1. Introduction

Let be a mapping. A point is called a fixed point of if Let be an arbitrary chosen point in Define a sequence in by a simple iterative method given by where Such a sequence is called a picard iterative sequence and its convergence plays a very important role in proving existence of fixed point of a mapping A self mapping on a metric space is said to be a Banach contraction mapping if, holds for all where Recently, many results appeared in literature related to fixed point results in complete metric spaces endowed with a partial ordering. Ran and Reurings ^[21] proved an analogue of Banach’s fixed point theorem in metric space endowed with partial order and gave applications to matrix equations. Subsequently, Nieto et. al. ^[16] extended the results of ^[21] for non decreasing mappings and applied this result to obtain a unique solution for a 1st order ordinary differential equation with periodic boundary conditions. Mustafa and Sims in ^[18] introduce the notion of a generalized metric space as a generalization of the usual metric space. Mustafa and others studied fixed point theorems for mappings satisfying different contractive conditions. Further useful results can be seen in ^{[3, 12, 13, 14, 19, 20, 27, 28]}. Recently, Arshad et. al. ^[4] proved a result concerning the existence of fixed points of a mapping satisfying a contractive condition on closed ball in a complete dislocated metric space. For further results on closed ball we refer the reader to see ^{[5, 6, 7, 24, 25, 26]}. The dominated mapping ^[2] which satis.es the condition occurs very naturally in several practical problems. For example 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 have obtained fixed point results for dominated self- mappings in an ordered complete dislocated symmetric G_d-metric space on a closed ball satisfying Hardy Roger type contractive condition. In the process we extend and improve several recent and classical fixed point results. We have used weaker contractive condition and weaker restrictions to obtain unique fixed point. Our results do not exists even yet in metric spaces. An example is given to show the validity of our result.

Definition 1.1. Let be a nonempty set and let be a function satisfying the following axioms

(i) If then

(ii) for all (rectangle inequality).

Then the pair is called the dislocated quasi -metric space. It is clear that if

then from (i) But if then may not be 0: It is observed that if for all then becomes a dislocated -metric space.

Definition 1.2. If X be a set of non-negative real numbers, then defines a dislocated quasi metric on

Definition 1.3. A -metric space is called symmetric if for all

Definition 1.4. Let be a -metric space, and let be a sequence of points in a point in is said to be the limit of the sequence if and one says that sequence is -convergent to Thus, if in a -metric space then for any there exist such that for all

Definition 1.5. Let be a metric space. A sequence is called -Cauchy sequence if, for each there exists a positive integer such that for all i.e. if as

Definition 1.6. A -metric space is said to be -complete if every -Cauchy sequence in is -convergent in

Lemma 1.7. Let be a -metric space, then the following are equivalent:

(i) is Gd convergent to

(ii) as

(iii) as

(iv) as

Definition 1.8. Let be a -metric space then for the -ball with centre and radius is,

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

Definition 1.10. ^[2] Let be a partially ordered set. A self mapping on is called dominated if for each in

Example 1.11. ^[2] Let be endowed with usual ordering and be defined by for some Since for all therefore is a dominated map.

2. Main Result

Theorem 2.1: Let be an ordered complete dislocated symmetric metric space, and be a dominated mapping. Suppose there exists and such that and for all comparable elements and in

(2.1)

and

(2.2)

where

If for a nonincreasing sequence in implies that then there exists a point in such that and

Proof. Consider a picard sequence with initial guess . As for all Now by using inequality (2.2) we have,

it implies that Now by using inequality (2.1) we have,

(2.3)

Now by using (2.2) and (2.3) we get,

it implies that Let for some Now by using inequality (2.1) we have,

(2.4)

Similarly we get,

(2.5)

By using (2.3), (2.4) and (2.5) we get,

which further implies Hence by induction for all Using inequality (2.5) we get

Hence the sequence is Cauchy sequence in Therefore, there exists a point such that

(2.6)

Now,

By assumption therefore

Taking on both sides and by using (2.6) we have,

Also,

Hence,

If we take in inequality then we obtain the following corollary.

Corollary 2.2. Let be an ordered complete dislocated symmetric metric space, and be a dominated mapping and be any arbitrary point in Suppose there exists with,

and

If for a nonincreasing sequence implies that Then there exists a point in such that and Moreover if for any three points and in such that there exists a point such that and that is, every three of elements in has a lower bound, then the point is unique.

Similarly if we take in inequality then we obtain the following corollary.

Corollary 2.3. Let be an ordered complete dislocated symmetric -metric space be a mapping and be an arbitrary point in Suppose there exists with

for all elements and

where If for nonincreasing sequence implies that Then there exists a point in such that and

If we take in inequality then we obtain the following Corollary.

Corollary 2.4. Let be an ordered complete dislocated symmetric metric space, , and be a dominated mapping. Suppose there exists and for all elements and in

where and and

If for a nonincreasing sequence in implies that there exists a point in such that and

Example 2.5. Let be endowed with usual order and be a complete dislocated symmetric metric space defined by,

Then is a complete dislocated symmetric metric space. Let be defined by,

Clearly, is a dominated mapping. Take and where and

Also if and We assume that then

So the contractive condition does not holds in Now if and then,

Hence it satisfies all the requirements of Corollary 2.4 and 0 is the fixed point of S.

Theorem 2.6: Let be an ordered complete dislocated symmetric metric space, and be a dominated map and be an arbitrary point in Suppose there exists with,

(2.7)

for all comparable elements and in If, for a nonincreasing sequence in implies that then there exists a point in such that and Moreover, is unique, for every triple of elements and in if there exist a point such that and

Proof. From the proof of Theorem 2.1, we can find such that Now if and are not comparable then there exists a point which is a lower bound of both and that is and As then by inequality 2.7, we have

Which implies that,

(2.8)

As then,

(2.9)

Now let,

(2.10)

Using inequality (2.10) we get

Continuing in this way we get

(2.11)

On taking limit and by using (2.8) we get,

Also

Similarly

Now,

Also

Hence

Competing Interests

The authors declare that they have no competing interests.

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