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We introduce some new generalization of fixed point theorems in complete metric spaces endowed with

Recently, extensive researches have been performed on various topics in different spaces equipped

In 2014, Du and Khojasteh [

Moreover, the notion of

Very recently, in order to extend the previous families of functions, Roldán López de Hierro and Shahzad [

In this work, we introduce some new generalization of fixed point theorems in complete metric spaces endowed with

In the following, we assert some notions which are needed in continuance.

Let

(1)

(2) for any

(3) for any

Denote

Let

if

if

if

if

A symmetric

(SY):

A function

for all

A mapping

Meir and Keeler [

A

if

Many examples of simulation functions can be found in [

Let

If

If

In some cases, for a given function

If

In the present research, we will introduce our main result and demonstrate that many of known results which have been invented in the set of metric spaces, with or without

The following definition plays the crucial role in our main result.

Let

In what follows, we assume that

Let

The function

Let

On the contrary, assume that

If we consider a sequence

Now suppose that

On the other hand, let

Now consider the following set:

for all

Moreover, if

The following items are some examples of

every simulation function

It is sufficient to show that

Let

It means that

and so taking limit on both sides of (

By ratio test,

Let

By taking limit on both sides of (

which is a contradiction. Thus,

Let

so

Let

Since

Hence,

Taking limit on both sides of (

Let

and so

Taking limit on both sides of (

Let

so

Let

It means that

Therefore,

Let

Also

Since

Since

Let

so

Suppose that

Let

and this is a contradiction. So

Let

which is a contradiction. So

Let

First of all, we show that, for

Let

which implies that

Let

and so

Hence

Let

and so

Hence,

Let

Let

Hence,

Therefore,

Suppose that

which is a contradiction. So

Let

Therefore,

Hence,

Letting

Let

so

Let

Hence,

Therefore,

Suppose that

which is a contradiction. So

Let

Therefore,

Hence,

Letting

Let

so

Let

Thus,

Therefore,

Suppose that

which is a contradiction. So

Let

Therefore,

Hence,

Letting

Let

so

The following corollary is the generalization of Reich’s theorem (see [

Let

Defining

The following corollary is the generalization of Geraghty fixed point theorem (see [

Let

Defining

The following corollary is the generalization of Meir-Keeler fixed point theorem (also see [

Let

Defining

The following corollary is the generalization of Khojasteh’s result [

Let

Applying

Let

Considering

In the following, we are going to present some new corollaries to show that our results are constructive and new results can be built by

Let

Defining

Let

Considering

Let

Defining

The authors declare that they have no competing interests.

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Farshid Khojasteh would like to thank to Science and Research Branch of Islamic Azad University to support this research.