Coupled Fixed Points for (φ,ψ) - Contractive Mappings in Partially Ordered Mod

Abstract

The Banach contraction principle is the most famous fixed point theorem. Many authors presented some new results for contractions in partially ordered metric spaces. Fixed point theorems in modular spaces, generalizing the classical Banach fixed point theorem in metric spaces, have been studied extensively by many mathematicians. The aim of this paper is to determine some coupled fixed point theorems for nonlinear contractive mappings in the framework of a modular space endowed with a partial order. Our results are generalizations of the fixed point theorems due to M. Mursaleen, S.A. Mohiuddine and R.P. Agarwal.

1. Introduction

In 1922, Banach established the most famous fundamental fixed point theorem, so-called the Banach contraction principle [1], which has played an important role in various fields of applied mathematical analysis. Fixed point theory is one of the most important theory in mathematics. The Banach contraction mapping principle has many applications to very different type of problems arise in different branches. Many authors have obtained many interesting extensions and generalizations (cf. [2, 3, 4, 5, 6, 7, 8]).

The more generalization was given by Nakano [9] in 1950 based on replacing the particular integral form of the functional by an abstract one. This functional was called modular. In 1959, this idea, which was the basis of the theory of modular spaces and initiated by Nakano, was refined and generalized by Musielak and Orlicz [10]. Modular spaces have been studied for almost 40 years and there is a large set of known applications of them in various parts of analysis. For more details about modular spaces, we refer the reader to [11, 12].

Fixed point theorems in modular spaces, generalizing the classical Banach fixed point theorem in metric spaces, have been studied extensively by many mathematicians, see [13, 14, 15, 16, 17, 18].

The author [19] has investigated some coupled coincidence and coupled common fixed point theorems for mixed g-monotone nonlinear contractive mappings in partially ordered modular spaces.

The aim of this paper is to determine some coupled fixed point theorems for $\left(\phi ,\psi \right)$- contractive mappings in the framework of partially ordered complete modular spaces. Our results are generalizations of the fixed point theorems due to M. Mursaleen, S.A. Mohiuddine and R.P. Agarwal [20]. First, we recall some basic definitions and notations about modular spaces from [11].

Definition 1.1. Let $\mathcal{X}$ be a vector space over $\mathbb{F}\left(=\mathbb{R}\text{or}\mathrm{ℂ}\right)$.

A functional $\rho :\mathcal{X}\to \left[0,\infty \right]$ is said to be modular if for all $x,y\in \mathcal{X}$,

(i) $\rho \left(x\right)=0$ if and only if $x=0$,

(ii) $\rho \left(\mathit{\alpha x}\right)=\rho \left(x\right)$ for every $\alpha \in \mathbb{F}$ such that $\mid \alpha \mid =1$,

(iii) $\rho \left(\mathit{\alpha x}+\mathit{\beta y}\right)\le \rho \left(x\right)+\rho \left(y\right)$ if $\alpha ,\beta \ge 0$ and $\alpha +\beta =1$.

Definition 1.2. If in Definition 1.1, $\left(\mathit{iii}\right)$ is replaced by

$$\rho \left(\mathit{\alpha x}+\mathit{\beta y}\right)\le {\alpha }^s\rho \left(x\right)+{\beta }^s\rho \left(y\right),$$E1

for $\alpha ,\beta \ge 0$, $\alpha +\beta =1$ with an $s\in \left(0,1\right]$, then we say that $\rho$ is an $s$-convex modular, and if $s=1$, $\rho$ is said to be a convex modular.

Let $\rho$ be a modular, we define the corresponding modular space, i.e. the vector space ${\mathcal{X}}_{\rho }$ given by

$${\mathcal{X}}_{\rho }=\left\{x\in \mathcal{X}:\rho \left(\mathit{\lambda x}\right)\to 0\mathrm{as}\lambda \to 0\right\}.$$E2

The modular space ${\mathcal{X}}_{\rho }$ is a normed space with the Luxemburg norm, defined by

$$\parallel x{\parallel }_{\rho }=inf\left\{\lambda >0;\rho \left(\frac{x}{\lambda }\right)\le 1\right\}.$$E3

Definition 1.3. We say a function modular $\rho$ satisfies the ${\mathrm{\Delta }}_2$–condition if there exists $\kappa >0$ such that for any $x\in {\mathcal{X}}_{\rho }$, we have $\rho \left(2x\right)\le \mathit{\kappa \rho }\left(x\right)$.

Definition 1.4. Let ${\mathcal{X}}_{\rho }$ be a modular space and suppose $\left\{x_n\right\}$ and $x$ are in ${\mathcal{X}}_{\rho }$. Then.

1. $\left\{x_n\right\}$ is $\rho$–convergent to $x$ and write $x_n\stackrel{\rho }{\to }x$ if $\rho \left(x_n-x\right)\to 0$ as $n\to \infty$.

2. $\left\{x_n\right\}$ is $\rho$–Cauchy if $\rho \left(x_n-x_m\right)\to 0$ as $n,m\to \infty$.

3. A subset $\mathcal{S}$ of ${\mathcal{X}}_{\rho }$ is called $\rho$–complete if any $\rho$–Cauchy sequence is $\rho$–convergent to an element of $\mathcal{S}$.

4. The modular $\rho$ has the Fatou property if $\rho \left(x\right)\le {liminf}_{n\to \infty }\rho \left(x_n\right)$ whenever $x_n\stackrel{\rho }{\to }x$.

Remark 1.5. $(\mathit{ii}$) A $\rho$-convergent sequence is $\rho$-cauchy if and only if $\rho$ satisfies the ${\mathrm{\Delta }}_2$–condition. $\left(\mathit{ii}\right)\rho \left(.x\right)$ is an non-decreasing function, for any $x\in \mathcal{X}$. Fro this, let $0<a<b$, putting $y=0$ in $\left(\mathit{iii}\right)$ of Definition 1.1 implies that

$$\rho \left(\mathit{ax}\right)=\rho \left(\frac{a}{b}\mathit{bx}\right)\le \rho \left(\mathit{bx}\right),$$

for all $x\in \mathcal{X}$. Also, if $\rho$ is a convex modular on $\mathcal{X}$ and $\mid \alpha \mid \le 1$, then $\rho \left(\mathit{\alpha x}\right)\le \mathit{\alpha \rho }\left(x\right)$ and $\rho \left(x\right)\le \frac{1}{2}\rho \left(2x\right)$ for all $x\in \mathcal{X}$.

We end this section with a notion of a coupled fixed point introduced by Bhaskar and Lakshmikantham [5].

Definition 1.6. An element $\left(x,y\right)\in \mathcal{X}\times \mathcal{X}$ is called a coupled fixed point of the mapping $F:\mathcal{X}\times \mathcal{X}\to \mathcal{X}$ if

$$F\left(x,y\right)=x,F\left(y,x\right)=y.$$

2. Coupled fixed point theorems for nonlinear $\left(\phi ,\psi \right)$-contractive type mappings

In this section, we establish some coupled fixed point results by considering $\left(\phi ,\psi \right)$-contractive mappings on modular spaces endowed with a partial order. We assume that $\rho$ satisfies the ${\mathrm{\Delta }}_2$-condition with $\kappa <1$.

Let ${\mathrm{\Psi }}^{\prime }$ be the family of non-decreasing functions $\psi :\left[0,+\infty \right)\to \left[0,+\infty \right)$ such that

$$\sum _{n=1}^{\infty }{\psi }^n\left(t\right)<\infty ,{\psi }^{-1}\left(\left\{0\right\}\right)=\left\{0\right\},\psi \left(t\right)<t\mathrm{a}\mathit{nd}\underset{r\to t^{+}}{lim}\psi \left(r\right)<t,\mathrm{f}\mathit{\text{orallt}}>0.$$E4

The following results are generalizations of the fixed point theorems due to M. Mursaleen, S.A. Mohiuddine and R.P. Agarwal [20] in partially ordered modular spaces.

Definition 2.1. Let $\left({\mathcal{X}}_{\rho },\le \right)$ be a partially ordered modular space and $F:\mathcal{X}\times \mathcal{X}\to \mathcal{X}$ be a mapping. Then a map $F$ is said to be $\left(\phi ,\psi \right)$-contractive if there exist two functions $\phi :{\mathcal{X}}^2\times {\mathcal{X}}^2\to \left[0,\infty \right)$ and $\psi \in \mathrm{\Psi }$ and there exist $\alpha ,\beta >0$ with $\alpha >\beta$ such that

$$\phi \left(\left(x,y\right),\left(z,w\right)\right)\rho \left(\alpha \left(F\left(x,y\right)-F\left(z,w\right)\right)\right)\le \psi \left(\frac{\rho \left(\beta \left(x-z\right)\right)+\rho \left(\beta \left(y-w\right)\right)}{2}\right)$$E5

for all $x,y,z,w\in \mathcal{X}$ with $x\ge z$ and $y\le w$.

Definition 2.2. Let $F:\mathcal{X}\times \mathcal{X}\to \mathcal{X}$ and $\phi :{\mathcal{X}}^2\times {\mathcal{X}}^2\to \left[0,\infty \right)$ be two mappings. Then $F$ is called $\phi$-admissible if

$$\phi \left(\left(x,y\right),\left(z,w\right)\right)\ge 1\Rightarrow \phi \left(\left(F\left(x,y\right),F\left(y,x\right)\right),\left(F\left(z,w\right),F\left(w,z\right)\right)\right)\ge 1$$E6

for all $x,y,z,w\in \mathcal{X}$.

In the following theorem, we give some requirements that a $\phi$-admissible mapping has a coupled fixed point.

Theorem 2.3. Let $\left(\mathcal{X},\le ,\rho \right)$ be a complete ordered modular function space. Let $F:\mathcal{X}\times \mathcal{X}\to \mathcal{X}$ be a $\left(\phi ,\psi \right)$-contractive mapping having the mixed monotone property of $\mathcal{X}$. Suppose that.

1. $F$ is $\phi$-admissible,

2. there exist $x_0,y_0\in \mathcal{X}$ such that $x_0\le F\left(x_0,y_0\right)$ and $y_0\ge F\left(y_0,x_0\right)$, also

   $$\phi \left(\left(x_0,y_0\right),\left(F\left(x_0,y_0\right),F\left(y_0,x_0\right)\right)\right)\ge 1\mathit{\text{and}}\phi \left(\left(y_0,x_0\right),\left(F\left(y_0,x_0\right),F\left(x_0,y_0\right)\right)\right)\ge 1,$$E7

3. if $\left\{x_n\right\}$ and $\left\{y_n\right\}$ are sequences in $\mathcal{X}$ such that

$$\phi \left(\left(x_n,y_n\right),\left(x_{n+1},y_{n+1}\right)\right)\ge 1\mathit{\text{and}}\phi \left(\left(y_n,x_n\right),\left(y_{n+1},x_{n+1}\right)\right)\ge 1$$E8

for all $n$ and ${lim}_{n\to \infty }{x}_n=x$ and ${lim}_{n\to \infty }{y}_n=y$, then

$$\phi \left(\left(x_n,y_n\right),\left(x,y\right)\right)\ge 1\mathit{\text{and}}\phi \left(\left(y_n,x_n\right),\left(y,x\right)\right)\ge 1.$$E9

Then $F$ has a coupled fixed point.

Proof. Let $x_0,y_0\in \mathcal{X}$ be such that

$$\phi \left(\left(x_0,y_0\right),\left(F\left(x_0,y_0\right),F\left(y_0,x_0\right)\right)\right)\ge 1\text{and}\phi \left(\left(y_0,x_0\right),\left(F\left(y_0,x_0\right),F\left(x_0,y_0\right)\right)\right)\ge 1$$E10

and $x_0\le F\left(x_0,y_0\right)$ and $y_0\ge F\left(y_0,x_0\right)$. Put $x_1=F\left(x_0,y_0\right)$ and $y_1=F\left(y_0,x_0\right)$. Let $x_2,y_2\in \mathcal{X}$ be such that $x_2=F\left(x_1,y_1\right)$ and $y_2=F\left(y_1,x_1\right)$. Continuing this process, we can construct two sequences $\left\{x_n\right\}$ and $\left\{y_n\right\}$ in $\mathcal{X}$ such that

$$x_{n+1}=F\left(x_n,y_n\right)\text{and}{y}_{n+1}=F\left(y_n,x_n\right)\left(n\ge 0\right).$$E11

Using the mathematical induction, we will show that

$$x_n\le x_{n+1}\text{and}{y}_n\ge y_{n+1}\left(n\ge 0\right).$$E12

By assumption, (12) hold for $n=0$. Now suppose that (12) hold for some fixed $n\ge 0$. Then by the mixed monotone property of $F$, we have

$$x_{n+2}=F\left(x_{n+1},y_{n+1}\right)\ge F\left(x_n,y_{n+1}\right)\ge F\left(x_n,y_n\right)=x_{n+1}$$E13

and

$$y_{n+2}=F\left(y_{n+1},x_{n+1}\right)\le F\left(y_n,x_{n+1}\right)\le F\left(y_n,x_n\right)=y_{n+1}.$$E14

Hence (12) hold for $n\ge 0$. If for some $n$, $\left(x_{n+1},y_{n+1}\right)=\left(x_n,y_n\right)$, then $F\left(x_n,y_n\right)=x_n$ and $F\left(y_n,x_n\right)=y_n$, and so $F$ has a coupled fixed point. Thus we assumed that $\left(x_{n+1},y_{n+1}\right)\ne \left(x_n,y_n\right)$ for all $n\ge 0$. Since $F$ is $\phi$-admissible, we have

$$\phi \left(\left(x_0,y_0\right),\left(x_1,y_1\right)\right)=\phi (\left(\left(x_0,y_0\right),\left(F\left(x_0,y_0\right),F\left(y_0,x_0\right)\right)\right)\ge 1$$E15

hence

$$\phi \left(\left(F\left(x_0,y_0\right),F\left(y_0,x_0\right)\right),\left(F\left(x_1,y_1\right),F\left(y_1,x_1\right)\right)\right)=\phi \left(\left(x_1,y_1\right),\left(x_2,y_2\right)\right)\ge 1.$$E16

Therefore by induction we get

$$\phi \left(\left(x_n,y_n\right),\left(x_{n+1},y_{n+1}\right)\right)\ge 1\text{and}\phi \left(\left(y_n,x_n\right),\left(y_{n+1},x_{n+1}\right)\right)\ge 1$$E17

for all $n\in \mathbb{N}$. Since $F$ is $\left(\phi ,\psi \right)$-contractive, using (35) and (17), we obtain

$$\begin{array}{l}\rho \left(\alpha \left(x_n-x_{n+1}\right)\right)=\rho \left(\alpha \left(F\left(x_{n-1},y_{n-1}\right),F\left(x_n,y_n\right)\right)\right)\\ \le \phi \left(\left(\left(x_{n-1},y_{n-1}\right),\left(x_n,y_n\right)\right)\right)\rho \left(\alpha \left(F\left(x_{n-1},y_{n-1}\right),F\left(x_n,y_n\right)\right)\right)\\ \le \psi \left(\frac{\rho \left(\beta \left(x_{n-1}-x_n\right)\right)+\rho \left(\beta \left(y_{n-1}-y_n\right)\right)}{2}\right),\end{array}$$E18

and

$$\begin{array}{l}\rho \left(\alpha \left(y_n-y_{n+1}\right)\right)=\rho \left(\alpha \left(F\left(y_{n-1},x_{n-1}\right),F\left(y_n,x_n\right)\right)\right)\\ \le \phi \left(\left(\left(y_{n-1},x_{n-1}\right),\left(y_n,x_n\right)\right)\right)\rho \left(\alpha \left(F\left(y_{n-1},x_{n-1}\right),F\left(y_n,x_n\right)\right)\right)\\ \le \psi \left(\frac{\rho \left(\beta \left(y_{n-1}-y_n\right)\right)+\rho \left(\beta \left(x_{n-1}-x_n\right)\right)}{2}\right).\end{array}$$E19

Adding (18) and (23), we obtain

$$\frac{\rho \left(\alpha \left(x_n-x_{n+1}\right)\right)+\rho \left(\alpha \left(y_n-y_{n+1}\right)\right)}{2}\le \psi \left(\frac{\rho \left(\beta \left(x_{n-1}-x_n\right)\right)+\rho \left(\beta \left(y_{n-1}-y_n\right)\right)}{2}\right)$$E20

Since $\beta <\alpha$ and $\psi$ is non-decreasing, repeating the above process, we get

$$\frac{\rho \left(\alpha \left(x_n-x_{n+1}\right)\right)+\rho \left(\alpha \left(y_n-y_{n+1}\right)\right)}{2}\le {\psi }^n\left(\frac{\rho \left(\beta \left(x_0-x_1\right)\right)+\rho \left(\beta \left(y_0-y_1\right)\right)}{2}\right)$$E21

for all $n\in \mathbb{N}$. Given $\epsilon >0$ there exists $N\in \mathbb{N}$ such that

$$\sum _{n\ge N}{\psi }^n\left(\frac{\rho \left(\beta \left(x_0-x_1\right)\right)+\rho \left(\beta \left(y_0-y_1\right)\right)}{2}\right)<\frac{\epsilon }{2}.$$E22

Let $m,n\in \mathbb{N}$ and ${\alpha }_0\in {\mathbb{R}}^{+}$ be such that $m>n>N$ and $\frac{\beta }{\alpha }+\frac{1}{{\alpha }_0}=1$. Then we have

$$\begin{array}{l}\rho \left(\beta \left(x_n-x_m\right)\right)\le \rho \left(\alpha \left(x_n-x_{n+1}\right)\right)+\rho \left({\alpha }_0\beta \left(x_{n+1}-x_m\right)\right)\\ \le \rho \left(\alpha \left(x_n-x_{n+1}\right)\right)+\mathit{\kappa \rho }\left(\beta \left(x_{n+1}-x_m\right)\right)\\ \le \rho \left(\alpha \left(x_n-x_{n+1}\right)\right)+\rho \left(\alpha \left(x_{n+1}-x_{n+2}\right)\right)+\left({\alpha }_0\beta \left(x_{n+2}-x_m\right)\right)\\ \le \rho \left(\alpha \left(x_n-x_{n+1}\right)\right)+\rho \left(\alpha \left(x_{n+1}-x_{n+2}\right)\right)+\left(\beta \left(x_{n+2}-x_m\right)\right)\\ \le \cdots \end{array}$$

$$\le \sum _{i=n}^{m-1}\rho \left(\alpha \left(x_i-x_{i+1}\right)\right),$$E23

similarly we obtain

$$\rho \left(\beta \left(y_n-y_m\right)\right)\le \sum _{i=n}^{m-1}\rho \left(\alpha \left(y_i-y_{i+1}\right)\right).$$E24

Adding (23) and (24) we obtain

$$\begin{array}{l}\frac{\rho \left(\beta \left(x_n-x_m\right)\right)+\rho \left(\beta \left(y_n-y_m\right)\right)}{2}\le \sum _{i=n}^{m-1}\frac{\rho \left(\alpha \left(x_i-x_{i+1}\right)\right)+\rho \left(\alpha \left(y_i-y_{i+1}\right)\right)}{2}\\ \le \sum _{i=n}^{m-1}{\psi }^n\left(\frac{\rho \left(\beta \left(x_0-x_1\right)\right)+\rho \left(\beta \left(y_0-y_1\right)\right)}{2}\right)\\ <\frac{\epsilon }{2}.\end{array}$$E25

Consequently

$$\rho \left(\beta \left(x_n-x_m\right)\right)\le \rho \left(\beta \left(x_n-x_m\right)\right)+\rho \left(\beta \left(y_n-y_m\right)\right)<\epsilon$$E26

and

$$\rho \left(\beta \left(y_n-y_m\right)\right)\le \rho \left(\beta \left(x_n-x_m\right)\right)+\rho \left(\beta \left(y_n-y_m\right)\right)<\epsilon ,$$E27

therefore $\left\{x_n\right\}$ and $\left\{y_n\right\}$ are cauchy sequences in complete modular space $\left(\mathcal{X},\rho \right)$, and so $\left\{x_n\right\}$ and $\left\{y_n\right\}$ are convergent in $\left(\mathcal{X},\rho \right)$. Thus there exist $x,y\in \mathcal{X}$ such that

$$\underset{n\to \infty }{lim}{x}_n=x\text{and}\underset{n\to \infty }{lim}{y}_n=y.$$

Now from (17) and hypothesis $\left(\mathit{iii}\right)$, we get

$$\phi \left(\left(x_n,y_n\right),\left(x_{n+1},y_{n+1}\right)\right)\ge 1\text{and}\phi \left(\left(y_n,x_n\right),\left(y_{n+1},x_{n+1}\right)\right)\ge 1$$E28

for all $n\in \mathbb{N}$. From (28) and the condition $\left(\mathit{iii}\right)$ of the modular $\rho$ we obtain

$$\begin{array}{l}\rho \left(\beta \left(F\left(x,y\right)-x\right)\right)\le \rho \left(\alpha \left(F\left(x,y\right)-F\left(x_n,y_n\right)\right)\right)+\rho \left({\alpha }_0\beta \left(x_{n+1}-x\right)\right)\\ \le \phi \left(\left(x_n,y_n\right),\left(x,y\right)\right)\rho \left(\alpha \left(F\left(x,y\right)-F\left(x_n,y_n\right)\right)\right)+\rho \left({\alpha }_0\beta \left(x_{n+1}-x\right)\right)\\ \le \psi \left(\frac{\rho \left(\beta \left(x_n-x\right)\right)+\rho \left(\beta \left(y_n-y\right)\right)}{2}\right)+\rho \left({\alpha }_0\beta \left(x_{n+1}-x\right)\right)\\ <\frac{\rho \left(\beta \left(x_n-x\right)\right)+\rho \left(\beta \left(y_n-y\right)\right)}{2}+\rho \left({\alpha }_0\beta \left(x_{n+1}-x\right)\right)\end{array}$$E29

similarly, we get

$$\begin{array}{l}\rho \left(\beta \left(F\left(y,x\right)-y\right)\right)\le \rho \left(\alpha \left(F\left(y,x\right)-F\left(y_n,x_n\right)\right)\right)+\rho \left({\alpha }_0\beta \left(y_{n+1}-y\right)\right)\\ \le \phi \left(\left(y_n,x_n\right),\left(y,x\right)\right)\rho \left(\alpha \left(F\left(y,x\right)-F\left(y_n,x_n\right)\right)\right)+\rho \left({\alpha }_0\beta \left(y_{n+1}-y\right)\right)\\ \le \psi \left(\frac{\rho \left(\beta \left(y_n-y\right)\right)+\rho \left(\beta \left(x_n-x\right)\right)}{2}\right)+\rho \left({\alpha }_0\beta \left(y_{n+1}-y\right)\right)\\ <\frac{\rho \left(\beta \left(y_n-y\right)\right)+\rho \left(\beta \left(x_n-x\right)\right)}{2}+\rho \left({\alpha }_0\beta \left(y_{n+1}-y\right)\right).\end{array}$$E30

Taking the limit as $n\to \infty$, we obtain

$$\rho \left(\beta \left(F\left(x,y\right)-x\right)\right)=0\text{and}\rho \left(\beta \left(F\left(y,x\right)-y\right)\right)=0.$$E31

Therefore $F\left(x,y\right)=x$ and $F\left(y,x\right)=y$, that is $F$ has a coupled fixed point. □

Remark 2.4. If in Theorem 2.3, we replace the property $\left(\mathit{iii}\right)$ with the continuity of $F$, then the result holds that is $F$ has a coupled fixed point. In fact, since $F$ is continuous and $x_{n+1}=F\left(x_n,y_n\right)$ and $y_{n+1}=F\left(y_n,x_n\right)$, we get

$$x=\underset{n\to \infty }{lim}{x}_n=\underset{n\to \infty }{lim}F\left(x_{n-1},y_{n-1}\right)=F\left(x,y\right)$$E32

and

$$y=\underset{n\to \infty }{lim}{y}_n=\underset{n\to \infty }{lim}F\left(y_{n-1},x_{n-1}\right)=F\left(y,x\right).$$E33

Hence $F$ has a coupled fixed point.

As the proof of Theorem 2.3, one can prove the following theorem.

Theorem 2.5. In addition to the hypothesis of Theorem 2.3, suppose that for every $\left(x,y\right),\left(z,w\right)$ in $\mathcal{X}\times \mathcal{X}$, there exists $\left(s,t\right)$ in $\mathcal{X}\times \mathcal{X}$ such that

$$\phi \left(\left(x,y\right),\left(s,t\right)\right)\ge 1\mathit{\text{and}}\phi \left(\left(z,w\right),\left(s,t\right)\right)\ge 1,$$E34

and assume that $\left(s,t\right)$ is comparable to $\left(x,y\right)$ and $\left(z,w\right)$. Then $F$ has a unique coupled fixed point.

If we put $\psi \left(t\right)=\mathit{mt}$ for $m\in \left[0,1\right)$ in Theorem 2.3, we obtain the following corollary.

Corollary 2.6. Let $\left(\mathcal{X},\le ,\rho \right)$ be a complete ordered modular function space. Let $F:\mathcal{X}\times \mathcal{X}\to \mathcal{X}$ be a $\left(\phi ,\psi \right)$-contractive mapping having the mixed monotone property of $\mathcal{X}$. Suppose that there exist $\alpha ,\beta >0$ with $\alpha >\beta$ such that

$$\phi \left(\left(x,y\right),\left(z,w\right)\right)\rho \left(\alpha \left(F\left(x,y\right)-F\left(z,w\right)\right)\right)\le \frac{m}{2}\left(\rho \left(\beta \left(x-z\right)\right)+\rho \left(\beta \left(y-w\right)\right)\right)$$E35

for all $x,y,z,w\in \mathcal{X}$ with $x\ge z$ and $y\le w$. Also if.

$\left(\mathit{ii}\right)$ there exist $x_0,y_0\in \mathcal{X}$ such that $x_0\le F\left(x_0,y_0\right)$ and $y_0\ge F\left(y_0,x_0\right)$, also

$$\phi \left(\left(x_0,y_0\right),\left(F\left(x_0,y_0\right),F\left(y_0,x_0\right)\right)\right)\ge 1\mathit{\text{and}}\phi \left(\left(y_0,x_0\right),\left(F\left(y_0,x_0\right),F\left(x_0,y_0\right)\right)\right)\ge 1,$$E36

$\left(\mathit{ii}\right)$ if $\left\{x_n\right\}$ and $\left\{y_n\right\}$ are sequences in $\mathcal{X}$ such that

$$\phi \left(\left(x_n,y_n\right),\left(x_{n+1},y_{n+1}\right)\right)\ge 1\mathit{\text{and}}\phi \left(\left(y_n,x_n\right),\left(y_{n+1},x_{n+1}\right)\right)\ge 1$$E37

for all $n$ and ${lim}_{n\to \infty }{x}_n=x$ and ${lim}_{n\to \infty }{y}_n=y$, then

$$\phi \left(\left(x_n,y_n\right),\left(x,y\right)\right)\ge 1\mathit{\text{and}}\phi \left(\left(y_n,x_n\right),\left(y,x\right)\right)\ge 1$$E38

Then $F$ has a coupled fixed point.

3. Conclusion

In the present paper, nonlinear contractive mappings in the framework of a modular space endowed with a partial order have been given, then some well-known coupled fixed point theorems in ordered metric spaces are extended to these mappings in modular spaces endowed with a partial order.

2010 Mathematics Subject Classification:

Primary 47H10, 54H25; Secondary 46A80