Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 178306, 8 pages
doi:10.1155/2011/178306
Research Article
Fixed Point Results in Quasimetric Spaces
Abdul Latif and Saleh A. Al-Mezel
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Abdul Latif, [email protected]
Received 21 August 2010; Accepted 5 October 2010
Academic Editor: Qamrul Hasan Ansari
Copyright q 2011 A. Latif and S. A. Al-Mezel. 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.
In the setting of quasimetric spaces, we prove some new results on the existence of fixed points for
contractive type maps with respect to Q-function. Our results either improve or generalize many
known results in the literature.
1. Introduction and Preliminaries
Let X be a metric space with metric d.WeuseSX to denote the collection of all nonempty
subsets of X, ClX for the collection of all nonempty closed subsets of X, CBX for the
collection of all nonempty closed bounded subsets of X, and H for the Hausdorff metric
on CBX, that is,
H
A, B
max
sup
a∈A
d
x, y
, 1.2
bT is weakly contractive 2 if there exist constants h, b ∈ 0,
1,h<b, such that for
any x ∈ X, there is y ∈ I
x
b
satisfying
d
y, T
y
≤ hd
x, y
, 1.3
where I
x
b
{y ∈ Tx : bdx, y ≤ dx, Tx}.
2 Fixed Point Theory and Applications
Apointx ∈ X is called a fixed point of a multivalued map T : X → SX if x ∈ Tx.
We denote FixT{x ∈ X : x ∈ Tx}.
A sequence {x
n
} in X is called an orbit of T at x
w2 ω is lower semicontinuous in its second variable;
w3 for any ε>0, there exists δ>0, such that ωz, x ≤ δ and ωz, y ≤ δ imply
dx, y ≤ ε.
Note that in general for x,y ∈ X, ωx, y
/
ωy, x and not either of the implications
ωx, y0 ⇔ x y necessarily holds. Clearly, the metric d is a w-distance on X. Many other
examples and properties of w-distances are given in 3.
In 4, Suzuki and Takahashi improved Nadler contraction principle Theorem 1.1 as
follows.
Theorem 1.3. Let X, d
be a complete metric space and let T : X → ClX.Ifthereexistaw-
distance ω on X and a constant λ ∈ 0, 1, such that for each x, y ∈ X and u ∈ Tx,thereis
v ∈ Ty satisfying
ω
u, v
≤ λω
x, y
, 1.4
then T has a fixed point.
Recently, Latif and Albar 5 generalized Theorem 1.2 with respect to w-distance see,
Theorem 3.3in5,andLatif6 proved a fixed point result with respect to w-distance see,
Theorem 2.2in6 which contains Theorem 1.3 as a special case.
A nonempty set X together with a quasimetric d i.e., not necessarily symmetric
is called a quasimetric space. In the setting of a quasimetric spaces, Al-Homidan et al. 7
n
,y ≤ α
n
and qx
n
,z ≤ β
n
for all n ≥ 1, then y z; in particular, if qx, y0
and qx, z0,theny z;
ii if qx
n
,y
n
≤ α
n
and qx
n
,z ≤ β
n
for all n ≥ 1, then {y
n
} converges to z;
iii if qx
n
,x
m
≤ α
n
for any n, m ≥ 1 with m>n,then {x
n
b
satisfying
q
y, T
y
≤ hq
x, y
, 1.6
where J
x
b
{y ∈ Tx : bqx, y ≤ qx, Tx} and qx, Tx inf{qx, y : y ∈ Tx};
d T is generalized q-contractive if there exists a Q-function q on X, such that for each
x, y ∈ X and u ∈ Tx, there is v ∈ Ty satisfying
q
u, v
≤ k
q
x, y
q
,v
o
0, then v
0
is a fixed point of
T.
Proof. Let x
o
∈ X. Since T is weakly contractive, there is x
1
∈ J
x
o
b
⊆ Tx
o
, such that
q
x
1
,T
x
1
≤ hq
x
o
,n 0, 1, 2, 2.2
Since bqx
n
,x
n1
≤ qx
n
,Tx
n
and h<b<1, thus we get
q
x
n1
,T
x
n1
≤ q
x
n
,T
x
n
. 2.3
≤ a
n
q
x
o
,T
x
0
,n 0, 1, 2, , 2.5
and since 0 <a<1, hence the sequence {fx
n
} {qx
n
,Tx
n
}, which is decreasing,
converges to 0. Now, we show that {x
n
} is a Cauchy sequence. Note that
q
x
n
,x
n1
≤ a
n2
··· q
x
m−1
,x
m
≤ a
n
q
x
o
,x
1
a
n1
q
x
o
,x
1
··· a
m−1
q
. Now, since f is lower semicontinuous, we have
0 ≤ f
v
o
≤ lim inf
n →∞
f
x
n
0, 2.8
and thus, fv
o
qv
o
,Tv
o
0. It follows that there exists a sequence {v
n
} in Tv
0
,
such that qv
0
,v
n
→ 0. Now, if qv
o
be an arbitrary but fixed element of X and let x
1
∈ Tx
0
. Since T is generalized
as a q-contractive, there is x
2
∈ Tx
1
, such that
q
x
1
,x
2
≤ k
q
x
o
,x
1
q
q
x
n−1
,x
n
. 2.10
Thus, for all n ≥ 1, we have
q
x
n
,x
n1
<q
x
n−1
,x
n
. 2.11
Write t
n
qx
n
,x
n1
} is a Cauchy sequence. Let α
lim sup
r → 0
kr < 1. There exists real number β such that α<β<1. Then for sufficiently
large n, kt
n
<β,andthusforsufficiently large n, we have t
n
<βt
n−1
. Consequently, we
obtain t
n
<β
n
t
0
,thatis,
q
x
n
,x
n1
<β
n
q
x
m−1
,x
m
<β
n
q
x
o
,x
1
β
n1
q
x
o
,x
1
··· β
m−1
q
x
o
,x
0
∈ X such that lim
n →∞
x
n
v
o
. Let n be arbitrary fixed positive integer then for all
positive integers m with m>n, we have
q
x
n
,x
m
≤
β
n
1 − β
q
x
o
,x
1
. 2.16
Let M β
n
n
1 − β
q
x
o
,x
1
, for all positive integer n. 2.18
Note that qx
n
,v
o
converges to 0. Now, since x
n
∈ Tx
n−1
and T is a generalized q-
contractive map, then there is u
n
∈ Tv
0
, such that
q
x
n
,u
n
q
x
n−1
,v
0
q
x
n−1
,v
0
<βq
x
n−1
,v
0
, 2.20
thus, we get
q
x
n
,u
n
.
Corollary 2.4. Let X, d be a complete quasimetric space and q a Q-function on X.LetT : X →
ClX be a multivalued map, such that for any x, y ∈ X and u ∈ Tx,thereisv ∈ Ty with
q
u, v
≤ k
q
x, y
q
x, y
, 2.22
where k is a monotonic increasing function from 0, ∞ to 0, 1,thenT has a fixed point.
Finally, we conclude with the following remarks concerning our results related to the
known fixed point results.
Remark 2.5. 1Theorem 2.1 generalizes Theorem 1.2 according to Feng and Liu 2 and Latif
and Albar 5, Theorem 3.3.
2Theorem 2.3
generalizes Theorem 1.3 according to Suzuki and Takahashi 4 and
Theorem 1.5 according to Al-Homidan et al. 7 and contains Latif’s Theorem 2.2in6.
3Theorem 2.3 also generalizes Theorem 2.1in8 in several ways.
4Corollary 2.4 improves and generalizes Theorem 1 in 9.
Acknowledgments
The authors thank t he referees for their kind comments. The authors also thank King
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