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We introduce the concept of the generalized

It is well known that many problems in many branches of mathematics can be transformed to a fixed point problem of the form

In 1969, Nadler [

On the other hand, Kada et al. [

The purpose of this work is to introduce the generalized

In this section, we recall some definitions and lemmas of

Let

A point

A point

A point

Let

Let

Let

a mapping

for any

Let us give some examples of

Let

Let

Let

Let

We obtain that in general for

Let

For more details of other examples and properties of the

Let

if

if

if

if

Next, we give the definition of some type of mapping. Before giving next definition, we give the following notation. Let

Let

Let

It is easy to see that

In this section, we introduce the new mapping, the so-called generalized

Let

Let

there exist

if for every

For

By repeating (

For

Let

for each

where

for every

Setting

Next, we give the notion of

Let

Let

there exist

for every

We obtain that this result can be proven by using similar method in Theorem

First of all, we introduce the following concept.

Let

If

Let

If

Next, we give useful lemma of Haghi et al. [

Let

Now, we apply our result in Section

Let

there exist

for all

Consider the mapping

Since

Let

Finally, we obtain a common fixed point result. Before giving our results, we need a few definitions.

Let

Suppose that all the hypotheses of Theorem

From Theorem

If we set

In this section, we give the existence of fixed point theorems on a metric space endowed with an arbitrary binary relation.

Before presenting our results, we need a few definitions. Let

Let

Let

Let

there exist

for all

Consider the mapping

Next, we deduce Theorem

Let

For partially ordered metric space

Let

Let

Let

there exist

for all

Since

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.