Jinwei Zhao^*, Qianqian Zhao, Bo Jin, Linan Zhong
Department of Mathematics, Yanbian University, Yanji, China.
DOI: 10.4236/oalib.1104061   PDF    HTML   XML   858 Downloads   3,002 Views   Citations

Abstract

In this paper, we introduce the concept of weakly C-contraction mapping in modular metric spaces. And we established some fixed point results in w-complete spaces. Our results encompass various generalizations of Banach contraction.

Keywords

Modular Metric Spaces Weakly C-Contraction Fixed Point Theory

Share and Cite:

1. Introduction

Fixed point theory has absorbed many mathematicians since 1922 with the celebrated Banach contraction principle (see [1] ). It is one of the most useful results in nonlinear analysis, functional analysis and topology. Due to its application in mathematics, the Banach contraction principle has been generalized in many directions (see [2] [3] [4] ).

Chatteriea in [5] introduced the notion of C-contraction which is a generalization of the Banach contraction.

Definition 1.1. [5] A mapping $T\mathrm{<mo>\; :\; </mo>}X\to X$ where $\left(X\mathrm{,}d\right)$ is a metric space

is said to be a C-contraction if there exists $\alpha \in \left[\mathrm{0,}\frac{1}{2}\right)$ such that for all

$x\mathrm{,}y\in X$ the following inequality holds:

$d\left(Tx\mathrm{,}Ty\right)\le \alpha \left(d\left(x\mathrm{,}Ty\right)\mathrm{,}d\left(y\mathrm{,}Tx\right)\right)$ (1)

Chatteriea in [5] proved that if X is complete, then every C-contraction mapping have a unique fixed point.

The notion of C-contraction was generalized to a weak C-contraction by Choudhury in [6] .

Definition 1.2. [6] Let $\left(X\mathrm{,}d\right)$ be a metric space and $T\mathrm{<mo>\; :\; </mo>}X\to X$ be a map. Then T is called a weakly C-contraction (or a weak C-contraction) if there exists $\phi \mathrm{<mo>\; :\; </mo>}{\left[0\to \infty \right)}^2\to \left[0\to \infty \right)$ which is continuous, and $\phi \left(x,y\right)=0$ if and only if $x=y=0$ such that

$d\left(Tx\mathrm{,}Ty\right)\le \frac{1}{2}\left[d\left(x\mathrm{,}Ty\right)+d\left(y\mathrm{,}Tx\right)\right]-\phi \left(d\left(x\mathrm{,}Ty\right)\mathrm{,}d\left(y\mathrm{,}Tx\right)\right)\mathrm{,}$ (2)

for all $x\mathrm{,}y\in X$ .

In [6] the author proved that if X is a complete metric space, then every weakly C-contraction has a unique fixed point. This fixed point theory was generalized to a complete, partially ordered metric space in [7] and a ordered 2-metric space in [8] .

In 2006, Chistyakov introduced the notion of modular metric space in [9] . Recently, there have been many interesting results in the field of existence and uniqueness of fixed point in complete modular metric (see [10] [11] ). In this paper, we will establish fixed point theorems for weakly C-contraction in modular metric space. The presented results extend some recent results in the literature.

2. Preliminaries

Throughout this paper $\mathbb{N}$ will denote the set of natural numbers.

The notion of modular metric space was introduced by Chistyakov in [9] [12] [13] , who proved some fixed point results in such kind of spaces.

Let X be a nonempty set. Throughout this paper, for a function $w\mathrm{<mo>\; :\; </mo>}\left(\mathrm{0,}\infty \right)\times X\times X\to \left[\mathrm{0,}\infty \right)$ , we write

$w_{\lambda }\left(x,y\right)=w\left(\lambda ,x,y\right),$ (3)

for all $\lambda >0$ and $x\mathrm{,}y\in X$ .

Definition 2.1. [9] Let X be a nonempty set. A function $w\mathrm{<mo>\; :\; </mo>}\left(\mathrm{0,}\infty \right)\times X\times X\to \left[\mathrm{0,}\infty \right)$ is said to be a metric modular on X if it satisfies, for all $x\mathrm{,}y\mathrm{,}z\in X$ , the following condition:

1) $w_{\lambda }\left(x,y\right)=0$ for all $\lambda >0$ if and only if $x=y$ ;

2) $w_{\lambda }\left(x,y\right)=w_{\lambda }\left(y,x\right)$ for all $\lambda >0$ ;

3) $w_{\lambda +\mu }\left(x\mathrm{,}y\right)\le w_{\lambda }\left(x\mathrm{,}z\right)+w_{\mu }\left(z\mathrm{,}y\right)$ for all $\lambda ,\mu >0$ .

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

$w_{\lambda }\left(x,x\right)=0\text{forall}\lambda >0,\mathrm{<mi>\; </mi>}x\in X,$

then w is said to be a pseudomodular (metric) on X.

An important property of the (metric) pseudomodular on set X is that the mapping $\lambda \mapsto w_{\lambda }\left(x\mathrm{,}y\right)$ is non increasing for all $x\mathrm{,}y\in X$ .

Definition 2.2. [9] Let w is a pseudomodular on X. Fixed $x_0\in X$ . The set

$X_w=X_w\left(x_0\right)=\left\{x\in X:w_{\lambda }\left(x,x_0\right)\to 0\text{as}\mathrm{<mi>\; </mi>}\lambda \to \infty \right\}$

is said to be a modular metric space (around $x_0$ ).

Definition 2.3. [14] Let $X_w$ be a modular metric space.

1) The sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ in $X_w$ is said to be w-convergent to $x\in X_w$ if and only if $w_{\lambda }\left(x_n\mathrm{,}x\right)\to 0$ , as $n\to \infty$ for some $\lambda >0$ ;

2) The sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ in $X_w$ is said to be w-Cauchy if $w_{\lambda }\left(x_m\mathrm{,}{x}_n\right)\to 0$ as $m\mathrm{,\; <mi>\; </mi>}n\to \infty$ for some $\lambda >0$ ;

3) A subset C of $X_w$ is said to be w-complete if any w-Cauchy sequence in C is a convergent sequence and its limit is in C.

Definition 2.4. [15] Let w be a metric modular on X and $X_w$ be a modular metric space induced by w. If $X_w$ is a w-complete modular metric space and $T\mathrm{<mo>\; :\; </mo>}{X}_w\to X_w$ be an arbitrary mapping T is called a contraction if for each $x\mathrm{,}y\in X_w$ and for all $\lambda >0$ there exists $0\le k<1$ such that

$w_{\lambda }\left(Tx\mathrm{,}Ty\right)\le k{w}_{\lambda }\left(x\mathrm{,}y\right)\mathrm{.}$ (4)

In [15] Chirasak proved that if $X_w$ is a w-complete modular metric space, then contraction mapping T has a unique fixed point. At the same time, the author proved the following theorem.

Theorem 2.5. [15] Let w be a metric modular on X, $X_w$ be a w-complete modular metric space induced by w and $T\mathrm{<mo>\; :\; </mo>}{X}_w\to X_w$ . If

$w_{\lambda }\left(Tx\mathrm{,}Ty\right)\le k\left(w_{2\lambda }\left(Tx\mathrm{,}x\right)+w_{2\lambda }\left(Ty\mathrm{,}y\right)\right)\mathrm{,}$ (5)

for all $x\mathrm{,}y\in X_w$ and for all $\lambda >0$ , where $k\in \left[\mathrm{0,}\frac{1}{2}\right)$ , then T has a unique

fixed point in $X_w$ . Moreover, for any $x\in X_w$ , iterative sequence $\left\{T^{n}x\right\}$ converges to the fixed point.

3. Main Results

Theorem 3.1. Let w be a metric modular on X, $X_w$ be a w-complete modular metric space induced by w and $T\mathrm{<mo>\; :\; </mo>}{X}_w\to X_w$ . If

$w_{\lambda }\left(Tx\mathrm{,}Ty\right)\le k\left(w_{2\lambda }\left(x\mathrm{,}Ty\right)+w_{2\lambda }\left(y\mathrm{,}Tx\right)\right)\mathrm{,}$ (6)

for all $x\mathrm{,}y\in X_w$ and for all $\lambda >0$ , where $k\in \left[\mathrm{0,}\frac{1}{2}\right)$ , then T has a unique

fixed point in $X_w$ .

Proof. Let $x_0$ be an arbitrary point in $X_w$ and we write $x_1=T{x}_0$ ,

$x_2=T{x}_1=T^2{x}_0$ , and in general, $x_n=T{x}_{n-1}=T^2{x}_0$ for all $n\in \mathbb{N}$ . If $T{x}_{n_0-1}=T{x}_{n_0}$ for some $n_0\in \mathbb{N}$ , then $T{x}_{n_0}=x_{n_0}$ . Thus $x_{n_0}$ is a fixed point of T. Suppose that

$T{x}_{n-1}\ne T{x}_n$ for all $n\in \mathbb{N}$ . For $k\in \left[\mathrm{0,}\frac{1}{2}\right)$ , we have

$\begin{array}{c}{w}_{\lambda }\left(x_{n+1},x_n\right)=w_{\lambda }\left(T{x}_n,T{x}_{n-1}\right)\\ \le k\left(w_{2\lambda }\left(x_n,T{x}_{n-1}\right)+w_{2\lambda }\left(x_{n-1},T{x}_n\right)\right)\\ =k{w}_{2\lambda }\left(x_{n-1},x_{n+1}\right)\\ \le k\left(w_{\lambda }\left(x_{n-1},x_n\right)+w_{\lambda }\left(x_n,x_{n+1}\right)\right),\end{array}$ (7)

for all $\lambda >0$ and all $n\in \mathbb{N}$ . Hence,

$w_{\lambda }\left(x_{n+1}\mathrm{,}{x}_n\right)\le \frac{k}{1-k}{w}_{\lambda }\left(x_n\mathrm{,}{x}_{n-1}\right)\mathrm{,}$ (8)

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

and hence

$w_{\lambda }\left(x_{n+1},x_n\right)\le \beta w_{\lambda }\left(x_n,x_{n-1}\right)\le {\beta }^2{w}_{\lambda }\left(x_{n-1},x_{n-2}\right)\le \cdots \le {\beta }^n{w}_{\lambda }\left(x_1,x_0\right),$ (9)

for all $\lambda >0$ and each $n\in \mathbb{N}$ . Therefore, $\underset{n\to \infty }{\lim }{w}_{\lambda }\left(x_{n+1},x_n\right)=0$ for all $\lambda >0$ . So for each $\lambda >0$ , we have for all $\epsilon >0$ there exists $n_0\in \mathbb{N}$ such that $w_{\lambda }\left(x_{n+1},x_n\right)<\epsilon$ for all $n\in \mathbb{N}$ with $n\ge n_0$ . Without loss of generality, suppose

$m\mathrm{,}n\in \mathbb{N}$ and $m>n$ . Observe that, for $\frac{\lambda }{m-n}>0$ and for above-mentioned

$\epsilon$ , there exists $n_{\lambda /\left(m-n\right)}\in \mathbb{N}$ such that

$w_{\frac{\lambda }{m-n}}\left(x_{n+1},x_n\right)<\frac{\epsilon }{m-n},$ (10)

for all $n\ge n_{\lambda /\left(m-n\right)}$ . Now we have

$\begin{array}{c}{w}_{\lambda }\left(x_n,x_m\right)\le w_{\frac{\lambda }{m-n}}\left(x_n,x_{n+1}\right)+w_{\frac{\lambda }{m-n}}\left(x_{n+1},x_{n+2}\right)+\cdots +w_{\frac{\lambda }{m-n}}\left(x_{m-1},x_m\right)\\ <\frac{\epsilon }{m-n}+\frac{\epsilon }{m-n}+\cdots +\frac{\epsilon }{m-n}=\epsilon ,\end{array}$ (11)

for all $m\mathrm{,\; <mi>\; </mi>}n\ge n_{\lambda /\left(m-n\right)}\in \mathbb{N}$ . This implies ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ is a Cauchy sequence. By the completeness of $X_w$ , there exists point $x\in X_w$ , such that $x_n\to x$ as $n\to \infty$ .

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

$\begin{array}{c}{w}_{\lambda }\left(Tx,x\right)\le w_{\frac{\lambda }{2}}\left(Tx,T{x}_n\right)+w_{\frac{\lambda }{2}}\left(T{x}_n,x\right)\\ \le k\left(w_{\lambda }\left(x,T{x}_n\right)+w_{\lambda }\left(x_n,Tx\right)\right)+w_{\frac{\lambda }{2}}\left(T{x}_n,x\right)\\ =k\left(w_{\lambda }\left(x,x_{n+1}\right)+w_{\lambda }\left(x_n,Tx\right)\right)+w_{\frac{\lambda }{2}}\left(x_{n+1},x\right),\end{array}$ (12)

for all $\lambda >0$ and for all $n\in \mathbb{N}$ . Taking $n\to \infty$ in inequality (12), we obtained that

$w_{\lambda }\mathrm{<mo\; stretchy="false">\; (\; </mo>}Tx\mathrm{,}x\mathrm{<mo\; stretchy="false">\; )\; </mo>}\le k{w}_{\lambda }\mathrm{<mo\; stretchy="false">\; (\; </mo>}Tx\mathrm{,}x\mathrm{<mo\; stretchy="false">\; )\; </mo>.}$ (13)

Since $k\in \left[\mathrm{0,}\frac{1}{2}\right)$ , 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

$\begin{array}{c}{w}_{\lambda }\left(x,z\right)=w_{\lambda }\left(Tx,Tz\right)\\ \le k\left(w_{2\lambda }\left(x,Tz\right)+w_{2\lambda }\left(z,Tx\right)\right)\\ \le k\left(w_{\lambda }\left(x,z\right)+w_{\lambda }\left(z,Tz\right)+w_{\lambda }\left(z,x\right)+w_{\lambda }\left(x,Tx\right)\right)\\ =2k{w}_{\lambda }\left(x,z\right),\end{array}$ (14)

for all $\lambda >0$ . Therefore we have

$\mathrm{<mo\; stretchy="false">\; (\; </mo>1}-2k\mathrm{<mo\; stretchy="false">\; )\; </mo>}{w}_{\lambda }\mathrm{<mo\; stretchy="false">\; (\; </mo>}x\mathrm{,}z\mathrm{<mo\; stretchy="false">\; )\; </mo>}\le 0.$

Since $1-2k>0$ , we can imply that $x=z$ . Therefore, x is a unique fixed point of T. $\square$

Next, we will introduce the notion of weakly C-contraction in modular metric space.

Definition 3.2. Let w be a metric modular on X, $X_w$ be a modular metric space induced by w. A mapping $T\mathrm{<mo>\; :\; </mo>}{X}_w\to X_w$ is said to be a weak C-contraction in $X_w$ if for all $x\mathrm{,}y\in X_w$ and for all $\lambda >0$ , the following inequality holds:

$w_{\lambda }\left(Tx,Ty\right)\le \frac{1}{2}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right)-\phi \left(w_{\lambda }\left(x,Ty\right),w_{\lambda }\left(y,Tx\right)\right),$ (15)

where $\phi {\left[\mathrm{0,}\infty \right)}^2\to \left[\mathrm{0,}\infty \right)$ is a continuous mapping such that $\phi \left(x,y\right)=0$ if and only if $x=y$ .

Theorem 3.3. Let w be a metric modular on X, $X_w$ be a w-complete modular metric space induced by w. Let $T\mathrm{<mo>\; :\; </mo>}{X}_w\to X_w$ be a weak C-contraction in $X_w$ such that T is continuous and non-decreasing. Then T has a unique fixed point.

Proof. Let $x_0$ be an arbitrary point in $X_w$ and we write $x_1=T{x}_0$ , $x_2=T{x}_1=T^2{x}_0$ , and in general, $x_n=T{x}_{n-1}=T^2{x}_0$ for all $n\in \mathbb{N}$ . If $T{x}_{n_0-1}=T{x}_{n_0}$ for some $n_0\in \mathbb{N}$ , then $T{x}_{n_0}=x_{n_0}$ . Thus $x_{n_0}$ is a fixed point of T. Suppose that $T{x}_{n-1}\ne T{x}_n$ for all $n\in \mathbb{N}$ , we have

$\begin{array}{l}{w}_{\lambda }\left(x_{n+1},x_n\right)=w_{\lambda }\left(T{x}_n,T{x}_{n-1}\right)\\ \le \frac{1}{2}\left(w_{2\lambda }\left(x_n,T{x}_{n-1}\right)+w_{2\lambda }\left(x_{n-1},T{x}_n\right)\right)-\phi \left(w_{\lambda }\left(x_n,T{x}_{n-1}\right),w_{\lambda }\left(x_{n-1},T{x}_n\right)\right)\\ =\frac{1}{2}\left(w_{2\lambda }\left(x_n,x_n\right)+w_{2\lambda }\left(x_{n-1},x_{n+1}\right)\right)-\phi \left(w_{\lambda }\left(x_n,x_n\right),w_{\lambda }\left(x_{n-1},x_{n+1}\right)\right)\\ =\frac{1}{2}{w}_{2\lambda }\left(x_{n-1},x_{n+1}\right)-\phi \left(0,w_{\lambda }\left(x_{n-1},x_{n+1}\right)\right)\\ \le \frac{1}{2}{w}_{2\lambda }\left(x_{n-1},x_{n+1}\right)\le \frac{1}{2}\left(w_{\lambda }\left(x_{n-1},x_n\right)+w_{\lambda }\left(x_n,x_{n+1}\right)\right),\end{array}$ (16)

for all $\lambda >0$ . The last inequality gives us

$w_{\lambda }\left(x_n\mathrm{,}{x}_{n+1}\right)\le w_{\lambda }\left(x_{n-1}\mathrm{,}{x}_n\right)\mathrm{,}$

for all $\lambda >0$ and for all $n\in \mathbb{N}$ . Thus $\left\{w_{\lambda }\left(x_n\mathrm{,}{x}_{n+1}\right)\right\}$ is a decreasing sequence of nonnegative real numbers and hence it is convergent.

For each $\lambda >0$ , let

$\underset{n\to \infty }{\lim }{w}_{\lambda }\left(x_n,x_{n+1}\right)=r.$ (17)

Letting $n\to \infty$ in (16) we have

$r\le {\lim }_{n\to \infty }\frac{1}{2}{w}_{\lambda }\left(x_{n-1},x_{n+1}\right)\le \frac{1}{2}\left(r+r\right)=r.$ (18)

or, equivalently,

${\lim }_{n\to \infty }{w}_{\lambda }\left(x_{n-1},x_{n+1}\right)=2r.$ (19)

Again, making $n\to \infty$ in (17), (19) and the continuity of $\phi$ we have

$r\le \frac{1}{2}2r-\phi \left(0,2r\right)=r-\phi \left(0,2r\right)\le r.$ (20)

And, consequently, $\phi \left(0,2r\right)=0$ . This gives us that $r=0$ by our assumption about $\phi$ .

Thus, for all $\lambda >0$ , we have

${\lim }_{n\to \infty }{w}_{\lambda }\left(x_n,x_{n+1}\right)=0.$ (21)

From the proof of theorem 3.1, we can prove that $\left\{x_n\right\}$ is a w-Cauchy sequence. By the completeness of $X_w$ , there exists a point $x\in X_w$ , such that $x_n\to x$ as $n\to \infty$ .

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

$\begin{array}{c}{w}_{\lambda }\left(Tx,x\right)\le w_{\frac{\lambda }{2}}\left(Tx,T{x}_n\right)+w_{\frac{\lambda }{2}}\left(T{x}_n,x\right)\\ \le \frac{1}{2}\left(w_{\lambda }\left(x,T{x}_n\right)+w_{\lambda }\left(x_n,Tx\right)\right)\\ -\phi \left(w_{\lambda }\left(x,T{x}_n\right),w_{\lambda }\left(x_n,Tx\right)\right)+w_{\frac{\lambda }{2}}\left(T{x}_n,x\right)\\ =\frac{1}{2}\left(w_{\lambda }\left(x,x_{n+1}\right)+w_{\lambda }\left(x_n,Tx\right)\right)\\ -\phi \left(w_{\lambda }\left(x,x_{n+1}\right),w_{\lambda }\left(x_n,Tx\right)\right)+w_{\frac{\lambda }{2}}\left(x_{n+1},x\right),\end{array}$ (22)

for all $\lambda >0$ and for all $n\in \mathbb{N}$ . Taking $n\to \infty$ by (22), we obtained that

$w_{\lambda }\left(Tx\mathrm{,}x\right)\le \frac{1}{2}{w}_{\lambda }\left(Tx\mathrm{,}x\right)-\phi \left(\mathrm{0,}{w}_{\lambda }\left(Tx\mathrm{,}x\right)\right)\mathrm{.}$ (23)

This prove that $x=Tx$ . Thus x is a fixed point of T. Next, we prove that x is a unique fixed point. Suppose that z and x are different fixed points of T, then from (15), we have

$\begin{array}{c}{w}_{\lambda }\left(z\mathrm{,}x\right)=w_{\lambda }\left(Tz\mathrm{,}Tx\right)\\ \le \frac{1}{2}\left(w_{2\lambda }\left(z\mathrm{,}Tx\right)+w_{2\lambda }\left(x\mathrm{,}Tz\right)\right)-\phi \left(w_{\lambda }\left(z\mathrm{,}Tx\right)\mathrm{,}{w}_{\lambda }\left(x\mathrm{,}Tz\right)\right)\\ \le w_{2\lambda }\left(x\mathrm{,}z\right)-\phi \left(w_{\lambda }\left(z\mathrm{,}x\right)\mathrm{,}{w}_{\lambda }\left(x\mathrm{,}z\right)\right)\mathrm{,}\end{array}$ (24)

for all $\lambda >0$ By the property of the $\phi$ , we have $x=z$ . Hence x is a unique fixed point of T. $\square$

Example 3.4 Let $X=\left\{\left(a,0\right)\in R^2|a\ge 0\right\}\cup \left\{\left(0,b\right)\in R^2|b\ge 0\right\}$ . Defined the mapping $w\mathrm{<mo>\; :\; </mo>}\left(\mathrm{0,}\infty \right)\times X\times X\to \left[\mathrm{0,}\infty \right)$ by

$w_{\lambda }\left(\left(a_1\mathrm{,0}\right)\mathrm{,}\left(a_2\mathrm{,0}\right)\right)=\frac{3\left|a_1-a_2\right|}{\lambda }\mathrm{,}$

$w_{\lambda }\left(\left(\mathrm{0,}{b}_1\right)\mathrm{,}\left(\mathrm{0,}{b}_2\right)\right)=\frac{\left|b_1-b_2\right|}{\lambda }\mathrm{,}$

and

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

We note that if we take $\lambda \to \infty$ , then we see that $X=X_w$ and also T and $\phi$ is define by

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

$T\left(\left(\mathrm{0,}b\right)\right)=\left(\frac{b}{24}\mathrm{,0}\right)\mathrm{.}$

and

$\phi \left(x,y\right)=\frac{1}{20}\left(x+y\right).$

We can imply that

$w_{\lambda }\left(Tx,Ty\right)\le \frac{1}{2}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right)-\phi \left(w_{\lambda }\left(x,Ty\right),w_{\lambda }\left(y,Tx\right)\right)$ for all $x\mathrm{,}y\in X$ and all $\lambda >0$ .

Indeed, case1. let $x=\left(a_1,0\right),\mathrm{<mi>\; </mi>}y=\left(a_2,0\right)$ , then

$w_{\lambda }\left(Tx,Ty\right)=w_{\lambda }\left(T\left(a_1,0\right),T\left(a_2,0\right)\right)=w_{\lambda }\left(\left(0,\frac{a_1}{2}\right),\left(0,\frac{a_2}{2}\right)\right)=\frac{\left|a_1-a_2\right|}{2\lambda },$ (25)

$w_{2\lambda }\left(x,Ty\right)=w_{2\lambda }\left(\left(a_1,0\right),T\left(a_2,0\right)\right)=w_{2\lambda }\left(\left(a_1,0\right),\left(0,\frac{a_2}{2}\right)\right)=\frac{3a_1}{2\lambda }+\frac{a_2}{4\lambda },$ (26)

$w_{2\lambda }\left(y,Tx\right)=w_{2\lambda }\left(\left(a_2,0\right),T\left(a_1,0\right)\right)=w_{2\lambda }\left(\left(a_2,0\right),\left(0,\frac{a_1}{2}\right)\right)=\frac{3a_2}{2\lambda }+\frac{a_1}{4\lambda },$ (27)

$w_{\lambda }\left(Tx\mathrm{,}Ty\right)\le \frac{2}{7}\left(w_{2\lambda }\left(x\mathrm{,}Ty\right)+w_{2\lambda }\left(y\mathrm{,}Tx\right)\right)\mathrm{.}$ (28)

Case 2. let $x=\left(0,b_1\right),\mathrm{<mi>\; </mi>}y=\left(0,b_2\right)$ , we have

$w_{\lambda }\left(Tx,Ty\right)=w_{\lambda }\left(T\left(0,b_1\right),T\left(0,b_2\right)\right)=w_{\lambda }\left(\left(\frac{b_1}{24},0\right),\left(\frac{b_2}{24},0\right)\right)=\frac{\left|b_1-b_2\right|}{8\lambda },$ (29)

$w_{2\lambda }\left(x,Ty\right)=w_{2\lambda }\left(\left(0,b_1\right),T\left(0,b_2\right)\right)=w_{2\lambda }\left(\left(0,b_1\right),\left(\frac{b_2}{24},0\right)\right)=\frac{b_2}{16\lambda }+\frac{b_1}{2\lambda },$ (30)

$w_{\lambda }\left(Tx,Ty\right)\le \frac{2}{9}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right).$ (31)

Case 3. Let $x=\left(a,0\right),\mathrm{<mi>\; </mi>}y=\left(0,b\right)$ , then

$w_{\lambda }\left(Tx,Ty\right)=w_{\lambda }\left(T\left(a,0\right),T\left(0,b\right)\right)=w_{\lambda }\left(\left(0,\frac{a}{2}\right),\left(\frac{b}{24},0\right)\right)=\frac{b}{8\lambda }+\frac{a}{2\lambda },$ (32)

$w_{2\lambda }\left(x,Ty\right)=w_{2\lambda }\left(\left(a,0\right),T\left(0,b\right)\right)=w_{2\lambda }\left(\left(a,0\right),\left(\frac{b}{24},0\right)\right)=\left|\frac{b}{16\lambda }-\frac{3a}{2\lambda }\right|,$ (33)

$w_{2\lambda }\left(y,Tx\right)=w_{2\lambda }\left(\left(0,b\right),T\left(a,0\right)\right)=w_{2\lambda }\left(\left(0,b\right),\left(0\frac{a}{2}\right)\right)=\left|\frac{b}{2\lambda }-\frac{a}{4\lambda }\right|,$ (34)

$w_{\lambda }\left(Tx,Ty\right)\le \frac{2}{5}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right).$ (35)

$\begin{array}{c}\phi \left(w_{\lambda }\left(x,Ty\right),w_{\lambda }\left(y,Tx\right)\right)=\frac{1}{20}\left(w_{\lambda }(\left(x,Ty\right)+w_{\lambda }\left(y,Tx\right)\right)\\ =\frac{1}{20}\left[2\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right)\right]\\ =\frac{1}{10}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right).\end{array}$ (36)

Hence we have

$w_{\lambda }\left(Tx,Ty\right)\le \frac{2}{5}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right),$ (37)

for all $\lambda >0$ and $x\mathrm{,}y\in X$ . And

$\begin{array}{l}\frac{1}{2}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right)-\phi \left(w_{\lambda }\left(x,Ty\right),w_{\lambda }\left(y,Tx\right)\right)\\ =\frac{1}{2}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right)-\frac{1}{10}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right)\\ =\frac{2}{5}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right),\end{array}$ (38)

for all $\lambda >0$ and $x\mathrm{,}y\in X$ . We can get

$w_{\lambda }\left(Tx,Ty\right)\le \frac{1}{2}\left(w_{2\lambda }\left(x,Ty\right)+w_{2\lambda }\left(y,Tx\right)\right)-\phi \left(w_{\lambda }\left(x,Ty\right),w_{\lambda }\left(y,Tx\right)\right),$ (39)

for all $x\mathrm{,}y\in X$ and all $\lambda >0$ . Thus T is a weakly C-contractive mapping. Therefore, T has a unique fixed point that is $\left(\mathrm{0,0}\right)\in X_w$ .

On the Euclidean metric d on $X_w$ , we see that

$\begin{array}{c}d\left(T\left(1,0\right),T\left(0,\frac{1}{2}\right)\right)>\frac{1}{2}\left(d\left(T\left(1,0\right),T\left(0,\frac{1}{2}\right)\right)+d\left(\left(0,\frac{1}{2}\right),T\left(1,0\right)\right)\right)\\ -\phi \left(d\left(\left(1,0\right),T\left(0,\frac{1}{2}\right)\right),d\left(\left(0,\frac{1}{2}\right),T\left(1,0\right)\right)\right).\end{array}$ (40)

Thus, T is not a weak C-contraction on standard metric space.

4. Conclution

In this paper, we extend the fixed point results for the weakly C-contraction in modular metric space. Moreover, as example, we give a unique fixed point theorem for a mapping satisfying a weak C-contractive condition in modular metric space rather than in standard metric space. The main results of this article generalize and unify some recent results given by some authors.

NOTES

*Co-first authors.

^#Corresponding authors.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Banach, S. (1992) Sur les opérations dans les ensembles abtraits et leur applications aux équations intégrales. Fundamenta Mathematicae, 3, 133-181. [CrossRef]
[2]	Bakhtin, I.A. (1989) The Contraction Mapping Principle in Almost Metric Space. Functional Analysis, 30, 26-37.
[3]	Du, W.S. (2010) A Note on Cone Metric Spaces and Its Equivalence. Nonlinear Analysis: Theory, Methods & Applications, 72, 2259-2261. [CrossRef]
[4]	Azam, A., Fssher, B. and Khan, M. (2011) Common Fixed Points Theorems in Complete Valued Metric Spaces. Numerical Functional Analysis and Optimization, 32, 243-253.
[5]	Chatterjea, S.K. (1972) Fixed Point Theorems. Comptes Rendus de l’Academie bulgare des Sciences, 25, 727-730.
[6]	Choudhury, B.S. (2009) Unique Fixed Point Theorem for Weak C-Contractive Mapping. Journal of Engineering & Technology, 5, 6-13.
[7]	Harjani, J., López, B. and Sadarangani, K. (2011) Fixed Point Theorems for Weakly C-Contractive Mappings in Ordered Metric Spaces. Computers and Mathematics with Applications, 61, 790-769. [CrossRef]
[8]	Nguyen, V.D. and Vo Tie, L.H. (2013) Fixed Point Theorems for Weakly C-Contractions in Ordered 2-Metric Spaces. Fixed Point Theory and Applications, 2013, 161.
[9]	Chistyakov, W. (2006) Metric Modular Spaces and Their Application. Doklady Mathematics, 73, 32-35. [CrossRef]
[10]	Afrah, A.A. and Mohamed, A.K. (2013) Fixed Point Results of Pointwise Contractions in Modular Metric Spaces. Fixed Point Theory and Applications, 2013, 163.
[11]	Chirasak, M., Wutiphol, S. and Poom, K. (2011) Fixed Point Theorems for Contraction Mappings in Modular Metric Spaces. Fixed Point Theory and Applications, 2011, 93.
[12]	Chistyakov, W. (2010) Modular Metric Spaces, I: Basic Concepts. Nonlinear Analysis: Theory, Methods & Applications, 72, 1-14. [CrossRef]
[13]	Chistyakov, W. (2008) Modular Metric Spaces Generated by F-Modular. Folia Mathematica, 14, 3-25.
[14]	Padcharoen, A., Gopal, D., Chaipunya, P. and Kumam, P. (2016) Fixed Point and Periodic Point Results for α-Type F-Contractions in Modular Metric Spaces. Fixed Point Theory and Applications, 2016, 39. [CrossRef]
[15]	Chirasak, M., Wutiphol, S. and Poom, K. (2011) Fixed Point Theorems for Contraction Mappings in Modular Metric Spaces. Fixed Point Theory and Applications, 2011, 93.