Abstract

In this article, we find fixed point results for a pairs of mappings satisfying the locally contractive conditions on a closed ball for a new generalized rational type contraction in complete dislocated metric space. Example has been constructed to demonstrate the novelty of our results.

1. Introduction and Preliminaries

Fixed point theory plays a fundamental role in functional analysis. Banach ⁴ proved significant result in the manipulation of contractive mappings in analysis. Many authors have presented a large number of theorems related to fixed point theorems. These authors have made different generalizations of the Banach’s result. After that, huge number of fixed point theorems have been established by various authors and they made different generalizations of the Banach’s result. Let be a mapping. A point is called a fixed point of if In literature, there are many results about the fixed point of mappings that are contractive over all the theories. It is possible that is not a contraction but is a contraction, where is a closed ball in It is possible for one to get fixed point results for mappings under different condition. It has been shown by. Hussain et al. ⁸ related to the results concerning the presence of fixed points of a mapping that ful.lls the conditions on closed ball (see also ^{1, 2, 3, 5, 15, 16, 17}). The idea of dislocated is proved to be useful for logic programming semantics (see ⁷). Dislocated metric space (metric-like space) (see ^{10, 13}) is an induction of partial metric space (see ¹¹).

In 1994, Matthews ¹¹ proposed the idea of partial metric spaces and got numerous fixed point results. In particular, he brought about the short connection between partial metric spaces and quasi-metric spaces, and showed a partial metric generalization of Banach’s contraction mapping theorem. We can observe many findings about fixed point results on cone metric spaces (see ^{9, 12, 14}). The idea of dislocated topologies presented by Hitzler and Seda ⁷ is called a generalized metric a dislocated metric. They have presented also fixed point results in complete dislocated metric spaces to draw conclusion from the well known Banach contraction principle. The aim of this paper is to strengthen the conclusion of fixed point results for a pair of mappings proving the contractive conditions on subspace for a new generalized rational type contraction in complete dislocated metric space.

Definition 1.1. ⁷ Let be a nonempty set. A mapping is called a dislocated metric (or simply -metric) if the following conditions hold, for any

(1) If then

(2) ;

(3) .

Then is called a dislocated metric on and the pair is called dislocated metric space or metric space. It is clear that if then from (i), But if may not be

Example 1.2. If then defines a dislocated metric on

Definition 1.3. ⁷ A sequence in -metric space is called Cauchy sequence if for given there corresponds such that for all we have

Definition 1.4. ⁷ A sequence in -metric space converges with respect to if there exists such that as In this case, is called limit of and we write

Every metric space is a dislocated metric, but the converse may not be true.

Example 1.5. Let and defined by for all

Note that dl is a dislocated metric, but not a metric since

Definition 1.6. Let be a dislocated metric space then for the closed ball with centre and radius is,

Definition 1.7. ⁷ A -metric space is called complete if every Cauchy sequence in converges to a point in

Example 1.8. Let and Then the pair is dislocated metric space, but it is not a metric space.

Definition 1.9. ⁷ Let be a dislocated metric space. A mapping is called contraction if there exists such that

Then has a unique fixed point in

The purpose of this paper is to prove common fixed point theorems for generalized rational contractions on dislocated metric spaces. We provide an example to validate our results.

2. The Results

In this section, we will prove the existance of common fixed points of two self mappings involving rational expressions in complete dislocated metric space.

Theorem 2.1. Let be a complete dislocated metric space and be any arbitrary point in let the mappings satisfy:

(2.1)

for all and with and

(2.2)

where and are non negative reals with Then is a non increasing sequence in for all and Then and have common fixed point in

Proof: Let be an arbitrary point in and define and such that

Then

As (owing to triangular inequality),

Where

Hence

Where Now,

This implies that Similarly, by repeating the same process for

we get

Consequently, for some If where we get

(2.3)

Similarly, if where we have

(2.4)

Now, (2.3) implies that

(2.5)

Also, (2.4) implies that

(2.6)

Now, by combining (2.5) and (2.6), we have

(2.7)

Now,

Thus, Hence for all therefore is a sequence in Now, the inequality (2.7) can be written as

(2.8)

Hence for any

And

This implies that is a Cauchy sequence in Since is closed and complete, there exists a point such that It follows that otherwise and we would then have

This implies that

which on making gives rise a contradiction so that Similarly, one can show that

Example 2.2. Let be a dislocated metric space defined by

Let defined by

And defined by

Take then We have with

and

Also if then

Taking and then clearly and we have

Now,

Hence, clearly whole space does not satisfy the contractive condition. Also if then

Hence, all contractive conditions of theorem 2.1 are satisfied.

Corollary 2.3. Let be a complete dislocated metric space and be any arbitrary point in X let the mappings satisfy:

for all with and

where and, are non negative reals with If, is a non increasing sequence in for all and Then S and T have common fixed point h in

Proof: By putting in Theorem 2.1, we get the required result:

Theorem 2.4. Let be a complete dislocated metric space and be any arbitrary point in let the mappings satisfy:

(2.9)

for all and with

(2.10)

where and are nonnegative reals with If, is a non increasing sequence in for all and Then S and T have common fixed point u in

Proof: Let be an arbitrary point in X and define and such that

Then

This implies that

Where Now

this implies that similarly,

This implies that

Consequently, for some If where we get

(2.11)

Similarly, if where we have

(2.12)

Now, (2.11) implies that

(2.13)

Also, (2.12) implies that

(2.14)

Now, by combining (2.13) and (2.14), we have

(2.15)

Now,

Thus Hence for all therefore is a sequence in Now, the inequality (2.15) can be written as

(2.16)

To prove that is a Cauchy sequence, we have for any m > n,

This implies that is a Cauchy sequence in Since is closed and complete, there exists a point such that and suppose Therefore we have

letting and we get,

which implies that It follows similarly that Now, we show that and have a unique common fixed point. For this, assume that in is a second common fixed point of and Then

This implies that,

This implies that completing the proof of the theorem.

Example 2.5. Let be a dislocated metric space defined by

Let defined by

And defined by

Take then We have with

and

Also if then

Then and then clearly and we have

Hence, clearly whole space does not satisfy the contractive condition. Also if then

Hence, the contractive conditions of Theorem (2.4) are satisfied.

As an application of Theorems (2.1) and (2.4) we prove the following theorem for two finite families of mappings.

Theorem 2.6. If and are two finite pair wise commuting finite families of self mappings de.ned on complete dislocated metric space such that the mappings T and S satisfy the conditions of theorems (2.1) and (2.4), then the component maps of the two families and have a unique common fixed point.

Proof. In view of theorems (2.1) and (2.4), one can infer that and have a unique common fixed point i.e. Now we are required to show that is common fixed point of all the components maps of both the families. In view of pairwise commutativity of families of and (for every ) we can write

which shows that (for every ) is also a common fixed point of and By using the uniqueness of common fixed point, we can write (for every ) which shows that is the common fixed point of the family Using the foregoing arguments, one can also shows that (for every )

This completes the proof of the theorem.

Corollary 2.7. If be a self mapping de.ned on a complete dislocated metric space satisfying the condition

for all with

where and a, b, c, d are non negative reals with Then is a non increasing sequence in for all and Then S has common fixed point h in

Corollary 2.8. If be a self mapping defined on a complete on dislocated metric space satisfying the condition:

for all with

where and are non negative reals with If, is a non increasing sequence in for all and Then has common fixed point in

Conflict of Interests

The authors declare that they have no competing interests.

References