Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered dislocated metric space

Author Affiliations

1 Department of Mathematics, International Islamic University, Islamabad, 44000, Pakistan

2 Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan

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Fixed Point Theory and Applications 2013, 2013:115  doi:10.1186/1687-1812-2013-115

 Received: 5 December 2012 Accepted: 15 April 2013 Published: 29 April 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

MSC: 46S40, 47H10, 54H25.

Keywords:
common fixed point; contractive mapping; closed ball; dominated mapping; dislocated metric space

1 Introduction and preliminaries

Let be a mapping. A point is called a fixed point of T if . Let be an arbitrarily chosen point in X. Define a sequence in X by a simple iterative method given by

Such a sequence is called a Picard iterative sequence, and its convergence plays a very important role in proving the existence of a fixed point of a mapping T. A self-mapping T on a metric space X is said to be a Banach contraction mapping if

holds for all , where .

Fixed points results of mappings satisfying a certain contractive condition on the entire domain have been at the center of rigorous research activity (for example, see [1-12]) and they have a wide range of applications in different areas such as nonlinear and adaptive control systems, parameter estimation problems, computing magnetostatic fields in a nonlinear medium and convergence of recurrent networks (see [13-15]).

From the application point of view, the situation is not yet completely satisfactory because it frequently happens that a mapping T is a contraction not on the entire space X but merely on a subset Y of X. However, if Y is closed and a Picard iterative sequence in X converges to some x in X, then by imposing a subtle restriction on the choice of , one may force the Picard iterative sequence to stay eventually in Y. In this case, the closedness of Y coupled with some suitable contractive condition establishes the existence of a fixed point of T. Azam et al.[16] proved a significant result concerning the existence of fixed points of a mapping satisfying contractive conditions on a closed ball of a complete metric space. Recently, many results related to the fixed point theorem in complete metric spaces endowed with a partial ordering ⪯ appeared in literature. Ran and Reurings [17] proved an analogue of Banach’s fixed point theorem in a metric space endowed with a partial order and gave applications to matrix equations. In this way, they weakened the usual contractive condition. Subsequently, Nieto et al.[18] extended this result in [17] for non-decreasing mappings and applied it to obtain a unique solution for a first-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 [17-23]). Indeed, they all deal with monotone mappings (either order-preserving or order-reversing) such that for some , either or , where f is a self-map on a metric space. To obtain a unique solution, they used an additional restriction that each pair of elements has a lower bound and an upper bound. We have not used these conditions in our results. In this paper we introduce a new condition of partial order.

On the other hand, the notion of a partial metric space was introduced by Matthews in [24]. In partial metric spaces, the distance of a point from itself may not be zero. He also proved a partial metric version of the Banach fixed point theorem. Karapınar et al.[25] have proved a common fixed point in partial metric spaces. Partial metric spaces have applications in theoretical computer science (see [26]). Altun et al.[20], Aydi [27], Samet et al.[28] and Paesano et al.[29] used the idea of a partial metric space and partial order and gave some fixed point theorems for the contractive condition on ordered partial metric spaces. Further useful results can be seen in [28]. To generalize a partial metric, Hitzler and Seda [30] introduced the concept of dislocated topologies and its corresponding generalized metric, named a dislocated metric, and established a fixed point theorem in complete dislocated metric spaces to generalize the celebrated Banach contraction principle. The notion of dislocated topologies has useful applications in the context of logic programming semantics (see [31]). Further useful results can be seen in [32-35]. The dominated mapping, which satisfies the condition , occurs very naturally in several practical problems. For example, if x 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 exploit this concept for contractive mappings [36] to generalize, extend and improve some classical fixed point results for two, three and four mappings in the framework of an ordered complete dislocated metric space X. Our results not only extend some primary theorems to ordered dislocated metric spaces, but also restrict the contractive conditions on a closed ball only. The concept of a dominated mapping has been applied to approximate the unique solution of nonlinear functional equations.

Consistent with [30,32,34] and [35], the following definitions and results will be needed in the sequel.

Definition 1.1 Let X 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 0.

Recently Sarma and Kumari [34] proved the results that establish the existence of a topology induced by a dislocated metric and the fact that this topology is metrizable. This topology has as a base the family of sets , where is an open ball and for some and . Also, is a closed ball.

Also, Harandi [37] defined the concept of a metric-like space which is similar to a dislocated metric space. Each metric-like σ on X generates a topology on X whose base is the family of open σ-balls

Definition 1.2 Let , where X is a nonempty set. p is said to be a partial metric on X if for any :

(P1) if and only if ,

(P2) ,

(P3) ,

(P4) .

The pair is then called a partial metric space.

Each partial metric p on X induces a topology p on X which has as a base the family of open balls , where for all and .

It is clear that any partial metric is a -metric. A basic example of a partial metric space is the pair , where for all . It is also a -metric. An example of a -metric space which is not a partial metric is given below.

Example 1.3 If , then defines a dislocated metric on X. Note that this metric is not a partial metric as (P2) is not satisfied.

From the examples and definitions, it is clear that any partial metric is a -metric, whereas a -metric may not be a partial metric. We also remark that for those -metrics which are also partial metrics, we have . Also, for any -metric, . Thus it is better to find a fixed point on a closed ball defined by Hitzler in a -metric because we restrict ourselves to applying the contractive condition on the smallest closed ball. In this way, we also weaken the contractive condition.

Definition 1.4[30]

A sequence in a -metric space is called a Cauchy sequence if given , there corresponds such that for all , we have or .

Definition 1.5[30]

A sequence in a -metric space converges with respect to if there exists such that as . In this case, x is called the limit of , and we write .

Definition 1.6[30]

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

In Harandi’s sense, a sequence in the metric-like space converges to a point if and only if . The sequence of elements of X is called σ-Cauchy if the limit exists and is finite. The metric-like space is called complete if for each σ-Cauchy sequence , there is some such that

Romaguera [38] has given the idea of a 0-Cauchy sequence and a 0-complete partial metric space. Using his idea, we can observe the following:

(a) Every Cauchy sequence with respect to Hitzler is a Cauchy sequence with respect to Harandi.

(b) Every complete metric space with respect to Harandi is complete with respect to Hitzler. The following example shows that the converse assertions of (a) and (b) do not hold.

Example 1.7 Let and let be defined by . Note that is a Cauchy sequence with respect to Harandi, but it is not a Cauchy sequence with respect to Hitzler. Also, every Cauchy sequence (with respect to Hitzler) in X converges to a point ‘0’ in X. Hence X is complete with respect to Hitzler, but X is not complete with respect to Harandi as .

Definition 1.8 Let X be a nonempty set. Then is called an ordered dislocated metric space if (i) is a dislocated metric on X and (ii) ⪯ is a partial order on X.

Definition 1.9 Let be a partial ordered set. Then are called comparable if or holds.

Definition 1.10[39]

Let be a partially ordered set. A self-mapping f on X is called dominated if for each x in X.

Example 1.11[39]

Let be endowed with the usual ordering and be defined by for some . Since for all , therefore f is a dominated map.

Definition 1.12 Let X be a nonempty set and . A point is called a point of coincidence of T and f if there exists a point such that . The mappings T, f are said to be weakly compatible if they commute at their coincidence point (i.e., whenever ).

For , we denote by the set of all limit points of A and closure of A in X. We state without proof the following simple facts due to [34].

Lemma 1.13A subset ofAof a dislocated metric space is closed if and only if.

Lemma 1.14The topology induced by a dislocated metric is a Hausdorff topology.

Lemma 1.15Every closed ball in a complete dislocated metric space is complete.

We also need the following results for subsequent use.

Lemma 1.16[40]

LetXbe a nonempty set and letbe a function. Then there exists a subsetsuch thatandis one-to-one.

Lemma 1.17[1]

LetXbe a nonempty set and let the mappingshave a unique point of coincidencevinX. Ifandare weakly compatible, thenS, T, fhave a unique common fixed point.

Theorem 1.18 [[36], p.303]

Letbe a complete metric space, letbe a mapping, letandbe an arbitrary point inX. Suppose that there existswith

and. Then there exists a unique pointinsuch that.

2 Fixed points of contractive mappings

Theorem 2.1Letbe an ordered complete dislocated metric space, letbe dominated maps and letbe an arbitrary point inX. Suppose that forand for, we have

(2.1)

and

(2.2)

If for a non-increasing sequencein, implies that, then there existssuch thatand. Also if, for any two pointsx, yin, there exists a pointsuch thatand, that is, every pair of elements has a lower bound, thenis a unique common fixed point in.

Proof Choose a point in X such that . As , so and let . Now gives . Continuing this process, we construct a sequence of points in X such that

First we show that for all . Using inequality (2.2), we have

It follows that

Let for some . If , then , where . So, using inequality (2.1), we obtain

(2.3)

If , then as and (). We obtain

(2.4)

Thus from inequalities (2.3) and (2.4), we have

(2.5)

Now

Thus . Hence for all . It implies that

(2.6)

It implies that

Notice that the sequence is a Cauchy sequence in . Therefore there exists a point with . Also,

(2.7)

Now,

On taking limit as and using the fact that when , we have

By equation (2.7), we obtain

and hence . Similarly, by using

we can show that . Hence S and T have a common fixed point in . Now,

This implies that

For uniqueness, assume that y is another fixed point of T and S in . If and y are comparable, then

This shows that . Now if and y are not comparable, then there exists a point such that and . Choose a point in X such that . As , so and let . Now gives . Continuing this process and having chosen in X such that

we obtain that . As and , it follows that and for all . We will prove that for all by using mathematical induction. For ,

It follows that . Let for some . Note that if j is odd, then

and if j is even, then

Now

which implies that

Thus . Hence for all . As and , it follows that , , and for all as and for all . If n is odd, then

So, . Similarly, we can show that if n is even. Hence is a unique common fixed point of T and S in . □

Theorem 2.1 extends Theorem 1.18 to ordered complete dislocated metric spaces.

Example 2.2 Let be endowed with the order if , . Let be defined by

and

Clearly, S and T are dominated mappings. Let be defined by . Then it is easy to prove that is a complete dislocated metric space. Let , , then

with ,

Also, for all comparable elements such that and , we have

So, the contractive condition does not hold on . Now if , then

Therefore, all the conditions of Theorem 2.1 are satisfied. Moreover, is the common fixed point of S and T. Also, note that for any metric d on , the respective condition does not hold on since

Moreover, is not complete for any metric d on .

Remark 2.3 If we impose a Banach-type contractive condition for a pair of mappings on a metric space , that is,

then it follows that for all (that is, S and T are equal). Therefore the above condition fails to find common fixed points of S and T. However, the same condition in a dislocated metric space does not assert that , which is seen in Example 2.2. Hence Theorem 2.1 cannot be obtained from a metric fixed point theorem.

Theorem 2.4Letbe an ordered complete dislocated metric space, letbe a dominated map and letbe an arbitrary point inX. Suppose that there existswith

and

If, for a non-increasing sequencein, implies that, and also, for any two pointsx, yin, there exists a pointsuch that every pair of elements has a lower bound, then there exists a unique fixed pointofSin. Further, .

Proof By following similar arguments to those we have used to prove Theorem 2.1, one can easily prove the existence of a unique fixed point of S in . □

In Theorem 2.1, condition (2.2) is imposed to restrict condition (2.1) only for x, y in and Example 2.2 explains the utility of this restriction. However, the following result relaxes condition (2.2) but imposes condition (2.1) for all comparable elements in the whole space X.

Theorem 2.5Letbe an ordered complete dislocated metric space, letbe the dominated map and letbe an arbitrary point inX. Suppose that forand for, we have

Also, if for a non-increasing sequenceinX, implies that, and for any two pointsx, yinX, there exists a pointsuch thatand, then there exists a unique pointinXsuch that. Further, .

In Theorem 2.1, the condition ‘for a non-increasing sequence, implies that ’ and the existence of z or a lower bound is imposed to restrict condition (2.1) only for comparable elements. However, the following result relaxes these restrictions but imposes condition (2.1) for all elements in . In Theorem 2.1, it may happen that S has more fixed points, but these fixed points of S are not the fixed points of T, because a common fixed point of S and T is unique, whereas without order we can obtain a unique fixed point of S and T separately, which is proved in the following theorem.

Theorem 2.6Letbe a complete dislocated metric space, letbe self-maps and letbe an arbitrary point inX. Suppose that forand for, we have

and

Then there exists a uniquesuch thatand. Further, SandThave no fixed point other than.

Proof By Theorem 2.1, . Let y be another point such that . Then

This shows that . Thus T has no fixed point other than . Similarly, S has no fixed point other than . □

Now we apply our Theorem 2.1 to obtain a unique common fixed point of three mappings on a closed ball in an ordered complete dislocated metric space.

Theorem 2.7Letbe an ordered dislocated metric space, letS, Tbe self-mappings and letfbe a dominated mapping onXsuch that, , , and letbe an arbitrary point inX. Suppose that forand for, we have

(2.8)

for all comparable elements; and

(2.9)

If for a non-increasing sequence, implies that, and for any two pointszandxin, there exists a pointsuch thatand, that is, every pair of elements inhas a lower bound in; iffXis a complete subspace ofXandandare weakly compatible, thenS, Tandfhave a unique common fixed pointfzin. Also, .

Proof By Lemma 1.16, there exists such that and is one-to-one. Now, since , we define two mappings by and , respectively. Since f is one-to-one on E, then g, h are well defined. As implies that and implies that , therefore g and h are dominated maps. Now . Then . Let , choose a point in fX such that . As , so and let . Now gives . Continuing this process and having chosen in fX such that

then for all . Following similar arguments of Theorem 2.1, . Also, by inequality (2.9),

Note that for , where fx, fy are comparable. Then by using inequality (2.8), we have

As fX is a complete space, all the conditions of Theorem 2.1 are satisfied, we deduce that there exists a unique common fixed point of g and h. Also, . Now or . Thus fz is the point of coincidence of S, T and f. Let be another point of coincidence of f, S and T, then there exists such that , which implies that , a contradiction as is a unique common fixed point of g and h. Hence . Thus S, T and f have a unique point of coincidence . Now, since and are weakly compatible, by Lemma 1.17 fz is a unique common fixed point of S, T and f. □

In a similar way, we can apply our Theorems 2.5 and 2.6 to obtain a unique common fixed point of three mappings in an ordered complete dislocated metric space and a unique common fixed point of three mappings on a closed ball in a complete dislocated metric space, respectively.

In the following theorem, we use Theorem 2.6 to establish the existence of a unique common fixed point of four mappings on a closed ball in a complete dislocated metric space. One cannot prove the following theorem for an ordered dislocated metric space in a way similar to that of Theorem 2.7. In order to prove the unique common fixed point of four mappings on a closed ball in an ordered dislocated metric space, we should prove that S and T have no fixed point other than in Theorem 2.1.

Theorem 2.8Letbe a dislocated metric space and letS, T, gandfbe self-mappings onXsuch that. Assume that for, an arbitrary point inX, and forand for, the following conditions hold:

(2.10)

for all elements; and

(2.11)

IffXis a complete subspace ofX, then there existssuch that. Also, ifandare weakly compatible, thenS, T, fandghave a unique common fixed pointfzin.

Proof By Lemma 1.16, there exist such that , , are one-to-one. Now define the mappings by and , respectively. Since f, g are one-to-one on and , respectively, then the mappings A, B are well defined. As fX is a complete space, all the conditions of Theorem 2.6 are satisfied, we deduce that there exists a unique common fixed point of A and B. Further, A and B have no fixed point other than fz. Also, . Now or . Thus fz is a point of coincidence of f and S. Let be another point of coincidence of S and f, then there exists such that , which implies that , a contradiction as is a unique fixed point of A. Hence . Thus S and f have a unique point of coincidence . Since are weakly compatible, by Lemma 1.17 fz is a unique common fixed point of S and f. As , then there exists such that . Now, as , thus gv is the point of coincidence of T and g. Now, if , a contradiction. This implies that . As are weakly compatible, we obtain gv, a unique common fixed point for T and g. But . Thus S, T, g and f have a unique common fixed point . □

Corollary 2.9Letbe an ordered dislocated metric space, letS, Tbe self-mappings and letfbe a dominated mapping onXsuch that, , , and letbe an arbitrary point inX. Suppose that forand for, we have

for all comparable elements; and

If for a non-increasing sequence, implies that, and for any two pointszandxin, there exists a pointsuch thatand; iffXis a complete subspace ofX, thenS, Tandfhave a unique point of coincidence. Also, .

In a similar way, we can obtain a coincidence point result of four mappings as a corollary of Theorem 2.8.

A partial metric version of Theorem 2.1 is given below.

Theorem 2.10Letbe an ordered complete partial metric space, letbe dominated maps and letbe an arbitrary point inX. Suppose that forand for,

and

Then there existssuch that. Also, if for a non-increasing sequencein, implies that, and for any two pointsx, yin, there exists a pointsuch thatand, then there exists a unique pointinsuch that.

A partial metric version of Theorem 2.7 is given below.

Theorem 2.11Letbe an ordered partial metric space, letS, Tbe self-mappings and letfbe a dominated mapping onXsuch thatand. Assume that for, an arbitrary point inX, and forand for, the following conditions hold:

for all comparable elements; and

If for a non-increasing sequence, implies that, also for any two pointszandxin, there exists a pointsuch thatand; iffXis complete subspace ofXandandare weakly compatible, thenS, Tandfhave a unique common fixed pointfzin. Also, .

Remark 2.12 We can obtain a partial metric version as well as a metric version of other theorems in a similar way.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

IB gave the idea. MA and AS wrote the initial draft. IB and MA finalized the manuscript. Correspondence was mainly done by IB. All authors read and approved the final manuscript.

Acknowledgements

The authors sincerely thank the learned referee for a careful reading and thoughtful comments. The present version of the paper owes much to the precise and kind remarks of three anonymous referees.

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