- Open Access
Fixed points of -contractive mappings in partially ordered b-metric spaces and application to quadratic integral equations
© Aghajani and Arab; licensee Springer. 2013
Received: 16 April 2013
Accepted: 20 August 2013
Published: 7 November 2013
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.
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  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 .
Definition 1.1 
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 ).
Example 1.2 
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 .
Example 1.3 
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  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 
In , Lakshmikantham and Ćirić introduced the concept of mixed g-monotone property as follows.
Definition 1.5 
Note that if g is an identity mapping, then F is said to have the mixed monotone property (see also ).
Definition 1.6 
Similarly, note that if g is an identity mapping, then is called a coupled fixed point of the mapping F (see also ).
Definition 1.7 
Definition 1.8 
Definition 1.9 
whenever and are sequences in X such that and for all .
2 Main results
ψ is continuous,
if and only if .
ϕ is lower semi-continuous,
if and only if ,
and Θ the set of all continuous functions with if and only if .
Now, we introduce the following definition.
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.
For this purpose, consider the following three cases.
so (2.11) obviously holds.
which is a contradiction.
which is again a contradiction.
This implies that is a coupled coincidence point of T and g. This completes the proof. □
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.
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
From Theorem 2.2, it follows that the set of coupled coincidences is nonempty.
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 .
a contradiction. Therefore, , that is, T and g have a common fixed point. □
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
γ is non-decreasing and for all .
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.
It is easy to see that is a complete b-metric space with .
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 .
This proves that the operator T satisfies the contractive condition (2.31) appearing in Corollary 2.4.
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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