The notion of a modular metric on an arbitrary set and the corresponding
modular spaces, generalizing classical modulars over linear spaces
like Orlicz spaces, were recently introduced. In this paper we introduced and
study the concept of one-local retract in modular metric space. In particular,
we investigate the existence of common fixed points of modular nonexpansive
mappings defined on nonempty

The purpose of this paper is to give an outline of a common fixed-point theory for nonexpansive mappings (i.e., mappings with the modular Lipschitz constant 1) on some subsets of modular metric spaces which are natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii, and many other spaces. Modular metric spaces were introduced in [

In recent years, there was an uptake interest in the study of electrorheological fluids, sometimes referred to as “smart fluids” (for instance, lithium polymethacrylate). For these fluids, modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces,

In many cases, particularly in applications to integral operators, approximation, and fixed point results, modular type conditions are much more natural as modular type assumptions can be more easily verified than their metric or norm counterparts. In recent years, there was a strong interest to study the fixed point property in modular function spaces after the first paper [

In this paper, we study the concept of one-local retract in more general setting in modular metric space; therefore, we prove the existence of common fixed points for a family of modular nonexpansive mappings defined on nonempty

For more on metric fixed point theory, the reader may consult the book [

Let

A function

Note that, for a metric pseudomodular

Let

It is clear that

Let

The sequence

The sequence

A subset

A subset

Let

A subset

In general if

Let

Let

We will say that

We will say that

Note that if

Let

In [

Let

Next we present the analog of Kirk’s fixed point theorem [

Let

Let

The result in [

Let

Theorem

Now, we discuss some properties of one-local retract subsets.

Let

Let us prove

Next, we prove that

For the rest of this work, we will need the following technical result.

Let

Let us first prove

As an application of this lemma we have the following result.

Let

Using the definition of one-local retract, it is easy to see that

The following result has found many application in metric spaces. Most of the ideas in its proof go back to Baillon’s work [

Let

Consider the family

The next theorem will be useful to prove the main result of the next section.

Let

Consider the family

In this section we discuss the existence of a common fixed point of a family of commutative

First, we will need to discuss the case of finite families.

Let

First, let us prove that

The following result extends [

Let

Let

The author would like to thank Professor Mohamed A. Khamsi with whom the author had many fruitful discussions regarding this work.