- Open Access
Fixed points of cyclic weakly -contractive mappings in ordered b-metric spaces with applications
© Hussain et al.; licensee Springer. 2013
Received: 18 June 2013
Accepted: 23 September 2013
Published: 7 November 2013
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.
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.  in 2003 by introducing the following notion of cyclic representation.
Definition 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 .
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  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. ) and ‘almost generalized contractive condition’ for two maps (Ćirić et al. ), and for four maps (Aghajani et al. ). See also a note by Pacurar . Here, we recall one of the respective definitions.
Definition 2 
for all .
Khan et al.  introduced the concept of an altering distance function as follows.
Definition 3 
φ is continuous and non-decreasing.
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 , 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  obtained some results on KKM mappings in cone b-metric spaces.
Definition 4 
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 ,
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 [, 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., ).
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 ) illustrates this fact.
that is, , but as .
Aghajani et al.  proved the following simple lemma about the b-convergent sequences.
The existence of fixed points for mappings in partially ordered metric spaces was first investigated in 2004 by Ran and Reurings , and then by Nieto and Lopez . Afterwards, this area was a field of intensive study of many authors.
Shatanawi and Postolache proved in  the following common fixed point results for cyclic contractions in the framework of ordered metric spaces.
Theorem 2 
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
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.  and Roshan et al. . In the last mentioned paper, the following common fixed point results for contractions in ordered b-metric spaces were proved.
Theorem 3 
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.
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)
Definition 6 
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 .
the pair is an ordered cyclic weakly -contraction;
f or g is continuous.
Then f and g have a common fixed point .
Proof Let us divide the proof into two parts.
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).
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.
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).
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.
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. , 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.
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)
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  (Theorem 2 in this paper).
Taking for in Corollary 1, we get the following.
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 .
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)
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.
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)
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.
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.
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.
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).
Hence, this result cannot be applied in the context of b-metric spaces without order.
4 Application to existence of solutions of integral equations
where , and are continuous functions.
Clearly, the space is regular.
Theorem 6 Under the assumptions (4.2)-(4.7), the integral equation (4.1) has a solution in the set .
for all . Thus, we have .
for all . Hence, we have . Thus, (4.8) holds.
for all . That is, . Hence, is increasing.
Now, by the conditions (4.6) and (4.7), we have for all and for all
That is, is the solution to (4.1). □
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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