Abstract
We prove some coupled coincidence and coupled common fixed point theorems for mappings satisfying -contractive conditions in partially ordered complete b-metric spaces. The obtained results extend and improve many existing results from the literature. As an application, we prove the existence of a unique solution to a class of nonlinear quadratic integral equations.
MSC:47H10, 54H25.
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1 Introduction and preliminaries
In [1, 2], Czerwik introduced the notion of a b-metric space, which is a generalization of the usual metric space, and generalized the Banach contraction principle in the context of complete b-metric spaces. After that, many authors have carried out further studies on b-metric spaces and their topological properties (see, e.g., [1–14]). In this paper, some coupled coincidence and coupled common fixed point theorems for mappings satisfying -contractive conditions in partially ordered complete b-metric spaces are proved. Also, we apply our results to study the existence of a unique solution to a large class of nonlinear quadratic integral equations. There are many papers in the literature concerning coupled fixed points introduced by Bhaskar and Lakshmikantham [15] and their applications in the existence and uniqueness of solutions for boundary value problems. A number of articles on this topic have been dedicated to the improvement and generalization; see [16–20] and references therein. Also, to see some results on common fixed points for generalized contraction mappings, we refer the reader to [21–23]. For the sake of convenience, some definitions and notations are recalled from [1, 3, 24] and [25].
Definition 1.1 [1]
Let X be a (nonempty) set and be a given real number. A function is said to be a b-metric space iff for all , the following conditions are satisfied:
-
(i)
iff ,
-
(ii)
,
-
(iii)
.
The pair is called a b-metric space with the parameter s.
It should be noted that the class of b-metric spaces is effectively larger than that of metric spaces since a b-metric is a metric when .
The following example shows that, in general, a b-metric need not necessarily be a metric (see also [14]).
Example 1.2 [3]
Let be a metric space and , where is a real number. Then ρ is a b-metric with . However, if is a metric space, then is not necessarily a metric space. For example, if is the set of real numbers and is the usual Euclidean metric, then is a b-metric on ℝ with , but is not a metric on ℝ.
Also, the following example of a b-metric space is given in [26].
Example 1.3 [26]
Let X be the set of Lebesgue measurable functions on such that . Define by . As is a metric on X, then, from the previous example, D is a b-metric on X, with .
Khamsi [27] also showed that each cone metric space over a normal cone has a b-metric structure.
Since, in general, a b-metric is not continuous, we need the following simple lemma about the b-convergent sequences in the proof of our main result.
Lemma 1.4 [3]
Let be a b-metric space with , and suppose that and are b-convergent to x, y, respectively. Then we have
In particular, if , then we have . Moreover, for each , we have
In [25], Lakshmikantham and Ćirić introduced the concept of mixed g-monotone property as follows.
Definition 1.5 [25]
Let be a partially ordered set and and . We say F has the mixed g-monotone property if F is non-decreasing g-monotone in its first argument and is non-increasing g-monotone in its second argument, that is, for any ,
and
Note that if g is an identity mapping, then F is said to have the mixed monotone property (see also [15]).
Definition 1.6 [25]
An element is called a coupled coincidence point of a mapping and a mapping if
Similarly, note that if g is an identity mapping, then is called a coupled fixed point of the mapping F (see also [15]).
Definition 1.7 [24]
An element is called a common fixed point of a mapping and if
Definition 1.8 [25]
Let X be a nonempty set and and . One says that F and g are commutative if for all ,
Definition 1.9 [28]
The mappings F and g, where and , are said to be compatible if
and
whenever and are sequences in X such that and for all .
2 Main results
Throughout the paper, let Ψ be a family of all functions satisfying the following conditions:
-
(a)
ψ is continuous,
-
(b)
ψ non-decreasing,
-
(c)
if and only if .
We denote by Φ the set of all functions satisfying the following conditions:
-
(a)
ϕ is lower semi-continuous,
-
(b)
if and only if ,
and Θ the set of all continuous functions with if and only if .
Let be a partially ordered b-metric space, and let and be two mappings. Set
and
Now, we introduce the following definition.
Definition 2.1 Let be a partially ordered b-metric space and , and . We say that is an almost generalized -contractive mapping with respect to if there exists such that
for all with and .
Now, we establish some results for the existence of a coupled coincidence point and a coupled common fixed point of mappings satisfying almost generalized -contractive condition in the setup of partially ordered b-metric spaces. The first result in this paper is the following coupled coincidence theorem.
Theorem 2.2 Suppose that is a partially ordered complete b-metric space. Let be an almost generalized -contractive mapping with respect to , and T and g are continuous such that T has the mixed g-monotone property and commutes with g. Also, suppose . If there exists such that and , then T and g have coupled coincidence point in X.
Proof By the given assumptions, there exists such that and . Since , we can define such that and , then and . Also, there exists such that and . Since T has the mixed g-monotone property, we have
and
Continuing in this way, we construct two sequences and in X such that
for which
From (2.2) and (2.3) and inequality (2.1) with and , we obtain
where
and
Since
and
then we get
By (2.4) and (2.5), we have
Similarly, we can show that
Now, denote
Combining (2.6), (2.7) and the fact that for , we have
So, using (2.6), (2.7), (2.8) together with (2.9), we obtain
Now we prove that for all ,
For this purpose, consider the following three cases.
Case 1. If , then by (2.10) we have
so (2.11) obviously holds.
Case 2. If , then by (2.6) we have
which is a contradiction.
Case 3. If , then from (2.7) we have
which is again a contradiction.
Thus, in all the cases, (2.11) holds for each . It follows that the sequence is a monotone decreasing sequence of nonnegative real numbers and, consequently, there exists such that
We show that . Suppose, on the contrary, that . Taking the limit as in (2.12) and using the properties of the function ϕ, we get
which is a contradiction. Therefore , that is,
which implies that
Now, we claim that
Assume, on the contrary, that there exist and subsequences , of and , of with such that
Additionally, corresponding to , we may choose such that it is the smallest integer satisfying (2.16) and . Thus,
Using the triangle inequality in a b-metric space and (2.16) and (2.17), we obtain that
Taking the upper limit as and using (2.14), we obtain
Similarly, we obtain
Also,
So, from (2.14) and (2.18), we have
Similarly, we obtain
Also,
So, from (2.14) and (2.18), we have
In a similar way, we obtain
Also,
So, from (2.14) and (2.22), we have
Similarly, we obtain
Linking (2.14), (2.18), (2.19), (2.20), (2.21), (2.22) together with (2.23), we get
So,
Similarly, we have
and
Since , from (2.2) we have
Thus,
Since ψ is a non-decreasing function, we have
Taking the upper limit as and using (2.25) and (2.26), we get
which implies that
so
a contradiction to (2.27). Therefore, (2.15) holds and we have
Since X is a complete b-metric space, there exist such that
From the commutativity of T and g, we have
Now, we shall show that
Letting in (2.30), from the continuity of T and g, we get
This implies that is a coupled coincidence point of T and g. This completes the proof. □
Corollary 2.3 Let be a partially ordered complete b-metric space, and let be a continuous mapping such that T has the mixed monotone property. Suppose that there exist , , and such that
where
and
for all with and . If there exists such that and , then T has a coupled fixed point in X.
Proof Take and apply Theorem 2.2. □
The following result is the immediate consequence of Corollary 2.3.
Corollary 2.4 Let be a partially ordered complete b-metric space. Let be a continuous mapping such that T has the mixed monotone property. Suppose that there exists such that
where
for all with and . If there exists such that and , then T has a coupled fixed point in X.
3 Uniqueness of a common fixed point
In this section we shall provide some sufficient conditions under which T and g have a unique common fixed point. Note that if is a partially ordered set, then we endow the product with the following partial order relation, for all ,
From Theorem 2.2, it follows that the set of coupled coincidences is nonempty.
Theorem 3.1 By adding to the hypotheses of Theorem 2.2, the condition: for every and in , there exists such that is comparable to and to , then T and g have a unique coupled common fixed point; that is, there exists a unique such that
Proof We know, from Theorem 2.2, that there exists at least a coupled coincidence point. Suppose that and are coupled coincidence points of T and g, that is, , , and . We shall show that and . By the assumptions, there exists such that is comparable to and to . Without any restriction of the generality, we can assume that
Put , and choose such that
For , continuing this process, we can construct sequences and such that
Further, set , and , and in the same way define sequences , and , . Then it is easy to see that
for all . Since is comparable to , then it is easy to show . Recursively, we get that
Thus from (2.1) we have
where
It is easy to show that
and
Hence,
Similarly, one can prove that
Combining (3.3), (3.4) and the fact that for , we have
Using the non-decreasing property of ψ, we get that
implies that is a non-increasing sequence. Hence, there exists such that
Passing the upper limit in (3.5) as , we obtain
which implies that , and then . We deduce that
which concludes
Similarly, one can prove that
From (3.6) and (3.7), we have and . Since and , by the commutativity of T and g, we have
Denote and . Then from (3.8) we have
Thus, is a coupled coincidence point. It follows that and , that is,
From (3.9) and (3.10), we obtain
Therefore, is a coupled common fixed point of T and g. To prove the uniqueness of the point , assume that is another coupled common fixed point of T and g. Then we have
Since is a coupled coincidence point of T and g, we have and . Thus and , which is the desired result. □
Theorem 3.2 In addition to the hypotheses of Theorem 3.1, if and are comparable, then T and g have a unique common fixed point, that is, there exists such that .
Proof Following the proof of Theorem 3.1, T and g have a unique coupled common fixed point . We only have to show that . Since and are comparable, we may assume that . By using the mathematical induction, one can show that
where and are defined by (2.2). From (2.29) and Lemma 1.4, we have
a contradiction. Therefore, , that is, T and g have a common fixed point. □
Remark 3.3 Since a b-metric is a metric when , from the results of Jachymski [29], the condition
is equivalent to
where , and is continuous, for all and if and only if . So, our results can be viewed as a generalization and extension of the corresponding results in [15, 25, 30–32] and several other comparable results.
4 Application to integral equations
Here, in this section, we wish to study the existence of a unique solution to a nonlinear quadratic integral equation, as an application to our coupled fixed point theorem. Consider the nonlinear quadratic integral equation
Let Γ denote the class of those functions which satisfy the following conditions:
-
(i)
γ is non-decreasing and for all .
-
(ii)
There exists such that for all .
For example, , where and are in Γ.
We will analyze Eq. (4.1) under the following assumptions:
(a1) () are continuous functions, and there exist two functions such that ().
(a2) is monotone non-decreasing in x and is monotone non-increasing in y for all and .
(a3) is a continuous function.
(a4) () are continuous in for every and measurable in for all such that
and .
(a5) There exist constants () and such that for all and ,
(a6) There exist such that
(a7) .
Consider the space of continuous functions defined on with the standard metric given by
This space can also be equipped with a partial order given by
Now, for , we define
It is easy to see that is a complete b-metric space with [3].
Also, is a partially ordered set if we define the following order relation:
For any and each , and belong to X and are upper and lower bounds of x, y, respectively. Therefore, for every , one can take which is comparable to and . Now, we formulate the main result of this section.
Theorem 4.1 Under assumptions (a1)-(a7), Eq. (4.1) has a unique solution in .
Proof We consider the operator defined by
By virtue of our assumptions, T is well defined (this means that if , then ). Firstly, we prove that T has the mixed monotone property. In fact, for and , we have
Similarly, if and , then . Therefore, T has the mixed monotone property. Also, for , that is, and , we have
Since the function γ is non-decreasing and and , we have
and
hence
Then we can obtain
and using the fact that for and , we have
This proves that the operator T satisfies the contractive condition (2.31) appearing in Corollary 2.4.
Finally, let α, β be the functions appearing in assumption (a6); then, by (a6), we get
Theorem 3.1 gives us that T has a unique coupled fixed point . Since , Theorem 3.2 says that and this implies . So, is the unique solution of Eq. (4.1) and the proof is complete. □
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Aghajani, A., Arab, R. Fixed points of -contractive mappings in partially ordered b-metric spaces and application to quadratic integral equations. Fixed Point Theory Appl 2013, 245 (2013). https://doi.org/10.1186/1687-1812-2013-245
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DOI: https://doi.org/10.1186/1687-1812-2013-245