Abstract
We introduce the notion of almost generalized -contractive mappings, and establish the coincidence and common fixed point results for this class of mappings in partially ordered complete b-metric spaces. Our results extend and improve several known results from the context of ordered metric spaces to the setting of ordered b-metric spaces. As an application, we prove the existence of a unique solution to a class of nonlinear quadratic integral equations.
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1 Introduction
Fixed points theorems in partially ordered metric spaces were firstly obtained in 2004 by Ran and Reurings [1], and then by Nieto and Rodríguez-López [2]. In this direction several authors obtained further results under weak contractive conditions (see, e.g., [3–8]). Berinde initiated in [9] the concept of almost contractions and obtained several interesting fixed point theorems. This has been a subject of intense study since then; see, e.g., [10–20]. Some authors used related notions as ‘condition (B)’ (Babu et al. [21]) and ‘almost generalized contractive condition’ for two maps (Ćirić et al. [22]), and for four maps (Aghajani et al. [23]). See also a note by Pacurar [15]. On the other hand, the concept of b-metric space was introduced by Czerwik in [24]. After that, several interesting results of the existence of fixed point for single-valued and multivalued operators in b-metric spaces have been obtained (see [25–40]). Pacurar [41] proved some results on sequences of almost contractions and fixed points in b-metric spaces. Recently, Hussain and Shah [42] obtained results on KKM mappings in cone b-metric spaces. Using the concepts of partially ordered metric spaces, almost generalized contractive condition, and b-metric spaces, we define a new concept of almost generalized -contractive condition. In this paper, some coincidence and common fixed point theorems for mappings satisfying almost generalized -contractive condition in the setup of partially ordered complete b-metric spaces are proved. Consistent with [43] and [[40], p.264], the following definitions and results will be needed in the sequel.
Definition 1.1 [43]
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 does not necessarily need to be a metric (see, also, [40]).
Example 1.1 [44]
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 it is not a metric on ℝ.
Also, the following example of a b-metric space is given in [45].
Example 1.2 [45]
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 , where the b-metric D is defined with , d is a cone metric (also see [46–49]).
Khamsi [50] also showed that each cone metric space over a normal cone has a b-metric structure.
Definition 1.2 [6]
We shall say that the mapping T is g-nondecreasing if
2 Main results
Throughout the paper, let Ψ be the family of all functions satisfying the following conditions:
-
(a)
ψ is continuous,
-
(b)
ψ is nondecreasing,
-
(c)
for every .
We denote by Φ the set of all functions satisfying the following conditions:
-
(i)
φ is right continuous,
-
(ii)
φ is nondecreasing,
-
(iii)
for every .
Let be a partially ordered b-metric space and and be two mappings. Set
and
Now, we introduce the following definition.
Definition 2.1 Let be a partially ordered b-metric space. We say that is an almost generalized -contractive mapping with respect to for some , , and if
for all with .
Now, we establish some results for the existence of coincidence point and 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 coincidence point theorem.
Theorem 2.1 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 is a monotone g-nondecreasing mapping, commutative with g and . If there exists such that , then T and g have a coincidence point in X.
Proof By the given assumptions, there exists such that . Since , we can define such that , then . Also there exists such that . Since T is a monotone g-nondecreasing mapping, we have
Continuing in this way, we construct a sequence in X such that for all ,
for which
If there exists such that , then . This means that is a coincidence point of T, g, and the proof is finished. Thus, for all . From (2.2) and (2.3) and the inequality (2.1) with , we have
where
and
Since
then we get
By (2.4) and (2.5), we have
Suppose that for some , then by (2.6)
a contradiction. Hence,
and thus
Thus, we get
for all . Now, from
and the property of φ, we obtain , and consequently
Now, we shall prove that is a Cauchy sequence in . Suppose, on the contrary, that is not a Cauchy sequence. Then there exist and subsequences , of with such that
Additionally, corresponding to , we may choose such that it is the smallest integer satisfying (2.8) and . Thus,
Using the triangle inequality in b-metric space and (2.8) and (2.9) we obtain
Taking the upper limit as and using (2.7) we obtain
Also
So from (2.7) and (2.10), we have
Also
So from (2.7) and (2.10), we have
Also
so from (2.7) and (2.12), we have
Linking (2.7), (2.10), (2.11) together with (2.12) we get
So,
Similarly, we have
Since from (2.2), we have
Thus,
Passing to the upper limit as , and using (2.13), (2.14), and (2.15), we get
which is a contradiction. Thus, we proved that is a Cauchy sequence in . Since X is a complete b-metric space, there exists such that
From the commutativity of T and g, we have
Letting in (2.17) and from the continuity of T and g, we get
This implies that x is a coincidence point of T and g. This completes the proof. □
Now, we will prove the following result.
Theorem 2.2 Suppose that is a partially ordered complete b-metric space. Let be an almost generalized -contractive mapping with respect to , T is a g-nondecreasing mapping and . Also suppose
Also suppose gX is closed. If there exists such that , then T and g have a coincidence. Further, if T and g commute at their coincidence points, then T and g have a common fixed point.
Proof As in the proof of Theorem 2.1, we can show that is a Cauchy sequence. Since gX is a closed, there exists such that
Now we show that x is a coincidence point of T and g. Since from (2.18) and (2.19) we have for all n, then by the triangle inequality in a b-metric space and (2.1), we get
Indeed,
and
Hence , that is, . Thus we proved that T and g have a coincidence. Suppose now that T and g commute at x. Set . Then
Since from (2.18) we have and as and , from (2.1) we obtain
Hence , that is, . Therefore, . Thus we proved that T and g have a common fixed point. □
In the following, we deduce some fixed point theorems from our main results given by Theorems 2.1 and 2.2.
Corollary 2.3 Let be a partially ordered complete b-metric space and is a nondecreasing mapping. Suppose there exist , , and such that
where
and
for all with . Also suppose either
-
(a)
if is a nondecreasing sequence with in X, then , for all n, holds, or
-
(b)
T is continuous.
If there exists such that , then T has a fixed point in X.
Example 2.1 Let X be the set of Lebesgue measurable functions on such that . Define by
Then D is a b-metric on X, with . Also, this space can also be equipped with a partial order given by
The operator defined by
Now, we prove that T has a fixed point. For all with , we have
Now, if we define , , and . Thus, by Corollary 2.3 we see that T has a fixed point.
Remark 2.1 Corollary 2.3 extends and generalizes many existing fixed point theorems in the literature [2, 3, 51, 52].
The following result is the immediate consequence of Corollary 2.3.
Corollary 2.4 Let be a partially ordered complete b-metric space and is a nondecreasing mapping. Suppose there exists such that
for all with . Also suppose either
-
(a)
if is a nondecreasing sequence with in X, then , for all n, holds, or
-
(b)
T is continuous.
If there exists such that , then T has a fixed point in X.
Remark 2.2 Corollary 2.4 is a generalization to [[3], Theorem 1.3].
Taking , , in Corollary 2.4 we obtain the following generalization of the results in [1, 53].
Corollary 2.5 Let be a partially ordered complete b-metric space and is a nondecreasing mapping. Suppose there exists such that
for all with . Also suppose either
-
(a)
if is a nondecreasing sequence with in X, then , for all n, holds, or
-
(b)
T is continuous.
If there exists such that , then T has a fixed point in X.
Corollary 2.6 Let be a partially ordered complete b-metric space and is a nondecreasing mapping. Suppose there exist and such that
for all with . Also suppose either
-
(a)
if is a nondecreasing sequence with in X, then , for all n, holds, or
-
(b)
T is continuous.
If there exists such that , then T has a fixed point in X.
3 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 the our fixed point theorem. Consider the integral equation
Let Γ denote the class of those functions for which and , for all .
For example, , where and are in Γ.
We will analyze (3.1) under the following assumptions:
(a1) is continuous monotone nondecreasing in x, and there exist constant and such that for all and
(a2) is a continuous function.
(a3) is continuous in for every and measurable in for all such that
and .
(a4) There exists such that
(a5) .
We 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 [44].
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 are comparable to x, y. Now, we formulate the main result of this section.
Theorem 3.1 Under assumptions (a1)-(a5), (3.1) 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 ). For , and we have
Therefore, T has the monotone nondecreasing property. Also, for , we have
Since the function γ is nondecreasing and , we have
hence
Then we obtain
This proves that the operator T satisfies the contractive condition (2.21) appearing in Corollary 2.4. Also, let α, β be the functions appearing in assumption (a4); then, by (a4), we get . So, (3.1) has a solution and the proof is complete. □
Example 3.1 Consider the following functional integral equation:
for . Observe that this equation is a special case of (3.1) with
Indeed, by using we see that and , for all . Further, for arbitrarily fixed such that and for we obtain
Thus, the function f satisfies assumption (a1) with . It is also easily seen that h is a continuous function. Further, notice that the function k is continuous in for every and measurable in for all and . Moreover, we have
If we put , we have
This shows that assumption (a4) holds. Taking , and , then inequality appearing in assumption (a5) has the following form:
It is easily seen that each number satisfies the above inequality. Consequently, all the conditions of Theorem 3.1 are satisfied. Hence the integral equation (3.2) has a unique solution in .
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Allahyari, R., Arab, R. & Shole Haghighi, A. A generalization on weak contractions in partially ordered b-metric spaces and its application to quadratic integral equations. J Inequal Appl 2014, 355 (2014). https://doi.org/10.1186/1029-242X-2014-355
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DOI: https://doi.org/10.1186/1029-242X-2014-355