Abstract
In this paper, we first introduce the concepts of generalized -expansive mappings and generalized -weakly expansive mappings designed for three mappings. Then we establish some common fixed point results for such two new types of mappings in partial b-metric spaces. These results generalize and extend the main results of Karapınar et al. (J. Inequal. Appl. 2014:22, 2014), Nashine et al. (Fixed Point Theory Appl. 2013:203, 2013) and many comparable results from the current literature. Moreover, some examples and an application to a system of integral equations are given here to illustrate the usability of the obtained results.
MSC:47H10, 54H25.
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1 Introduction and preliminaries
Fixed point theory in metric spaces is an important branch of nonlinear analysis, which is closely related to the existence and uniqueness of solutions of differential equations and integral equations.
There are many generalizations of the concept of metric spaces in the literature. In particular, Matthews [1] introduced the concept of a partial metric space as a part of the study of denotational data for networks and proved that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. After that, fixed point results in partial metric spaces have been studied by many authors (see [2–7]). On the other hand, the concept of a b-metric space was introduced and studied by Bakhtin [8] and Czerwik [9]. Since then, several papers have been published on the fixed point theory of the variational principle for single-valued and multi-valued operators in b-metric spaces (see [8–15] and the references therein). We begin with the definition of b-metric spaces.
Definition 1.1 ([8])
Let X be a nonempty set and be a given real number. A function is said to be a b-metric on X if, for all , the following conditions are satisfied:
(b1) if and only if ,
(b2) ,
(b3) .
In this case, the pair is called a b-metric space.
Recently, Shukla [16] introduced the notion of a partial b-metric space as a generalization of partial metric spaces and b-metric spaces.
Definition 1.2 ([16])
Let X be a nonempty set and be a given real number. A mapping is said to be a partial b-metric on X if for all , the following conditions are satisfied:
() if and only if ,
() ,
() ,
() .
A partial b-metric space is a pair such that X is a nonempty set and is a partial b-metric on X. The number is called the coefficient of .
In [17], Mustafa et al. introduced a new concept of partial b-metric by modifying Definition 1.2 in order to guarantee that each partial b-metric can induce a b-metric. The advantage of the new 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 partial b-metric . The new concept of partial b-metric is as follows.
Definition 1.3 ([17])
Let X be a nonempty set and be a given real number. A mapping is said to be a partial b-metric on X if for all , the following conditions are satisfied:
() if and only if ,
() ,
() ,
() .
The pair is called a partial b-metric space with coefficient .
Since , from (), we have
Hence, a partial b-metric in the sense of Definition 1.3 is also a partial b-metric in the sense of Definition 1.2.
In a partial b-metric space , if , then () and () imply that . But the converse does not hold always. It is clear that every partial metric space is a partial b-metric space with coefficient and every b-metric is a partial b-metric space with same coefficient and zero distance. However, the converse of these facts need not hold. The following example shows that a partial b-metric on X might be neither a partial metric, nor a b-metric on X.
Example 1.1 ([17])
Let , be a constant and be defined by
for all . Then is a partial b-metric space with the coefficient , but it is neither a b-metric nor a partial metric space.
Each partial b-metric on X generates a topology on X, which has a subbase of the family of open -balls , where , for all and . The topology space is , but does not need to be . The topology on X is called a -metric topology.
Definition 1.4 ([17])
A sequence in a partial b-metric space is said to be:
-
(1)
-convergent to a point if .
-
(2)
a -Cauchy sequence if exists and is finite.
-
(3)
A partial b-metric space is said to be -complete if every -Cauchy sequence in X -converges to a point such that .
It should be noted that the limit of a convergent sequence in a partial b-metric space may not be unique (see [[16], Example 2]).
In [17], using Definition 1.3, Mustafa et al. proved the fact if is a partial b-metric on X, then the function given by defines a b-metric on X. Using Definition 1.3, Mustafa et al. also obtained the following lemma which is the key to the proof of our theorems.
Lemma 1.1 ([17])
Let be a partial b-metric space. Then:
-
(1)
A sequence in X is a -Cauchy sequence in if and only if it is a b-Cauchy sequence in 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 .
It should be noted that in general a partial b-metric function for is not jointly continuous for all variables. The following example illustrates this fact.
Example 1.2 Let , and let be defined by
Then considering all possible cases, it can be checked that, for all , we have
Thus, is a partial b-metric space (with ). Let for each . Then as , that is, , but .
Since in general a partial b-metric is not continuous, we need the following simple lemma about the -convergent sequences in the proof of our results.
Lemma 1.2 ([17])
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
Jungck [18] introduced the concept of weakly compatible mappings as follows.
Definition 1.5 ([18])
Let X be a nonempty set, A and be two self-maps. A and T are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e., if for some , then .
It is worth mentioning that most of the preceding references concerned with fixed point results of contractions in partial metric spaces and b-metric spaces, but we rarely see fixed point results of expansions in such two types of spaces. Recently, in [19], Karapınar et al. considered a generalized expansive mapping and proved the fixed point theorem in metric spaces. Nashine et al. [20] introduced -contractive mappings and proved some fixed point theorems in ordered metric spaces. Here, we recall the relevant definition.
Definition 1.6 ([20])
Let be an ordered metric space, and let . The mappings S, T are said to be -contractive if
for all , where is a strictly increasing and continuous function in each coordinate, and for all , , , , and .
Inspired by the notions of -contractive mappings of [20], we first introduce the concepts of generalized -expansive mappings and generalized -weakly expansive mappings. Then we establish some common fixed point theorems for these classes of mappings in complete partial b-metric spaces. The obtained results generalize and extend the main results of [15–23]. We also provide some examples to show the generality of our results. Finally, an application is given to illustrate the usability of the obtained results.
2 Main results
The study of expansive mappings is a very interesting research area in fixed point theory (see [19, 21–23]). In this section, inspired by the notion of -contractive mappings of [20], we first introduce the notions of generalized -expansive mappings and generalized -weakly expansive mappings in partial b-metric spaces.
For convenience, we denote by Ψ the class of functions satisfying the following conditions:
-
(i)
ψ is a nondecreasing and continuous function in each coordinate;
-
(ii)
for , , , where ;
-
(iii)
and , where .
The following are some easy examples of functions from class Ψ:
, for ;
, for ;
, for , , and ;
, for , , and ;
, where is a nondecreasing and continuous function, and if and only if .
Definition 2.1 Let be a partial b-metric space with the coefficient , A, S, and be three mappings. Then A, S, and T are said to be generalized -expansive mappings if
for all , where , is a nondecreasing and continuous function, , and for all , , where or .
Definition 2.2 Let be a partial b-metric space with the coefficient , A, S, and be three mappings. Then A, S, and T are said to be generalized -weakly expansive mappings if
for all , where , are continuous and nondecreasing functions, , if and only if , and for all , .
It is easy to acquire the following example of generalized -expansive mappings or generalized -weakly expansive mappings.
Example 2.1 Let be endowed with the partial b-metric given by
for , where . Let and be given by
for all , and be given by
Then A, S, and T are generalized -expansive mappings. In fact, if are defined by
for all , where . Then A, S, and T are also generalized -weakly expansive mappings.
Now, we first prove some fixed point results for generalized -expansive mappings in -complete partial b-metric spaces.
Theorem 2.1 Let be a -complete partial b-metric space, A, S, and be three mappings satisfying the generalized -expansive condition (2.1). Suppose that the following conditions are satisfied:
-
(i)
, , and is a closed subset of ;
-
(ii)
A is an injective and A and T are weakly compatible.
Then A, S, and T have a unique common fixed point in X.
Proof Let be an arbitrary point in X. Since , there exists an such that . Since , there exists an such that . Continuing this process, we can construct a sequence in X such that
We will complete the proof in three steps.
Step 1. We prove that
Suppose that for some . In the case that , gives . Indeed, by (2.1), we have
which implies that , that is, . Similarly, if , then . Consequently, for . Hence, .
Now, suppose that , for each n. By (2.1), we have
If , then we have . It follows from (2.4) and the properties of ψ and f that
Since f is nondecreasing, we get , which is a contradiction. Thus,
Hence, we deduce that, for each , . Similarly, we can prove that , for all . Therefore, is a decreasing sequence of nonnegative real numbers. So, there exists such that .
From Definition 1.3(), we have
It follows from (2.5) that and are two bounded sequences. Hence, the sequence has a subsequence which converges to a real number , and the sequence has a subsequence which converges to a real number . By (2.4), we have
Letting in the above inequality, by the properties of ψ and f, we have , which implies that . Hence, .
Step 2. We show that is a -Cauchy sequence.
Indeed, we first prove that . Because of (2.3), it is sufficient to show that . Suppose on the contrary, then there exists for which we can find two subsequences and of such that is the smallest index for which
This means that
From (2.6), using the triangular inequality, we can see that
and
By means of (2.3), taking the lower limit as in the above inequality, we get
On the other hand, we have
With the help of (2.3) and (2.7) and taking the upper limit as in the above inequality, it is not difficult to see that
From (2.1), we have
Now, taking upper limit as in the above inequality, the properties of ψ, f, and (2.8)-(2.11) guarantee that
which implies that . Thus, , a contradiction. Hence, , that is, is a -Cauchy sequence.
Step 3. We will show that A, S, and T have a unique common fixed point.
Since is a -Cauchy sequence in , and thus it is also b-Cauchy sequence in the b-metric space by Lemma 1.1. Since is -complete, from Lemma 1.1, is also b-complete, so the sequence is b-convergent in the b-metric space . Therefore, there exists such that . Then .
Since is a closed set of , , and , we get . Hence, there exists such that . This together with (2.1) ensures that
Since and , we can find by Lemma 1.1 that
Taking the upper limit as in (2.12), using the properties of ψ and f, (2.13), and (2.14), it is clear that
which implies that . Hence, . Similarly, since , there exists such that , we have . Hence, . Since A is an injective, we get .
Let . Then . Since A and T are weakly compatible, it is obvious that . By (2.1), we get
which implies that . Thus, . Since A is an injective, we get . Hence, , that is, z is a common fixed point of A, S, and T.
Now, we prove the uniqueness of common fixed points of A, S, and T. Suppose that such that and . By means of (2.1), we have
which implies that . Hence, . This completes the proof. □
Remark 2.1 Let I be the identity mappings on X. Taking , , for all in Theorem 2.1, we have the following corollary, which extends and generalizes Theorem 2.1 in [19] and Theorem 2 in [20].
Corollary 2.1 Let be a -complete partial b-metric space, S and be two bijective mappings. Suppose that
for all , where . Then S and T have a unique common fixed point in X.
Now, in order to support the usability of our results, we present the following example.
Example 2.2 Let be the set of all real continuous functions defined on and . Define a partial b-metric by
It is easy to see that is a -complete partial b-metric space with . Let be defined by
Then it is easy to show that all the conditions (i)-(ii) of Theorem 2.1 are satisfied. Define and by , .
Now, we consider following cases:
Case 1. If , then .
Case 2. If , then
Case 3. If and , then
Case 4. If and , then
That is,
for all . Thus, all conditions of Theorem 2.1 are satisfied. Hence, A, S, and T have a unique common fixed point .
Now, we state and prove some fixed point results for generalized -weakly expansive mappings in partial b-metric spaces.
Theorem 2.2 Let be a -complete partial b-metric space, A, S, and be three mappings satisfying the generalized -weakly expansive condition (2.2). Suppose that the following conditions are satisfied:
-
(i)
, , and is a closed subset of ;
-
(ii)
A is an injective and A and T are weakly compatible.
Then A, S, and T have a unique common fixed point in X.
Proof Let . Repeating the proof of Theorem 2.1, we know that there exists a sequence in X such that and , for .
We will complete the proof in three steps.
Step 1. We prove that .
Suppose first that for some . Then . In the case that , then gives . Indeed, by (2.2), we have
where
Thus, , implies that . Similarly, if , then . Consequently, for . Hence, .
Now, suppose that , for each n. By (2.2), we have
where
If , then we have . It follows from (2.15) and the properties of ϕ, g, h that
Since g is nondecreasing, we get , which is a contradiction. Thus, . From (2.15), using the properties of ϕ, g, h, we have
Hence, we deduce that, for each , . Similarly, we can prove that , for all . Therefore, is a decreasing sequence of nonnegative real numbers. So, there exists such that . Following the proof of Theorem 2.1, we know that the sequence has a subsequence which converges to a real number . By (2.15), we get
Letting in the above inequality, using the properties of ϕ, g, h, we can see that
which implies that . Therefore, .
Step 2. We now show that is a -Cauchy sequence.
Indeed, we first prove that . Since , it is sufficient to show that . Suppose on the contrary, then there exists for which we can find two subsequences and of such that is the smallest index, for which , , for every k. This means that .
Repeating to the proof of Theorem 2.1, we also have (2.8)-(2.11). By means of (2.2), we get
where
Taking the lower limit as , using (2.8), (2.9), and (2.10), it is clear that
Taking the upper limit as in (2.16), using the properties of ϕ, g, h, (2.11), and (2.17), we obtain
which implies that , a contradiction. Hence, we obtain , that is, is a -Cauchy sequence.
Step 3. We will show that A, S, and T have a unique common fixed point.
Since is a -Cauchy sequence in . Similar to the proof of Theorem 2.1, we know that there exists such that .
Since is a closed set of , , and , we get . Hence, there exists such that . This together with (2.2) ensures that
where
Taking the upper limit as in (2.18), using the properties of ϕ, g, h, (2.13), and (2.14), we get
which implies that . Hence, . Similarly, since , there exists such that , we have . Hence, . Since A is an injective, we get .
Let . Then . Since A and T are weakly compatible, it is obvious that . Then we can find by (2.2) that
where
Thus,
which implies that . Thus, . Since A is an injective, we get . Thus, and z is a common fixed point of A, S, and T.
Now, we prove the uniqueness of common fixed points of A, S, and T. Suppose that such that and . By means of (2.2), we have
where
Hence,
which implies that . Hence, . This completes the proof. □
Remark 2.2 Taking in Theorem 2.2, we have the following corollary, which extends and generalizes Theorem 2.1 in [15] and Theorem 1 in [17].
Corollary 2.2 Let be a -complete partial b-metric space, A, S, and be three mappings. Suppose that the following conditions are satisfied:
-
(i)
, , and is a closed subset of ;
-
(ii)
A is an injective and A and T are weakly compatible;
-
(iii)
for all , we have
where , ϕ, g are the same as in Definition 2.2.
Then A, S, and T have a unique common fixed point in X.
In the sequel, we will take an example to support our results of Theorem 2.2.
Example 2.3 Let . Define a partial b-metric by
It is easy to see that is a -complete partial b-metric space with . Let be defined by
Then it is easy to show that all the conditions (i)-(ii) of Theorem 2.2 are satisfied. Define by , , for all . Without loss of generality, we assume that . Then
That is,
for all , where . Thus, all conditions of Theorem 2.2 are satisfied. Hence, A, S, and T have a unique common fixed point .
3 An application
In this section, we establish the existence theorem for the solutions of a class of system of integral equations.
Consider the system of integral equations
for , where , is a continuous function and are also continuous functions.
Let be the set of all real continuous functions defined on I. We endowed X with the partial b-metric
for all , where and . It is not difficult to prove that is a -complete partial b-metric space with coefficient .
Now, we define S and by
for all . Then x is a solution of (3.1) if and only if it is a common fixed point of S and T.
We shall prove the existence of common fixed point of S and T under certain conditions.
Theorem 3.1 Suppose that the following hypotheses hold:
-
(i)
there exist a continuous function and such that
for all , where , for all .
-
(ii)
.
Then the system of integral equations (3.1) has a solution .
Proof Let . From the conditions (i) and (ii), we have
for all . Thus, for any , we get the inequality of Corollary 2.1. Hence, all the hypotheses of Corollary 2.1 are satisfied. Then S and T have a common fixed point , that is, is a solution of the system of integral equations (3.1). □
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Acknowledgements
The authors thank the editor and the referees for their valuable comments and suggestions. The research was supported by the National Natural Science Foundation of China (11071108, 11361042, 11326099) and the Provincial Natural Science Foundation of Jiangxi, China (2010GZS0147, 20114BAB201007, 20142BAB211004, 20142BAB201007), and supported partly by the Provincial Graduate Innovation Foundation of Jiangxi, China (YC2012-B004).
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Zhu, C., Xu, W., Chen, C. et al. Common fixed point theorems for generalized expansive mappings in partial b-metric spaces and an application. J Inequal Appl 2014, 475 (2014). https://doi.org/10.1186/1029-242X-2014-475
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DOI: https://doi.org/10.1186/1029-242X-2014-475