Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 941371, 11 pages
doi:10.1155/2010/941371
Research Article
Fixed Point Results for Multivalued Maps in Cone
Metric Spaces
Abdul Latif and Fawzia Y. Shaddad
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Abdul Latif, latifmath@yahoo.com
Received 4 February 2010; Accepted 16 April 2010
Academic Editor: Mohamed Amine Khamsi
Copyright q 2010 A. Latif and F. Y. Shaddad. 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 prove some fixed point theorems for multivalued maps in cone metric spaces. We improve
and extend a number of known fixed point results including the corresponding recent fixed point
results of Feng and Liu 1996 and Chifu and Petrusel 1997. The remarks and example provide
improvement in the mentioned results.
1. Introduction
The well-known Banach contraction principle and its several generalizations in the setting
of metric spaces play a central role for solving many problems of nonlinear analysis.
For example, see 1–5. Using the concept of the Hausdorff metric, Nadler 6 obtained
a multivalued version of the Banach contraction principle. Without using the concept of
the Hausdorff metric, recently, Feng and Liu 7 obtained a new fixed point theorem for
nonlinear contractions in metric spaces, extending Nadler’s result. Recently, Chifu and
Petrusel obtained a fixed point result 18, Theorem 2.1 which contains 7, Theorem 3.1.
In 1980, Rzepecki 8 introduced a generalized metric by replacing the set of real
numbers with normal cone of the Banach space. In 1987, Lin 9 introduced the notion of Kmetric spaces by replacing the set of real numbers with cone in the metric function. Zabrejko
10 studied new revised version of the fixed point theory in K-metric and K-normed linear
spaces. Most recently, Huang and Zhang 11 announced the notion of cone metric spaces,
replacing the set of real numbers by an ordered Banach spaces. They proved some basic
properties of convergence of sequences and also obtained various fixed point theorems for
contractive single-valued maps in such spaces. For more fixed point results in cone metric
spaces, see 12–17.
2
Fixed Point Theory and Applications
In this paper, first we prove a useful lemma in the setting of cone metric spaces.
Then, we prove some results on the existence of fixed points for multivalued maps in cone
metric spaces. Consequently, our results improve and extend a number of known fixed point
results including the corresponding recent main fixed point results of Chifu and Petrusel 18,
Theorems 2.1 and 2.5.
2. Preliminaries
Let E be a real Banach space and P a subset of E. P is called a cone if and only if
i P is closed, nonempty, and P /
{0};
ii a, b ∈ R, a, b ≥ 0, and x, y ∈ P imply that ax by ∈ P ;
iii x ∈ P and −x ∈ P imply that x 0.
For a given cone P ⊂ E, we define partial ordering on E with respect to P by the
following: for x, y ∈ E, we say that x y if and only if y − x ∈ P. Also,we write x y if
y − x ∈ intP, where int P denotes the interior of P.
The cone P is called normal if there is a constant K > 0 such that for all x, y ∈ E
0 x y implies x ≤ K y.
2.1
The least positive number K satisfying the above inequality is called the normal constant of
P ; for details see 3, 11.
In the sequel, E is a real Banach space, P is a cone in E, and is partial ordering with
respect to P .
Definition 2.1. Let X be a nonempty set. Suppose that the map d : X × X → E satisfies
i 0 dx, y for all x, y ∈ X, and dx, y 0 if and only if x y;
ii dx, y dy, x for all x, y ∈ X;
iii dx, y dx, z dz, y for all x, y, z ∈ X.
Then d is called a cone metric on X, and X, d is called a cone metric space 11.
Example 2.2 see 11. Let E R2 , P {x, y ∈ E : x, y ≥ 0} ⊂ R2 , X R, and d : X × X → E
defined by dx, y |x − y|, α|x − y|, where α ≥ 0 is a constant. Then X, d is a cone metric
space.
Example 2.3 see 16. Let E 1 , P {{xn }n≥1 ∈ E : xn ≥ 0, for all n}, X, ρ a metric space,
and d : X × X → E defined by dx, y {ρx, y/2n }n≥1 . Then X, d is a cone metric space.
Clearly, the above examples show that class of cone metric spaces contains the class of
metric spaces.
Now, we recall some basic definitions of sequences in cone metric spaces see, 11, 17.
Let X, d be a cone metric space and {xn } a sequence in X. Then
i {xn } converges to x ∈ X whenever for every c ∈ E with 0 c, there is a natural
number N such that dxn , x c for all n ≥ N; we denote this by limn → ∞ xn x or
xn → x;
Fixed Point Theory and Applications
3
ii {xn } is a Cauchy sequence whenever for every c ∈ E with 0 c, there is a natural
number N such that dxn , xm c for all n, m ≥ N;
iii X, d is said to be complete space if every Cauchy sequence in X is convergent in
X;
iv A set A ⊆ X is said to be closed if for any sequence {xn } ⊂ A converges to x, we
have x ∈ A;
v A map f : X → R is called lower semicontinuous if for any sequence {xn } ⊂ X such
that xn → x ∈ X, we have fx ≤ lim infn → ∞ fxn .
Lemma 2.4 see 11. Let X, d be a cone metric space, and let P be a normal cone with normal
constant K. Let {xn } be any sequence in X. Then
a {xn } converges to x ∈ X if and only if dxn , x → 0, as n → ∞;
b {xn } is a Cauchy sequence if and only if dxn , xm → 0, as n, m → ∞.
Let X, d be a cone metric space. We denote 2X as a collection of nonempty subsets of X, and
ClX as a collection of nonempty closed subsets of X. An element x ∈ X is called a fixed point of a
multivalued map T : X → 2X if x ∈ T x. Denote FixT {x ∈ X : x ∈ T x}.
For T : X → ClX, and x ∈ X, one denotes
Dx, T x {dx, z : z ∈ T x}.
2.2
c y ∈ X : d x, y c .
Bx,
2.3
For c ∈ E with 0 c, one denotes
c is closed [16, Lemma 2.3].
The set Bx,
3. The Results
First, we prove our key lemma.
Lemma 3.1. Let X, d be a cone metric space and let P be a normal cone with normal constant K. If
there exist a sequence {xn } in X and a real number γ ∈ 0, 1 such that for every n ∈ N,
dxn1 , xn γdxn , xn−1 ,
3.1
then {xn } is a Cauchy sequence.
Proof. Let x0 ∈ X be an arbitrary but fixed. Note that
dxn1 , xn γ n dx1 , x0 .
3.2
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Fixed Point Theory and Applications
Now, for all m > n,
dxn , xm dxn , xn1 dxn1 , xn2 · · · dxm−1 , xm γ n γ n1 · · · γ m−1 dx1 , x0 3.3
n
γ
dx1 , x0 .
1−γ
Since P is a normal cone with the normal constant K, we have
dxn , xm ≤ K
γn
dx1 , x0 ;
1−γ
3.4
taking limit as n, m → ∞, we get dxn , xm → 0. Thus {xn } is a Cauchy sequence.
Applying Lemma 3.1, we prove the following result.
Theorem 3.2. Let X, d be a complete cone metric space, P a normal cone with normal constant K,
and T : X → ClX. Suppose that the following hold for arbitrary but fixed x0 ∈ X, u0 ∈ Dx0 , T x0 ,
and c ∈ E with c
0:
0 , c and for any
i there exist constants a, b ∈ 0, 1 with a < b such that for each x ∈ Bx
u ∈ Dx, T x there exist y ∈ T x and v ∈ Dy, T y satisfying
bd x, y u,
v ad x, y ;
3.5
ii u0 1 − a/bbc;
iii the function f : X → R defined by fx infy∈T x dx, y is lower semicontinuous.
0 , c Then FixT ∩ Bx
/ ∅.
0 , c and u0 ∈ Dx0 , T x0 , it follows from i and ii that there exist
Proof. Since x0 ∈ Bx
x1 ∈ T x0 and u1 ∈ Dx1 , T x1 satisfying
a
bdx0 , x1 u0 1 −
bc,
b
3.6
u1 adx0 , x1 .
3.7
a
c c,
dx0 , x1 1 −
b
3.8
Note that
Fixed Point Theory and Applications
5
0 , c. From 3.6 and 3.7 it follows that
and thus x1 ∈ Bx
u1 a
u0 .
b
3.9
0 , c and u1 ∈ Dx1 , T x1 , there exist x2 ∈ T x1 and u2 ∈ Dx2 , T x2 such
Now, since x1 ∈ Bx
that
bdx1 , x2 u1 ,
3.10
u2 adx1 , x2 .
3.11
Using 3.9, 3.10 3.11 we obtain
u2 a
2
u0 .
3.12
a
a
a
dx0 , x1 1−
c.
b
b
b
3.13
b
From 3.7, 3.8 and 3.10 it follows that
dx1 , x2 Note that
dx0 , x2 dx0 , x1 dx1 , x2 a
a
a
1−
c
1−
c
b
b
b
a 2
1−
c
b
3.14
c,
0 , c. Continuing this process, we obtain xn ∈ Bx
0 , c and un ∈ Dxn , T xn and so x2 ∈ Bx
such that xn1 ∈ T xn and un1 ∈ Dxn1 , T xn1 satisfying
bdxn , xn1 un adxn−1 , xn ,
3.15
and we get
dxn , xn1 a
dxn−1 , xn b
for n ∈ N.
3.16
0 , c ⊂ X. Due to the
Thus by Lemma 3.1, {xn } is a Cauchy sequence in the closed set Bx
0 , c such that lim xn x∗ . Note that
0 , c, there exists x∗ ∈ Bx
completeness of Bx
n→∞
un1 adxn , xn1 a
a
un · · · b
b
n1
u0 ,
3.17
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Fixed Point Theory and Applications
and thus
0 un1 a
n1
b
3.18
u0.
From 3.18 and the fact the cone P is normal, we have
un1 ≤ K
a
b
n1
u0 ,
3.19
and thus un1 → 0 as n → ∞; it follows that {un } is convergent to 0. Since un ∈ Dxn , T xn ,
there exists a sequence {zn } such that zn ∈ T xn and un dxn , zn . Now by the convergence
of the sequence {un } and by assumption iii we obtain
inf ∗ d x∗ , y ≤ lim inf inf d xn , y ≤ lim inf dxn , zn 0.
3.20
inf ∗ d x∗ , y 0.
3.21
y∈T x
n → ∞ y∈T xn
n→∞
Thus
y∈T x
From 3.21, it follows that there exists a sequence {yn } ⊂ T x∗ such that limn → ∞ dx∗ , yn 0, and thus dx∗ , yn → 0 as n → ∞. Hence, yn → x∗ . Since T x∗ is closed, we get x∗ ∈ T x∗ .
0 , c Thus, FixT ∩ Bx
/ ∅.
Remark 3.3. Our Theorem 3.2 extends the main fixed point result of Chifu and Petrusel 18,
Theorem 2.1 to the setting of cone metric spaces, and thus the result of Feng and Liu 7,
Theorem 2.1 follows from our Theorem 3.2 as well. Theorem 3.2 also extends some results
from 2, 5, 6.
Another fixed point result is the following.
Theorem 3.4. Let X, d be a complete cone metric space, P a normal cone with normal constant K,
and T : X → ClX. Suppose that the following hold for arbitrary but fixed x0 ∈ X, u0 ∈ Dx0 , T x0 ,
and c ∈ E with c
0:
0 , c and for any
i there exist a, b, s ∈ 0, 1 with a sb < b such that for each x ∈ Bx
u ∈ Dx, T x there exist y ∈ T x and v ∈ Dy, T y satisfying
bd x, y u,
v ad x, y su;
3.22
ii u0 1 − a/b sbc;
iii the function f : X → R defined by fx infy∈T x dx, y is lower semicontinuous.
0 , c /
Then FixT ∩ Bx
∅.
Fixed Point Theory and Applications
7
0 , c and u0 ∈ Dx0 , T x0 , there exist x1 ∈ T x0 and u1 ∈ Dx1 , T x1 Proof. Since x0 ∈ Bx
satisfying
3.23
3.24
bdx0 , x1 u0 ,
u1 adx0 , x1 su0 .
From 3.23 and 3.24 we have
u1 a
b
s u0 .
3.25
Using 3.23 and ii, we get
dx0 , x1 1 −
a
b
s
c c,
3.26
0 , c. Therefore, there exist x2 ∈ T x1 and u2 ∈ Dx2 , T x2 satisfying
and so x1 ∈ Bx
bdx1 , x2 u1
3.27
u2 adx1 , x2 su1 .
3.28
Using 3.25, 3.27, and 3.28, we get
u2 a
b
s u1 a
b
2
s u0 .
3.29
Now, using 3.23, 3.24, 3.27, and ii, we have
dx1 , x2 a
a
1a
s u0 s 1−
s
b b
b
b
3.30
c.
Note that
dx0 , x2 dx0 , x1 dx1 , x2 a
a
a
s c
s 1−
s
1−
b
b
b
a
2
s
1−
c
b
c
3.31
c.
0 , c. Continuing this process, we get xn ∈ Bx
0 , c and un ∈ Dxn , T xn such
Thus, x2 ∈ Bx
that xn1 ∈ T xn and un1 ∈ Dxn1 , T xn1 satisfying
bdxn , xn1 un a
b
s
n
u0 .
3.32
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Fixed Point Theory and Applications
Thus, we get
dxn , xn1 a
1
un s
b
b
n
u0 .
3.33
Now, for m > n, we have
dxn , xm dxn , xn1 dxn1 , xn2 · · · dxm−1 , xm 1 n n1
t t · · · tm−1 u0
b
1 tn
u0 ,
b1−t
3.34
where t a/b s < 1. Since P is normal, we have
dxn , xm ≤ K
tm
u0 ,
b1 − t
3.35
0 , c, there exists x∗ ∈
and hence {xn } is a Cauchy sequence. Due to the completeness of Bx
∗
Bx0 , c, such that limn → ∞ xn x . Also, note that
un1 ≤ K
a
b
s
n1
u0 ,
3.36
and thus,
lim un1 0.
n→∞
3.37
The rest of the proof runs as the proof of Theorem 3.2, and hence we get FixT ∩
0 , c /
∅.
Bx
Remark 3.5. Theorem 3.4 extends the fixed point result of Chifu and Petrusel 18, Theorem 2.5 to cone metric spaces.
Most recently, Asadi et al. 13, Lemma 2.1 proved the closedness of the set FixT in
complete cone metric spaces without the normality assumption. In the following remark, we
obtain the same conclusion without normality and completeness assumptions.
Remark 3.6. Let X, d be a cone metric space, and let T : X → ClX be any multivalued
map. If the function f : X → R defined by fx infy∈T x dx, y is lower semicontinuous,
then the set FixT is closed.
Fixed Point Theory and Applications
9
Indeed, let zn ∈ FixT be such that zn → z∗ as n → ∞. Clearly, fzn infy∈T zn dzn , y 0 because zn ∈ T zn . Using the lower semicontinuity of the function f,
we get
inf d z∗ , y ≤ lim inf inf d zn , y 0.
3.38
inf ∗ d z∗ , y 0.
3.39
n → ∞ y∈T zn
y∈T z∗
Thus
y∈T z
So, there exists a sequence {yn } ⊂ T z∗ such that limn → ∞ dz∗ , yn 0. Hence yn → z∗ ∈
T z∗ .
Example 3.7. Let X 0, 1, E R2 a Banach space with the maximum norm, and P {x, y ∈
E : x, y ≥ 0} a normal cone. Define d : X × X → E by
d x, y x − y, βx − y ,
β ∈ 0, 1.
3.40
Then the pair X, d is a complete cone metric space. Now, define the map T : X → ClX by
⎧
⎪
{0, 1}, x 0,
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 1 x , x ∈ 0, 1,
T x 2
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
⎩ , 1 , x 1.
2
3.41
⎧
⎪1
⎨ x, x ∈ 0, 1,
fx inf d x, y 2
⎪
y∈T x
⎩0, x ∈ {0, 1}
3.42
Note that the map
is lower semicontinuous. Now, if we take c 1/2, 1/2 ∈ E, x0 0 ∈ X, we get
0 , c {x ∈ X, dx0 , x ≤ c} 0, 1 .
Bx
2
3.43
Now, for the case x ∈ 0, 1/2 and y 1/2x ∈ T x, we obtain
Dx, T x D y, T y 1
1
x, β x
2
2
1
1
x, β x
4
4
,
.
3.44
10
Fixed Point Theory and Applications
Now, taking a 1/2 and b 1, we get
bd x, y ≤ u,
v ≤ ad x, y ,
for each u ∈ Dx, T x,
for v 1
1
x, β x
4
4
∈ D y, T y .
3.45
Now, for the case x 0, y 0 ∈ T x, we have
Dx, T x 0, 0, 1, β ,
D y, T y 0, 0, 1, β .
3.46
bd x, y ≤ u, for each u ∈ Dx, T x,
v ≤ ad x, y , for v 0 ∈ D y, T y .
3.47
And also, for this case we get
Further, for u0 0, 0 ∈ Dx0 , T x0 , we have
a
bc.
u0 ≤ 1 −
b
3.48
0 , c Therefore, all the assumptions of Theorem 3.2 are satisfied, and note that FixT ∩ Bx
{0}.
Acknowledgments
The authors are grateful to Professor Sh. Rezapour for providing a copy of the paper 16.
Also, the authors are thankful to the referees for their valuable suggestions to improve this
paper. Finally, the first author thanks the Deanship of Scientific Research, King Abdulaziz
University for the research Grant no. 3-62/429.
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