Abstract
Considering α-admissible mappings in the setup of partial b-metric spaces, we establish some fixed and common fixed point results for ordered cyclic weakly -contractive mappings in complete ordered partial b-metric spaces. Our results extend several known results in the literature. Examples are also provided in support of our results.
MSC:47H10, 54H25.
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1 Introduction
There are a lot of generalizations of the concept of metric space. The concepts of b-metric space and partial metric space were introduced by Czerwik [1] and Matthews [2], respectively. Combining these two notions, Shukla [3] introduced another generalization which is called a partial b-metric space. Also, in [4], Mustafa et al. introduced a modified version of partial b-metric spaces. In fact, the advantage of their definition of partial b-metric is that by using it one can define a dependent b-metric which is called the b-metric associated with the partial b-metric.
Definition 1.1 [4]
Let X be a (nonempty) set and be a given real number. A function is a partial b-metric if, for all , the following conditions are satisfied:
,
,
,
.
The pair is called a partial b-metric space.
Example 1.2 [3]
Let , be a constant, and be defined by
Then is a partial b-metric space with the coefficient , but it is neither a b-metric nor a partial metric space.
Some more examples of partial b-metrics can be constructed with the help of following propositions.
Proposition 1.3 [3]
Let X be a nonempty set and let p be a partial metric and d be a b-metric with the coefficient on X. Then the function defined by , for all , is a partial b-metric on X with the coefficient s.
Proposition 1.4 [3]
Let be a partial metric space and . Then is a partial b-metric space with the coefficient , where is defined by .
Proposition 1.5 [4]
Every partial b-metric defines a b-metric , where
for all .
Now, we recall some definitions and propositions in a partial b-metric space.
Definition 1.6 [4]
Let be a partial b-metric space. Then for an and an , the -ball with center x and radius ϵ is
Proposition 1.7 [4]
Let be a partial b-metric space, , and . If then there exists a such that .
Thus, from the above proposition the family of all -balls
is a base of a topology on X which we call the -metric topology.
The topological space is , but it does not need to be .
Definition 1.8 [4]
A sequence in a partial b-metric space is said to be:
-
(i)
-convergent to a point if .
-
(ii)
A -Cauchy sequence if exists (and is finite).
-
(iii)
A partial b-metric space is said to be -complete if every -Cauchy sequence in X -converges to a point such that .
Lemma 1.9 [4]
-
(1)
A sequence is a -Cauchy sequence in a partial b-metric space if and only if it is a b-Cauchy sequence in the b-metric space .
-
(2)
A partial b-metric space is -complete if and only if the b-metric space is b-complete. Moreover, if and only if
Definition 1.10 [4]
Let and be two partial b-metric spaces and let be a mapping. Then f is said to be -continuous at a point if for a given , there exists such that and imply that . The mapping f is -continuous on X if it is -continuous at all .
Proposition 1.11 [4]
Let and be two partial b-metric spaces. Then a mapping is -continuous at a point if and only if it is -sequentially continuous at x; that is, whenever is -convergent to x, is -convergent to .
Definition 1.12 A triple is called an ordered partial b-metric space if is a partially ordered set and is a partial b-metric on X.
The following crucial lemma is useful in proving our main results.
Lemma 1.13 [4]
Let be a partial b-metric space with the coefficient and suppose that and are convergent to x and y, respectively. Then we have
In particular, if , then we have .
Moreover, for each we have
In particular, if , then we have
One of the interesting generalizations of the Banach contraction principle was given by Kirk et al. [5] in 2003 by introducing the notion of cyclic representation.
Definition 1.14 [5]
Let A and B be nonempty subsets of a metric space and . Then T is called a cyclic map if and .
The following interesting theorem for a cyclic map was given in [5].
Theorem 1.15 [5]
Let A and B be nonempty closed subsets of a complete metric space . Suppose that is a cyclic map such that
for all and , where is a constant. Then T has a unique fixed point u and .
Berinde initiated in [6, 7] the concept of almost contractions and obtained several interesting fixed point theorems for Ćirić strong almost contractions. Babu et al. introduced in [8] the class of mappings which satisfy ‘condition (B)’. Moreover, they proved the existence of fixed points for such mappings on complete metric spaces. Finally, Ćirić et al. in [9], and Aghajani et al. in [10] introduced the concept of almost generalized contractive conditions (for two, resp. four mappings) and proved some important results in ordered metric spaces. Let us recall one of these definitions.
Definition 1.16 [9]
Let f and g be two self-mappings on a metric space . They are said to satisfy almost generalized contractive condition, if there exist a constant and some such that
for all .
Definition 1.17 [11]
A function is called an altering distance function, if the following properties hold:
-
(1)
φ is continuous and nondecreasing.
-
(2)
if and only if .
Definition 1.18 [12]
Let be a partially ordered set and A and B be closed subsets of X with . Let be two mappings. The pair is said to be -weakly increasing if , for all and , for all .
In [13], Hussain et al. introduced the notion of ordered cyclic weakly -contractive pair of self-mappings as follows.
Definition 1.19 [13]
Let be an ordered b-metric space, let be two mappings, and let A and B be nonempty closed subsets of X. The pair is called an ordered cyclic weakly -contraction if
-
(1)
is a cyclic representation of X w.r.t. the pair ; that is, and ;
-
(2)
there exist two altering distance functions ψ, φ and a constant , such that for arbitrary comparable elements with and , we have
where
and
Also, in [13] the authors proved the following results.
Theorem 1.20 [13]
Let be a complete ordered b-metric space and A and B be closed subsets of X. Let be two -weakly increasing mappings with respect to ⪯. Suppose that:
-
(a)
the pair is an ordered cyclic weakly -contraction;
-
(b)
f or g is continuous.
Then f and g have a common fixed point .
An ordered b-metric space is called regular if for any nondecreasing sequence in X such that , as , one has for all .
Theorem 1.21 [13]
Let the hypotheses of Theorem 1.20 be satisfied, except that condition (b) is replaced by the assumption
(b′) the space is regular.
Then f and g have a common fixed point in X.
In this paper, first we prove some fixed point results for α-admissible mappings in the context of partial b-metric spaces. Then we express some common fixed point results for cyclic generalized almost contractive mappings. Our results extend and generalize some recent results in [4] and [13]. In fact, they are cyclic variants of the results in [4].
2 Fixed point results via α-admissible mappings in partial b-metric spaces
Samet et al. [14] defined the notion of α-admissible mappings and proved the following result.
Definition 2.1 [14]
Let T be a self-mapping on X and be a function. We say that T is an α-admissible mapping if
Denote by the family of all nondecreasing functions such that for all , where is the n th iterate of ψ.
Theorem 2.2 [14]
Let be a complete metric space and T be an α-admissible mapping. Assume that
where . Also, suppose that the following assertions hold:
-
(i)
there exists such that ;
-
(ii)
either T is continuous or for any sequence in X with for all such that as , we have for all .
Then T has a fixed point.
We now recall the concept of -comparison function which was introduced by Berinde [15].
Definition 2.3 (Berinde [15])
A function is said to be a -comparison function if
() φ is increasing,
() there exist , , and a convergent series of nonnegative terms such that , for and any .
Later, Berinde [16] introduced the notion of -comparison function as a generalization of a -comparison function.
Definition 2.4 (Berinde [16])
Let be a real number. A mapping is called a -comparison function if the following conditions are fulfilled:
-
(1)
φ is monotone increasing;
-
(2)
there exist , , and a convergent series of nonnegative terms such that , for and any .
Let be the class of -comparison functions . It is clear that the notion of -comparison function coincides with -comparison function for .
We now recall the following lemma, which will simplify the proofs.
Lemma 2.5 (Berinde [17])
If is a -comparison function, then we have the following.
-
(1)
the series converges for any ;
-
(2)
the function , defined by , , is increasing and continuous at 0.
Theorem 2.6 Let be a -complete partial b-metric space, f be a continuous α-admissible mapping on X, there exists such that and if any sequence in X -converges to a point x, where for all n, then we have . Assume that
for all , where and
Then f has a fixed point.
Proof Let be such that . Define a sequence by for all . Since f is an α-admissible mapping and , we deduce that . Continuing this process, we get that for all .
Now, we will finish the proof in the following steps.
First, we prove that
for each .
If , for some , then . Thus, is a fixed point of f. Therefore, we assume that , for all .
Using condition (2.2) as for all , we obtain
Here,
If , then
which yields
a contradiction.
Hence,
So (2.3) holds.
By induction, we get
Then, by the triangular inequality and (2.4), we get
as .
Since is a -Cauchy sequence in the -complete partial b-metric space X, from Lemma 1.9, is a b-Cauchy sequence in the b-metric space . -Completeness of shows that is also b-complete. Then there exists such that
Since , from Lemma 1.9
From the continuity of f we have
and hence we get
So, we get . As , we have
Hence, . Thus, , that is, . □
In Theorem 2.6, we omit the continuity of the mapping f and we replace instead of and rearrange it as follows.
Theorem 2.7 Let be a -complete partial b-metric space and f be an α-admissible mapping on X such that
for all , where . Assume that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
if is a sequence in X such that for all n and as , then for all .
Then f has a fixed point.
Proof Let be such that and define a sequence in X by for all . Following the proof of Theorem 2.6, we have for all and there exists such that as which . Hence, from (ii) we deduce that for all . Therefore, by (2.7), we obtain
Here,
Taking the upper limit as in the above inequality from Lemma 1.13 we obtain
which implies that . □
Definition 2.8 [18]
Let and . We say that f is a triangular α-admissible mapping if
(T1) implies , ,
(T2) implies , .
Example 2.9 [18]
Let , , and , then f is a triangular α-admissible mapping. Indeed, if , then which implies that , that is, . Also, if , then , that is, and therefore .
Example 2.10 [18]
Let , , and . Hence, f is a triangular α-admissible mapping. Indeed, if then which implies that , that is, .
Moreover, if , then and hence .
Example 2.11 [18]
Let , , and
Then f is a triangular α-admissible mapping. In fact, if
then . Hence, , that is, . Also,
Thus, . Now, if , then .
Example 2.12 [18]
Let , , and . Then f is a triangular α-admissible mapping.
Lemma 2.13 [18]
Let f be a triangular α-admissible mapping. Assume that there exists such that . Define the sequence by . Then
A mapping is called a comparison function if it is increasing and , as for any .
Lemma 2.14 (Berinde [15], Rus [19])
If is a comparison function, then:
-
(1)
each iterate of ψ, , is also a comparison function;
-
(2)
ψ is continuous at 0;
-
(3)
, for any .
Denote by Ψ the family of all continuous comparison functions .
In the sequel, , is a function and
Theorem 2.15 Let be a -complete partial b-metric space, f be a continuous triangular α-admissible mapping on X, there exists such that and if any sequence in X -converges to a point x, where for all n, then we have . Assume that
for all . Then f has a fixed point.
Proof Let be such that . Define a sequence by for all . Since f is an α-admissible mapping and , we deduce that . Continuing this process, we get for all .
Now, we will finish the proof in the following steps.
Step I. We will prove that
First, we prove that
for each .
If , for some , then . Thus, is a fixed point of f. Therefore, we assume that , for all .
Using condition (2.8) as for all , we obtain
Here,
If , then
which yields
a contradiction.
Hence,
So (2.9) holds.
By induction, we get
As , we conclude that
So by we get
Step II. We will show that is a -Cauchy sequence in X. For this, we have to show that is a b-Cauchy sequence in (see Lemma 1.9). Suppose the contrary; that is, is not a b-Cauchy sequence. Then there exists for which we can find two subsequences and of such that is the smallest index for which
This means that
From (2.13) and using the triangular inequality, we get
Using (2.11), (2.12), and from the definition of and (2.14), and taking the upper limit as , we get
Also,
Further,
and
On the other hand, by the definition of and (2.12)
Hence, by (2.15),
Similarly,
and
From (2.8) and Lemma 2.13 as , we have
where
Taking the upper limit as in (2.24) and using (2.11), (2.19), (2.20), and (2.22), we get
Now, taking the upper limit as in (2.23) and using (2.21) and (2.25), we have
a contradiction.
Step III. There exists z such that .
Since is a -Cauchy sequence in the -complete partial b-metric space X, from Lemma 1.9, is a b-Cauchy sequence in the b-metric space . -Completeness of shows that is also b-complete. Then there exists such that
Since , from the definition of and (2.12), we get
Again, from Lemma 1.9,
From the continuity of f we have
and hence we get
So, we get . As , we have
Hence, . Thus, , that is, . □
If in Theorem 2.15 we take then we deduce the following corollary.
Corollary 2.16 Let be a -complete partial b-metric space and f be a continuous mapping on X. Assume that
for all . Then f has a fixed point.
In Theorem 2.15, we omit the continuity of the mapping f and we replace instead of and rearrange it as follows.
Theorem 2.17 Let be a -complete partial b-metric space and f be a triangular α-admissible mapping on X such that
for all , where . Assume that the following conditions hold:
-
(i)
there exists such that ;
-
(ii)
if is a sequence in X such that for all n and as , then for all .
Then f has a fixed point.
Proof Let be such that and define a sequence in X by for all . Following the proof of Theorem 2.15, we have for all and there exists such that as which . Hence, from (ii) we deduce that for all . Therefore, by (2.29), we obtain
Here,
Taking the upper limit as in the above inequality from Lemma 1.13 we obtain
which implies that . □
Example 2.18 Let and be a -metric on X. Define by and by
and for all . Now, we prove that all the hypotheses of Theorem 2.17 are satisfied and hence f has a fixed point.
First, we see that is a -complete partial b-metric space. Let . If , then . On the other hand, for all , we have and hence . This implies that f is a triangular α-admissible mapping on X. Obviously, .
Now, if is a sequence in X such that for all and as , it is easy to see that .
Using the Mean Value Theorem for the function for any , we have
Thus, all the conditions of Theorem 2.17 are satisfied and therefore f has a fixed point .
3 Common fixed points of generalized almost cyclic weakly -contractive mappings
In this section, we consider the notion of ordered cyclic weakly -contractions in the setup of ordered partial b-metric spaces and then obtain some common fixed point theorems for these cyclic contractions in the setup of complete ordered partial b-metric spaces. Our results extend some fixed point theorems from the framework of ordered metric spaces and ordered b-metric spaces, in particular Theorems 1.20 and 1.21.
We shall call an ordered partial b-metric space regular if for any nondecreasing sequence in X such that , as , one has , for all .
Definition 3.1 Let be an ordered partial b-metric space, let be two mappings, and let A and B be nonempty closed subsets of X. The pair is called an ordered cyclic almost generalized weakly -contraction if
-
(1)
is a cyclic representation of X w.r.t. the pair ; that is, and ;
-
(2)
there exist two altering distance functions ψ, φ and a constant , such that for arbitrary comparable elements with and , we have
(3.1)
where
and
Theorem 3.2 Let be a -complete ordered partial b-metric space and A and B be two nonempty closed subsets of X. Let be two -weakly increasing mappings with respect to ⪯. Suppose that the pair is an ordered cyclic almost generalized weakly -contraction. Then f and g have a common fixed point .
Proof First, note that is a fixed point of f if and only if u is a fixed point of g. Indeed, suppose that u is a fixed point of f. As and , by (3.1), we have
It follows that . Therefore, and hence . Similarly, we can show that if u is a fixed point of g, then u is a fixed point of f.
Let and let . Since , we have . Also, let . Since , we have . Continuing this process, we can construct a sequence in X such that , , and . Since f and g are -weakly increasing, we have
If , for some , then . Thus is a fixed point of f. By the first part of the proof, we conclude that is also a fixed point of g. Similarly, if , for some , then . Thus, is a fixed point of g. By the first part of the proof, we conclude that is also a fixed point of f. Therefore, we assume that , for all . Now, we complete the proof in the following steps.
Step 1. We will prove that
As and are comparable and and , by (3.1), we have
where
and
Hence, we have
If
then (3.4) becomes
which gives a contradiction. So,
and hence (3.4) becomes
Similarly, we can show that
By (3.5) and (3.6), we see that is a nonincreasing sequence of positive numbers. Hence, there is such that
Letting in (3.5), we get
which implies that and hence . So, we have
Step 2. We will prove that is a -Cauchy sequence. Because of (3.7), it is sufficient to show that is a -Cauchy sequence. By Lemma 1.9, we should show that is b-Cauchy in . Suppose the contrary, i.e., that is not a b-Cauchy sequence in . Then there exists for which we can find two subsequences and of such that is the smallest index for which
This means that
From (3.8) and using the triangular inequality, we get
Using (3.7) and from the definition of and taking the upper limit as , we get
On the other hand, we have
Using (3.7), (3.9), and taking the upper limit as , we get
Again, using the triangular inequality, we have
and
Taking the upper limit as in the above inequalities, and using (3.7), (3.9), and (3.11) we get
and
From the definition of and (3.7), (3.10), (3.11), (3.12), and (3.13) we have the following relations:
Since and are comparable, using (3.1) we have
where
and
Taking the upper limit in (3.19) and (3.20), and using (3.7) and (3.14)-(3.17), we get
and
Now, taking the upper limit as in (3.18) and using (3.14), (3.21), and (3.22), we have
which implies that . By (3.19), it follows that
which is in contradiction with (3.8). Thus, we have proved that is a b-Cauchy sequence in the metric space . Since is -complete, from Lemma 1.9, is a b-complete b-metric space. Therefore, the sequence converges to some , that is, . Since , from the definition of and (3.7), we get
Again, from Lemma 1.9,
Step 3. In the above steps, we constructed an increasing sequence in X such that , for some . As A and B are closed subsets of X, we have . Using the regularity assumption on X, we have , for all . Now, we show that . By (3.1), we have
where
and
Letting in (3.24) and (3.25), and using Lemma 1.13, we get
and . Now, taking the upper limit as in (3.23), and using Lemma 1.13 and (3.26) we get
It follows that , and hence, by (3.24), that . Thus, z is a fixed point of g. On the other hand, from the first part of the proof, . Hence, z is a common fixed point of f and g. □
Theorem 3.3 Let be a -complete ordered partial b-metric space and A and B be nonempty closed subsets of X. Let be two -weakly increasing mappings with respect to ⪯. Suppose that
Also, let f and g be continuous. Then f and g have a common fixed point .
Proof Repeating the proof of Theorem 3.2, we construct an increasing sequence in X such that , for some . As A and B are closed subsets of X, we have . Now, we show that .
Using the triangular inequality, we get
and
Letting and using continuity of f and g, we get
Therefore,
From (3.27) as , we have
where
As ψ is nondecreasing, we have . Hence, by (3.28) we obtain . But then, using (3.29), we get . Thus, we have and z is a common fixed point of f and g. □
As consequences, we have the following results.
By putting in Theorems 3.2 and 3.3 and in Theorem 3.2, we obtain the main results (Theorems 3 and 4) of Mustafa et al. [4].
Taking , in Theorem 3.2, we get the following.
Corollary 3.4 Let be a -complete ordered partial b-metric space and A and B be closed subsets of X. Let be two -weakly increasing mappings with respect to ⪯. Suppose that:
-
(a)
is a cyclic representation of X w.r.t. the pair ;
-
(b)
there exist , , and an altering distance function ψ such that for any comparable elements with and , we have
(3.30)
where and are given by (3.2) and (3.3), respectively;
-
(c)
f and g are continuous, or
(c′) the space is regular.
Then f and g have a common fixed point .
Taking and in Corollary 3.4, we obtain the partial version of Theorems 2.1 and 2.2 of Shatanawi and Postolache [12].
In Definitions 1.18 and 3.1 and Theorems 3.2 and 3.3, if we take , then we have the following definitions and results.
Definition 3.5 Let be a partially ordered set and A and B be closed subsets of X with . The mapping is said to be -weakly increasing if , for all and , for all .
Definition 3.6 Let be an ordered partial b-metric space, let be a mapping, and let A and B be nonempty closed subsets of X. The mapping f is called an ordered cyclic almost generalized weakly -contraction if
-
(1)
is a cyclic representation of X w.r.t. f; that is, and ;
-
(2)
there exist two altering distance functions ψ, φ and a constant , such that for arbitrary comparable elements with and , we have
where
and
Corollary 3.7 Let be a -complete ordered partial b-metric space and A and B be two nonempty closed subsets of X. Let be a -weakly increasing mapping with respect to ⪯. Suppose that the mapping f is an ordered cyclic almost generalized weakly -contraction. Then f has a fixed point .
Corollary 3.8 Let be a -complete ordered partial b-metric space and A and B be nonempty closed subsets of X. Let be a -weakly increasing mapping with respect to ⪯. Suppose that
Also, let f be continuous. Then f has a fixed point .
We illustrate our results with the following example.
Example 3.9 Consider the partial b-metric space by . Define an order ⪯ on X by
Obviously, is a -complete ordered -metric space. Indeed, if we have , for some , then we have
So, we have , which convergence holds in the case of the usual metric in X. Now, it is easy to see that .
Let be given by
and for all . Also, let and . In order to check the conditions of Corollary 3.8, take such that and consider the following two possible cases.
1∘ . Then obviously also and . It is easy to check that
2∘ . Then and
Hence, all the conditions of Corollary 3.8 are satisfied and f has a fixed point (which is ).
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks DSR for technical and financial support.
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Latif, A., Roshan, J.R., Parvaneh, V. et al. Fixed point results via α-admissible mappings and cyclic contractive mappings in partial b-metric spaces. J Inequal Appl 2014, 345 (2014). https://doi.org/10.1186/1029-242X-2014-345
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DOI: https://doi.org/10.1186/1029-242X-2014-345