^{1}

^{2}

^{1}

^{2}

The notion of modular metric spaces being a natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, and Calderon-Lozanovskii spaces was recently introduced. In this paper we investigate the
existence of fixed points of generalized

Chistyakov introduced the notion of modular metric spaces in [

For the study of electrorheological fluids (for instance lithium polymethacrylate), modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces,

Let

A function

Following example presented by Abdou and Khamsi [

Let

We say that

We say that

The associated modular function space is the vector space

The following formula defines a norm in

Other easy examples may be found in [

Let

The sequence

The sequence

A subset

A subset

A subset

In 2012, Samet et al. [

Let

Let

Let

Let

A mapping

Let

In this paper, we investigate existence and uniqueness of fixed points of generalized

Let us first start this section with a definition of a family of functions.

Assume that

if

Notice that here we denote with

Let

Let

Let

there exists

assume that there exists

where

Let

Now since

Let

there exists

assume that there exists

where

Let

there exists

if

holds for all

condition (iv) of Theorem

As in proof of Theorem

By using Example

Let

there exists

if

holds for all

for all

where

Let

there exists

if

holds for all

for all

where

Let

there exists

if

holds for all

for all

where

Let

Let

Let

Let

there exists

assume that there exists

where

Let

there exists

if

holds for all

assume that there exists

where

Let

there exists

if

holds for all

assume that for all

where

there exists

if

holds for all

assume that for all

where

there exists

if

holds for all

assume that for all

where

In 2008, Suzuki proved a remarkable fixed point theorem, that is, a generalization of the Banach contraction principle and characterizes the metric completeness. Consequently, a number of extensions and generalizations of this result appeared in the literature (see [

Let

Define

Let

Define

Therefore,

Let

Recently, Azadifar et al. [

Let

there exists

assume that there exists

where

there exists

if

holds for all

condition (iv) of Theorem

Let

Let

In 1988, Grabiec [

A 3-tuple

Let

For all

As an application of Lemma

Let

Let

As an application of Lemma

Let

there exists

assume that there exists

where

Let

there exists

if

holds for all

condition (iv) of Theorem

Let

there exists

assume that there exists

where

Let

there exists

if

holds for all

assume that there exists

where

Let

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR, KAU for financial support.