → X are such that SX ⊂ FX and that there exist nonnegative constants a
i
,
i 1, ,5, satisfying
a
1
a
2
a
3
2K max
{
a
4
,a
5
}
< 1,a
3
d
Fy,Sy
a
4
d
Fx,Sy
a
5
d
Fy,Sx
,
3.49
holds. If one of SX or FX is complete subspace of X,thenS and F have a unique point of coincidence
in X.Moreover,if{F, S}is a weakly compatible pair, then F and S have a unique common fixed point.
Acknowledgments
The authors are indebted to the referees for their valuable suggestions, which have
contributed to improve the presentation of the paper. The first two authors were supported in
part by the Serbian Ministry of Science and Technological Developments Grant no. 174015.
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