Fixed Point Results for Weakly C-Contraction Mapping in Modular Metric Spaces
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.

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 $T:X\to X$ where $\left(X,d\right)$ is a metric space

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

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

$d\left(Tx,Ty\right)\le \alpha \left(d\left(x,Ty\right),d\left(y,Tx\right)\right)$ (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 $\left(X,d\right)$ be a metric space and $T:X\to X$ be a map. Then T is called a weakly C-contraction (or a weak C-contraction) if there exists $\phi :{\left[0\to \infty \right)}^{2}\to \left[0\to \infty \right)$ which is continuous, and $\phi \left(x,y\right)=0$ if and only if $x=y=0$ such that

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

for all $x,y\in X$ .

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 $ℕ$ 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 $w:\left(0,\infty \right)×X×X\to \left[0,\infty \right)$ , we write

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

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

Definition 2.1. [9] Let X be a nonempty set. A function $w:\left(0,\infty \right)×X×X\to \left[0,\infty \right)$ is said to be a metric modular on X if it satisfies, for all $x,y,z\in X$ , 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) ${w}_{\lambda +\mu }\left(x,y\right)\le {w}_{\lambda }\left(x,z\right)+{w}_{\mu }\left(z,y\right)$ for all $\lambda ,\mu >0$ .

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

${w}_{\lambda }\left(x,x\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{forall}\text{\hspace{0.17em}}\lambda >0,x\in X,$

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 $\lambda ↦{w}_{\lambda }\left(x,y\right)$ is non increasing for all $x,y\in X$ .

Definition 2.2. [9] Let w is a pseudomodular on X. Fixed ${x}_{0}\in X$ . The set

${X}_{w}={X}_{w}\left({x}_{0}\right)=\left\{x\in X:{w}_{\lambda }\left(x,{x}_{0}\right)\to 0\text{as}\lambda \to \infty \right\}$

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 ${\left\{{x}_{n}\right\}}_{n\in ℕ}$ in ${X}_{w}$ is said to be w-convergent to $x\in {X}_{w}$ if and only if ${w}_{\lambda }\left({x}_{n},x\right)\to 0$ , as $n\to \infty$ for some $\lambda >0$ ;

2) The sequence ${\left\{{x}_{n}\right\}}_{n\in ℕ}$ in ${X}_{w}$ is said to be w-Cauchy if ${w}_{\lambda }\left({x}_{m},{x}_{n}\right)\to 0$ as $m,n\to \infty$ 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 $T:{X}_{w}\to {X}_{w}$ be an arbitrary mapping T is called a contraction if for each $x,y\in {X}_{w}$ and for all $\lambda >0$ there exists $0\le k<1$ such that

${w}_{\lambda }\left(Tx,Ty\right)\le k{w}_{\lambda }\left(x,y\right).$ (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 $T:{X}_{w}\to {X}_{w}$ . If

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

for all $x,y\in {X}_{w}$ and for all $\lambda >0$ , where $k\in \left[0,\frac{1}{2}\right)$ , 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 $T:{X}_{w}\to {X}_{w}$ . If

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

for all $x,y\in {X}_{w}$ and for all $\lambda >0$ , where $k\in \left[0,\frac{1}{2}\right)$ , 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 $n\in ℕ$ . If $T{x}_{{n}_{0}-1}=T{x}_{{n}_{0}}$ for some ${n}_{0}\in ℕ$ , 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 $n\in ℕ$ . For $k\in \left[0,\frac{1}{2}\right)$ , 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 $n\in ℕ$ . Hence,

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

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

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

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

$\epsilon$ , there exists ${n}_{\lambda /\left(m-n\right)}\in ℕ$ 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 $m,n\ge {n}_{\lambda /\left(m-n\right)}\in ℕ$ . This implies ${\left\{{x}_{n}\right\}}_{n\in ℕ}$ 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 $n\in ℕ$ . Taking $n\to \infty$ in inequality (12), we obtained that

${w}_{\lambda }\left(Tx,x\right)\le k{w}_{\lambda }\left(Tx,x\right).$ (13)

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

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

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

$\left(1-2k\right){w}_{\lambda }\left(x,z\right)\le 0.$

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 $T:{X}_{w}\to {X}_{w}$ is said to be a weak C-contraction in ${X}_{w}$ if for all $x,y\in {X}_{w}$ 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 $\phi {\left[0,\infty \right)}^{2}\to \left[0,\infty \right)$ 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 $T:{X}_{w}\to {X}_{w}$ 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 $n\in ℕ$ . If $T{x}_{{n}_{0}-1}=T{x}_{{n}_{0}}$ for some ${n}_{0}\in ℕ$ , 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 $n\in ℕ$ , 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

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

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

For each $\lambda >0$ , let

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

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

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

or, equivalently,

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

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

$\begin{array}{c}{w}_{\lambda }\left(Tx,x\right)\le {w}_{\frac{\lambda }{2}}\left(Tx,T{x}_{n}\right)+{w}_{\frac{\lambda }{2}}\left(T{x}_{n},x\right)\\ \le \frac{1}{2}\left({w}_{\lambda }\left(x,T{x}_{n}\right)+{w}_{\lambda }\left({x}_{n},Tx\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\phi \left({w}_{\lambda }\left(x,T{x}_{n}\right),{w}_{\lambda }\left({x}_{n},Tx\right)\right)+{w}_{\frac{\lambda }{2}}\left(T{x}_{n},x\right)\\ =\frac{1}{2}\left({w}_{\lambda }\left(x,{x}_{n+1}\right)+{w}_{\lambda }\left({x}_{n},Tx\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\phi \left({w}_{\lambda }\left(x,{x}_{n+1}\right),{w}_{\lambda }\left({x}_{n},Tx\right)\right)+{w}_{\frac{\lambda }{2}}\left({x}_{n+1},x\right),\end{array}$ (22)

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

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

$\begin{array}{c}{w}_{\lambda }\left(z,x\right)={w}_{\lambda }\left(Tz,Tx\right)\\ \le \frac{1}{2}\left({w}_{2\lambda }\left(z,Tx\right)+{w}_{2\lambda }\left(x,Tz\right)\right)-\phi \left({w}_{\lambda }\left(z,Tx\right),{w}_{\lambda }\left(x,Tz\right)\right)\\ \le {w}_{2\lambda }\left(x,z\right)-\phi \left({w}_{\lambda }\left(z,x\right),{w}_{\lambda }\left(x,z\right)\right),\end{array}$ (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 $w:\left(0,\infty \right)×X×X\to \left[0,\infty \right)$ by

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

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

and

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

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

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

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

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 $x,y\in X$ and all $\lambda >0$ .

Indeed, case1. let $x=\left({a}_{1},0\right),y=\left({a}_{2},0\right)$ , then

${w}_{\lambda }\left(Tx,Ty\right)={w}_{\lambda }\left(T\left({a}_{1},0\right),T\left({a}_{2},0\right)\right)={w}_{\lambda }\left(\left(0,\frac{{a}_{1}}{2}\right),\left(0,\frac{{a}_{2}}{2}\right)\right)=\frac{|{a}_{1}-{a}_{2}|}{2\lambda },$ (25)

${w}_{2\lambda }\left(x,Ty\right)={w}_{2\lambda }\left(\left({a}_{1},0\right),T\left({a}_{2},0\right)\right)={w}_{2\lambda }\left(\left({a}_{1},0\right),\left(0,\frac{{a}_{2}}{2}\right)\right)=\frac{3{a}_{1}}{2\lambda }+\frac{{a}_{2}}{4\lambda },$ (26)

${w}_{2\lambda }\left(y,Tx\right)={w}_{2\lambda }\left(\left({a}_{2},0\right),T\left({a}_{1},0\right)\right)={w}_{2\lambda }\left(\left({a}_{2},0\right),\left(0,\frac{{a}_{1}}{2}\right)\right)=\frac{3{a}_{2}}{2\lambda }+\frac{{a}_{1}}{4\lambda },$ (27)

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

Case 2. let $x=\left(0,{b}_{1}\right),y=\left(0,{b}_{2}\right)$ , we have

${w}_{\lambda }\left(Tx,Ty\right)={w}_{\lambda }\left(T\left(0,{b}_{1}\right),T\left(0,{b}_{2}\right)\right)={w}_{\lambda }\left(\left(\frac{{b}_{1}}{24},0\right),\left(\frac{{b}_{2}}{24},0\right)\right)=\frac{|{b}_{1}-{b}_{2}|}{8\lambda },$ (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 $x=\left(a,0\right),y=\left(0,b\right)$ , 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)

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

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

$\begin{array}{c}\phi \left({w}_{\lambda }\left(x,Ty\right),{w}_{\lambda }\left(y,Tx\right)\right)=\frac{1}{20}\left({w}_{\lambda }\left(\left(x,Ty\right)+{w}_{\lambda }\left(y,Tx\right)\right)\\ =\frac{1}{20}\left[2\left({w}_{2\lambda }\left(x,Ty\right)+{w}_{2\lambda }\left(y,Tx\right)\right)\right]\\ =\frac{1}{10}\left({w}_{2\lambda }\left(x,Ty\right)+{w}_{2\lambda }\left(y,Tx\right)\right).\end{array}$ (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 $x,y\in X$ . 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 $x,y\in X$ . 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 $x,y\in X$ and all $\lambda >0$ . Thus T is a weakly C-contractive mapping. Therefore, T has a unique fixed point that is $\left(0,0\right)\in {X}_{w}$ .

On the Euclidean metric d on ${X}_{w}$ , we see that

$\begin{array}{c}d\left(T\left(1,0\right),T\left(0,\frac{1}{2}\right)\right)>\frac{1}{2}\left(d\left(T\left(1,0\right),T\left(0,\frac{1}{2}\right)\right)+d\left(\left(0,\frac{1}{2}\right),T\left(1,0\right)\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\phi \left(d\left(\left(1,0\right),T\left(0,\frac{1}{2}\right)\right),d\left(\left(0,\frac{1}{2}\right),T\left(1,0\right)\right)\right).\end{array}$ (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.

Cite this paper: Zhao, J. , Zhao, Q. , Jin, B. and Zhong, L. (2018) Fixed Point Results for Weakly C-Contraction Mapping in Modular Metric Spaces. Open Access Library Journal, 5, 1-9. doi: 10.4236/oalib.1104061.
References

[1]   Banach, S. (1992) Sur les opérations dans les ensembles abtraits et leur applications aux équations intégrales. Fundamenta Mathematicae, 3, 133-181.
https://doi.org/10.4064/fm-3-1-133-181

[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.
https://doi.org/10.1016/j.na.2009.10.026

[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.
https://doi.org/10.1016/j.camwa.2010.12.027

[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.
https://doi.org/10.1134/S106456240601008X

[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.
https://doi.org/10.1016/j.na.2009.04.057

[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.
https://doi.org/10.1186/s13663-016-0525-4

[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.

Top