## 1 Introduction

Let (X, d) be a metric space. A mapping T : XX is a contraction if

$d\left(T\left(x\right),T\left(y\right)\right)\le kd\left(x,y\right),$
(1.1)

for all x, yX, where 0 ≤ k < 1. The Banach Contraction Mapping Principle appeared in explicit form in Banach's thesis in 1922 [1]. Since its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions, see [210]. The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano [11] and was intensively developed by Koshi, Shimogaki, Yamamuro [1113] and others. Further and the most complete development of these theories are due to Luxemburg, Musielak, Orlicz, Mazur, Turpin [1418] and their collaborators. A lot of mathematicians are interested fixed points of Modular spaces, for example [4, 1926].

In 2008, Chistyakov [27] introduced the notion of modular metric spaces generated by F-modular and develop the theory of this spaces, on the same idea he was defined the notion of a modular on an arbitrary set and develop the theory of metric spaces generated by modular such that called the modular metric spaces in 2010 [28].

In this article, we study and prove the existence of fixed point theorems for contraction mappings in modular metric spaces.

## 2 Preliminaries

We will start with a brief recollection of basic concepts and facts in modular spaces and modular metric spaces (see [14, 15, 2729] for more details).

Definition 2.1. Let X be a vector space over(or). A functional ρ : X → [0, ∞] is called a modular if for arbitrary x and y, elements of X satisfies the following three conditions :

(A.1) ρ(x) = 0 if and only if x = 0;

(A.2) ρ(αx) = ρ(x) for all scalar α with |α| = 1;

(A.3) ρ(αx + βy) ≤ ρ(x) + ρ(y), whenever α, β ≥ 0 and α + β = 1.

If we replace (A.3) by

(A.4) ρ(αx + βy) ≤ αs ρ(x) + βs ρ(y), for α, β ≥ 0, αs + βs = 1 with an s ∈ (0, 1], then the modular ρ is called s-convex modular, and if s = 1, ρ is called a convex modular.

If ρ is modular in X, then the set defined by

$\begin{array}{c}\hfill {X}_{\rho }=\left\{x\in X:\rho \left(\lambda x\right)\to 0\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}\lambda \to {0}^{+}\right\}\hfill \end{array}$
(2.1)

is called a modular space. X ρ is a vector subspace of X it can be equipped with an F-norm defined by setting

$\begin{array}{c}\hfill {∥x∥}_{\rho }=\text{inf}\left\{\lambda >0:\rho \left(\frac{x}{\lambda }\right)\le \lambda \right\},\phantom{\rule{1em}{0ex}}x\in {X}_{\rho }.\hfill \end{array}$
(2.2)

In addition, if ρ is convex, then the modular space X ρ coincides with

(2.3)

and the functional ${∥x∥}_{\rho }^{*}=\text{inf}\left\{\lambda >0:\rho \left(\frac{x}{\lambda }\right)\le 1\right\}$is an ordinary norm on ${X}_{\rho }^{*}$ which is equivalence to ${∥x∥}_{\rho }$(see [16]).

Let X be a nonempty set, λ ∈ (0, ∞) and due to the disparity of the arguments, function w : (0, ∞) × X × X → [0, ∞] will be written as wλ(x, y) = w(λ, x, y) for all λ > 0 and x, yX.

Definition 2.2. [[28], Definition 2.1] Let X be a nonempty set. A function w : (0, ∞) × X × X → [0, ∞] is said to be a metric modular on X if satisfying, for all x, y, zX the following condition holds:

(i) w λ (x, y) = 0 for all λ > 0 if and only if x = y;

(ii) w λ (x, y) = w λ (y, x) for all λ > 0;

(iii) wλ + μ(x, y) ≤ w λ (x, z) + w μ (z, y) for all λ, μ > 0.

If instead of (i), we have only the condition

(i') w λ (x, x) = 0 for all λ > 0, then w is said to be a (metric) pseudomodular on X.

The main property of a (pseudo) modular w on a set X is a following: given x, yX, the function 0 < λw λ (x, y) ∈ [0, ∞] is a nonincreasing on (0, ∞).

In fact, if 0 < μ < λ, then (iii), (i') and (ii) imply

$\begin{array}{ccc}\hfill {w}_{\lambda }\left(x,y\right)\hfill & \hfill \le {w}_{\lambda -\mu }\left(x,x\right)+{w}_{\mu }\left(x,y\right)\hfill & \hfill ={w}_{\mu }\left(x,y\right).\hfill \end{array}$
(2.4)

It follows that at each point λ > 0 the right limit ${w}_{\lambda +0}\left(x,y\right):=\underset{\epsilon \to +0}{\text{lim}}{w}_{\lambda +\epsilon }\left(x,y\right)$ and the left limit ${w}_{\lambda -0}\left(x,y\right):=\underset{\epsilon \to +0}{\text{lim}}{w}_{\lambda -\epsilon }\left(x,y\right)$ exists in [0, ∞] and the following two inequalities hold :

$\begin{array}{ccc}\hfill {w}_{\lambda +0}\left(x,y\right)\hfill & \hfill \le {w}_{\lambda }\left(x,y\right)\hfill & \hfill \le {w}_{\lambda -0}\left(x,y\right).\hfill \end{array}$
(2.5)

Definition 2.3. [[28], Definition 3.3] A function w : (0, ∞) × X × X → [0, ∞] is said to be a convex (metric) modular on X if it is satisfies the conditions (i) and (ii) from Definition 2.2 as well as this condition holds;

(iv) ${w}_{\lambda +\mu }\left(x,y\right)=\frac{\lambda }{\lambda +\mu }{w}_{\lambda }\left(x,z\right)+\frac{\mu }{\lambda +\mu }{w}_{\mu }\left(z,y\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}\lambda ,\mu >0\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}x,y,z\in X.$

If instead of (i), we have only the condition (i') from Definition 2.2, then w is called a convex(metric) pseudomodular on X.

From [27, 28], we know that, if x0X, the set ${X}_{w}=\left\{x\in X:\underset{\lambda \to \infty }{\text{lim}}{w}_{\lambda }\left(x,{x}_{0}\right)=0\right\}$ is a metric space, called a modular space, whose metric is given by ${d}_{w}^{\circ }\left(x,y\right)=\text{inf}\left\{\lambda >0:{w}_{\lambda }\left(x,y\right)\le \lambda \right\}$ for all x, yX w . Moreover, if w is convex, the modular set X w is equal to ${X}_{w}^{*}=\left\{x\in X:\exists \lambda =\lambda \left(x\right)>0$ such that w λ (x, x0) <∞} and metrizable by ${d}_{w}^{*}\left(x,y\right)=\text{inf}\left\{\lambda >0:{w}_{\lambda }\left(x,y\right)\le 1\right\}$for all $x,y\in {X}_{w}^{*}$. We know that (see [[28], Theorem 3.11]) if X is a real linear space, ρ : X → [0, ∞] and

(2.6)

then ρ is modular (convex modular) on X in the sense of (A.1)-(A.4) if and only if w is metric modular (convex metric modular, respectively) on X. On the other hand, if w satisfy the following two conditions (i) w λ (μx, 0) = w λ/μ (x, 0) for all λ, μ > 0 and xX, (ii) w λ (x + z, y + z) = w λ (x, y) for all λ > 0 and x, y, zX, if we set ρ(x) = w1(x, 0) with (2.6) holds, where xX, then

1. (i)

X ρ = X w is a linear subspace of X and the functional ${∥x∥}_{\rho }={d}_{w}^{\circ }\left(x,0\right)$, xX ρ , is an F-norm on X ρ ;

2. (ii)

if w is convex, ${X}_{\rho }^{*}\equiv {X}_{w}^{*}\left(0\right)={X}_{\rho }$ is a linear subspace of X and the functional ${∥x∥}_{\rho }={d}_{w}^{*}\left(x,0\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}x\in {X}_{\rho }^{*}$, is an norm on ${X}_{\rho }^{*}$.

Similar assertions hold if replace the word modular by pseudomodular. If w is metric modular in X, we called the set X w is modular metric space.

By the idea of property in metric spaces and modular spaces, we defined the following:

Definition 2.4. Let X w be a modular metric space.

(1) The sequence (x n )n∈ℕ in X w is said to be convergent to xX w if w λ (x n , x) → 0, as n →for all λ > 0.

(2) The sequence (x n ) n∈ℕ in X w is said to be Cauchy if w λ ( x m , x n ) → 0, as m, n →for all λ > 0.

(3) A subset C of X w is said to be closed if the limit of a convergent sequence of C always belong to C.

(4) A subset C of X w is said to be complete if any Cauchy sequence in C is a convergent sequence and its limit is in C.

(5) A subset C of X w is said to be bounded if for all λ > 0 δ w (C) = sup{w λ (x, y); x, yC} <∞.

## 3 Main results

In this section, we prove the existence of fixed points theorems for contraction mapping in modular metric spaces.

Definition 3.1. Let w be a metric modular on X and X w be a modular metric space induced by w and T : X w → X w be an arbitrary mapping. A mapping T is called a contraction if for each x, yX w and for all λ > 0 there exists 0 ≤ k < 1 such that

${w}_{\lambda }\left(Tx,Ty\right)\le k{w}_{\lambda }\left(x,y\right).$
(3.1)

Theorem 3.2. Let w be a metric modular on X and X w be a modular metric space induced by w. If X w is a complete modular metric space and T : X w → X w is a contraction mapping, then T has a unique fixed point in X w . Moreover, for any xX w , iterative sequence {Tnx} converges to the fixed point.

Proof. Let x0 ba an arbitrary point in X w and we write x1 = Tx0, x2 = Tx1 = T2x0, and in general, x n = Txn-1= Tnx0 for all n ∈ ℕ. Then,

$\begin{array}{ll}\hfill {w}_{\lambda }\left({x}_{n+1},{x}_{n}\right)& ={w}_{\lambda }\left(T{x}_{n},T{x}_{n-1}\right)\phantom{\rule{2em}{0ex}}\\ \le k{w}_{\lambda }\left({x}_{n},{x}_{n-1}\right)\phantom{\rule{2em}{0ex}}\\ =k{w}_{\lambda }\left(T{x}_{n-1},T{x}_{n-2}\right)\phantom{\rule{2em}{0ex}}\\ \le {k}^{2}{w}_{\lambda }\left({x}_{n-1},{x}_{n-2}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}⋮\phantom{\rule{2em}{0ex}}\\ \le {k}^{n}{w}_{\lambda }\left({x}_{1},{x}_{0}\right)\phantom{\rule{2em}{0ex}}\end{array}$

for all λ > 0 and for each n ∈ ℕ. Therefore, $\underset{n\to \infty }{\text{lim}}{w}_{\lambda }\left({x}_{n+1},{x}_{n}\right)=0$ for all λ > 0. So for each λ > 0, we have for all ∊ > 0 there exists n0 ∈ ℕ such that w λ (x n , xn+1) < ∊ for all n ∈ ℕ with n ≥ n0. Without loss of generality, suppose m, n ∈ ℕ and m > n. Observe that, for $\frac{\lambda }{m-n}>0$, there exists nλ/(m-n)∈ ℕ such that

${w}_{\frac{\lambda }{m-n}}\left({x}_{n},{x}_{n+1}\right)<\frac{\epsilon }{m-n}$

for all n ≥ nλ/(m-n). Now, we have

$\begin{array}{ll}\hfill {w}_{\lambda }\left({x}_{n},{x}_{m}\right)& \le {w}_{\frac{\lambda }{m-n}}\left({x}_{n},{x}_{n+1}\right)+{w}_{\frac{\lambda }{m-n}}\left({x}_{n+1},{x}_{n+2}\right)+\cdots +{w}_{\frac{\lambda }{m-n}}\left({x}_{m-1},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{x}_{m}\right)\phantom{\rule{2em}{0ex}}\\ <\frac{\epsilon }{m-n}+\frac{\epsilon }{m-n}+\cdots +\frac{\epsilon }{m-n}\phantom{\rule{2em}{0ex}}\\ =\epsilon \phantom{\rule{2em}{0ex}}\end{array}$

for all m, n ≥ nλ/(m-n). This implies {x n }n∈ℕis a Cauchy sequence. By the completeness of X w , there exists a point xX w such that x n → × as n → ∞.

By the notion of metric modular w and the contraction of T, we get

$\begin{array}{ccc}\hfill {w}_{\lambda }\left(Tx,x\right)\hfill & \hfill \le \hfill & \hfill {w}_{\frac{\lambda }{2}}\left(Tx,T{x}_{n}\right)+{w}_{\frac{\lambda }{2}}\left(T{x}_{n},x\right)\hfill \\ \hfill \le \hfill & \hfill k{w}_{\frac{\lambda }{2}}\left(x,{x}_{n}\right)+{w}_{\frac{\lambda }{2}}\left({x}_{n+1},x\right)\hfill \end{array}$
(3.2)

for all λ > 0 and for each n ∈ ℕ. Taking n → ∞ in (3.2) implies that w λ (Tx, x) = 0 for all λ > 0 and thus Tx = x. Hence, x is a fixed point of T. Next, we prove that x is a unique fixed point. Suppose that z is another fixed point of T. We see that

$\begin{array}{ccc}\hfill {w}_{\lambda }\left(x,z\right)\hfill & \hfill =\hfill & \hfill {w}_{\lambda }\left(Tx,Tz\right)\hfill \\ \hfill \le \hfill & \hfill k{w}_{\lambda }\left(x,z\right)\hfill \end{array}$

for all λ > 0. Since 0 ≤ k < 1, we get w λ (x, z) = 0 for all λ > 0 this implies that x = z. Therefore, x is a unique fixed point of T and the proof is complete.   □

Theorem 3.3. Let w be a metric modular on X and X w be a modular metric space induced by w. If X w is a complete modular metric space and T : X w → X w is a contraction mapping. Suppose x*X w is a fixed point of T, {ε n } is a sequence of positive numbers for which $\underset{n\to \infty }{\text{lim}}{\epsilon }_{n}=0$, and {y n } ⊆ X w satisfies

${w}_{\lambda }\left({y}_{n+1},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}T{y}_{n}\right)\le {\epsilon }_{n}$

for all λ > 0. Then, $\underset{n\to \infty }{\text{lim}}{y}_{n}={x}^{*}$.

Proof. For each m ∈ ℕ, we observe that

$\begin{array}{ll}\hfill {w}_{\lambda }\left({T}^{m+1}x,{y}_{m+1}\right)& ={w}_{\frac{\lambda \cdot m}{m}}\left({T}^{m+1}x,{y}_{m+1}\right)\phantom{\rule{2em}{0ex}}\\ \le {w}_{\frac{\lambda \cdot \left(m-1\right)}{m}}\left({T}^{m+1}x,T{y}_{m}\right)+{w}_{\frac{\lambda }{m}}\left(T{y}_{m},{y}_{m+1}\right)\phantom{\rule{2em}{0ex}}\\ \le k{w}_{\frac{\lambda \cdot \left(m-1\right)}{m}}\left({T}^{m}x,{y}_{m}\right)+{\epsilon }_{m}\phantom{\rule{2em}{0ex}}\\ \le k{w}_{\frac{\lambda \cdot \left(m-2\right)}{m}}\left({T}^{m}x,T{y}_{m-1}\right)+k{w}_{\frac{\lambda }{m}}\left(T{y}_{m-1}x,{y}_{m}\right)+{\epsilon }_{m}\phantom{\rule{2em}{0ex}}\\ \le {k}^{2}{w}_{\frac{\lambda \cdot \left(m-2\right)}{m}}\left({T}^{m-1}x,{y}_{m-1}\right)+k{\epsilon }_{m-1}+{\epsilon }_{m}\phantom{\rule{2em}{0ex}}\\ ⋮\phantom{\rule{2em}{0ex}}\\ \le \sum _{i=0}^{m}{k}^{m-i}{\epsilon }_{i}\phantom{\rule{2em}{0ex}}\end{array}$
(3.3)

for all λ > 0. Thus, we get

$\begin{array}{ll}\hfill {w}_{\lambda }\left({y}_{m+1},{x}^{*}\right)& \le {w}_{\frac{\lambda }{2}}\left({y}_{m+1},{T}^{m+1}x\right)+{w}_{\frac{\lambda }{2}}\left({T}^{m+1}x,{x}^{*}\right)\phantom{\rule{2em}{0ex}}\\ \le \sum _{i=0}^{m}{k}^{m-i}{\epsilon }_{i}+{w}_{\frac{\lambda }{2}}\left({T}^{m+1}x,{x}^{*}\right).\phantom{\rule{2em}{0ex}}\end{array}$
(3.4)

Next, we claimed that $\underset{m\to \infty }{\text{lim}}{w}_{\lambda }\left({y}_{m+1},{x}^{*}\right)=0$ for all λ > 0.

Now let ε > 0. Since $\underset{n\to \infty }{\text{lim}}{\epsilon }_{n}=0$, there exists N ∈ ℕ such that for m ≥ N, ε m ε. Thus,

$\begin{array}{ll}\hfill \sum _{i=0}^{m}{k}^{m-i}{\epsilon }_{i}& =\sum _{i=0}^{N}{k}^{m-i}{\epsilon }_{i}+\sum _{i=N+1}^{m}{k}^{m-i}{\epsilon }_{i}\phantom{\rule{2em}{0ex}}\\ \le {k}^{m-N}\sum _{i=0}^{N}{k}^{N-i}{\epsilon }_{i}+\epsilon \sum _{i=N+1}^{m}{k}^{m-i}.\phantom{\rule{2em}{0ex}}\end{array}$
(3.5)

Taking limit as m → ∞ in (3.5), we have

$\underset{m\to \infty }{\text{lim}}\sum _{i=0}^{m}{k}^{m-i}{\epsilon }_{i}=0.$
(3.6)

Since x0 is a fixed point of T and using result of Theorem 3.2, we get the sequence {Tnx} converge to x*. This implies that

$\underset{m\to \infty }{\text{lim}}{w}_{\frac{\lambda }{2}}\left({T}^{m+1}x,{x}^{*}\right)=0$
(3.7)

for all λ > 0. From (3.4), (3.6) and (3.7), we have

$\underset{m\to \infty }{\text{lim}}{w}_{\lambda }\left({y}_{m+1},{x}^{*}\right)=0$
(3.8)

for all λ > 0 which implies that $\underset{n\to \infty }{\text{lim}}{y}_{n}={x}^{*}$.   □

Theorem 3.4. Let w be a metric modular on X and X w be a modular metric space induced by w. If X w is a complete modular metric space and T : X w X w is a mapping, which TN is a contraction mapping for some positive integer N. Then, T has a unique fixed point in X w .

Proof. By Theorem 3.2 , TN has a unique fixed point uX w . From TN(T u ) = TN+1u = T(TNu) = Tu, so Tu is a fixed point of TN. By the uniqueness of fixed point of TN, we have Tu = u. Thus, u is a fixed point of T. Since fixed point of T is also fixed point of TN, we can conclude that T has a unique fixed point in X w .   □

Theorem 3.5. Let w be metric modular on X, X w be a complete modular metric space induced by w and for x*X w we define

${B}_{w}\left({x}^{*},\gamma \right):=\left\{x\in {X}_{w}|{w}_{\lambda }\left(x,{x}^{*}\right)\le \gamma \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\lambda >0\right\}.$

If T : B w (x*, γ) → X w is a contraction mapping with

${w}_{\frac{\lambda }{2}}\left(T{x}^{*},{x}^{*}\right)\le \left(1-k\right)\gamma$
(3.9)

for all λ > 0, where 0 ≤ k < 1. Then, T has a unique fixed point in B w (x*, γ).

Proof. By Theorem 3.2 , we only prove that B w (x*, γ) is complete and TxB w (x*, γ), for all xB w (x*, γ). Suppose that {x n } is a Cauchy sequence in B w (x*, γ), also {x n } is a Cauchy sequence in X w . Since X w is complete, there exists xX w such that

$\underset{n\to \infty }{\text{lim}}{w}_{\frac{\lambda }{2}}\left({x}_{n},x\right)=0$
(3.10)

for all λ > 0. Since for each n ∈ ℕ, x n B w (x*, γ), using the property of metric modular, we get

$\begin{array}{c}{w}_{\lambda }\left({x}^{*},x\right)\le {w}_{\frac{\lambda }{2}}\left({x}^{*},{x}_{n}\right)+{w}_{\frac{\lambda }{2}}\left({x}_{n},x\right)\\ \le \gamma +{w}_{\frac{\lambda }{2}}\left({x}_{n},{x}^{*}\right)\end{array}$
(3.11)

for all λ > 0. It follows the inequalities (3.10) and (3.11), we have w λ (x*, x) ≤ γ which implies that xB w (x*, γ). Therefore, {x n } is convergent sequence in B w (x*, γ) and also B w (x*, γ) is complete.

Next, we prove that TxB w (x*, γ) for all xB w (x*, γ). Let xB w (x*, γ). From the inequalities (3.9), using the contraction of T and the notion of metric modular, we have

$\begin{array}{ll}\hfill {w}_{\lambda }\left({x}^{*},Tx\right)& \le {w}_{\frac{\lambda }{2}}\left({x}^{*},T{x}^{*}\right)+{w}_{\frac{\lambda }{2}}\left(T{x}^{*},Tx\right)\phantom{\rule{2em}{0ex}}\\ \le \left(1-k\right)\gamma +k{w}_{\frac{\lambda }{2}}\left({x}^{*},x\right)\phantom{\rule{2em}{0ex}}\\ \le \left(1-k\right)\gamma +k\gamma \phantom{\rule{2em}{0ex}}\\ =\gamma .\phantom{\rule{2em}{0ex}}\end{array}$

Therefore, TxB w (x*, γ) and the proof is complete.

Theorem 3.6. Let w be a metric modular on X, X w be a complete modular metric space induced by w and T : X w X w . If

${w}_{\lambda }\left(Tx,Ty\right)\le k\left({w}_{2\lambda }\left(Tx,x\right)+{w}_{2\lambda }\left(Ty,y\right)\right)$
(3.12)

for all x, yX w and for all λ > 0, where $k\in \left[0,\frac{1}{2}\right)$, then T has a unique fixed point in X w . Moreover, for any xX w , iterative sequence {Tnx} converges to the fixed point.

Proof. Let x0 be an arbitrary point in X w and we write x1 = Tx0, x2 = Tx1 = T2x0, and in general, x n = Txn-1= Tnx0 for all n ∈ ℕ. If $T{x}_{{n}_{0}-1}=T{x}_{{n}_{0}}$for some n0 ∈ ℕ, then $T{x}_{{n}_{0}}={x}_{{n}_{0}}$. Thus, ${x}_{{n}_{0}}$ is a fixed point of T. Suppose that Txn-1Tx n for all n ∈ ℕ. For $k\in \left[0,\frac{1}{2}\right)$, we have

$\begin{array}{ll}\hfill {w}_{\lambda }\left({x}_{n+1},{x}_{n}\right)& ={w}_{\lambda }\left(T{x}_{n},T{x}_{n-1}\right)\phantom{\rule{2em}{0ex}}\\ \le k\left({w}_{2\lambda }\left(T{x}_{n},{x}_{n}\right)+{w}_{2\lambda }\left(T{x}_{n-1},{x}_{n-1}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le k\left({w}_{\lambda }\left({x}_{n+1},{x}_{n}\right)+{w}_{\lambda }\left({x}_{n},{x}_{n-1}\right)\right)\phantom{\rule{2em}{0ex}}\end{array}$
(3.13)

for all λ > 0 and for all n ∈ ℕ. Hence,

${w}_{\lambda }\left({x}_{n+1},{x}_{n}\right)\le \frac{k}{1-k}{w}_{\lambda }\left({x}_{n},{x}_{n-1}\right)$
(3.14)

for all λ > 0 and for all n ∈ ℕ. Put $\beta :=\frac{k}{1-k}$, since $k\in \left[0,\frac{1}{2}\right)$, we get β ∈ [0, 1) and hence

$\begin{array}{ccc}\hfill {w}_{\lambda }\left({x}_{n+1},{x}_{n}\right)\hfill & \hfill \le \hfill & \hfill \beta {w}_{\lambda }\left({x}_{n},{x}_{n-1}\right)\hfill \\ \hfill \le \hfill & \hfill {\beta }^{2}{w}_{\lambda }\left({x}_{n-1},{x}_{n-2}\right)\hfill \\ \hfill ⋮\hfill \\ \hfill \le \hfill & \hfill {\beta }^{n}{w}_{\lambda }\left({x}_{1},{x}_{0}\right)\hfill \end{array}$
(3.15)

for all λ > 0 and for all n ∈ ℕ. Similar to the proof of Theorem 3.2, we can conclude that {x n } is a Cauchy sequence and by the completeness of X w there exists a point xX w such that x n x as n → ∞. By the property of metric modular and the inequality (3.12), we have

$\begin{array}{ll}\hfill {w}_{\lambda }\left(Tx,x\right)& \le {w}_{\frac{\lambda }{2}}\left(Tx,T{x}_{n}\right)+{w}_{\frac{\lambda }{2}}\left(T{x}_{n},x\right)\phantom{\rule{2em}{0ex}}\\ \le k\left({w}_{\lambda }\left(Tx,x\right)+{w}_{\lambda }\left(T{x}_{n},{x}_{n}\right)\right)+{w}_{\frac{\lambda }{2}}\left(T{x}_{n},x\right)\phantom{\rule{2em}{0ex}}\\ \le k\left({w}_{\lambda }\left(Tx,x\right)+{w}_{\frac{\lambda }{2}}\left(T{x}_{n},x\right)+{w}_{\frac{\lambda }{2}}\left(x,{x}_{n}\right)\right)+{w}_{\frac{\lambda }{2}}\left(T{x}_{n},x\right)\phantom{\rule{2em}{0ex}}\\ =k\left({w}_{\lambda }\left(Tx,x\right)+{w}_{\frac{\lambda }{2}}\left({x}_{n+1},x\right)+{w}_{\frac{\lambda }{2}}\left(x,{x}_{n}\right)\right)+{w}_{\frac{\lambda }{2}}\left({x}_{n+1},x\right)\phantom{\rule{2em}{0ex}}\end{array}$
(3.16)

for all λ > 0 and for all n ∈ ℕ. Taking n → ∞ in the inequality (3.16), we obtained that

${w}_{\lambda }\left(Tx,x\right)\le k{w}_{\lambda }\left(Tx,x\right).$
(3.17)

Since $k\in \left[0,\frac{1}{2}\right)$, we have Tx = x. Thus, x is a fixed point of T. Next, we prove that x is a unique fixed point. Suppose that z be another fixed point of T. We note that

$\begin{array}{ll}\hfill {w}_{\lambda }\left(x,z\right)& ={w}_{\lambda }\left(Tx,Tz\right)\phantom{\rule{2em}{0ex}}\\ \le k\left({w}_{\frac{\lambda }{2}}\left(Tx,x\right)+{w}_{\frac{\lambda }{2}}\left(Tz,z\right)\right)\phantom{\rule{2em}{0ex}}\\ =0\phantom{\rule{2em}{0ex}}\end{array}$

for all λ > 0, which implies that x = z. Therefore, x is a unique fixed point of T.   □

Now, we shall give a validate example of Theorem 3.2 .

Example 3.7. Let X = {(a, 0) ∈ ℝ2|0 ≤ a ≤ 1} ∪ {(0, b) ∈ ℝ2|0 ≤ b ≤ 1}.

Defined the mapping w : (0, ∞) × X × X → [0, ∞] by

${w}_{\lambda }\left(\left({a}_{1},0\right),\left({a}_{2},0\right)\right)=\frac{4|{a}_{1}-{a}_{2}|}{3\lambda },$
${w}_{\lambda }\left(\left(0,{b}_{1}\right),\left(0,{b}_{2}\right)\right)=\frac{|{b}_{1}-{b}_{2}|}{\lambda },$

and

${w}_{\lambda }\left(\left(a,0\right),\left(0,b\right)\right)=\frac{4a}{3\lambda }+\frac{b}{\lambda }={w}_{\lambda }\left(\left(0,b\right),\left(a,0\right)\right).$

We note that if we take λ → ∞, then we see that X = X w and also X w is a complete modular metric space. We let a mapping T : X w X w is define by

$T\left(\left(a,0\right)\right)=\left(0,a\right)$

and

$T\left(\left(0,b\right)\right)=\left(\frac{b}{2},0\right).$

Simple computations show that

${w}_{\lambda }\left(T\left(\left({a}_{1},{b}_{1}\right)\right),T\left(\left({a}_{2},{b}_{2}\right)\right)\right)\le \phantom{\rule{2.77695pt}{0ex}}\frac{3}{4}{w}_{\lambda }\left(\left({a}_{1},{b}_{1}\right),\left({a}_{2},{b}_{2}\right)\right)$

for all (a1, b1), (a2, b2) ∈ X w . Thus, T is a contraction mapping with constant $k=\frac{3}{4}$. Therefore, T has a unique fixed point that is (0, 0) ∈ X w .

On the Euclidean metric d on X w , we see that

$d\left(T\left(\left(0,0\right)\right),T\left(\left(1,0\right)\right)\right)=d\left(\left(0,0\right),\left(0,1\right)\right)=1>k=kd\left(\left(0,0\right),\left(1,0\right)\right)$

for all k ∈ [0, 1). Thus, T is not a contraction mapping and then the Banach contraction mapping cannot be applied to this example.