^{1}

^{*}

^{1}

^{*}

In this paper, first we introduce notions of ( α , Ψ)-contractive and ( α)-admissible for a pair of map and prove a coupled coincidence point theorem for compatible mappings using these notions. Our work extends and generalizes the results of Mursaleen et al. [1]. At the end, we will provide an example in support of our result.

Fixed point theorems give the conditions under which maps have solutions.

Fixed point theory is a beautiful mixture of Analysis, Topology and Geometry. Fixed points Theory has been playing a vital role in the study of nonlinear phenomena. In particular, fixed point techniques have been applied in diverse fields as Biology, Chemistry, and Economics, Engineering, Game theory and Physics. The usefulness of the concrete applications has increased enormously due to the development of accurate techniques for computing fixed points.

The fixed point theory has many important applications in numerical methods like Newton-Raphson Method and establishing Picard’s Existence Theorem regarding existence and uniqueness of solution of first order differential equation, existence of solution of integral equations and a system of linear equations. The credit of making the concept of fixed point theory useful and popular goes to polish mathematician Stefan Banach. In 1922, Banachproved a fixed point theorem, which ensures the existence and uniqueness of a fixed point under appropriate conditions. This result of Banach is known as Banach fixed point theoremor contraction mapping principle, “Let x be any non empty set and

1) What functions/maps have a fixed point?

2) How do we determine the fixed point?

3) Is the fixed point unique?

Currently, fixed point theory has been receiving much attention on in partially ordered metric spaces; that is, metric spaces endowed with a partial ordering. Turinici [

Choudhury and Kundu [

In this paper, we will generalize the results of Mursaleen et al. [

In order to obtain our results we need to consider the followings.

Definition 2.1. [

Definition 2.2. [

Definition 2.3. [

Definition 2.4. [

Choudhury et al. [

Definition 2.5. [

whenever

In order to obtain our results we need to consider the followings.

Definition 2.6. [

_{n} is the nth iterate of ψ satisfying

1)

2)

3)

Lemma 2.7. [

Definition 2.8. [

Definition 2.9. [

Now, we will introduce our notions:

Definition 2.10. Let

with

Definition 2.11. Let

Then F and g are said to be (α)-admissible if

Recently, Mursaleen et al. [

Theorem 3.1 [

Such that for

Suppose also that

1) F is (a)-admissible.

2) There exists

3) F is continuous.

If there exists

Then F has a coupled fixed point, that is, there exist,

Now we are ready to prove our results for compatible mappings.

Theorem 3.2 Let

For all

Suppose also that

1) F and g are (a)-admissible.

2) There exists

3)

4) F is continuous.

If there exists

Then F and g has coupled coincidence point that is there exist,

Proof: Let

and

Let

Continuing this process, we can construct two sequences

Now we will show that

For

and as

We have,

Thus (3.4) holds for

Now suppose that (3.4) holds for some fixed

Then, since

Therefore, by g-mixed monotone property of F, we have

From above, we conclude that

Thus, by mathematical induction, we conclude that (3.4) holds for all

If following holds for some

Then obviously,

Now, we assume that

Since, F and g a-admissible, we have

implies,

Thus by mathematical induction, we have

Similarly, we have

From (3.3) and conditions 1) and 2) of hypothesis, we get

Similarly, we have

On adding (3.7) and (3.8), we get

Repeating the above process, we get

For

Let

that is;

Since,

Hence,

Since,

There exists,

Since, F and g are compatible mappings; therefore, we have

Next we will show that

For all

Taking limit

Similarly, we have

Thus

Hence, we have proved that F and g has coupled coincidence point.

Now, we will replace continuity of F in the theorem 3.2 by a condition on sequences.

Theorem 3.3. Let

1) Inequality (3.3) and conditions 1), 2) and 3) hold.

2) if

for all n and

If there exists

Then F and g has coupled coincidence point, that is, there exist,

Proof. Proceeding along the same lines as in the proof of Theorem 3.2, we know that

are Cauchy sequences in the complete metric space

Similarly,

Using the triangle inequality, (3.11) and the property of

Similarly, on using (3.12), we have

Proceeding limit

Thus,

Remark. On putting

Example 3.4. Let

Then

Let

Let

Let

Let,

Then obviously,

Now, for all_{ }

Then it follows that,

Hence, the mappings F and g are compatible in X.

Consider a mapping

Thus (3.3) holds for

mixed monotone property. Let

is a coupled coincidence point of g and F in X.

 Preeti,SanjayKumar, (2015) Coupled Fixed Point for (α, Ψ)-Contractive in Partially Ordered Metric Spaces Using Compatible Mappings. Applied Mathematics,06,1380-1388. doi: 10.4236/am.2015.68130