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.

Subject Areas:

Mathematical Analysis

Keywords:

Modular Metric Spaces Weakly C-Contraction Fixed Point Theory

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.

Cite this paper

Zhao, J.W., Zhao, Q.Q., Jin, B. and Zhong, L.N. (2018) Fixed Point Results for Weakly C-Contraction Mapping in Modular Metric Spaces. Open Access Library Journal, 5: e4061. https://doi.org/10.4236/oalib.1104061

References

NOTES

*Co-first authors.

^#Corresponding authors.