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The aim of this paper is to give fixed point theorems for

Let

The two most important results in fixed point theory, are without contest, the Banach contraction principle (BCP for short) and Tarski’s fixed point theorem. Since their appearances, they were subject of many generalizations, either by extending the contractive condition for the B.C.P., or changing the structure of the space itself. For example, in the case of B.C.P., Ran and Reurings in [

In the beginning of 1930’s, Orlicz and Birnbaum considered the space

where

Fixed point theory in modular function spaces was first studied by Khamsi et al. in [

On the other hand, the combination of metric fixed point theory and graph theory allows Jachymski in [

In this paper, we generalize all the results obtained by Alfuraidan in [

Let

If

The following definitions will be needed in the sequel.

Let

We say that

A set

A set

A set

If

Let

It is known that the

The following lemma will be very useful along this work.

Let

where

Let

If

We recall the definition of digraph, the interested reader can consult the book [

A directed graph or digraph

If whenever

A digraph

A dipath of

A finite dipath of length

A closed directed path of length

A digraph is connected if there is a finite (di)path joining any two of its vertices and it is weakly connected if

It seems that the graph theory when coupled with the classical metric fixed point theory leads to a new interesting theory, following the works of [

Let

We say that

We say that

Note that if

Let

We start this section by the following result which will be useful in the sequel.

Let

Since

By induction we construct a sequence

for all

If

Let

If

Let

In the same way, there exists

And then clearly,

Note that as

Let

Then,

Consider the digraph

for every

It is clear that this binary relation defines a reflexive digraph without being a partially order (i.e., without antisymmetric condition).

Let

For all

and the

Now, if

For the (

Furthermore,

implies that

Let

Indeed for such

Recall the notion of approximated fixed point sequence.

We say that

A digraph

If the reflexive digraph

Let

Let

Let

for any

Moreover,

Let

thus,

i.e.,

Choosing

Now with more restrictions on the graph

Let

Notice that if

Let

and

Let

thus, it comes clearly that

as

Clearly

By induction on

we get the existence of a fixed point

We then apply Lemma

Let

The preceding lemma (Lemma

Since

Now, since

Consider the modular function space

for every

Let

It is clear that

Then there exists a subsequence

Moreover, for each

then for all

Let

The

Let

A set

A set

Let

If

If

We need the following definition of the growth function.

Let

is called the growth of

The growth function has the following properties.

Let

We then get the following lemma.

Let

It is obvious that for

and then,

as

The same arguments give:

Let

for every

The following relaxed definition replace the (

Let

If for any sequence

If we substitute “(

We conclude this paper by the

Let

By Lemma

for every

As

Now the

Moreover, as

So, by letting

Then if

which is a contradiction. Then necessarily

No data were used to support this study.

The authors declare that they have no conflicts of interest.

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