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We establish the existence and uniqueness of coupled common fixed point for symmetric

The structure of

Let

Let

Thus, if

In [

Let

If

If

the sequence

for every

If

If

Let

Let

A

Let

Recently, fixed point theorems under different contractive conditions in metric spaces endowed with the partial ordering have been studied by various authors. Works noted in [

Lakshmikantham and

Let

Let

An element

An element

An element

The mappings

Let

Choudhury and Maity [

Various authors extended and generalized the results of Choudhury and Maity [

Mohiuddine and Alotaibi [

Interestingly, for

On the other hand, Karapinar et al. [

Present work extend and generalize several results present in the literature of fixed point theory of

Before proving our results, we need the following.

Denote by

We note that

Denote by

Let

Our first result is the following.

Let

If there exist two elements

Without loss of generality, assume that there exist

Continuing this process, we can construct sequences

Suppose that (

If for some

Using (

On the contrary, suppose that

Taking limit as

If possible, suppose that at least one of

In the next theorem, we omit the continuity hypotheses of the mapping

Let

if a nondecreasing sequence

if a nonincreasing sequence

Let

Proceeding exactly as in Theorem

Next we give an example in support of Theorem

Let

Define

Clearly

Indeed, for

Now, putting

Let

The following example shows that the contractive condition (

Let

Define

Then

Next, we prove that (

This shows that

Finally, we prove that (

By Corollary

Let

Note that if

The following result provides us the recent result of Karapinar et. al. [

Let

Define

The choice of functions

Now, putting

Let

The choice of function

Next we prove the existence and uniqueness of the coupled common fixed point for our main result.

In addition to the hypotheses of Theorem

From Theorem

Then, as in the proof of Theorem (

Further, set

We shall show that

Hence, it follows that

Similarly, we can show that

By uniqueness of limit, it follows that

Since

Then from (

To prove the uniqueness, assume that

In addition to the hypotheses of Theorem

Proceeding exactly as in Theorem

Motivated by the works of Aydi et al. [

Consider the integral equations in the following system:

Let

there exists

We shall analyze the system (

there exist

we suppose that

there exist continuous functions

Under assumptions (i)–(vi), the system (

Consider the operator

In fact, for

First we estimate the quantity

Similarly, we can obtain that

Let

In the setup of ordered