A NEW CONCEPT OF (α, Fd )-CONTRACTION ON QUASI
METRIC SPACE
I. ALTUN, N. AL ARIFI, M. JLELI, A. LASHIN, AND B. SAMET
Abstract. In the present paper, we introduce a new concept of (α, Fd )contraction on quasi metric space. Then we provide some new fixed
point theorems for such type mappings on left K, left M and left Smythcomplete quasi metric spaces.
1. Introduction and Preliminaries
A quasi-pseudo metric on a nonempty set X is a function d : X ×X → R+
such that for all x, y, z ∈ X
• d(x, x) = 0,
• d(x, y) ≤ d(x, z) + d(z, y).
If a quasi-pseudo metric d satisfies
• d(x, y) = d(y, x) = 0 ⇒ x = y,
then d is said to be quasi metric, in addition if a quasi metric d satisfies
• d(x, y) = 0 ⇒ x = y,
then d is said to be T1 -quasi metric. It is clear that, every metric is a
T1 -quasi metric, every T1 -quasi metric is a quasi metric and every quasi
metric is a quasi-pseudo metric. In this case the pair (X, d) is said to be
quasi-pseudo (resp. quasi, T1 -quasi) metric space.
Let (X, d) be a quasi-pseudo metric space. Given a point x0 ∈ X and a
real constant ε > 0, the sets
Bd (x0 , ε) = {y ∈ X : d(x0 , y) < ε}
and
Bd [x0 , ε] = {y ∈ X : d(x0 , y) ≤ ε}
are called open ball and closed ball, respectively, with center x0 and radius
ε.
Each quasi-pseudo metric d on X generates a topology τd on X which has
a base the family of open balls {Bd (x, ε) : x ∈ X and ε > 0}. The closure
of a subset A of X with respect to τd is denoted by cld (A). If d is a quasi
metric on X, then τd is a T0 topology, and if d is a T1 -quasi metric, then τd
is a T1 topology on X.
2000 Mathematics Subject Classification. Primary 54H25; Secondary, 47H10.
Key words and phrases. Quasi metric space, left K-Cauchy sequence, left Kcompleteness, fixed point.
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2
I. ALTUN, N. AL ARIFI, M. JLELI, A. LASHIN, AND B. SAMET
If d is a quasi metric and τd is T1 topology, then d is T1 -quasi metric.
If d is a quasi-pseudo metric on X, then the functions d−1 , ds and d+
defined by
d−1 (x, y) = d(y, x),
ds (x, y) = max d(x, y), d−1 (x, y)
and
d+ (x, y) = d(x, y) + d−1 (x, y)
are also quasi-pseudo metrics on X. If d is a quasi metric, then ds and d+
are (equivalent) metrics on X. Further, if a quasi-pseudo metric d satisfies
x 6= y ⇒ d(x, y) + d−1 (x, y) > 0,
then d+ (and also ds ) is a metric on X.
Let (X, d) be a quasi metric space and x ∈ X. The convergence of a
sequence {xn } to x with respect to τd called d-convergence and denoted by
d
xn → x, is defined
d
xn → x ⇔ d(x, xn ) → 0.
Similarly, the convergence of a sequence {xn } to x with respect to τd−1 called
d−1
d−1 -convergence and denoted by xn → x, is defined
d−1
xn → x ⇔ d(xn , x) → 0.
Finally, the convergence of a sequence {xn } to x with respect to τds called
ds
ds -convergence and denoted by xn → x, is defined
ds
xn → x ⇔ ds (xn , x) → 0.
ds
d
d−1
It is clear that xn → x ⇔ xn → x and xn → x. More and detailed
information about some important properties of quasi metric spaces and
their topological structures can be found in [8, 14, 15].
Definition 1 ([22]). Let (X, d) be a quasi metric space. A sequence {xn }
in X is called
• left K-Cauchy (or forward Cauchy) if for every ε > 0, there exists
n0 ∈ N such that
∀n, k, n ≥ k ≥ n0 , d(xk , xn ) < ε,
• right K-Cauchy (or backward Cauchy) if for every ε > 0, there exists
n0 ∈ N such that
∀n, k, n ≥ k ≥ n0 , d(xn , xk ) < ε,
• ds -Cauchy if for every ε > 0, there exists n0 ∈ N such that
∀n, k ≥ n0 , d(xn , xk ) < ε.
(α, Fd )-CONTRACTION ON QUASI METRIC SPACE
3
It is clear that {xn } is ds -Cauchy if and only if it is both left K-Cauchy
and right K-Cauchy. If a sequence is left K-Cauchy with respect to d, then
it is right K-Cauchy with respect to d−1 . If {xn } is a sequence in a quasi
metric space (X, d) such that
∞
X
d(xn , xn+1 ) < ∞,
n=1
then it is left K-Cauchy sequence.
It is well known that every convergent sequence is Cauchy in a metric
space. In general, this situation is not valid in a quasi metric space. That
is, d-convergent or d−1 -convergent sequences may not be Cauchy (in the
sense of ds , left K and right K) in a quasi metric space (see [22]).
Definition 2 ([4, 22]). Let (X, d) be a quasi metric space. Then (X, d) is
said to be
• bicomplete if every ds -Cauchy sequence is ds -convergent,
• left (right) K-complete if every left (right) K-Cauchy sequence is
d-convergent,
• left (right) M-complete if every left (right) K-Cauchy sequence is
d−1 -convergent,
• left (right) Smyth complete if every left (right) K-Cauchy sequence
is ds -convergent.
Remark 1. If (X, d) is both left K-complete and left M-complete, then it
is bicomplete.
One can find more detailed information about some kind of Cauchyness,
completeness and some important properties of quasi metric space in [4, 22,
23].
On the other hand, α-admissibility and F -contractivity of a mapping
are popular concepts in recent metrical fixed point theory. The concept of
α-admissibility of a mapping on a nonempty set has been introduced by
Samet et al [24]. Let X be a nonempty set, T be a self mapping of X and
α : X × X → [0, ∞) be a function. Then T is said to be α-admissible if
α(x, y) ≥ 1 ⇒ α(T x, T y) ≥ 1.
Using α-admissibility of a mapping, Samet et al [24] provided some general fixed point results including many known theorems on complete metric
spaces.
The concept of F -contraction was introduced by Wardowski [25]. Let F be
the family of all functions F : (0, ∞) → R satisfying the following conditions:
(F1) F is strictly increasing,
(F2) For each sequence {λn } of positive numbers limn→∞ λn = 0 if and
only if limn→∞ F (λn ) = −∞,
(F3) There exists k ∈ (0, 1) such that limλ→0+ λk F (λ) = 0.
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I. ALTUN, N. AL ARIFI, M. JLELI, A. LASHIN, AND B. SAMET
Let T be a self mapping of a metric space (X, d) and F ∈ F. Then, T is
said to be F -contraction if for all x, y ∈ X with d(T x, T y) > 0, there exists
τ > 0 such that
τ + F (d(T x, T y)) ≤ F (d(x, y)).
Various fixed point results for α-admissible mappings and F -contractions
on complete metric space can be found in [3, 10, 12, 13, 20] and [5, 6, 7, 9,
19, 21, 26], respectively.
The aim of this paper is to establish several new fixed point results
on some kind of complete quasi metric spaces by taking into account αadmissibility and a quasi metric version of F -contractivity of a mapping.
We can find some recent fixed point results for single valued and multivalued mappings on quasi metric spaces in [1, 2, 11, 16, 17, 18].
2. Fixed Point Results
Let (X, d) be a quasi metric space, T : X → X be a mapping and α :
X×X → [0, ∞) be a function. We will consider the following set to introduce
a quasi metric version of F -contractivity of a mapping:
Tα = {(x, y) ∈ X × X : α(x, y) ≥ 1 and d(T x, T y) > 0}.
Definition 3. Let (X, d) be a quasi metric space, T : X → X be a mapping
satisfying
d(x, y) = 0 ⇒ d(T x, T y) = 0,
(2.1)
α : X × X → [0, ∞) and F ∈ F be two functions. Then T is said to be an
(α, Fd )-contraction if there exists τ > 0 such that
τ + F (d(T x, T y)) ≤ F (d(x, y)),
(2.2)
for all (x, y) ∈ Tα .
The following example shows the importance of condition (2.1).
Example 1. Let X = {0, 1} ∪ A, where A = {2, 3, · · · } and let d be the
quasi metric on X given by
d(n, n) = 0 for all n ∈ X
d(0, n) = d(n, 1) = 0 for all n ∈ X\{0}
d(n, 0) = 1 for all n ∈ X\{0}
1
1
d(n, m) =
+ m+1 , otherwise.
n+1
2
2
Now define T : X → X as T n = n + 1 for all n ∈ X and α : X × X → [0, ∞)
as

 1 , (n, m) ∈ A × A
α(n, m) =
.

0 ,
otherwise
(α, Fd )-CONTRACTION ON QUASI METRIC SPACE
5
Note that Tα = {(n, m) : n, m ∈ A and n 6= m}. Now, we show that
inequality (2.2) is satisfied with F (λ) = ln λ and τ = ln 2 for all (n, m) ∈ Tα .
Indeed, for n, m ∈ A with n 6= m, we have
ln d(T n, T m) = ln d(n + 1, m + 1)
1
1
= ln
+
2n+2 2m+2
1
1
+
= − ln 2 + ln
2n+1 2m+1
= − ln 2 + d(n, m).
However, T is not (α, Fd )-contraction, since the condition (2.1) does not
hold. Indeed, d(0, 1) = 0 but d(T 0, T 1) > 0.
Remark 2. If (X, d) is a T1 -quasi metric space, then every mapping T :
X → X satisfies the condition (2.1).
Remark 3. It is clear from Definition 3 that if T is an (α, Fd )-contraction
on a quasi metric space (X, d), then
d(T x, T y) ≤ d(x, y)
for all x, y ∈ X with α(x, y) ≥ 1.
Our main fixed point result as follows:
Theorem 1. Let (X, d) be a Hausdorff left K-complete quasi metric space,
T : X → X be an (α, Fd )-contraction. Assume that T is α-admissible and
τd -continuous. If there exists x0 ∈ X such that α(x0 , T x0 ) ≥ 1, then T has
a fixed point in X.
Proof. Let x0 ∈ X be such that α(x0 , T x0 ) ≥ 1. Define a sequence {xn } in X
by xn = T xn−1 for all n ∈ N. Since T is α-admissible, then α(xn , xn+1 ) ≥ 1
for all n ∈ N. Now, let dn = d(xn , xn+1 ) for all n ∈ N. If there exists k ∈ N
with dk = d(xk , xk+1 ) = 0, then xk is a fixed point of T since d is T1 -quasi
metric. Suppose dn > 0 for all n ∈ N. Since T is (α, Fd )-contraction, we get
F (dn ) ≤ F (dn−1 ) − τ ≤ F (dn−2 ) − 2τ ≤ · · · ≤ F (d0 ) − nτ.
(2.3)
From (2.3), we get limn→∞ F (dn ) = −∞.Thus, from (F2), we have
lim dn = 0.
n→∞
From (F3) there exists k ∈ (0, 1) such that
lim dk F (dn )
n→∞ n
= 0.
By (2.3), the following holds for all n ∈ N
dkn F (dn ) − dkn F (d0 ) ≤ −dkn nτ ≤ 0.
(2.4)
Letting n → ∞ in (2.4), we obtain that
lim ndkn = 0.
n→∞
(2.5)
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I. ALTUN, N. AL ARIFI, M. JLELI, A. LASHIN, AND B. SAMET
From (2.5), there exits n1 ∈ N such that ndkn ≤ 1 for all n ≥ n1 . So, we
have, for all n ≥ n1
1
dn ≤ 1/k .
(2.6)
n
∞
P
Therefore
dn < ∞. Now let m, n ∈ N with m > n ≥ n1 , then we get
n=1
d(xn , xm ) ≤ d(xn+1 , xn+2 ) + d(xn+1 , xn+2 ) + · · · + d(xm−1 , xm )
= dn + dn+1 + · · · + dm−1
∞
X
≤
dk .
k=n
Since
∞
P
dk is convergent, then we get {xn } is left K-Cauchy sequence in the
k=1
quasi metric space (X, d). Since (X, d) left K-complete, there exists z ∈ X
such that {xn } is d-converges to z, that is, d(z, xn ) → 0 as n → ∞. Since T
is τd -continuous, then d(T z, T xn ) = d(T z, xn+1 ) → 0 as n → ∞. Since X is
Hausdorff we get z = T z.
The following example shows that the Hausdorffness condition of X can
not be removed in above theorem.
Example 2. Let X = { n1 : n ∈ N} and

 0 , x=y
d(x, y) =
.

y , x 6= y
Then (X, d) is a left K-complete T1 -quasi metric space, but not Hausdorff
since τd is cofinite topology. Define a mapping T : X → X by T x = x2 , then
T is (α, Fd )-contraction with α(x, y) = 1, F (λ) = ln λ and τ = ln 2. Also T
is α-admissible and τd -continuous. However, T has no fixed point.
In Theorem 1, if we take τd−1 continuity of the mapping T instead of
τd -continuity, we can take left M-completeness of X.
Theorem 2. Let (X, d) be a Hausdorff left M-complete quasi metric space,
T : X → X be an (α, Fd )-contraction. Assume that T is α-admissible and
τd−1 -continuous. If there exists x0 ∈ X such that α(x0 , T x0 ) ≥ 1, then T
has a fixed point in X.
Proof. As in the proof of Theorem 1, we can get the iterative sequence {xn }
is left K-Cauchy. Since (X, d) left M-complete, there exists z ∈ X such
that {xn } is d−1 -converges to z, that is, d(xn , z) → 0 as n → ∞. By τd−1 continuity of T, we have d(T xn , T z) = d(xn+1 , T z) → 0 as n → ∞. Since X
is Hausdorff we get z = T z.
The following theorems show that if we take into account the left Smyth
completeness of X, we can remove the Hausdorffness condition. However,
we need the quasi metric d is still T1 -quasi metric.
(α, Fd )-CONTRACTION ON QUASI METRIC SPACE
7
Theorem 3. Let (X, d) be left Smyth complete T1 -quasi metric space, T :
X → X be an (α, Fd )-contraction. Assume that T is α-admissible and τd
or τd−1 -continuous. If there exists x0 ∈ X such that α(x0 , T x0 ) ≥ 1, then T
has a fixed point in X.
Proof. As in the proof of Theorem 1, we can get the iterative sequence {xn }
is left K-Cauchy. Since (X, d) left Smyth complete, there exists z ∈ X such
that {xn } is ds -converges to z, that is, ds (xn , z) → 0 as n → ∞.
If T is τd -continuous, then d(T z, T xn ) = d(T z, xn+1 ) → 0 as n → ∞.
Therefore, we get
d(T z, z) ≤ d(T z, xn+1 ) + d(xn+1 , z) → 0 as n → ∞.
If T is τd−1 -continuous, then d(T xn , T z) = d(xn+1 , T z) → 0 as n → ∞.
Therefore, we get
d(z, T z) ≤ d(z, xn+1 ) + d(xn+1 , T z) → 0 as n → ∞.
Since X is T1 -quasi metric space, we obtain z = T z.
One of the following properties of the spaces X, can be considered instead
of τd or τd−1 -continuity of T in Theorem 3.
It is said that the quasi metric space (X, d) has (Ad ) (resp. (Bd )) property
d
whenever {xn } is a sequence in X such that xn → z and α(xn , xn+1 ) ≥ 1
implies α(xn , z) ≥ 1 (resp. α(z, xn ) ≥ 1) for all n ∈ N.
Theorem 4. Let (X, d) be left Smyth complete T1 -quasi metric space, T :
X → X be an (α, Fd )-contraction. Assume that T is α-admissible and X
has (Ad ), (Ad−1 ), (Bd ) or (Bd−1 ) property. If there exists x0 ∈ X such that
α(x0 , T x0 ) ≥ 1, then T has a fixed point in X.
Proof. As in the proof of Theorem 1, we can get the iterative sequence {xn }
is left K-Cauchy. Since (X, d) left Smyth complete, there exists z ∈ X such
that {xn } is ds -converges to z, that is, ds (xn , z) → 0 as n → ∞.
If X has (Ad ) or (Ad−1 ) property, we get α(xn , z) ≥ 1. Therefore, considering Remark 3, we have
d(z, T z) ≤ d(z, xn+1 ) + d(T xn , T z)
≤ d(z, xn+1 ) + d(xn , z) → 0 as n → ∞.
If X has (Bd ) or (Bd−1 ) property, we get α(z, xn ) ≥ 1. Therefore, considering Remark 3, we have
d(T z, z) ≤ d(T z, T xn ) + d(xn+1 , z)
≤ d(z, xn ) + d(xn+1 , z) → 0 as n → ∞.
Since X is T1 -quasi metric space, we get z = T z.
The following example emphasizes the importances of the continuity of T
in Theorem 3 and (Ad ), (Ad−1 ), (Bd ) or (Bd−1 ) properties on X in Theorem
4.
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I. ALTUN, N. AL ARIFI, M. JLELI, A. LASHIN, AND B. SAMET
Example 3. Let X = [0, 1] and d(x, y) = |x − y| for all x, y ∈ X. Now
define T : X → X as

 1 , x=0
Tx =
 x
2 , x 6= 0
and α : X × X → [0, ∞) as

 1 , xy 6= 0
α(x, y) =
.

0 , otherwise
Since (X, d) is complete metric space then it is left Smyth complete T1 -quasi
metric space. Observe that T is α-admissible and also for x0 = 1, we have
α(x0 , T x0 ) = α(1, T 1) = α(1, 21 ) ≥ 1. On the other hand, since
Tα = {(x, y) ∈ X × X : α(x, y) ≥ 1 and d(T x, T y) > 0}
= {(x, y) ∈ X × X : xy 6= 0 and x 6= y},
then
1
d(T x, T y) = d(x, y)
2
for all (x, y) ∈ Tα . That is, T is (α, Fd )-contraction with F (λ) = ln λ and
τ = ln 2.
Note that T is not τd (and τd−1 )-continuous. Also note that X has not
(Ad ) (and (Bd )) property. To see this we can consider the sequence xn = n1 .
Remark 4. Taking into account the right completeness (in the sense of K,
M and Smyth), we can provide similar fixed point results on quasi metric
space.
Acknowledgements.
The authors extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University
(Saudi Arabia).
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10
I. ALTUN, N. AL ARIFI, M. JLELI, A. LASHIN, AND B. SAMET
(I. Altun) Adress 1: King Saud University, College of Science, Riyadh, Saudi
Arabia
Adress 2: Department of Mathematics, Faculty of Science and Arts, Kirikkale
University, 71450 Yahsihan, Kirikkale, Turkey
E-mail address: ishakaltun@yahoo.com
(N. Al Arifi) College of Science, Geology and Geophysics Department, King
Saud University, Riyadh 11451, Saudi Arabia
E-mail address: nalarifi@ksu.edu.sa
(M. Jleli) Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
E-mail address: jleli@ksu.edu.sa
(A. Lashin) Address 1: College of Engineering, Petroleum and Natural Gas
Engineering Department, King Saud University, Riyadh 11421, Saudi Arabia
Address 2: Faculty of Science, Geology Department, Benha University, Benha
13518, Egypt
E-mail address: arlashin@ksu.edu.sa
(B. Samet) Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
E-mail address: bsamet@ksu.edu.sa