Fixed point theorems for contraction mappings in modular metric spaces

Abstract

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

AMS: 47H09; 47H10.

1 Introduction

Let (X, d) be a metric space. A mapping T : X → X 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, y ∈ X, 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 [2–10]. The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano [11] and was intensively developed by Koshi, Shimogaki, Yamamuro [11–13] and others. Further and the most complete development of these theories are due to Luxemburg, Musielak, Orlicz, Mazur, Turpin [14–18] and their collaborators. A lot of mathematicians are interested fixed points of Modular spaces, for example [4, 19–26].

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, 27–29] for more details).

Definition 2.1. Let X be a vector space over ℝ (or ℂ). 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) ≤ α^s ρ(x) + β^s ρ(y), for α, β ≥ 0, α^s + β^s = 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

$$\begin{array}{c}\hfill X_{\rho }=\left\{x\in X:\rho \left(\lambda x\right)\to 0\text{as}\lambda \to 0^{+}\right\}\hfill \end{array}$$

(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

$$\begin{array}{c}\hfill {∥x∥}_{\rho }=\text{inf}\left\{\lambda >0:\rho \left(\frac{x}{\lambda }\right)\le \lambda \right\},x\in X_{\rho }.\hfill \end{array}$$

(2.2)

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

$$\begin{array}{c}\hfill X_{\rho }^{*}=\left\{x\in X:\exists \lambda =\lambda \left(x\right)>0\text{such that}\rho \left(\lambda x\right)<\infty \right\}\hfill \end{array}$$

(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 w_λ(x, y) = w(λ, x, y) for all λ > 0 and x, y ∈ X.

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, z ∈ X 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) w_{λ + 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, y ∈ X, the function 0 < λ ↦ w _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) $w_{\lambda +\mu }\left(x,y\right)=\frac{\lambda }{\lambda +\mu }{w}_{\lambda }\left(x,z\right)+\frac{\mu }{\lambda +\mu }{w}_{\mu }\left(z,y\right)forall\lambda ,\mu >0andx,y,z\in X.$

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 x₀ ∈ X, 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, y ∈ X _w . Moreover, if w is convex, the modular set X _w is equal to $X_w^{*}=\{x\in X:\exists \lambda =\lambda \left(x\right)>0$ 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

$$w_{\lambda }\left(x,y\right)=\rho \left(\frac{x-y}{\lambda }\right)\text{for all }\lambda >0\text{and }x,y\in X,$$

(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 x ∈ X, (ii) w _λ (x + z, y + z) = w _λ (x, y) for all λ > 0 and x, y, z ∈ X, if we set ρ(x) = w₁(x, 0) with (2.6) holds, where x ∈ X, then

• (i) X _ρ = X _w is a linear subspace of X and the functional ${∥x∥}_{\rho }=d_w^{\circ }\left(x,0\right)$, x ∈ X _ρ , is an F-norm on X _ρ ;

• (ii) if w is convex, $X_{\rho }^{*}\equiv X_w^{*}\left(0\right)=X_{\rho }$ is a linear subspace of X and the functional ${∥x∥}_{\rho }=d_w^{*}\left(x,0\right),x\in X_{\rho }^{*}$, 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 x ∈ X _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, y ∈ C} <∞.

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, y ∈ X _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 x ∈ X _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 = Tx_n-1= Tⁿx₀ for all n ∈ ℕ. Then,

$$\begin{array}{ll}\hfill w_{\lambda }\left(x_{n+1},x_n\right)& =w_{\lambda }\left(T{x}_n,T{x}_{n-1}\right)\\ \le k{w}_{\lambda }\left(x_n,x_{n-1}\right)\\ =k{w}_{\lambda }\left(T{x}_{n-1},T{x}_{n-2}\right)\\ \le k^2{w}_{\lambda }\left(x_{n-1},x_{n-2}\right)\\ ⋮\\ \le k^n{w}_{\lambda }\left(x_1,x_0\right)\end{array}$$

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 , x_n+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 n_λ/(m-n)∈ ℕ such that

$$w_{\frac{\lambda }{m-n}}\left(x_n,x_{n+1}\right)<\frac{\epsilon }{m-n}$$

for all n ≥ n_λ/(m-n). Now, we have

$$\begin{array}{ll}\hfill w_{\lambda }\left(x_n,x_m\right)& \le w_{\frac{\lambda }{m-n}}\left(x_n,x_{n+1}\right)+w_{\frac{\lambda }{m-n}}\left(x_{n+1},x_{n+2}\right)+\cdots +w_{\frac{\lambda }{m-n}}\left(x_{m-1},x_m\right)\\ <\frac{\epsilon }{m-n}+\frac{\epsilon }{m-n}+\cdots +\frac{\epsilon }{m-n}\\ =\epsilon \end{array}$$

for all m, n ≥ n_λ/(m-n). This implies {x _n }_n∈ℕis a Cauchy sequence. By the completeness of X _w , there exists a point x ∈ X _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

$$w_{\lambda }\left(y_{n+1},T{y}_n\right)\le {\epsilon }_n$$

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

Proof. For each m ∈ ℕ, we observe that

$$\begin{array}{ll}\hfill w_{\lambda }\left(T^{m+1}x,y_{m+1}\right)& =w_{\frac{\lambda \cdot m}{m}}\left(T^{m+1}x,y_{m+1}\right)\\ \le w_{\frac{\lambda \cdot \left(m-1\right)}{m}}\left(T^{m+1}x,T{y}_m\right)+w_{\frac{\lambda }{m}}\left(T{y}_m,y_{m+1}\right)\\ \le k{w}_{\frac{\lambda \cdot \left(m-1\right)}{m}}\left(T^{m}x,y_m\right)+{\epsilon }_m\\ \le k{w}_{\frac{\lambda \cdot \left(m-2\right)}{m}}\left(T^{m}x,T{y}_{m-1}\right)+k{w}_{\frac{\lambda }{m}}\left(T{y}_{m-1}x,y_m\right)+{\epsilon }_m\\ \le k^2{w}_{\frac{\lambda \cdot \left(m-2\right)}{m}}\left(T^{m-1}x,y_{m-1}\right)+k{\epsilon }_{m-1}+{\epsilon }_m\\ ⋮\\ \le \sum _{i=0}^m{k}^{m-i}{\epsilon }_i\end{array}$$

(3.3)

for all λ > 0. Thus, we get

$$\begin{array}{ll}\hfill w_{\lambda }\left(y_{m+1},x^{*}\right)& \le w_{\frac{\lambda }{2}}\left(y_{m+1},T^{m+1}x\right)+w_{\frac{\lambda }{2}}\left(T^{m+1}x,x^{*}\right)\\ \le \sum _{i=0}^m{k}^{m-i}{\epsilon }_i+w_{\frac{\lambda }{2}}\left(T^{m+1}x,x^{*}\right).\end{array}$$

(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,

$$\begin{array}{ll}\hfill \sum _{i=0}^m{k}^{m-i}{\epsilon }_i& =\sum _{i=0}^N{k}^{m-i}{\epsilon }_i+\sum _{i=N+1}^m{k}^{m-i}{\epsilon }_i\\ \le k^{m-N}\sum _{i=0}^N{k}^{N-i}{\epsilon }_i+\epsilon \sum _{i=N+1}^m{k}^{m-i}.\end{array}$$

(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 T^N is a contraction mapping for some positive integer N. Then, T has a unique fixed point in X _w .

Proof. By Theorem 3.2 , T^N has a unique fixed point u ∈ X _w . From T^N(T _u ) = T^N+1u = T(T^Nu) = Tu, so Tu is a fixed point of T^N. By the uniqueness of fixed point of T^N, we have Tu = u. Thus, u is a fixed point of T. Since fixed point of T is also fixed point of T^N, 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

$$B_w\left(x^{*},\gamma \right):=\left\{x\in X_w|w_{\lambda }\left(x,x^{*}\right)\le \gamma forall\lambda >0\right\}.$$

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 Tx ∈ B _w (x*, γ), for all x ∈ B _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 x ∈ X _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 x ∈ B _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 Tx ∈ B _w (x*, γ) for all x ∈ B _w (x*, γ). Let x ∈ B _w (x*, γ). From the inequalities (3.9), using the contraction of T and the notion of metric modular, we have

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

Therefore, Tx ∈ B _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, y ∈ X _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 x ∈ X _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 = Tx_n-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 Tx_n-1≠ Tx _n for all n ∈ ℕ. For $k\in \left[0,\frac{1}{2}\right)$, we have

$$\begin{array}{ll}\hfill w_{\lambda }\left(x_{n+1},x_n\right)& =w_{\lambda }\left(T{x}_n,T{x}_{n-1}\right)\\ \le k\left(w_{2\lambda }\left(T{x}_n,x_n\right)+w_{2\lambda }\left(T{x}_{n-1},x_{n-1}\right)\right)\\ \le k\left(w_{\lambda }\left(x_{n+1},x_n\right)+w_{\lambda }\left(x_n,x_{n-1}\right)\right)\end{array}$$

(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 x ∈ X _w such that x _n → x as n → ∞. By the property of metric modular and the inequality (3.12), we have

$$\begin{array}{ll}\hfill w_{\lambda }\left(Tx,x\right)& \le w_{\frac{\lambda }{2}}\left(Tx,T{x}_n\right)+w_{\frac{\lambda }{2}}\left(T{x}_n,x\right)\\ \le k\left(w_{\lambda }\left(Tx,x\right)+w_{\lambda }\left(T{x}_n,x_n\right)\right)+w_{\frac{\lambda }{2}}\left(T{x}_n,x\right)\\ \le k\left(w_{\lambda }\left(Tx,x\right)+w_{\frac{\lambda }{2}}\left(T{x}_n,x\right)+w_{\frac{\lambda }{2}}\left(x,x_n\right)\right)+w_{\frac{\lambda }{2}}\left(T{x}_n,x\right)\\ =k\left(w_{\lambda }\left(Tx,x\right)+w_{\frac{\lambda }{2}}\left(x_{n+1},x\right)+w_{\frac{\lambda }{2}}\left(x,x_n\right)\right)+w_{\frac{\lambda }{2}}\left(x_{n+1},x\right)\end{array}$$

(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

$$\begin{array}{ll}\hfill w_{\lambda }\left(x,z\right)& =w_{\lambda }\left(Tx,Tz\right)\\ \le k\left(w_{\frac{\lambda }{2}}\left(Tx,x\right)+w_{\frac{\lambda }{2}}\left(Tz,z\right)\right)\\ =0\end{array}$$

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

$$w_{\lambda }\left(T\left(\left(a_1,b_1\right)\right),T\left(\left(a_2,b_2\right)\right)\right)\le \frac{3}{4}{w}_{\lambda }\left(\left(a_1,b_1\right),\left(a_2,b_2\right)\right)$$

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.