Abstract
In this paper, we characterize α-ψ contractive mappings in the setting of quasi-metric spaces and investigate the existence and uniqueness of a fixed point of such mappings. We notice that by using our result some fixed-point theorems in the context of G-metric space can be deduced.
MSC:46T99, 47H10, 54H25, 46J10.
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1 Introduction and preliminaries
Let Ψ be the family of functions satisfying the following conditions:
() ψ is nondecreasing;
() for all , where is the n th iterate of ψ.
These functions are known in the literature as (c)-comparison functions. One can easily deduce that if ψ is a (c)-comparison function, then for any .
Definition 1 Let X be a non-empty and let be a function which satisfies:
(d1) if and only if ,
(d2) . Then d called a quasi-metric and the pair is called a quasi-metric space.
Remark 2 Any metric space is a quasi-metric space, but the converse is not true in general.
Now, we give convergence and completeness on quasi-metric spaces.
Definition 3 Let be a quasi-metric space, be a sequence in X, and . The sequence converges to x if and only if
Definition 4 Let be a quasi-metric space and be a sequence in X. We say that is left-Cauchy if and only if for every there exists a positive integer such that for all .
Definition 5 Let be a quasi-metric space and be a sequence in X. We say that is right-Cauchy if and only if for every there exists a positive integer such that for all .
Definition 6 Let be a quasi-metric space and be a sequence in X. We say that is Cauchy if and only if for every there exists a positive integer such that for all .
Remark 7 A sequence in a quasi-metric space is Cauchy if and only if it is left-Cauchy and right-Cauchy.
Definition 8 Let be a quasi-metric space. We say that
-
(1)
is left-complete if and only if each left-Cauchy sequence in X is convergent.
-
(2)
is right-complete if and only if each right-Cauchy sequence in X is convergent.
-
(3)
is complete if and only if each Cauchy sequence in X is convergent.
Definition 9 Let be a quasi-metric space and be a given mapping. We say that T is an α-ψ contractive mapping if there exist two functions and such that
Remark 10 We easily see that any contractive mapping, that is, a mapping satisfying the Banach contraction, is an α-ψ contractive mapping with for all and , .
Definition 11 Let and . We say that T is α admissible if for all we have
2 Main results
We start this section by the following definition, which is a characterization of α-ψ contractive mappings [1] in the context of a quasi-metric space.
Definition 12 (cf. [2])
Let be a quasi-metric space and be a given mapping. We say that T is an α-ψ contractive mapping if there exist two functions and such that for all , we have
Theorem 13 Let be a complete quasi-metric space. Suppose that is a α-ψ contractive mapping which satisfies
-
(i)
T is α admissible;
-
(ii)
there exists such that and ;
-
(iii)
T is continuous.
Then T has a fixed point.
Proof By (ii), there exists such that and . Let us define a sequence in X by for all . If for some , then it is evident that is a fixed point of T. Consequently, throughout the proof, we suppose that for all . Regarding the assumption (i), we derive
Recursively, we get
Taking (2.1) and (2.3) into account, we find that
for all . Inductively, we obtain
By using the triangular inequality and (2.5), for all , we get
Letting in the above inequality, we derive . Hence, as . Therefore, is a left-Cauchy sequence in .
Analogously, we deduce that is a right-Cauchy sequence in . Indeed, by assumption (i), we obtain
Recursively, we find that
By combining (2.1) with (2.8), we find
for all . By iteration, we have
Due to the triangular inequality, together with (2.10), for all , we get
Consequently, is a right-Cauchy sequence in . By Remark 7, we deduce that is a Cauchy sequence in complete quasi-metric space . It implies that there exists such that
Then, by using the property (d1) together with the continuity of T, we obtain
and
Thus, we have
Keeping (2.12) and (2.15) in mind together with the uniqueness of the limit, we conclude that , that is, u is a fixed point of T. □
Theorem 14 Let be a complete quasi-metric space. Suppose that is an α-ψ contractive mapping which satisfies:
-
(i)
T is α admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then there exists a subsequence of such that for all k.
Then T has a fixed point.
Proof Following the lines of the proof of Theorem 13, we know that the sequence defined by , for all , converges for some . From (2.3) and condition (iii), there exists a subsequence of such that for all k. Applying (2.1), for all k, we get
Letting in the above equality, we obtain
Thus, we have , that is, . □
Definition 15 (cf. [3])
Let be a quasi-metric space and be a given mapping. We say that T is a generalized α-ψ contractive mapping of type A if there exist two functions and such that for all and we have
where
Definition 16 Let be a quasi-metric space and be a given mapping. We say that T is a generalized α-ψ contractive mapping of type B if there exist two functions and such that for all and we have
where
Theorem 17 Let be a complete quasi-metric space. Suppose that is a generalized α-ψ contractive mapping of type A and satisfies
-
(i)
T is α admissible;
-
(ii)
there exists such that and ;
-
(iii)
T is continuous.
Then T has a fixed point.
Proof By assumption (ii), there exists such that and . We construct a sequence in X in the following way:
If for some , then it is clear that is a fixed point of T. Hence, we assume that for all . Due to assumption (i), we have
If we continue in this way, we obtain
From (2.18) and (2.23), for all , we derive
where
Since ψ is a nondecreasing function, (2.24) implies that
for all . We shall examine two cases. Suppose that . Since , we obtain
a contradiction. Therefore, we find that . Since , (2.26) yields
for all . Recursively, we derive
Together with (2.29) and the triangular inequality, for all , we get
Therefore, is a left-Cauchy sequence in .
Analogously, we shall prove that is a right-Cauchy sequence in . Again by the assumption (i), we find that
Recursively, we obtain
From (2.18) and (2.32), for all , we deduce that
where
Since ψ is nondecreasing function, the inequality (2.33) turns into
for all . We shall examine three cases.
Case 1. Assume that . Since we get
a contradiction.
Case 2. Suppose that . Since , from (2.34) we find that
for all . Inductively, we get
By using the triangular inequality and taking (2.38) into consideration, for all , we get
Case 3. Assume that . Regarding and (2.35), we obtain
for all . From (2.18) and (2.23), for all , we derive
where
Since ψ is a nondecreasing function, (2.24) implies that
for all .
We shall examine two cases. Suppose that . Since , we obtain
a contradiction. Therefore, we find that . Since , (2.43) yields
for all . Recursively, we derive
If we combine the inequalities (2.40) with (2.46), we derive
Together with (2.47) and the triangular inequality, for all , we get
Therefore, by (2.39) and (2.48), we conclude that is a right-Cauchy sequence in .
From Remark 7, is a Cauchy sequence in complete quasi-metric space . This implies that there exists such that
Then, using property (d1) and the continuity of T, we obtain
and
Thus, we have
It follows from (2.49) and (2.52) that , that is, u is a fixed point of T. □
Corollary 18 Let be a complete quasi-metric space. Suppose that is a generalized α-ψ contractive mapping of type A and satisfies:
-
(i)
T is α admissible;
-
(ii)
there exists such that and ;
-
(iii)
T is continuous.
Then T has a fixed point.
The proof is evident due to Theorem 17. Indeed, ψ is nondecreasing and, hence,
where and are defined as in Definition 15 and Definition 16. The rest follows from Theorem 17.
In the following theorem we are able to remove the continuity condition for the α-ψ contractive mappings of type B.
Theorem 19 Let be a complete quasi-metric space. Suppose that is a generalized α-ψ contractive mapping of type B which satisfies:
-
(i)
T is α admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then there exists a subsequence of such that for all k.
Then T has a fixed point.
Proof Following the lines in the proof of Theorem 17, we know that the sequence defined by for all , converges for some . From (2.23) and condition (iii), there exists a subsequence of such that for all k. Applying (2.20), for all k, we get
Also, using (2.21) we find
Taking the limit as in the above equality, we obtain
Assume that . From (2.55), for k large enough, we have , which implies that . Then, from (2.53), we have
Taking the limit as in the above equality, we get
which is a contradiction. Therefore, we find , that is, . □
3 Consequences: fixed-point result on G-metric spaces
In this section, we note that some existing fixed-point results in the context of G-metric spaces are consequences of our main theorems. For the sake of completeness, we recollect some basic definitions and crucial results on the topic in the literature. For more details, see e.g. [4–6].
Definition 20 Let X be a non-empty set, be a function satisfying the following properties:
(G1) if ,
(G2) for all with ,
(G3) for all with ,
(G4) (symmetry in all three variables),
(G5) (rectangle inequality) for all .
Then the function G is called a generalized metric, or, more specifically, a G-metric on X, and the pair is called a G-metric space.
Note that every G-metric on X induces a metric on X defined by
For a better understanding of the subject we give the following examples of G-metrics.
Example 21 Let be a metric space. The function , defined by
for all , is a G-metric on X.
Example 22 Let . The function , defined by
for all , is a G-metric on X.
Definition 23 Let be a G-metric space, and let be a sequence of points of X. We say that is G-convergent to if
that is, for any , there exists such that , for all . We call x the limit of the sequence and write or .
Proposition 24 Let be a G-metric space. The following are equivalent:
-
(1)
is G-convergent to x,
-
(2)
as ,
-
(3)
as ,
-
(4)
as .
Definition 25 Let be a G-metric space. A sequence is called a G-Cauchy sequence if, for any , there exists such that for all , that is, as .
Proposition 26 Let be a G-metric space. Then the following are equivalent:
-
(1)
the sequence is G-Cauchy,
-
(2)
for any , there exists such that , for all .
Definition 27 A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
For more details of G-metric space, we refer e.g. to [7–9].
Theorem 28 Let be a G-metric space. Let be the function defined by . Then
-
(1)
is a quasi-metric space;
-
(2)
is G-convergent to if and only if is convergent to x in ;
-
(3)
is G-Cauchy if and only if is Cauchy in ;
-
(4)
is G-complete if and only if is complete.
Every quasi-metric induces a metric, that is, if is a quasi-metric space, then the function defined by
is a metric on X.
As an immediate consequence of the definition above and Theorem 28, the following theorem is obtained.
Theorem 29 Let be a G-metric space. Let be the function defined by . Then
-
(1)
is a quasi-metric space;
-
(2)
is G-convergent to if and only if is convergent to x in ;
-
(3)
is G-Cauchy if and only if is Cauchy in ;
-
(4)
is G-complete if and only if is complete.
Now, we state the characterization of Definition 9 and Definition 11 in the context of G-metric space.
Definition 30 (See e.g. [10, 11])
Let be a G-metric space and be a given mapping. We say that T is a β-ψ contractive mapping of type I if there exist two functions and such that for all , we have
Definition 31 (See e.g. [10, 11])
Let and . We say that T is β admissible if for all we have
Lemma 32 Let where X is non-empty set. It is clear that the self-mapping T is β admissible if and only if T is α admissible.
Proof It is sufficient to let . □
Theorem 33 Let be a complete G-metric space. Suppose that is a β-ψ contractive mapping which satisfies:
-
(i)
T is β admissible;
-
(ii)
there exists such that and ;
-
(iii)
T is continuous.
Then T has a fixed point.
Proof Consider the quasi-metric for all . Due to Lemma 32 and (3.2), we have
Then the result follows from Theorem 13. □
Definition 34 Let be a G-metric space and be a given mapping. We say that T is a generalized β-ψ contractive mapping of type A if there exist two functions and such that for all we have
where
Definition 35 Let be a G-metric space and be a given mapping. We say that T is a generalized β-ψ contractive mapping of type B if there exist two functions and such that for all we have
where
Remark 36 It is simple to see that every β-ψ contractive mapping is a generalized β-ψ contractive mapping of type A.
Similarly, every β-ψ contractive mapping is a generalized β-ψ contractive mapping of type B.
Theorem 37 Let be a complete G-metric space. Suppose that is a generalized β-ψ contractive mapping of type A and satisfies:
-
(i)
T is β admissible;
-
(ii)
there exists such that and ;
-
(iii)
T is continuous.
Then T has a fixed point.
Proof Consider the quasi-metric for all . From Lemma 32 together with (3.5) and (3.6), we deduce that
Then the result follows from Theorem 17. □
Theorem 38 Let be a complete G-metric space. Suppose that is a β-ψ contractive mapping which satisfies:
-
(i)
T is β admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then there exists a subsequence of such that for all k.
Then T has a fixed point.
Proof Consider the quasi-metric for all . By Lemma 32 and (3.2), we find that
Then the result follows from Theorem 14. □
Theorem 39 Let be a complete G-metric space. Suppose that is a generalized β-ψ contractive mapping of type B which satisfies:
-
(i)
T is β admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then there exists a subsequence of such that for all k.
Then T has a fixed point.
Proof Consider the quasi-metric for all . Regarding Lemma 32, (3.7), and (3.8), we derive
Then the result follows from Theorem 19. □
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Acknowledgements
The authors thank the Visiting Professor Programming at King Saud University for funding this work. The authors thank the anonymous referees for their remarkable comments, suggestion, and ideas that helped to improve this paper.
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Bilgili, N., Karapınar, E. & Samet, B. Generalized α-ψ contractive mappings in quasi-metric spaces and related fixed-point theorems. J Inequal Appl 2014, 36 (2014). https://doi.org/10.1186/1029-242X-2014-36
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DOI: https://doi.org/10.1186/1029-242X-2014-36