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The Scientific World Journal

Volume 2013 (2013), Article ID 194897, 8 pages

http://dx.doi.org/10.1155/2013/194897

## Fixed Point Results of Locally Contractive Mappings in Ordered Quasi-Partial Metric Spaces

^{1}Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan^{2}Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan

Received 30 April 2013; Accepted 18 June 2013

Academic Editors: A. Bekir, A. Ibeas, and A. M. Peralta

Copyright © 2013 Abdullah Shoaib et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Fixed point results for a self-map satisfying locally contractive conditions on a closed ball in an ordered 0-complete quasi-partial metric space have been established. Instead of monotone mapping, the notion of dominated mappings is applied. We have used weaker metric, weaker contractive conditions, and weaker restrictions to obtain unique fixed points. An example is given which shows that how this result can be used when the corresponding results cannot. Our results generalize, extend, and improve several well-known conventional results.

#### 1. Introduction and Preliminaries

Fixed points results of mappings satisfying certain contractive conditions on the entire domain have been at the centre of vigorous research activity, (see [1–3]) and it has a wide range of applications in different areas such as nonlinear and adaptive control systems, parameterize estimation problems, fractal image decoding, computing magnetostatic fields in a nonlinear medium, and convergence of recurrent networks (see [4–6]).

Recently, many results appeared related to fixed point theorem in complete metric spaces endowed with a partial ordering in the literature. Ran and Reurings [7] 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 contraction condition. Subsequently, Nieto and Rodríguez-López [8] extended the result in [7] 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 [7–11]).

On the other hand notion of a partial metric space was introduced by Matthews in [12]. In partial metric spaces, the distance of a point from itself may not be zero. Partial metric spaces have applications in theoretical computer science (see [13]). Altun and Erduran [14] and Paesano and Vetro [15] used the idea of partial metric space and partial order and gave some fixed point theorems for contractive condition on ordered partial metric spaces. Recently, Karapınar et al. [16] introduced the concept of quasi-partial metric space. Romaguera [17] has given the idea of -complete partial metric space. Nashine et al. [18] used this concept and proved some classical results.

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. [19] proved a significant result concerning the existence of fixed points of a mapping satisfying a contractive conditions on closed ball in a complete dislocated metric space. Other results on closed ball can be seen in [20, 21]. In this paper we have obtained fixed point theorems for dominated self-mappings in a -complete ordered quasi-partial metric space on closed ball under several contractive conditions to generalize, extend and improve some classical fixed point results. We have used weaker contractive condition and weaker restrictions to obtain unique fixed point. Our results do not exist even yet in metric spaces. An example shows how this result can be used when the corresponding results cannot.

Consistent with [16, 22, 23] the following definitions and results will be needed in the sequel.

*Definition 1 (see [16]). *A quasi-partial metric is a function satisfying the following:(i)if , then (equality);(ii) (small self-distances);(iii) (small self-distances);(iv) (triangle inequality), for all .The pair is called a quasi-partial metric space.

Note that if for all , then becomes a partial metric space . Moreover if for all , then and become a quasimetric space and a metric space, respectively. Also , is a partial metric on . The function defined by is a (usual) metric on . The ball , where , is a closed ball in quasi-partial metric space, for some and .

*Definition 2 (see [16]). *Let be a quasi-partial metric. Then, we have the following; (a)A sequence in converges to a point if and only if .(b)A sequence in is called a Cauchy sequence if the limits and exist (and are finite).(c)The space is said to be complete if every Cauchy sequence in converges to a point such that .

Lemma 3 (see [16]). *Let be a quasi-partial metric space, let be the corresponding partial metric space, and let be the corresponding metric space. These statements are equivalent. (i) The sequence is Cauchy in . (ii) The sequence is Cauchy in . (iii) The sequence is Cauchy in . These statements are also equivalent. (i) is complete. (ii) is complete. (iii) is complete.*

*Definition 4. *Let be a partial ordered set. Then are called comparable if or holds.

*Definition 5 (see [22]). *Let be a partially ordered set. A self-mapping on is called dominated if for each in .

*Example 6 (see [22]). *Let be endowed with the usual ordering and defined by for some . Since for all , therefore is a dominated map.

Theorem 7 (see [23]). *Let be a complete metric space, a mapping, and an arbitrary point in . Suppose there exists with
**
and . Then there exists a unique point in such that .*

#### 2. Fixed Points of Dominated Mapping

In this section we introduce the notion of a -complete quasi-partial metric spaces. We also prove some results in these spaces.

*Definition 8. *Let be a quasi-partial metric space.(a)A sequence in is called -Cauchy if or .(b)The space is called -complete if every -Cauchy sequence in converges to a point such that .

It is easy to see that every -Cauchy sequence in is Cauchy in and if is complete, then it is -complete but the converse assertions do not hold. For example, the space with is a -complete quasi-partial metric space but it is not complete (since and is not complete).

Moreover, if is complete quasimetric space which is also quasi-partial, then it is -complete, quasi-partial metric space.

*Definition 9. *Let be a nonempty set. Then is called an ordered quasi-partial metric space if (i) is a quasi-partial metric on and (ii) is a partial order on .

Theorem 10. *Let be a -complete ordered quasi-partial metric space, a dominated map, and an arbitrary point in . Suppose that there exists such that
**
And
**
If, for a nonincreasing sequence in , implies that , then there exists a point in such that and . Moreover, is unique, if for every pair of elements , in there exists a point such that and .*

*Proof. *Consider a Picard sequence with initial guess , as for all . We will prove that for all by mathematical induction. By using inequality (3), we have
Therefore . Now let for some . As , so using inequality (2), we obtain
which implies that,
Now
Thus Hence for all . Also for all . It implies that
It follows that
Notice that the sequence is a -Cauchy sequence in . As is closed so is -complete. Therefore there exists a point with
Now,
On taking limit as and using the fact that , when , we have,
Then by inequality (10), we have
Similarly,
and hence .

*Uniqueness*. Let be another point in such that . If and are comparable, then
This shows that . Now if and are not comparable then there exists a point which is lower bound of both and , that is, and . Moreover by assumption . Now we will prove that :
It follows that . Now let
It follows that . Hence for all . Nowas is dominated, it follows that and for all which further implies and for all as and for all :
Hence .

*Example 11. *Let be endowed with order if and let be the -complete ordered quasi-partial metric on defined by . Define
Clearly, is dominated mapping. Take ,; then ; we have , with
Also if , then,
So the contractive condition does not hold on in each case.

Now if then, Therefore, all the conditions of Theorem 13 are satisfied. Moreover, is the unique fixed point of and .

In Theorem 10, the condition (3) is imposed to restrict the condition (2) only for , in and Example 11 explains the utility of this restriction. However, the following result relaxes the condition (3) but imposes the condition (2) for all comparable elements in the whole space .

Theorem 12. *Let be a -complete ordered quasi-partial metric space, a dominated map, and an arbitrary point in . Suppose there exists with
**
If, for a nonincreasing sequence in , implies that , then there exists a point in such that and . Moreover, is unique, if for every pair of elements , in , there exists a point such that and .*

In Theorem 10, the existence of a lower bound and for a nonincreasing sequence in , implies that are imposed to restrict the condition (2) only for comparable elements. However, the following result relaxes these conditions but imposes the condition (2) for all elements in .

Theorem 13. *Let be a -complete partial metric space, a map, and an arbitrary point in . Suppose there exists with
**
Then there exists a unique point in such that . Further .*

Metric version of Theorem 10 is given below.

Theorem 14. *Let be a complete ordered metric space, a dominated map, and an arbitrary point in . Suppose that there exists such that
**
If, for a nonincreasing sequence in , implies that , then there exists a point in such that and . Moreover, is unique, if for every pair of elements , in there exists a point such that and .*

Theorem 15. *Let be a -complete ordered quasi-partial metric space, a dominated map, and an arbitrary point in . Suppose that there exists 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 . Moreover, is unique, if for any two points in there exists a point such that and , and
*

*Proof. *Consider a Picard sequence with initial guess , as for all . We will prove that for all by mathematical induction. By using inequality (28), we have
Therefore . Now let for some . As , so using inequality (27), we obtain
which implies that
Now,
Thus . Hence for all . Also for all . It implies that
It follows that
Notice that the sequence is a -Cauchy sequence in . As is closed, so is -complete. Therefore there exists a point with
Now,
On taking limit as and using the fact that , when , we have,
Similarly,
and hence . Now,
This implies that

*Uniqueness.* Let be another point in such that . If and are comparable, then
Using the fact that we have . Similarly, . Now if and are not comparable to then there exists a point which is a lower bound of both and . Moreover by assumption . Now by using inequality (27), we have
It follows that . Now, we will prove that , by using mathematical induction to apply inequality (27). Let for some . As ; then,
which implies that
Now,
Now,
It follows that and hence . Now inequality (46) can be written as
Now,
Using the fact that we have . Similarly, . Hence .

*In Theorem 15, the conditions (29), the existence of a lower bound and for a nonincreasing sequence in , implies that are imposed to restrict the condition (27) only for comparable elements. However, the following result relaxes these conditions but imposes the condition (27) for all elements in .*

*Theorem 16. Let be a complete dislocated quasimetric space, a map, and an arbitrary point in . Suppose there exists with
for all elements in and
where . Then there exists a unique point in such that and .*

*In Theorem 10, the conditions (28) and (29) are imposed to restrict the condition (27) only for comparable elements. However, the following result relaxes these conditions but imposes the condition (27) for all elements in .*

*Theorem 17. Let be a -complete ordered quasi-partial metric space and let be a dominated map. Suppose that there exists 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, is unique, if for any two points in there exists a point such that and .*

*Theorem 18. Let be a -complete ordered quasi-partial metric space, a dominated map, and an arbitrary point in . Suppose that there exists 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 . Further .

*Proof. *Consider a Picard sequence with initial guess . As for all . We will prove that for all by mathematical induction. By using inequality (55), we have
Therefore . Now let for some . As , so using inequality (54), we obtain
which implies that
Now
Thus Hence for all . Also for all . It implies that
It follows that
Notice that the sequence is a -Cauchy sequence in . As is closed so is -complete. Therefore there exists a point with
Now,
On taking limit as and using the fact that , when , we have
Similarly,
and hence .

*Remark 19. *We can obtain the partial metric, quasi-metric, and metric version of all theorems which are still not present in the literature.

*Conflict of Interests*

*The authors declare that they have no competing interests.*

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