C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat.
Volum e 68, N umb er 2, Pages 1819–1830 (2019) D O I: 10.31801/cfsuasm as.419821
ISSN 1303–5991 E-ISSN 2618-6470
http://com munications.science.ankara.edu.tr/index.php?series= A 1
ON TWO TYPES ALMOST ( ; Fd)-CONTRACTIONS ON QUASI METRIC SPACE
HATICE ASLAN HANÇER
Abstract. In this paper, …rst we introduce two new types almost contractions on quasi metric space named as almost ( ; Fd)-contraction of type (x) and of type (y). Then, taking into account both left and right completeness of quasi metric space, we present some …xed point results for these contractions. We also provide some illustrative and comparative examples.
1. Introduction and Preliminaries
Fundamentally, …xed point theory divides into three major subject which are topological, discrete and metric. Especially, it has been intensively improving on the metric case because of useful to applications. In general, metrical …xed point theory is related to contractive type mappings and it has been developed either taking into account the new type contractions or playing the structure of the space such as fuzzy metric space, quasi metric space, metric like space etc. A quasi metric space plays a crucial role in some …elds of theoretical computer service, asymmetric functional analysis and approximation theory. Now, we will recall some basic concepts of quasi metric space.
In quasi metric spaces there are many di¤erent types of Cauchyness, yielding even more notions of completeness. Another di¤erence comes from the fact that, in contrast to the metric case, in a quasi metric space a convergent sequence could not be Cauchy (see [3] for examples con…rming this situation).
Let X be nonempty set and d : X X ! R+ be a function. Consider the following conditions on d, for all x; y; z 2 X :
(qm1) d(x; x) = 0;
(qm2) d(x; y) d(x; z) + d(z; y);
(qm3) d(x; y) = d(y; x) = 0 ) x = y;
(qm4) d(x; y) = 0 ) x = y:
Received by the editors: Received: April 30, 2018; Accepted: September 17, 2018.
2010 Mathematics Subject Classi…cation. Primary 54H25; Secondary 47H10.
Key words and phrases. Quasi metric space, left K-Cauchy sequence, left K-completeness,
…xed point.
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If the function d satis…es conditions (qm1) and (qm2) then d is said to be a quasi-pseudo metric on X: Further, if a quasi-pseudo metric d satis…es condition (qm3), then d is said to be a quasi metric on X, and if a quasi metric d satis…es condition (qm4), then d is said to be a T1-quasi metric on X: In this case, the pair (X; d) is said to be a quasi-pseudo (resp. a quasi, a T1-quasi) metric space. It is clear that every metric space is a T1-quasi metric space.
Let (X; d) be a quasi-pseudo metric space. Given a point x0 2 X and a real constant " > 0, the set
Bd(x0; ") = fy 2 X : d(x0; y) < "g
is called open ball with center x0 and radius ": Each quasi-pseudo metric d on X generates a topology d on X which has a base the family of open balls fBd(x; ") : x 2 X and " > 0g: If d is a quasi metric on X, then dis a T0 topology, and if d is a T1-quasi metric, then d is a T1topology on X.
If d is a quasi-pseudo metric on X, then the function d 1 de…ned by d 1(x; y) = d(y; x)
is a quasi-pseudo metric on X and
ds(x; y) = max d(x; y); d 1(x; y)
is a quasi metric. If d is a quasi metric, then d 1 is also a quasi metric, and dsis a metric on X.
The convergence of a sequence fxng to x with respect to dcalled d-convergence and denoted by xn! x, is de…nedd
xn
! x , d(x; xd n) ! 0:
Similarly, the convergence of a sequence fxng to x with respect to d 1 called d 1-convergence and denoted by xn
d 1
! x, is de…ned xnd! x , d(x1 n; x) ! 0:
Finally, the convergence of a sequence fxng to x with respect to ds called ds- convergence and denoted by xn
ds
! x, is de…ned xn ! x , dds s(xn; x) ! 0:
It is clear that xn ds
! x , xn ! x and xd n d
1
! x. More and detailed informa- tion about some important properties of quasi metric spaces and their topological structures can be found in [10, 16, 17, 18].
De…nition 1 ([25]). Let (X; d) be a quasi metric space. A sequence fxng in X is called
left K-Cauchy (or forward Cauchy) if for every " > 0; there exists n0 2 N such that
8n; k, n k n0, d(xk; xn) < ";
right K-Cauchy (or backward Cauchy) if for every " > 0; there exists n02 N such that
8n; k, n k n0, d(xn; xk) < ";
ds-Cauchy if for every " > 0; there exists n02 N such that 8n; k n0, d(xn; xk) < ":
If a sequence is left K-Cauchy with respect to d, then it is right K-Cauchy with respect to d 1. A sequence is ds-Cauchy if and only if it is both left K-Cauchy and right K-Cauchy. Let fxng be a sequence in a quasi metric space (X; d) such that
X1 n=1
d(xn; xn+1) < 1;
then it is left K-Cauchy sequence (see [10]).
It is well known that a metric space is said to be complete if every Cauchy sequence is convergent. The completeness of a quasi metric space, however, can not be uniquely de…ned. Taking into account the convergence and the Cauchyness of sequences in a quasi metric space, one obtains several notions of completeness, most of them being already available in the literature (see [1, 2, 9, 10, 17, 25]) with di¤erent notations. It can be found a detailed classi…cation, some important properties and relations for completeness of quasi metric spaces in [3].
De…nition 2. Let (X; d) be a quasi metric space. Then (X; d) is said to be
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 3. It is clear that a quasi metric space (X; d) is left M -complete if and only if (X; d 1) is right K-complete. Also, a quasi metric space (X; d) is right M -complete if and only if (X; d 1) is left K-complete.
Remark 4. If a quasi metric space is left Smyht complete, then it is also left K-complete.
We will consider the sequential continuity of a mapping T in our results.
De…nition 5([28]). Let X be a nonempty set, d and be two quasi metrics on X and T : X ! X be a mapping. Then T is said to be sequentially d- -continuous at x 2 X, if for all sequence fxng in X such that d(x; xn) ! 0 implies (T x; T xn) ! 0:
If T is sequentially d- -continuous at all points of X, then T is said to be sequen- tially d- -continuous on X.
On the other hand, -admissibility and F -contractivity of a mapping are popular concepts in recent metrical …xed point theory. The concept of -admissibility of a mapping on a nonempty set has been introduced by Samet [27]. Let X be a nonempty set, T be a self mapping of X and : X X ! [0; 1) be a function.
Then T is said to be -admissible if
(x; y) 1 =) (T x; T y) 1:
Using -admissibilitiy of a mapping, Samet, et al [27] provided some general
…xed point results including many known theorems on complete metric spaces. We say that has (B) property on a metric space (X; d) whenever fxng is a sequence in X such that (xn; xn+1) 1 for all n 2 N and d(xn; x) ! 0; then (xn; x) 1 for all n 2 N:
The concept of F -contraction was introduced by Wardowski [29]. Let F be the family of all functions F : (0; 1) ! R satisfying the following conditions:
(F1) F is strictly increasing, i.e., for all ; 2 (0; 1) such that < ; F ( ) <
F ( );
(F2) For each sequence fang of positive numbers limn!1an = 0 if and only if limn!1F (an) = 1;
(F3) There exists k 2 (0; 1) such that lim !0+ kF ( ) = 0:
Many authors have extended …xed point results on metric space by considering the family F. For instance, in [4, 5, 6, 11, 26] be found some …xed point results for single valued and multivalued mappings on metric space [22]. Various …xed point results for -admissible mapping on complete metric space can be found in [7, 12, 14, 15, 19, 20].
De…nition 6 ([22]). Let (X; d) be a metric space and T : X ! X be a mapping.
Then T is said to be an almost F -contraction if F 2 F and there exist > 0 and L 0 such that
8x; y 2 X[d(T x; T y) > 0 =) + F (d(T x; T y)) F (d(x; y) + Ld(y; T x))]: (1) In order to check the almost F -contractiveness of a mapping T; it is necessary to check both (1) and
8x; y 2 X[d(T x; T y) > 0 =) + F (d(T x; T y)) F (d(x; y) + Ld(x; T y))]: (2) The aim of this paper is to present some new …xed point results on quasi metric space. To do this considering both almost and F -contractiveness of a self mapping on a quasi metric space, we introduce two type almost ( ; Fd)-contractions. Then by -admissibility of a mapping, we obtaine some …xed point results on some kind of complete quasi metric space.
We can …nd some recent …xed point results for single valued and multivalued mappings on quasi metric spaces in [8, 2, 13, 21, 23, 24]
2. Fixed Point Result
Let (X; d) be a quasi metric space, T : X ! X be a mapping and : X X ! [0; 1) be a function. We will consider the following set to introduce a quasi metric version of F -contractivity of a mapping:
T = f(x; y) 2 X X : (x; y) 1 and d(T x; T y) > 0g .
As a slight di¤erent from metric space, we say that has (Bd) (resp. (Bd 1)) property on a quasi metric space (X; d) whenever fxng is a sequence in X such that (xn; xn+1) 1 for all n 2 N and d(x; xn) ! 0 (resp. d(xn; x) ! 0), then (xn; x) 1 for all n 2 N. Similarly, we say that has (Cd) (resp. (Cd 1)) property on a quasi metric space (X; d) whenever fxng is a sequence in X such that (xn+1; xn) 1 for all n 2 N and d(x; xn) ! 0 (resp. d(xn; x) ! 0), then
(x; xn) 1 for all n 2 N:
De…nition 7. Let (X; d) be a quasi metric space. T : X ! X be a mapping, : X X ! [0; 1) and F 2 F be two functions. Then T is said to be an almost ( ; Fd)-contraction of type (y) if there exists > 0 and L 0 such that
+ F (d(T x; T y)) F (d(x; y) + Ld(y; T x)) (3) for all (x; y) 2 T :
De…nition 8. Let (X; d) be a quasi metric space. T : X ! X be a mapping, : X X ! [0; 1) and F 2 F be two functions. Then T is said to be an almost ( ; Fd)-contraction of type (x) if there exists > 0 and L 0 such that
+ F (d(T x; T y)) F (d(x; y) + Ld(x; T y)) (4) for all (x; y) 2 T :
The following example shows the di¤erences between the above two concepts.
Example 9. Let X = [0;14] [ f1g and d(x; y) = maxfy x; 0g for all x; y 2 X:
Then (X; d) is a quasi metric space. Consider the mappings T : X ! X; T x = x2; : X X ! [0; 1); (x; y) = 1. Now for x = 0 and y = 1, then we have
d(T x; T y) = d(T 0; T 1) = 1;
d(x; y) = d(0; 1) = 1;
and
d(y; T x) = d(1; 0) = 0:
Therefore we can not …nd > 0, L 0 and F 2 F satisfying (3). Thus T is not almost ( ; Fd)-contraction of type (y). However it is almost ( ; Fd)-contraction of type (x) with = ln85; L = 1 and F (t) = ln t. To see this we will consider the following cases: First note that
T = f(x; y) 2 X X : (x; y) 1 and d(T x; T y) > 0g
= f(x; y) 2 X X : x < yg .
Case 1: Let (x; y) 2 T and y = 1; then + F (d(T x; T y)) = ln8
5 + ln(1 x2)
= ln8
5 + ln[(1 + x)(1 x)]
ln8 5 + ln[5
4(1 x)]
= ln[2(1 x)]
= ln[(1 x) + (1 x)]
= F (d(x; y) + Ld(x; T y)) : Case 2: Let (x; y) 2 T and y 14; then
+ F (d(T x; T y)) = ln8
5+ ln(y2 x2)
= ln8
5+ ln[(y + x)(y x)]
ln8 5+ ln[1
2(y x)]
= ln[4
5(y x)]
ln(y x)
F (d(x; y) + Ld(x; T y)) : Therefore (4) is satis…ed and so almost ( ; Fd)-contraction of type (x).
Example 10. Let X = [0; 1] and d(x; y) = maxfy x; 0g for all x; y 2 X: Then (X; d) is quasi metric space. De…ne a map T : X ! X; T x = x, : X X ! [0; 1); (x; y) = 1 for x; y 2 X: Then, by the similar way we see that, T is not almost ( ; Fd)-contraction of type (y) but it is almost ( ; Fd)-contraction of type (x).
Now we present our main results.
Theorem 11. Let (X; d) be a left K-complete T1-quasi metric space, : X X ! [0; 1) be a function and T : X ! X be an -admissible and almost ( ; Fd)- contraction of type (y). Suppose there exists x0 2 X such that (x0; T x0) 1.
Then T has a …xed point in X provided that one of the following conditions holds:
(i) (X; d) is Hausdor¤ space and T is sequentially d-d continuous;
(ii) T is sequentially d-d 1 continuous;
(iii) (X; d) is left Smyth complete and has (Bd) or (Bd 1) property.
Proof. Let x0 2 X be such that (x0; T x0) 1: De…ne a sequence fxng in X by xn = T xn 1 for all n 2 N: Since T is -admissible, then (xn; xn+1) 1 for all n 2 N: Now, let dn = d(xn; xn+1) for all n 2 N: If there exists k 2 N with dk = d(xk; xk+1) = 0; then xk is a …xed point of T , since d is T1-quasi metric.
Suppose dn> 0 for all n 2 N: In this case (xn; xn+1) 2 T for all n 2 N. Since T is almost ( ; Fd)-contraction of type (y), we get
F (dn) = F (d(xn; xn+1))
= F (d(T xn 1; T xn))
F (d(xn 1; xn) + Ld(xn; T xn 1))
= F (d(xn 1; xn)) : Therefore we obtain that
F (dn) F (dn 1) F (dn 2) 2 F (d0) n : (5) From (5) , we get lim
n!1F (dn) = 1: Thus, from (F 2), we have
nlim!1dn= 0:
From (F 3), there exists k 2 (0; 1) such that
nlim!1dknF (dn) = 0:
By (5), the following holds for all n 2 N
dknF (dn) dknF (d0) dkn 0: (6) Letting n ! 1 in (6), we obtain that
nlim!1ndkn= 0: (7)
From (7), there exists n1 2 f1; 2; 3; :::g such that ndkn 1 for all n n1: So, we have, for all n n1
dn 1
n1k: (8)
Therefore P1
n=1
dn< 1: Now let m; n 2 N with m > n n1;then we get d(xn; xm) d(xn; xn+1) + d(xn+1; xn+2) + ::: + d(xm 1; xm)
= dn+ dn+1+ ::: + dm 1 X1
k=n
dk:
Since P1
k=1
dk is convergent, then we get fxng is left K-Cauchy sequence in the quasi metric space (X; d): Since (X; d) left K-complete, there exists z 2 X such that fxng is d-converges to z, that is, d(z; xn) ! 0 as n ! 1:
First, suppose (i) holds. Thus we have d(T z; T xn) ! 0; that is, d(T z; xn+1) ! 0:
Since (X; d) is Hausdor¤ we get z = T z:
Second, suppose (ii) holds. Then we have
d(z; T z) d(z; T xn) + d(T xn; T z)
= d(z; xn+1) + d(T xn; T z):
Since T is sequentially d-d 1continuous, we get d(T xn; T z) ! 0; that is, d(z; T z) = 0: Then z is a …xed point of T; since d is T1-quasi metric.
Third, suppose (iii) holds. Note that in this case since X is left Smyth complete, then ds(xn; x) ! 0 as n ! 1. On the other hand, from (F 2) and (3), it easy to conclude that
d(T x; T y) < d(x; y) + Ld(y; T x)
for all (x; y) 2 T : Therefore, for all x; y 2 X with (x; y) 1; we obtain
d(T x; T y) d(x; y) + Ld(y; T x): (9) Since has (Bd) or (Bd 1) property, then (xn; z) 1 for all n 2 X: By (9) we get
d(z; T z) d(z; xn+1) + d(T xn; T z)
d(z; xn+1) + d(xn; z) + Ld(z; xn+1): (10) Letting n ! 1 in (10), we obtain that d(z; T z) = 0, then z = T z:
Theorem 12. Let (X; d) be a right M -complete T1-quasi metric space, : X X ! [0; 1) be a function and T : X ! X be an -admissible and almost ( ; Fd)- contraction of type (x). Suppose there exists x0 2 X such that (T x0; x0) 1.
Then T has a …xed point in X provided that one of the following conditions holds:
(i) (X; d 1) is Hausdor¤ space and T is sequentially d 1-d 1 continuous;
(ii) T is sequentially d 1-d continuous;
(iii) (X; d) is right Smyth complete and has (Cd) or (Cd 1) property.
Proof. As in Theorem 11, we get the mentioned fxng is right K-Cauchy sequence in the quasi metric space (X; d): Since (X; d) right M -complete, there exists z 2 X such that fxng is d 1-converges to z, that is, d(xn; z) ! 0 as n ! 1:
First, supoose (i) holds. Thus we have d(T xn; T z) ! 0; that is, d(xn+1; T z) ! 0:
Since (X; d 1) is Hausdor¤ we get z = T z:
Second, suppose (ii) holds. Then we have
d(T z; z) d(T z; T xn) + d(T xn; z)
= d(T z; T xn) + d(xn+1; z):
Since T is sequentially d 1-d continuous, we get d(T z; T xn) ! 0; that is, d(T z; z) = 0: Then z is a …xed point of T , since d is T1-quasi metric.
Third, suppose (iii) holds. Note that in this case since X is right Smyth complete, then ds(xn; x) ! 0 as n ! 1. On the other hand, from (F 2) and (4) it easy to conclude that
d(T x; T y) < d(x; y) + Ld(x; T y)
for all (x; y) 2 T : Therefore, for all x; y 2 X with (x; y) 1 we obtain
d(T x; T y) d(x; y) + Ld(x; T y): (11)
Since has (Cd) or (Cd 1) property then (z; xn) 1 for all n 2 N: By (11) we get
d(T z; z) d(T z; T xn) + d(xn+1; z)
d(z; xn) + Ld(z; xn+1) + d(xn+1; z): (12) Letting n ! 1 in (12), we obtain that d(T z; z) = 0, then z = T z:
Theorem 13. Let (X; d) be a right K-complete T1-quasi metric space, : X X ! [0; 1) be a function and T : X ! X be an -admissible and almost ( ; Fd)- contraction of type (x). Suppose there exists x0 2 X such that (T x0; x0) 1.
Then T has a …xed point in X provided that one of the following conditions holds:
(i) (X; d) is Hausdor¤ space and T is sequentially d-d continuous;
(ii) T is sequentially d-d 1 continuous;
(iii) (X; d) is right Smyth complete and has (Cd) or (Cd 1) property.
Proof. Let x0 2 X be such that (T x0; x0) 1: De…ne a sequence fxng in X by xn = T xn 1 for all n 2 N: Since T is -admissible, then (xn+1; xn) 1 for all n 2 N: Now, let dn = d(xn+1; xn) for all n 2 N: If there exists k 2 N with dk = d(xk+1; xk) = 0; then xk is a …xed point of T , since d is T1-quasi metric.
Suppose dn> 0 for all n 2 N: In this case (xn+1; xn) 2 T for all n 2 N. Since T is almost ( ; Fd)-contraction of type (x), we get
F (dn) = F (d(xn+1; xn))
= F (d(T xn; T xn 1))
F (d(xn; xn 1) + Ld(xn; T xn 1))
= F (d(xn; xn 1)) : Therefore, we obtain that
F (dn) F (dn 1) F (dn 2) 2 ::: F (d0) n : (13) From (13), we get lim
n!1F (dn) = 1: Thus, from (F 2), we have
nlim!1dn= 0:
From (F 3), there exists k 2 (0; 1) such that
nlim!1dknF (dn) = 0:
By (13), the following holds for all n 2 N
dknF (dn) dknF (d0) dkn 0: (14) Letting n ! 1 in (14), we obtain that
nlim!1ndkn= 0: (15)
From (15), there exists n1 2 f1; 2; 3; :::g such that ndkn 1 for all n n1: So, we have, for all n n1
dn
1
n1k: (16)
Therefore P1
n=1
dn< 1: Now let m; n 2 N with m > n n1;then we get d(xm; xn) d(xm; xm 1) + d(xm 1; xm 2) + ::: + d(xn+1; xn)
dm+ dm 1+ ::: + dn+1 X1
k=n
dk:
Since P1
k=1
dk is convergent, then we get fxng is right K-Cauchy sequence in the quasi metric space (X; d): Since (X; d) right K-complete, there exists z 2 X such that fxng is d-converges to z, that is, d(z; xn) ! 0 as n ! 1:
First, suppose (i) holds. Thus we have d(T z; T xn) ! 0; that is, d(xn+1; T z) ! 0:
Since (X; d) is Hausdor¤ we get z = T z:
Second, suppose (ii) holds. Then we have
d(z; T z) d(z; xn+1) + d(T xn; T z):
Since T is sequentially d-d 1continuous, we get d(T xn; T z) ! 0; that is, d(z; T z) = 0: Then z is a …xed point of T , since d is T1-quasi metric.
Third, suppose (iii) holds. Note that in this case since X is right Smyth complete, then ds(xn; x) ! 0 as n ! 1. On the other hand, from (F 2) and (4) it easy to conclude that
d(T x; T y) < d(x; y) + Ld(x; T y)
for all (x; y) 2 T : Therefore, for all x; y 2 X with (x; y) 1 we obtain that d(T x; T y) d(x; y) + Ld(x; T y): (17) Since has (Cd) or (Cd 1) property then (z; xn) 1 for all n 2 N: By (17) we get
d(T z; z) d(T z; T xn) + d(xn+1; z)
d(z; xn) + Ld(z; xn+1) + d(xn+1; z): (18) Letting n ! 1 in (18), we obtain that d(T z; z) = 0, then z = T z:
Theorem 14. Let (X; d) be a left M -complete T1-quasi metric space, : X X ! [0; 1) be a function and T : X ! X be an -admissible and almost ( ; Fd)- contraction of type (y). Suppose there exists x0 2 X such that (x0; T x0) 1.
Then T has a …xed point in X provided that one of the following conditions holds:
(i) (X; d 1) is Hausdor¤ space and T is sequentially d 1-d 1 continuous;
(ii) T is sequentially d 1-d continuous;
(iii) (X; d) is left Smyth complete and has (Bd) or (Bd 1) property.
Proof. As in Theorem 13, we get the mentioned fxng is left K-Cauchy sequence in the quasi metric space (X; d): Since (X; d) left M -complete, there exists z 2 X such that fxng is d 1-converges to z, that is, d(xn; z) ! 0 as n ! 1:
First, supoose (i) holds. Thus we have d(T xn; T z) ! 0; that is, d(xn+1; T z) ! 0:
Since (X; d 1) is Hausdor¤ we get z = T z:
Second, suppose (ii) holds. Then we have
d(T z; z) d(T z; T xn) + d(xn+1; z):
Since T is sequentially d 1-d continuous, we get d(T z; T xn) ! 0; that is, d(T z; z) = 0: Then z is a …xed point of T , since d is T1-quasi metric.
Third, suppose (iii) holds. Note that in this case since X is left Smyth complete, then ds(xn; x) ! 0 as n ! 1. On the other hand, from (F 2) and (4) it easy to conclude that
d(T x; T y) < d(x; y) + Ld(y; T x)
for all (x; y) 2 T : Therefore, for all x; y 2 X with (x; y) 1 we obtain that d(T x; T y) d(x; y) + Ld(y; T x): (19) Since has (Bd) or (Bd 1) property then (xn; z) 1 for all n 2 N: By (19) we get
d(z; T z) d(z; xn+1) + d(T xn; T z)
d(z; xn+1) + d(xn; z) + Ld(z; xn+1): (20) Letting n ! 1 in (20), we obtain that d(z; T z) = 0, then z = T z:
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Current address : Hatice Aslan Hançer: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey.
E-mail address : haticeaslanhancer@gmail.com
ORCID Address: https://orcid.org/0000-0001-5928-9599
