Abstract
We introduce the notion of ordered cyclic weakly -contractive mappings, and we establish some fixed and common fixed point results for this class of mappings in complete ordered b-metric spaces. Our results extend several known results from the context of ordered metric spaces to the setting of ordered b-metric spaces. They are also cyclic variants of some very recent results in ordered b-metric spaces with even weaker contractive conditions. Some examples support our results and show that the obtained extensions are proper. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.
MSC:47H10, 54H25.
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1 Introduction and preliminaries
The Banach contraction principle is a very popular tool for solving problems in nonlinear analysis. One of the interesting generalizations of this basic principle was given by Kirk et al. [1] in 2003 by introducing the following notion of cyclic representation.
Definition 1 [1]
Let A and B be non-empty 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 [1].
Theorem 1 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 .
It should be noted that cyclic contractions (unlike Banach-type contractions) need not be continuous, which is an important gain of this approach. Following the work of Kirk et al., several authors proved many fixed point results for cyclic mappings, satisfying various (nonlinear) contractive conditions.
Berinde initiated in [2] 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., [3–7]. Some authors used related notions as ‘condition (B)’ (Babu et al. [8]) and ‘almost generalized contractive condition’ for two maps (Ćirić et al. [9]), and for four maps (Aghajani et al. [10]). See also a note by Pacurar [11]. Here, we recall one of the respective definitions.
Definition 2 [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 .
Khan et al. [12] introduced the concept of an altering distance function as follows.
Definition 3 [12]
A function is called an altering distance function if the following properties hold:
-
1.
φ is continuous and non-decreasing.
-
2.
if and only if .
So far, many authors have studied fixed point theorems, which are based on altering distance functions.
The concept of a b-metric space was introduced by Bakhtin in [13], and later used by Czerwik in [14, 15]. After that, several interesting results about the existence of fixed points for single-valued and multi-valued operators in b-metric spaces have been obtained (see, e.g., [16–28]). Recently, Hussain and Shah [29] obtained some results on KKM mappings in cone b-metric spaces.
Consistent with [15] and [28], the following definitions and results will be needed in the sequel.
Definition 4 [15]
Let X be a (nonempty) set, and let be a given real number. A function is a b-metric if for all , the following conditions hold:
(b1) iff ,
(b2) ,
(b3) .
In this case, the pair is called a b-metric space.
It should be noted that the class of b-metric spaces is effectively larger than the class of metric spaces, since a b-metric is a metric if (and only if) . Here, we present an easy example to show that in general, a b-metric need not necessarily be a metric (see also [[28], p.264]).
Example 1 Let be a metric space and , where is a real number. Then d is a b-metric with . Condition (b3) follows easily from the convexity of the function ().
The notions of b-convergent and b-Cauchy sequences, as well as of b-complete b-metric spaces are introduced in an obvious way (see, e.g., [18]).
It should be noted that in general, a b-metric function for need not be jointly continuous in both variables. The following example (corrected from [22]) illustrates this fact.
Example 2 Let , and let be defined by
Then considering all possible cases, it can be checked that for all , we have
Thus, is a b-metric space (with ). Let for each . Then
that is, , but as .
Aghajani et al. [16] proved the following simple lemma about the b-convergent sequences.
Lemma 1 Let be a b-metric space with , and suppose that and b-converge to x, y, respectively. Then we have
In particular, if , then . Moreover, for each , we have
The existence of fixed points for mappings in partially ordered metric spaces was first investigated in 2004 by Ran and Reurings [30], and then by Nieto and Lopez [31]. Afterwards, this area was a field of intensive study of many authors.
Shatanawi and Postolache proved in [32] the following common fixed point results for cyclic contractions in the framework of ordered metric spaces.
Theorem 2 [32]
Let be a complete ordered metric space, and let A, B be closed nonempty subsets of X with . Let be two mappings, which are -weakly increasing (see further Definition 6). Assume that
-
(a)
is a cyclic representation of X w.r.t. the pair , i.e., and ;
-
(b)
there exist and an altering distance function ψ such that for any two comparable elements with and , we have
-
(c)
f or g is continuous, or
(c′) the space is regular.
Then f and g have a common fixed point.
Here, the ordered metric space is called regular if for any non-decreasing sequence in X such that , as , one has for all .
By an ordered b-metric space, we mean a triple , where is a partially ordered set, and is a b-metric space. Fixed points in such spaces were studied, e.g., by Aghajani et al. [16] and Roshan et al. [27]. In the last mentioned paper, the following common fixed point results for contractions in ordered b-metric spaces were proved.
Theorem 3 [27]
Let be a complete ordered b-metric space, and let be two weakly increasing mappings. Suppose that there exist two altering distance functions ψ, φ and a constant such that the inequality
holds for all comparable , where
and
If either [f or g is continuous], or the space is regular, then f and g have a common fixed point.
In this paper, we introduce the notion of ordered cyclic weakly -contractions and then derive fixed point and common fixed point theorems for these cyclic contractions in the setup of complete ordered b-metric spaces. Our results extend some fixed point theorems from the framework of ordered metric spaces, in particular Theorem 2. On the other hand, they are cyclic variants of Theorem 3 with even weaker contractive conditions.
We show by examples that the obtained extensions are proper. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.
2 Common fixed point results
In this section, we introduce the notion of ordered cyclic weakly -contractive pair of self-mappings and prove our main results.
Definition 5 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
(2.1)
where
and
Definition 6 [32]
Let be a partially ordered set, and let 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 .
Theorem 4 Let be a complete ordered b-metric space, and let 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 .
Proof Let us divide the proof into two parts.
First part. We prove that is a fixed point of f if and only if u is a fixed point of g. Suppose that u is a fixed point of f. As and , by (2.1), 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.
Second part (construction of a sequence by iterative technique).
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 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 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 (2.1), we have
where
and
Hence, we have
If
then (2.4) becomes
which gives a contradiction. So,
and hence, (2.4) becomes
Similarly, we can show that
By (2.5) and (2.6), we get that is a non-increasing sequence of positive numbers. Hence, there is such that
Letting in (2.5), we get
which implies that , and hence . So, we have
Step 2. We will prove that is a b-Cauchy sequence. Because of (2.7), it is sufficient to show that is a b-Cauchy sequence. Suppose on the contrary, i.e., that 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 (2.8) and using the triangular inequality, we get
Using (2.7) and taking the upper limit as , we get
On the other hand, we have
Using (2.7), (2.9) 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 (2.7), (2.9) and (2.11), we get
and
Since and are comparable and and , using (2.1) we have
where
and
Taking the upper limit in (2.15) and using (2.7) and (2.11)-(2.13), we get
Hence, we have
and, from (2.16),
Now, taking the upper limit as in (2.14) and using (2.10), (2.17) and (2.18), we have
which implies that . By (2.15), it follows that , which is in contradiction with (2.8). Hence is a b-Cauchy sequence in X.
Step 3 (existence of a common fixed point).
As is a b-Cauchy sequence in X which is a b-complete b-metric space, there exists such that as , and
Now, without loss of generality, we may assume that f is continuous. Using the triangular inequality, we get
Letting , we get
Hence, we have . Thus, u is a fixed point of f and, since A and B are closed subsets of X, . By the first part of proof, we conclude that u is also a fixed point of g. □
The assumption of continuity of one of the mappings f or g in Theorem 4 can be replaced by another condition, which is often used in similar situations. Namely, we shall use the notion of a regular ordered b-metric space, which is defined analogously to the case of the standard metric (see the paragraph following Theorem 2).
Theorem 5 Let the hypotheses of Theorem 4 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.
Proof Repeating the proof of Theorem 4, we construct an increasing sequence in X such that for some . As A and B are closed subsets of X, we have . Using the assumption (b′) on X, we have for all . Now, we show that . By (2.1), we have
where
and
Letting in (2.20) and (2.21) and using Lemma 1, we get
and . Now, taking the upper limit as in (2.19) and using Lemma 1 and (2.22), we get
It follows that , and hence, by (2.20), that . Thus, u is a fixed point of g. On the other hand, similar to the first part of the proof of Theorem 4, we can show that . Hence, u is a common fixed point of f and g. □
3 Consequences and examples
As consequences, we have the following results.
By putting in Theorems 4 and 5, we obtain improvements of the main results (Theorems 5 and 6) of Roshan et al. [27], i.e., of Theorem 3 of the present paper (note that we have instead of in the contractive condition).
Taking , in Theorems 4 and 5, we get the following.
Corollary 1 Let be a complete ordered b-metric space, and let 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
(3.1)
where and are given by (2.2) and (2.3), respectively;
-
(c)
f or g is continuous, or
(c′) the space is regular.
Then f and g have a common fixed point .
Taking and in Corollary 1, we obtain Theorems 2.1 and 2.2 of Shatanawi and Postolache [32] (Theorem 2 in this paper).
Taking for in Corollary 1, we get the following.
Corollary 2 Let be a complete ordered b-metric space. Let A and B be nonempty closed subsets of X, and let be two -weakly increasing mappings with respect to ⪯ such that and . Suppose that there exist and such that
for all comparable elements with and . If either f or g is continuous, or the space is regular, then f and g have a common fixed point.
Putting in Theorems 4 and 5, the following corollary is obtained which extends and improves Theorems 3 and 4 in [27].
Corollary 3 Let be a complete ordered b-metric space, and let A and B be closed subsets of X. Let be a mapping such that f is non-decreasing with respect to ⪯. Assume the following:
-
(a)
is a cyclic representation of X w.r.t. f, that is, , ;
-
(b)
there exist two altering distance functions ψ, φ, and such that
(3.2)
for all comparable with and , where
and
-
(c)
f is continuous, or
(c′) the space is regular.
If there exists such that , then f has a fixed point.
Again, taking , in Corollary 3, we get the following.
Corollary 4 Let be a complete ordered b-metric space, let and A and B be closed subsets of X. Let be a non-decreasing map with respect to ⪯. Suppose that
-
(a)
is a cyclic representation of X w.r.t. f;
-
(b)
there exist , and an altering distance function ψ such that for any comparable elements with and , we have
(3.3)
where and are given in Corollary 3;
-
(c)
f is continuous, or
(c′) the space is regular.
Then f has a fixed point .
Remark 1 (Common) fixed points of the given mappings in Theorems 4 and 5 and Corollaries 3 and 4 need not be unique (see further Example 4). However, it is easy to show that they must be unique in the case that the respective sets of (common) fixed points are well ordered (recall that a subset W of a partially ordered set is said to be well ordered if every two elements of W are comparable).
We illustrate our results with the following two examples.
Example 3 Consider the b-metric space given in Example 2, ordered by natural ordering and a mapping given as
If and , then is a cyclic representation of X with respect to f. Take given as , (<1) and arbitrary. In order to check the contractive condition (3.3), consider the following cases.
If , then
and (3.3) holds. If and y is an even integer, then
Finally, if and y is an odd integer, then and (3.3) trivially holds.
Hence, all the conditions of Corollary 4 are satisfied. Obviously, f has a (unique) fixed point ∞, belonging to .
We now present an example showing that there are situations where our results can be used to conclude about the existence of (common) fixed points, while some other known results cannot be applied.
Example 4 Let be equipped with the following partial order:
Define a b-metric by
It is easy to see that is a b-complete b-metric space with . Set and , and define self-maps f and g by
It is easy to see that f and g are -weakly increasing mappings with respect to ⪯, and that f and g are continuous. Also, , and .
Define by . One can easily check that the pair satisfies the requirements of Corollary 1, with any δ and , as the left-hand side of the contractive condition (3.1) is equal to 0 for all comparable x, y such that and . Hence, f and g have a common fixed point. Indeed, 0 and 2 are two common fixed points of f and g. (Note that the ordered set is not well ordered).
However, take and (which are not comparable). Then
where and are arbitrary, since
and
Hence, this result cannot be applied in the context of b-metric spaces without order.
4 Application to existence of solutions of integral equations
Integral equations like (4.1) have been studied in many papers (see, e.g., [22, 33] and the references therein). In this section, we look for a nonnegative solution to (4.1) in .
Consider the integral equation
where , and are continuous functions.
Let be the set of real continuous functions on . We endow X with the b-metric
Clearly, is a complete b-metric space (with the parameter ). We endow X with the partial order ⪯ given by
Clearly, the space is regular.
Let and such that
Assume that for all , we have
and
Let for all , be a decreasing function, that is,
Assume that is such that
Define a mapping by
Suppose that for all and for all comparable with ( and ) or ( and ),
Theorem 6 Under the assumptions (4.2)-(4.7), the integral equation (4.1) has a solution in the set .
Proof Define closed subsets of X, and by
Consider the mapping defined above. We will prove that
Suppose that , that is,
Applying, condition (4.5), since for all , we obtain that
The above inequality with condition (4.3) implies that
for all . Thus, we have .
Similarly, let , that is,
Using condition (4.5), since for all , we obtain that
The above inequality with condition (4.4) implies that
for all . Hence, we have . Thus, (4.8) holds.
Now, let , that is, for all ,
This implies from condition (4.2) that for all ,
Also, if , then by (4.7), we have
for all . That is, . Hence, is increasing.
Now, by the conditions (4.6) and (4.7), we have for all and for all
comparable and ,
which implies that
with .
Now, all the conditions of Corollary 2 (with and ) hold, and has a fixed point z in
That is, is the solution to (4.1). □
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Acknowledgements
The authors are highly indebted to the referees of this paper who helped us to improve it in several places. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support. The fourth author is thankful to the Ministry of Education, Science and Technological Development of Serbia.
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Hussain, N., Parvaneh, V., Roshan, J.R. et al. Fixed points of cyclic weakly -contractive mappings in ordered b-metric spaces with applications. Fixed Point Theory Appl 2013, 256 (2013). https://doi.org/10.1186/1687-1812-2013-256
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DOI: https://doi.org/10.1186/1687-1812-2013-256