#### Abstract

Common fixed point results for mappings satisfying locally contractive conditions on a closed ball in an ordered complete dislocated metric space have been established. The notion of dominated mappings is applied to approximate the unique solution of nonlinear functional equations. Our results improve several well-known conventional results.

#### 1. Introduction and Preliminaries

Let be a mapping. A point is called a fixed point of if . Fixed points results of mappings satisfying certain contractive conditions on the entire domain have been at the centre of rigorous research activity, (e.g., see [1, 2]), and it has a wide range of applications in different areas such as nonlinear and adaptive control systems, parameterize estimation problems, computing magnetostatic fields in a nonlinear medium, and convergence of recurrent networks (see [3–5]).

Recently, many results appeared related to fixed point theorems in complete metric spaces endowed with a partial ordering. Ran and Reurings [6] proved an analogue of Banach's fixed point theorem in metric space endowed with a partial order and gave applications to matrix equations. In this way, they weakened the usual contractive condition. Subsequently, Nieto and Rodríguez-López [7] extended this result in [6] for nondecreasing mappings and applied it to obtain a unique solution for a 1st-order ordinary differential equation with periodic boundary conditions. Thereafter, many works related to fixed point problems have also been considered in partially ordered metric spaces (see [8–10]).

Hitzler and Seda [11] introduce the concept of a dislocated topology and its corresponding generalized metric named as dislocated metric and have established a fixed point theorem in complete dislocated metric spaces which generalizes the Banach contraction principle. The notion of dislocated topology has useful applications in the context of logic programming semantics (see [12]). Further useful results can be seen in [13–16].

From the application point of view the situation is not yet completely satisfactory because it frequently happens that a mapping is a contraction not on the entire space but merely on a subset of . However, if is closed and a sequence in converges to some in , then by imposing a subtle restriction on the choice of , one may force the sequence to stay eventually in . In this case, one can establish the existence of a fixed point of . Arshad et al. [17] proved a significant result concerning the existence of fixed points of a mapping satisfying contractive conditions on closed ball in a complete dislocated metric space. Other results on closed ball can be seen in [18–20]. The dominated mapping [21]which satisfies the condition occurs very naturally in several practical problems. For example, if denotes the total quantity of food produced over a certain period of time and gives the quantity of food consumed over the same period in a certain town, then we must have . In this paper, we will exploit this concept. We have obtained fixed point theorems for contractive dominated self-mappings in an ordered complete dislocated metric space on a closed ball to generalize, extend, and improve a classical fixed point result in [22]. We have used weaker contractive conditions and weaker restrictions to obtain unique fixed point. Our results do not exist even yet in metric spaces. We have given an example which shows how this result can be used when the corresponding results cannot.

Consistent with [12, 21, 22], the following definitions and results will be needed in the sequel.

*Definition 1. *Let be a nonempty set and let be a function, called a dislocated metric (or simply -metric) if the following conditions hold for any : (i)if , then ,(ii),
(iii). The pair is then called a dislocated metric space.

It is clear that if , then from (i), . But if , may not be .

*Remark 2. *From (iii) of Definition 1, we deduce
for all .

Recently Sarma and Kumari [15] proved results that establish existence of a topology induced by a dislocated metric and that this topology is metrizable. This topology has as a base the family of sets , where is an open ball and for some and . Examples of the -metric spaces are given below.

*Example 3. *If , then defines a dislocated metric on .

*Example 4. *If , then defines a dislocated metric on .

*Definition 5 (see [13]). *A sequence in a -metric space is called a Cauchy sequence if given , there exists such that for all , we have .

*Definition 6 (see [13]). *A sequence in a -metric space converges with respect to if there exists such that as . In this case, is called limit of , and we write .

*Definition 7. *A -metric space is called complete if every Cauchy sequence in converges to a point in .

*Definition 8. *Let be a nonempty set. Then is called an ordered dislocated metric space if (i) is a dislocated metric on and (ii) is a partial order on .

*Definition 9. *Let be a partial ordered set. Then are called comparable if or holds.

*Definition 10 (see [21]). *Let be a partially ordered set. A self-mapping on is called dominated if for each in .

*Example 11. *Let be endowed with the usual ordering and defined by for some . Since for all , therefore is a dominated mapping.

*Definition 12. *Let be a nonempty set and . A point is called point of coincidence of and if there exists a point such that . The mappings are said to be weakly compatible if they commute at their coincidence point (i.e., whenever . If is a partial order on , then the mapping is called -dominated if for all ; see [23].

We require the following lemmas for subsequent use.

Lemma 13 (see [24]). *Let be a nonempty set and a function. Then there exists a subset such that and is one-to-one. *

Lemma 14 (see [25]). *Let be a nonempty set and the mappings have a unique point of coincidence in . If and are weakly compatible, then have a unique common fixed point. *

Theorem 15 (see [22]). *Let be a complete metric space, a mapping, , and an arbitrary point in . Suppose that there exists with
**
and . Then there exists a unique point in such that .*

#### 2. Fixed Points of Hardy-Rogers Mappings

The Banach contraction principle in the setting of dislocated metric spaces was established in [26] (see also [11]).

Theorem 16. *Let be a complete dislocated metric space, and let be a contraction. Then has a unique fixed point. *

The following example shows that there exists a contraction in a dislocated metric space that is not a contraction in a metric space.

*Example 17 (see [27, Example 1]). *Let and . Fix , and define by

For all , we have
which ensures that is a contraction, and so, by Theorem 16, has a unique fixed point in .

Now, we show that is not a contraction in the complete metric space , where for all . In fact, if for odd we choose and , we have
which is not satisfied if . Consequently, is not a contraction if .

Thus, the above example gives a strong motivation for studying fixed point results in the setting of dislocated metric spaces.

For some results of fixed point in the setting of complete dislocated metric spaces for mappings that satisfy a contractive condition of Hardy and Rogers see [28]. The following theorem is one of our main results.

Theorem 18. *Let be an ordered complete dislocated metric space, , , and two dominated mappings. Suppose that there exists with such that
**
for all comparable elements in and
**
where . If for a nonincreasing sequence in , implies that , then there exists a point in such that and .*

*Proof. *Choose a point in such that . As so , and let . Now gives , continuing this process and choosing in such that
Clearly, , and this implies that the sequence is nonincreasing.

Now, we will prove that for all , by mathematical induction. By using inequality (7), we have
and so . Assume that for some . Now, if , by using inequality (6) and Remark 2, we obtain
and hence
Similarly, if , we deduce
By the previous inequalities, we get
Thus, from inequality (13), we have
Now, using (14) and (7), we get
Thus ; that is, for all . Since for all , then it follows that
So we have
Hence the sequence is a Cauchy sequence in . Now, is complete since it is closed in . Therefore there exists a point with
By assumptions as , we have
On taking limit as in the previous inequality, by using inequality (16) and (18), we get
which implies that . Similarly, from
we can obtain . Hence and have a common fixed point in . Now,
and this implies that

*Example 19. *Let be endowed with the order if , and define as . Then is an ordered complete dislocated metric space. Let be defined by
Clearly, and are dominated mappings. For , , , and , then , , and .

Now, if , then
and so
Thus, the contractive condition does not hold on . Now if , then
Therefore, all the conditions of Theorem 18 are satisfied. Moreover, is the common fixed point of and and .

If in Theorem 18 we choose , then we obtain the following corollary.

Corollary 20. *Let be an ordered complete dislocated metric space, , , and a dominated mapping. Suppose that there exist with such that
**
for all comparable elements in and
**
where . If for a nonincreasing sequence in implies that , then there exists a point in such that and .*

In Theorem 18, the condition “for a nonincreasing sequence implies that ” is imposed to restrict condition (6) only for comparable elements. However, the following result relaxes this restriction but imposes that condition (6) holds for all elements in . In Theorem 18, the common fixed point of and may not be unique, whereas without order we can obtain a unique fixed point of and separately, which is proved in the following theorem.

Theorem 21. *Let be a complete dislocated metric space, , , and two dominated mappings. Suppose that there exist with such that
**
for all elements in and
**
where . Then there exists a unique point in such that and . Moreover, and have no fixed point other than .*

*Proof. *By following similar arguments, as we used to prove Theorem 18, we can obtain a point in such that and . Let . Then, is a fixed point of , and it may not be the fixed point of . Then,
This shows that . Thus has no fixed point other than . Similarly has no fixed point other than .

In Theorem 18, condition (7) is imposed to restrict condition (6) only for in , and Example 19 explains the utility of this restriction. However, the following result relaxes condition (7) but imposes condition (6) for all comparable elements in the whole space . Moreover, we introduce a weaker restriction to obtain a unique common fixed point.

Theorem 22. *Let be an ordered complete dislocated metric space, , and two dominated mappings. Suppose that there exist with such that
**
for all comparable elements in . If for a nonincreasing sequence in implies that , then there exists a point in such that and . Moreover, the point is unique if for any two points in there exists a point such that and . *

*Proof. *By following similar arguments of Theorem 18, we can obtain a point in such that . By Theorem 21, is unique common fixed point for all comparable elements. Now if and are not comparable and , then there exists a point such that and . Choose a point in such that . As so , and let . Now gives , continuing this process and having chosen in such that
It follows that . Following similar arguments as we have used to prove inequality (16), we have
As and , it follows that and for all . Then, for ,
and so
where and .

Similarly, we deduce
Since , we get
and hence

On taking limit as in (40), we have
Similarly,
Now by using inequality (41) and (42), we have
So, .

*Remark 23. *In Theorem 18, the common fixed point of and may not be unique. However, fixed point is unique in Theorem 18, if for every pair of elements in there exists a point such that and and , where , .

Metric version of Theorem 18 is given below.

Theorem 24. *Let be an ordered complete metric space, , , and two dominated mappings. Suppose that there exist with such that
**
for all comparable elements in and
**
where . If for a nonincreasing sequence in , implies that , then there exists a point in such that .*

Now we apply our Theorem 22 to obtain a unique common fixed point of three mappings in ordered complete dislocated metric space. To this aim the innovative technique introduced in [29].

Theorem 25. *Let be an ordered dislocated metric space and such that are -dominated mappings and . Assume that the following condition holds for such that :
**
for all comparable elements .** Suppose that for a nonincreasing sequence in , implies that . Also for any two points and in there exists a point such that , . If fX is a complete subspace of and and are weakly compatible, then , and have a unique common fixed point in . Moreover .*

*Proof. *By Lemma 13, there exists such that and is one-to-one. Now since , we define two mappings by and , respectively. Since is one-to-one on , then are well defined. As implies that and implies that , therefore and are dominated mappings. Let ; choose a point in such that As , so , and let . Now gives . Continuing this process and having chosen in such that
then for all . Note that for , where and are comparable and such that , by using inequality (42), we have

As is a complete space, so that all conditions of Theorem 22 are satisfied, we deduce that there exists a unique common fixed point of and . Also . Now or . Thus is a point of coincidence of , and . Let be another point of coincidence of , and ; then there exists such that , which implies that . A contradiction as is a unique common fixed point of and . Hence . Thus , and have a unique point of coincidence . Now since and are weakly compatible, by Lemma 14 is a unique common fixed point of , and .

Unique common fixed point result of three mappings in complete dislocated metric space is given below, which can be proved with the help of Theorem 21.

Theorem 26. *Let be a dislocated metric space and, and self-mappings on such that . Assume that the following condition holds:
**
for all elements where such that and for ,
**
where. If , is a complete subspace of and and are weakly compatible, then , and have a unique common fixed point in . Moreover, .*

In the following theorem we use Theorem 21 to establish the existence of a unique common fixed point of four mappings on closed ball in complete dislocated metric space.

Theorem 27. *Let be a dislocated metric space and, , , and self-mappings on such that . Assume that the following condition holds:
**
for all elements , such that with and for ,
**
where. If is a complete subspace of and and are weakly compatible, then , , , and have a unique common fixed point in . Moreover, .*

*Proof. *We use the technique introduced in [29]. By Lemma 13, there exists such that , , and are one-to-one. Now define the mappings by and , respectively. Since are one-to-one on and , respectively, then the mappings are well defined. As , then . Let , choose a point in such that , and let ; continuing this process and chosen in such that
Following similar arguments of Theorem 18, we deduce that . Also by inequality (51), we have
By using inequality (46), for and we have
As is a complete space, all conditions of Theorem 21 are satisfied; we deduce that there exists a unique common fixed point of and . Furthermore and have no fixed point other than . Also . Now or . Thus is a point of coincidence of and . Let be another point of coincidence of and , then there exists such that , which implies that . A contradiction as is a unique fixed point of . Hence . Thus and have a unique point of coincidence . Since are weakly compatible, by Lemma 14?? is a unique common fixed point of and . As then there exists such that . Now as implies that , thus is the point of coincidence of and . Now if , then we have , a contradiction. This implies that . As are weakly compatible, we obtain that is the unique common fixed point for and . But . Thus , , , and have a unique common fixed point .

Theorem 28. *Let be an ordered dislocated metric space and such that are -dominated mappings and . Assume that the following condition holds for such that **
for all comparable elements . Suppose that for a nonincreasing sequence in , implies that . Also for any two points and in there exists a point such that , . If is a complete subspace of , then , and have a unique point of coincidence in . Moreover .*

Theorem 29. *Let be a dislocated metric space and, , , and self-mappings on such that . Assume that the following condition holds:
**
for all elements , with such that and for ,
**
where. If is a complete subspace of , then , , , and have a unique point of coincidence in . Moreover, . *

*Remark 30. *We can obtain the metric version of all theorems which are still not present in the literature.

#### 3. Conclusion

Azam et al. [18] very recently exploited the idea of fixed points and proved a significant result concerning the existence of fixed points for fuzzy mappings on closed ball in a complete metric space. We continue their investigations, and in this paper, some common fixed point theorems for mappings under Hardy Rogers contractive conditions on a closed ball in a ordered complete dislocated metric space have been discussed. Our analysis is based on the simple observation that fixed point results can be deduced from fixed point theory of mappings on closed balls. Practically speaking, there are many situations in which the mappings are not contractive on the whole space but instead they are contractive on its subsets. We feel that this aspect of finding the fixed points via closed balls was overlooked and our paper will bring a lot of interest into this area. Furthermore, we have applied the concept of dominated mappings in the process of investigating the existence of unique fixed point of contractive mappings on closed balls in the settings of ordered dislocated metric spaces.

#### Acknowledgment

Pasquale Vetro is supported by Università degli Studi di Palermo (Local University Project ex 60%).