# Fixed points of mappings satisfying contractive condition of integral type in modular spaces endowed with a graph

- Mahpeyker Öztürk
^{1}Email author, - Mujahid Abbas
^{2, 3}and - Ekber Girgin
^{1}

**2014**:220

https://doi.org/10.1186/1687-1812-2014-220

© Öztürk et al.; licensee Springer. 2014

**Received: **30 May 2014

**Accepted: **14 October 2014

**Published: **29 October 2014

## Abstract

Jachymski (Proc. Am. Math. Soc. 136:1359-1373, 2008) gave a modified version of a Banach fixed point theorem on a metric space endowed with a graph. The aim of this paper is to present fixed point results of mappings satisfying integral type contractive conditions in the framework of modular spaces endowed with a graph. Some examples are presented to support the results proved herein. Our results generalize and extend various comparable results in the existing literature.

**MSC:**47H10, 54H25, 54E50.

## Keywords

*G*-contractionintegral type contraction

## 1 Introduction

Fixed point theory for nonlinear mappings is an important subject of nonlinear functional analysis. One of the basic and the most widely applied fixed point theorem in all of analysis is the ‘Banach (or Banach-Caccioppoli) contraction principle’ due to Banach [1]. This Banach contraction principle [1] is a simple and powerful result with a wide range of applications, including iterative methods for solving linear, nonlinear, differential, integral, and difference equations. Due to its applications in mathematics and other related disciplines, the Banach contraction principle has been generalized in many directions.

The existence of fixed points in ordered metric spaces has been discussed by Ran and Reurings [2]. Recently, many researchers have obtained fixed point and common fixed point results for single valued maps defined on partially ordered metric spaces (see, *e.g.*, [3, 4]). Jachymski [5] investigated a new approach in metric fixed point theory by replacing an order structure with graph structure on a metric space. In this way, the results proved in ordered metric spaces are generalized (see for details [5] and the references therein). For further work in this direction, we refer to, *e.g.*, [6–8].

In 1968, Kannan [9] proved a fixed point theorem for a map satisfying a contractive condition that did not require continuity at each point. This paper led to the genesis for a multitude of fixed point papers over the next two decades. Since then, there have been many theorems dealing with mappings satisfying various types of contractive inequalities involving linear and nonlinear expressions. For a thorough survey, we refer to [10] and the references therein. On the other hand, Branciari [11] obtained a fixed point theorem for a single valued mapping satisfying an analog of Banach’s contraction principle for an integral type inequality. Recently, Akram *et al.* [12] introduced a new class of contraction maps, called *A*-contractions, which is a proper generalization of Kannan’s mappings [9], Bianchini’s mappings [13], and Reich type mappings [14].

The theory of modular spaces was initiated by Nakano [15] in connection with the theory of ordered spaces which was further generalized by Musielak and Orlicz [16] (see also [17–21]). The study of fixed point theory in the context of modular function spaces was initiated by Khamsi *et al.* [22] (see also [22–26]). Also, some fixed point theorems have been proved for mappings satisfying contractive conditions of integral type in modular space [27, 28].

In this paper, we introduce three new classes of mappings satisfying integral type contractive conditions in the setup of modular space endowed with graphs. We study the existence, uniqueness, and iterative approximations of fixed points for such mappings. Our results extend, unify, and generalize the comparable results in [5, 11, 12].

## 2 Preliminaries

A mapping *T* from a metric space $(X,d)$ into $(X,d)$ is called a Picard operator (PO) if *T* has a unique fixed point $z\in X$ and ${lim}_{n\to \mathrm{\infty}}{T}^{n}x=z$ for all $x\in X$.

Define Φ = {$\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$: *φ* is a Lebesgue integral mapping which is summable, nonnegative and satisfies ${\int}_{0}^{\u03f5}\phi (t)\phantom{\rule{0.2em}{0ex}}dt>0$, for each $\u03f5>0$}.

Let *A* = {$\alpha :{\mathbb{R}}_{+}^{3}\to {\mathbb{R}}_{+}$: *α* is continuous and $a\le kb$ for some $k\in [0,1)$ whenever $a\le \alpha (a,b,b)$ or $a\le \alpha (b,a,b)$ or $a\le \alpha (b,b,a)$ for all *a*, *b*}.

Let $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be a nondecreasing mapping which satisfies the following conditions:

(${\psi}_{1}$) $\psi (x)=0$ if and only if $x=0$;

(${\psi}_{2}$) for a sequence $\{{x}_{n}\}$ in ${\mathbb{R}}_{+}$, we have $\psi ({x}_{n})\to 0$ if and only if ${x}_{n}\to 0$ as $n\to \mathrm{\infty}$;

(${\psi}_{3}$) for every $x,y\in {\mathbb{R}}_{+}$, we have $\psi (x+y)\le \psi (x)+\psi (y)$.

The collection of all such mappings will be denoted by Ψ.

**Theorem 2.1** [11]

*Let*$(X,d)$

*be a complete metric space*, $\eta \in [0,1)$,

*and*$T:X\to X$

*a mapping*.

*Suppose that*

*is satisfied for every* $x,y\in X$, *where* $\phi \in \mathrm{\Phi}$. *Then* *T* *has a unique fixed point* $z\in X$ *and for each* $x\in X$, *we have* ${lim}_{n\to \mathrm{\infty}}{T}^{n}x=z$.

**Lemma 2.2** [29]

*Let*$(X,d)$

*be a metric space*, $\phi \in \mathrm{\Phi}$,

*and*$\{{x}_{n}\}$

*a nonnegative sequence*.

*Then*

- (a)
${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$

*implies that*${lim}_{n\to \mathrm{\infty}}{\int}_{0}^{{x}_{n}}\phi (t)\phantom{\rule{0.2em}{0ex}}dt={\int}_{0}^{x}\phi (t)\phantom{\rule{0.2em}{0ex}}dt$; - (b)
${lim}_{n\to \mathrm{\infty}}{\int}_{0}^{{x}_{n}}\phi (t)\phantom{\rule{0.2em}{0ex}}dt=0$

*if and only if*${lim}_{n\to \mathrm{\infty}}{x}_{n}=0$.

**Definition 2.3** [12]

*T*on a metric space

*X*is called an

*A*-contraction if for any $x,y\in X$ and for some $\alpha \in A$, the following condition holds:

Now, we recall some basic facts and notations as regards modular spaces. For more details the reader may consult [16].

**Definition 2.4** Let *X* be an arbitrary vector space. A functional $\rho :X\to [0,\mathrm{\infty}]$ is called modular if for any arbitrary *x*, *y* in *X*:

(m_{1}) $\rho (x)=0$ if and only if $x=0$;

(m_{2}) $\rho (\alpha x)=\rho (x)$ for every scalar *α* with $|\alpha |=1$;

(m_{3}) $\rho (\alpha x+\beta y)\le \rho (x)+\rho (y)$ if $\alpha +\beta =1$, $\alpha \ge 0$, $\beta \ge 0$.

If (m_{3}) is replaced by $\rho (\alpha x+\beta y)\le {\alpha}^{s}\rho (x)+{\beta}^{s}\rho (y)$ if ${\alpha}^{s}+{\beta}^{s}=1$, where $s\in (0,1]$, $\alpha \ge 0$, $\beta \ge 0$ then we say that *ρ* is *s*-convex modular. If $s=1$, then *ρ* is called convex modular.

$\rho :R\to [0,\mathrm{\infty}]$ defined by $\rho (x)=\sqrt{|x|}$ is a simple example of a modular functional.

is called a modular space. In general the modular *ρ* is not sub-additive and therefore does not behave as a norm or a distance. One can associate to a modular an *x*-norm.

**Remark 2.5** [28]

The following are immediate consequences of condition (m_{3}):

(r_{1}) if $a,b\in \mathbb{R}$ (set of all real numbers) with $|a|<|b|$, then $\rho (ax)<\rho (bx)$ for all $x\in X$;

_{2}) if ${a}_{1},\dots ,{a}_{n}$ are nonnegative real numbers with ${\sum}_{i=1}^{n}{a}_{i}=1$, then we have

*ρ*-ball, ${B}_{\rho}(x,r)$, centered at $x\in {X}_{\rho}$ with radius

*r*as

A point $x\in {X}_{\rho}$ is called *a fixed point of* $T:{X}_{\rho}\to {X}_{\rho}$ if $T(x)=x$.

A function modular is said to satisfy the ${\mathrm{\Delta}}_{2}$-type condition if there exists $K>0$ such that for any $x\in {X}_{\rho}$ we have $\rho (2x)\le K\rho (x)$. A modular *ρ* is said to satisfy the ${\mathrm{\Delta}}_{2}$-condition if $\rho (2{x}_{n})\to 0$ as $n\to \mathrm{\infty}$, whenever $\rho ({x}_{n})\to 0$ as $n\to \mathrm{\infty}$.

**Definition 2.6** Let ${X}_{\rho}$ be a modular space. The sequence $\{{x}_{n}\}\subset {X}_{\rho}$ is said to be:

(t_{1}) *ρ*-convergent to $x\in {X}_{\rho}$ if $\rho ({x}_{n}-x)\to 0$ as $n\to \mathrm{\infty}$;

(t_{2}) *ρ*-Cauchy if $\rho ({x}_{n}-{x}_{m})\to 0$ as *n* and $m\to \mathrm{\infty}$.

${X}_{\rho}$ is *ρ*-complete if any *ρ*-Cauchy sequence is *ρ*-convergent. Note that *ρ*-convergence does not imply *ρ*-Cauchy since *ρ* does not satisfy the triangle inequality. In fact, one can show that this will happen if and only if *ρ* satisfies the ${\mathrm{\Delta}}_{2}$-condition.

**Proposition 2.7** [30]

*Let* ${X}_{\rho}$ *be a modular space*. *If* $a,b\in {\mathbb{R}}_{+}$ *with* $b\ge a$, *then* $\rho (ax)\le \rho (bx)$.

**Proposition 2.8** [30]

*Suppose that* ${X}_{\rho}$ *is a modular space*, *ρ* *satisfies the* ${\mathrm{\Delta}}_{2}$-*condition and* ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ *is a sequence in* ${X}_{\rho}$. *If* $\rho (c({x}_{n}-{x}_{n-1}))\to 0$, *then* $\rho (\alpha l({x}_{n}-{x}_{n-1}))\to 0$, *as* $n\to \mathrm{\infty}$, *where* $c,l,\alpha \in {\mathbb{R}}_{+}$ *with* $c>l$ *and* $\frac{l}{c}+\frac{1}{\alpha}=1$. *Now we give some basic definitions from graph theory needed in the sequel*.

Throughout this paper, $\mathrm{\Delta}=\{(x,x):x\in X\}$ denotes the diagonal of $X\times X$, where *X* is any nonempty set. Let *G* be a directed graph such that the set $V(G)$ of its vertices coincides with *X* and $E(G)$ be the set of edges of the graph such that $\mathrm{\Delta}\subseteq E(G)$. Further assume that *G* has no parallel edge and *G* is a weighted graph in the sense that each edge is assigned a distance $d(x,y)$ between their vertices *x* and *y* and each vertex *x* is assigned a weight $d(x,x)$. The graph *G* is identified by the pair $(V(G),E(G))$.

If *x* and *y* are vertices of *G*, then a path in *G* from *x* to *y* of length $k\in N$ is a finite sequence $\{{x}_{n}\}$, $n\in \{0,1,2,\dots ,k\}$ of vertices such that $x={x}_{0},\dots ,{x}_{k}=y$ and $({x}_{i-1},{x}_{i})\in E(G)$ for $i\in \{1,2,\dots ,k\}$.

*G*is connected if there is a path between any two vertices and it is weakly connected if $\tilde{G}$ is connected, where $\tilde{G}$ denotes the undirected graph obtained from

*G*by ignoring the direction of edges. Denote by ${G}^{-1}$ the graph obtained from

*G*by reversing the direction of the edges. Thus

Let ${G}_{x}$ be the component of *G* consisting of all the edges and vertices which are contained in some path in *G* beginning at *x*. If *G* is such that $E(G)$ is symmetric, then for $x\in V(G)$, the equivalence class ${[x]}_{G}$ defined on $V(G)$ by the rule *R* ($xRy$ if there is a path in *G* from *x* to *y*) is such that $V({G}_{x})={[x]}_{G}$.

**Definition 2.9** [[5], Definition 2.1]

*G*-contraction if and only if:

- (a)
for each

*x*,*y*in*X*with $(x,y)\in E(G)$, we have $(T(x),T(y))\in E(G)$, that is,*T*preserves edges of*G*; - (b)there exists
*α*in $(0,1)$ such that for each $x,y\in X$ with $(x,y)\in E(G)$ implies$d(T(x),T(y))\le \alpha d(x,y).$(3)

That is, *T* decreases weights of edges of *G*.

For any $x,y\in {V}^{\prime}$, $(x,y)\in {E}^{\prime}$ such that ${V}^{\prime}\subseteq V(G)$, ${E}^{\prime}\subseteq E(G)$, $({V}^{\prime},{E}^{\prime})$ is called a subgraph of *G*.

## 3 Main results

*G*and let $T:{X}_{\rho}\to {X}_{\rho}$ be a mapping. Denote

**Definition 3.1**Let $\{{T}^{n}x\}$ be a sequence, there exists $C>0$ such that

Then a graph *G* is called a ${C}_{\rho}$-graph if there exists a subsequence $\{{T}^{{n}_{p}}x\}$ of $\{{T}^{n}x\}$ such that $({T}^{{n}_{p}}x,{x}^{\ast})\in E(G)$ for $p\in \mathbb{N}$.

**Definition 3.2**A mapping $T:{X}_{\rho}\to {X}_{\rho}$ is called orbitally ${G}_{\rho}$-continuous for all $x,y\in {X}_{\rho}$ and any sequence ${({n}_{p})}_{p\in \mathbb{N}}$ of positive integers, if there exists $C>0$ such that

as $p\to \mathrm{\infty}$.

**Definition 3.3** A mapping *T* is called a ${(G,A)}_{\rho}$-contraction if it satisfies the following conditions:

(A_{1}) *T* preserves edges of *G*;

_{2}) there exist nonnegative numbers

*l*,

*c*with $l<c$ such that

holds for each $(x,y)\in E(G)$, and some $\alpha \in A$ and $\phi \in \mathrm{\Phi}$.

**Remark 3.4**Let $T:{X}_{\rho}\to {X}_{\rho}$ be a ${(G,A)}_{\rho}$-contraction. If there exists ${x}_{0}\in {X}_{\rho}$ such that $T{x}_{0}\in {[{x}_{0}]}_{\tilde{G}}$, then

- (i)
*T*is both a ${({G}^{-1},A)}_{\rho}$-contraction and a ${(\tilde{G},A)}_{\rho}$-contraction, - (ii)
${[{x}_{0}]}_{\tilde{G}}$ is

*T*-invariant and $T{{|}_{[{x}_{0}]}}_{\tilde{G}}$ is a ${({\tilde{G}}_{{x}_{0}},A)}_{\rho}$-contraction.

**Lemma 3.5**

*Let*$T:{X}_{\rho}\to {X}_{\rho}$

*be a*${(G,A)}_{\rho}$-

*contraction*.

*If*$x\in {X}_{T}$,

*then there exists*$r(x,Tx)\ge 0$

*such that*

*holds for all* $n\in \mathbb{N}$, *where* $r(x,Tx)={\int}_{0}^{\rho (c(x-Tx))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt$.

*Proof*Let $x\in {X}_{T}$, that is, $(x,Tx)\in E(G)$. Then by induction, we have $({T}^{n}x,{T}^{n+1}x)\in E(G)$ for all $n\in \mathbb{N}$. Now, we have

*α*, we obtain

That is, ${\int}_{0}^{\rho (c({T}^{n}x-{T}^{n-1}x))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le {k}^{n}r(x,Tx)$ for all $n\in \mathbb{N}$, where $r(x,Tx)={\int}_{0}^{\rho (c(x-Tx))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt$. □

**Theorem 3.6** *Let* ${X}_{\rho}$ *be a* *ρ*-*complete modular space endowed with a graph* *G*, *where* *ρ* *satisfies the* ${\mathrm{\Delta}}_{2}$-*condition and let* $T:{X}_{\rho}\to {X}_{\rho}$ *be a* ${(\tilde{G},A)}_{\rho}$-*contraction*. *If the set* ${X}_{T}$ *is nonempty*, *the graph* *G* *is weakly connected and a* ${C}_{\rho}$-*graph*, *then* *T* *is a PO*.

*Proof*If $x\in {X}_{T}$, then $Tx\in {[x]}_{\tilde{G}}$ and $({T}^{n}x,{T}^{n+1}x)\in E(G)$ for all $n\in \mathbb{N}$. Let $m,n\in \mathbb{N}$ with $m>n$. Note that

It follows that $\{\frac{c}{m-n}{T}^{n}x\}$ is a *ρ*-Cauchy sequence in ${X}_{\rho}$. Since ${X}_{\rho}$ is *ρ*-complete, there exists a point ${x}^{\ast}\in {X}_{\rho}$ such that $\rho (\frac{c}{m-n}({T}^{n}x-{x}^{\ast}))\to 0$. Consequently, $\rho (l({T}^{n}x-{x}^{\ast}))\to 0$.

*T*. As $\rho (\frac{c}{m-n}({T}^{n}x-{x}^{\ast}))\to 0$, $({T}^{n}x,{T}^{n+1}x)\in E(G)$ for all $n\in \mathbb{N}$ and

*G*is a ${C}_{\rho}$-graph, there exists a subsequence $\{{T}^{{n}_{p}}x\}$ of $\{{T}^{n}x\}$ such that $({T}^{{n}_{p}}x,{x}^{\ast})\in E(G)$ for each $p\in \mathbb{N}$. Since $({T}^{{n}_{p}}x,{x}^{\ast})\in E(\tilde{G})$ and

*T*is a ${(\tilde{G},A)}_{\rho}$-contraction, it follows that

*α*, we have

From Lemma 2.2, it follows that $\rho (c({x}^{\ast}-T{x}^{\ast}))=0$ and $T{x}^{\ast}={x}^{\ast}$.

*T*has another fixed point ${y}^{\ast}\in {X}_{\rho}-\{{x}^{\ast}\}$. Since

*G*is a ${C}_{\rho}$-graph, there exists a subsequence $\{{T}^{{n}_{p}}x\}$ of $\{{T}^{n}x\}$ such that $({T}^{{n}_{p}}x,{x}^{\ast})\in E(G)$ and $({T}^{{n}_{p}}x,{y}^{\ast})\in E(G)$ for each $p\in \mathbb{N}$. Furthermore,

*G*is weakly connected, $({x}^{\ast},{y}^{\ast})\in E(\tilde{G})$, and we have

By the definition of *α* and Lemma 2.2, we have ${\int}_{0}^{\rho (c({x}^{\ast}-{y}^{\ast}))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le k\cdot 0=0$, $\rho (c({x}^{\ast}-{y}^{\ast}))=0$, and ${x}^{\ast}={y}^{\ast}$. □

In Theorem 3.6, if we replace the condition that *G* is a ${C}_{\rho}$-graph with orbitally ${G}_{\rho}$-continuity of *T*, then we have the following theorem.

**Theorem 3.7** *Let* ${X}_{\rho}$ *be a* *ρ*-*complete modular space endowed with a graph* *G*, *where* *ρ* *satisfies the* ${\mathrm{\Delta}}_{2}$-*condition and let* $T:{X}_{\rho}\to {X}_{\rho}$ *be a* ${(\tilde{G},A)}_{\rho}$-*contraction and orbitally* ${G}_{\rho}$-*continuous*. *If the set* ${X}_{T}$ *is nonempty and the graph* *G* *is weakly connected*, *then* *T* *is a PO*.

*Proof* If $x\in {X}_{T}$, then Theorem 3.6 implies that $\{\frac{c}{m-n}{T}^{n}x\}$ is a *ρ*-Cauchy sequence in ${X}_{\rho}$. Owing to *ρ*-completeness of ${X}_{\rho}$, there exists ${x}^{\ast}\in {X}_{\rho}$ such that $\rho (\frac{c}{m-n}({T}^{n}x-{x}^{\ast}))\to 0$. As $({T}^{n}x,{T}^{n+1}x)\in E(G)$ for all $n\in \mathbb{N}$ and *T* is orbitally ${G}_{\rho}$-continuous, we have $\rho (\frac{c}{m-n}(T({T}^{n}x)-T({x}^{\ast})))\to 0$, as $n\to \mathrm{\infty}$. That is, $T{x}^{\ast}={x}^{\ast}$. Assume that ${y}^{\ast}$ is another fixed point of *T*. Following arguments similar to those in the proof of Theorem 3.6, we obtain ${y}^{\ast}={x}^{\ast}$. □

**Corollary 3.8**

*Let*${X}_{\rho}$

*be a*

*ρ*-

*complete modular space endowed with a graph*

*G*,

*where*

*ρ*

*satisfies the*${\mathrm{\Delta}}_{2}$-

*condition and let*$T:{X}_{\rho}\to {X}_{\rho}$

*be edge*-

*preserving*,

*the set*${X}_{T}$

*nonempty and graph*

*G*

*be weakly connected and a*${C}_{\rho}$-

*graph*.

*If there exist nonnegative numbers*

*l*,

*c*

*with*$l<c$

*such that*

*holds for all* $(x,y)\in E(\tilde{G})$ *and some* $\alpha \in A$, *then* *T* *is a PO*.

Now, we introduce Hardy-Rogers type ${(G)}_{\rho}$-contraction and obtain related fixed point results.

**Definition 3.9** Let ${X}_{\rho}$ be a modular space. A mapping $T:{X}_{\rho}\to {X}_{\rho}$ is called a Hardy-Rogers type ${(G)}_{\rho}$-contraction if the following conditions hold:

(H_{1}) *T* preserves edges of *G*;

_{2}) there exist nonnegative numbers ${l}_{i}$,

*c*with ${l}_{i}<c$ for $i=1,\dots ,5$ such that

holds for each $(x,y)\in E(G)$ with nonnegative numbers *η*, *β*, *γ* such that $\eta +2\beta +2\gamma <1$ and $\phi \in \mathrm{\Phi}$.

**Remark 3.10**Let ${X}_{\rho}$ be a modular space endowed with a graph

*G*and let $T:{X}_{\rho}\to {X}_{\rho}$ be a Hardy-Rogers type ${(G)}_{\rho}$-contraction. If there exists ${x}_{0}\in {X}_{\rho}$ such that $T{x}_{0}\in {[{x}_{0}]}_{\tilde{G}}$, then

- (i)
*T*is both a Hardy-Rogers type ${({G}^{-1})}_{\rho}$-contraction and a Hardy-Rogers type ${(\tilde{G})}_{\rho}$-contraction, - (ii)
${[{x}_{0}]}_{\tilde{G}}$ is

*T*-invariant and $T{{|}_{[{x}_{0}]}}_{\tilde{G}}$ is a Hardy-Rogers type ${({\tilde{G}}_{{x}_{0}})}_{\rho}$-contraction.

**Theorem 3.11** *Let* ${X}_{\rho}$ *be a* *ρ*-*complete modular space endowed with a graph* *G*, *where* *ρ* *satisfies the* ${\mathrm{\Delta}}_{2}$-*condition and let* $T:{X}_{\rho}\to {X}_{\rho}$ *be a Hardy*-*Rogers type* ${(\tilde{G})}_{\rho}$-*contraction*. *Assume that the set* ${X}_{T}$ *is nonempty and the* ${C}_{\rho}$-*graph* *G* *is weakly connected*. *Then* *T* *is a PO*.

*Proof*If $x\in {X}_{T}$, then $Tx\in {[x]}_{\tilde{G}}$ and $({T}^{n}x,{T}^{n+1}x)\in E(G)$ for all $n\in \mathbb{N}$. Note that

which implies that ${lim}_{n}\rho (c({T}^{n}x-{T}^{n+1}x))=0$.

*ρ*-Cauchy sequence in ${X}_{\rho}$. Since ${X}_{\rho}$ is

*ρ*-complete, there exists a point ${x}^{\ast}\in {X}_{\rho}$ such that $\rho (\frac{c}{m-n}({T}^{n}x-{x}^{\ast}))\to 0$. As

*G*is a ${C}_{\rho}$-graph, there exists a subsequence $\{{T}^{{n}_{p}}x\}$ such that $({T}^{{n}_{p}}x,{x}^{\ast})\in E(G)$ for all $p\in \mathbb{N}$. Also, $({T}^{{n}_{p}}x,{x}^{\ast})\in E(G)$ for all $p\in \mathbb{N}$. Now we have

As $(\beta +\gamma )<1$, so $\rho (c({x}^{\ast}-T{x}^{\ast}))=0$ and ${x}^{\ast}=T{x}^{\ast}$.

*T*has another fixed point ${y}^{\ast}\in {X}_{\rho}-\{{x}^{\ast}\}$. Since

*G*is a ${C}_{\rho}$-graph, there exists a subsequence $\{{T}^{{n}_{p}}x\}$ of $\{{T}^{n}x\}$ such that $({T}^{{n}_{p}}x,{x}^{\ast})\in E(G)$ and $({T}^{{n}_{p}}x,{y}^{\ast})\in E(G)$ for each $p\in \mathbb{N}$. As

*G*is weakly connected, we have $({x}^{\ast},{y}^{\ast})\in E(\tilde{G})$, and

Since $(\eta +2\gamma )<1$, ${\int}_{0}^{\rho (c({x}^{\ast}-{y}^{\ast}))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt=0$. The result follows. □

In Theorem 3.11, if we replace the condition that *G* is a ${C}_{\rho}$-graph with orbitally ${G}_{\rho}$-continuity of *T*, then we have the following theorem.

**Theorem 3.12** *Let* ${X}_{\rho}$ *be a* *ρ*-*complete modular space endowed with a graph* *G*, *where* *ρ* *satisfies the* ${\mathrm{\Delta}}_{2}$-*condition and let* $T:{X}_{\rho}\to {X}_{\rho}$ *be a Hardy*-*Rogers type* ${(\tilde{G})}_{\rho}$-*contraction*, *which is orbitally* ${G}_{\rho}$-*continuous*. *Assume that the set* ${X}_{T}$ *is nonempty and the graph* *G* *is weakly connected*. *Then* *T* *is a PO*.

In the following suppose that ${X}_{\rho}$ is a *ρ*-complete modular space endowed with a graph *G*, where *ρ* satisfies the ${\mathrm{\Delta}}_{2}$-condition and $T:{X}_{\rho}\to {X}_{\rho}$ is edge-preserving such that the set ${X}_{T}$ is nonempty.

**Corollary 3.13**

*Assume*

- (i)
*the*${C}_{\rho}$-*graph**G**is weakly connected and* - (ii)
*there exist nonnegative numbers**l*,*c**with*$l<c$*such that*${\int}_{0}^{\rho (c(Tx-Ty))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \eta {\int}_{0}^{\rho (l(x-y))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt$

*holds for each* $(x,y)\in E(\tilde{G})$ *with* $\eta \in (0,1)$ *and* $\phi \in \mathrm{\Phi}$. *Then* *T* *is a PO*.

**Corollary 3.14**

*Assume*

- (i)
*the*${C}_{\rho}$-*graph**G**is weakly connected and* - (ii)
*there exist nonnegative numbers*${l}_{1}$, ${l}_{2}$,*c**with*${l}_{1},{l}_{2}<c$*such that*${\int}_{0}^{\rho (c(Tx-Ty))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \beta {\int}_{0}^{[\rho ({l}_{1}(x-Tx))+\rho ({l}_{2}(y-Ty))]}\phi (t)\phantom{\rule{0.2em}{0ex}}dt$

*holds for each* $(x,y)\in E(\tilde{G})$ *with* $\beta \in (0,\frac{1}{2})$ *and* $\phi \in \mathrm{\Phi}$. *Then* *T* *is a PO*.

**Corollary 3.15**

*Assume*

- (i)
*the*${C}_{\rho}$-*graph**G**is weakly connected and* - (ii)
*there exist nonnegative numbers*${l}_{1}$, ${l}_{2}$,*c**with*${l}_{1},{l}_{2}<c$*such that*${\int}_{0}^{\rho (c(Tx-Ty))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \gamma {\int}_{0}^{[\rho (\frac{{l}_{1}}{2}(x-Ty))+\rho (\frac{{l}_{2}}{2}(y-Tx))]}\phi (t)\phantom{\rule{0.2em}{0ex}}dt$

*for each* $(x,y)\in E(\tilde{G})$ *with* $\gamma \in (0,\frac{1}{2})$ *and* $\phi \in \mathrm{\Phi}$. *Then* *T* *is a PO*.

Now we introduce the ${(G,\varphi ,\psi )}_{\rho}$-contraction and obtain some fixed point results.

**Definition 3.16** A mapping $T:{X}_{\rho}\to {X}_{\rho}$ is called a ${(G,\varphi ,\psi )}_{\rho}$-contraction if the following conditions hold:

(Q_{1}) *T* preserves edges of *G*;

_{2}) there exist nonnegative numbers

*l*,

*c*with $l<c$ such that

holds for each $(x,y)\in E(G)$, where $\psi \in \mathrm{\Psi}$, $\varphi \in {\mathrm{\Phi}}_{1}$, and $\phi \in \mathrm{\Phi}$.

**Remark 3.17**Let $T:{X}_{\rho}\to {X}_{\rho}$ be a ${(G,\varphi ,\psi )}_{\rho}$-contraction. If there exists ${x}_{0}\in {X}_{\rho}$ such that $T{x}_{0}\in {[{x}_{0}]}_{\tilde{G}}$, then

- (i)
*T*is both a ${({G}^{-1},\varphi ,\psi )}_{\rho}$-contraction and a ${(\tilde{G},\varphi ,\psi )}_{\rho}$-contraction, - (ii)
${[{x}_{0}]}_{\tilde{G}}$ is

*T*-invariant and $T{{|}_{[{x}_{0}]}}_{\tilde{G}}$ is a ${({\tilde{G}}_{{x}_{0}},\varphi ,\psi )}_{\rho}$-contraction.

**Theorem 3.18** *Let* ${X}_{\rho}$ *be a* *ρ*-*complete modular space endowed with a graph* *G*, *where* *ρ* *satisfies the* ${\mathrm{\Delta}}_{2}$-*condition and let* $T:{X}_{\rho}\to {X}_{\rho}$ *be a* ${(\tilde{G},\varphi ,\psi )}_{\rho}$-*contraction*. *If* ${X}_{T}$ *is nonempty and* ${C}_{\rho}$-*graph* *G* *is weakly connected*, *then* *T* *is a PO*.

*Proof*If $x\in {X}_{T}$, then $({T}^{n}x,{T}^{n+1}x)\in E(G)$ for all $n\in \mathbb{N}$. First, we show that the sequence $\{\psi (\rho (c({T}^{n}x-{T}^{n+1}x)))\}$ converges to 0. From Definition 3.16, we have

*L*. If $L\ne 0$, we obtain

*ρ*-Cauchy sequence. By

*ρ*-completeness of ${X}_{\rho}$, there exists ${x}^{\ast}\in {X}_{\rho}$ such that $\rho (l({T}^{n}x-{x}^{\ast}))\to 0$ as $n\to \mathrm{\infty}$ and $({T}^{n}x,{T}^{n+1}x)\in E(G)$ for all $n\in \mathbb{N}$ and

*G*is a ${C}_{\rho}$-graph, then there exists a subsequence $\{{T}^{{n}_{p}}x\}$ such that $({T}^{{n}_{p}}x,{x}^{\ast})\in E(G)$ for each $p\in \mathbb{N}$. Also, $({T}^{{n}_{p}}x,{x}^{\ast})\in E(G)$ for each $p\in \mathbb{N}$. From Remark 2.5 and (${\psi}_{3}$), it follows that

Hence ${lim}_{n\to \mathrm{\infty}}\psi (\rho (\frac{c}{2}({x}^{\ast}-T{x}^{\ast})))=0$ and ${x}^{\ast}=T{x}^{\ast}$.

*T*has another fixed point ${y}^{\ast}\in {X}_{\rho}-\{{x}^{\ast}\}$. Since

*G*is a ${C}_{\rho}$-graph, there exists a subsequence $\{{T}^{{n}_{p}}x\}$ of $\{{T}^{n}x\}$ such that $({T}^{{n}_{p}}x,{x}^{\ast})\in E(G)$ and $({T}^{{n}_{p}}x,{y}^{\ast})\in E(G)$ for each $p\in \mathbb{N}$. Furthermore, as

*G*is weakly connected, $({x}^{\ast},{y}^{\ast})\in E(\tilde{G})$. We have

a contradiction. Hence, ${x}^{\ast}={y}^{\ast}$. □

In Theorem 3.18, if we replace the condition that *G* is a ${C}_{\rho}$-graph with orbitally ${G}_{\rho}$-continuity of *T*, then we have the following theorem.

**Theorem 3.19** *Let* ${X}_{\rho}$ *be a* *ρ*-*complete modular space endowed with a graph* *G*, *where* *ρ* *satisfies the* ${\mathrm{\Delta}}_{2}$-*condition and let* $T:{X}_{\rho}\to {X}_{\rho}$ *be a* ${(\tilde{G},\varphi ,\psi )}_{\rho}$-*contraction*, *which is orbitally* ${G}_{\rho}$-*continuous*. *Assume that the set* ${X}_{T}$ *is nonempty and the graph* *G* *is weakly connected*. *Then* *T* *is a PO*.

In the following corollaries, suppose that ${X}_{\rho}$ is a *ρ*-complete modular space endowed with a graph *G*, where *ρ* satisfies the ${\mathrm{\Delta}}_{2}$-condition and let $T:{X}_{\rho}\to {X}_{\rho}$ be edge-preserving and the set ${X}_{T}$ be nonempty.

**Corollary 3.20**

*Assume*

- (i)
*the*${C}_{\rho}$-*graph**G**is weakly connected and* - (ii)
*there exist nonnegative numbers**l*,*c**with*$l<c$*such that*${\int}_{0}^{\rho (c(Tx-Ty))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \varphi ({\int}_{0}^{\rho (l(x-y))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt)$

*hold for each* $(x,y)\in E(\tilde{G})$ *with* $\varphi \in {\mathrm{\Phi}}_{1}$ *and* $\phi \in \mathrm{\Phi}$. *Then* *T* *is a PO*.

**Corollary 3.21**

*Assume*

- (i)
*the*${C}_{\rho}$-*graph**G**is weakly connected and* - (ii)
*there exist nonnegative numbers**l*,*c**with*$l<c$*such that*${\int}_{0}^{\psi (\rho (c(Tx-Ty)))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \eta {\int}_{0}^{\psi (\rho (l(x-y)))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt$

*for each* $(x,y)\in E(\tilde{G})$ *with* $\eta \in (0,1)$ *and* $\phi \in \mathrm{\Phi}$. *Then* *T* *is a PO*.

Now we provide examples in support of our results.

**Example 3.22**Let ${X}_{\rho}=\{0,1,2,3,4,5\}$ and $\rho (x)=|x|$, for all $x\in {X}_{\rho}$. Consider

and $Tx=0$, $x\in {X}_{\rho}$. Then *G* is weakly connected and ${C}_{\rho}$-graph, ${X}_{T}$ is nonempty and *T* is a ${(\tilde{G},A)}_{\rho}$-contraction where $c=\frac{7}{5}$, $l=\frac{6}{5}$, $\phi (t)=1$. Hence, *T* is a PO.

**Example 3.23**Let ${X}_{\rho}=\{0,1,2,3\}$ and $\rho (x)=|x|$, for all $x\in {X}_{\rho}$. Consider

Then *G* is weakly connected and ${C}_{\rho}$-graph, ${X}_{T}$ is nonempty, and *T* is a Hardy-Rogers type ${(\tilde{G})}_{\rho}$-contraction where $c=4$, ${l}_{1}={l}_{2}={l}_{3}=3$, ${l}_{4}={l}_{5}=\frac{1}{2}$, $\eta =\frac{1}{3}$, $\beta =\frac{1}{4}$, $\gamma =0$, and $\phi (t)=1$. Moreover, 0 is a unique fixed point of *T*.

**Example 3.24**Let ${X}_{\rho}=[0,1]$ and $\rho (x)=|x|$, for all $x\in {X}_{\rho}$. Consider

and $Tx=\frac{x}{2}$, $x\in {X}_{\rho}$. Then *G* is weakly connected and a ${C}_{\rho}$-graph, ${X}_{T}$ is nonempty and *T* is a ${(\tilde{G},\varphi ,\psi )}_{\rho}$-contraction where $c=\frac{1}{2}$, $l=\frac{1}{3}$, $\phi (t)=1$, $\varphi (\xi )=\frac{\xi}{1+\xi}$, and $\psi (\omega )=\frac{\omega}{3}$. Thus all conditions of Theorem 3.18 are satisfied. Moreover, *T* is a PO.

**Remark 3.25** In the above examples, if we use $\rho (x)={x}^{2}$, the conclusions remain the same.

## Declarations

### Acknowledgements

The authors are thankful to the anonymous referees for their valuable comments and suggestions, which helped to improve the presentation of this paper.

## Authors’ Affiliations

## References

- Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales.
*Fundam. Math.*1922, 3: 133–181.Google Scholar - Ran ACM, Reuring MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations.
*Proc. Am. Math. Soc.*2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4View ArticleGoogle Scholar - Abbas M, Khamsi MA, Khan AR: Common fixed point and invariant approximation in hyperbolic ordered metric spaces.
*Fixed Point Theory Appl.*2011., 2011: Article ID 25 10.1186/1687-1812-2011-25Google Scholar - Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces.
*Appl. Math. Comput.*2011, 217: 5784–5789. 10.1016/j.amc.2010.12.060View ArticleMathSciNetGoogle Scholar - Jachymski J: The contraction principle for mappings on a metric space endowed with a graph.
*Proc. Am. Math. Soc.*2008, 136: 1359–1373.View ArticleMathSciNetGoogle Scholar - Abbas M, Nazir T: Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph.
*Fixed Point Theory Appl.*2013., 2013: Article ID 20Google Scholar - Öztürk M, Girgin E: On some fixed-point theorems for
*ψ*-contraction on metric space involving a graph.*J. Inequal. Appl.*2014., 2014: Article ID 39 10.1186/1029-242X-2014-39Google Scholar - Samreen M, Kamran T: Fixed point theorems for integral
*G*-contractions.*Fixed Point Theory Appl.*2013., 2013: Article ID 149Google Scholar - Kannan R: Some results on fixed points.
*Bull. Calcutta Math. Soc.*1968, 60: 71–76.MathSciNetGoogle Scholar - Rhaodes BE: A comparison of various definitions of contractive mappings.
*Trans. Am. Math. Soc.*1977, 26: 257–290.View ArticleGoogle Scholar - Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type.
*Int. J. Math. Math. Sci.*2002, 29: 531–536. 10.1155/S0161171202007524View ArticleMathSciNetGoogle Scholar - Akram M, Zafar AA, Siddiqui AA: A general class of contractions:
*A*-contractions.*Novi Sad J. Math.*2002, 38(1):25–33.MathSciNetGoogle Scholar - Bianchini R: Su un problema di S. Reich riguardante la teoria dei punti fissi.
*Boll. Unione Mat. Ital.*1972, 5: 103–108.MathSciNetGoogle Scholar - Reich S: Kannan’s fixed point theorem.
*Boll. Unione Mat. Ital.*1971, 5: 1–11.Google Scholar - Nakano H Tokyo Math. Book Ser. 1. In
*Modulared Semi-Ordered Linear Spaces*. Maruzen, Tokyo; 1950.Google Scholar - Musielak T, Orlicz W: On modular spaces.
*Stud. Math.*1959, 18: 49–65.MathSciNetGoogle Scholar - Koshi S, Shimogaki T: On
*F*-norms of quasi-modular spaces.*J. Fac. Sci. Hokkaido Univ., Ser. I*1961, 15(3–4):202–218.MathSciNetGoogle Scholar - Mazur S, Orlicz W: On some classes of linear spaces.
*Stud. Math.*1958, 17: 97–119. (Reprinted in Wladyslaw Orlicz Collected Papers, pp. 981–1003, PWN, Warszawa (1988))MathSciNetGoogle Scholar - Yamamuro S: On conjugate spaces of Nakano spaces.
*Trans. Am. Math. Soc.*1959, 90: 291–311. 10.1090/S0002-9947-1959-0132378-1View ArticleMathSciNetGoogle Scholar - Turpin PH: Fubini inequalities and bounded multiplier property in generalized modular spaces.
*Comment. Math.*1978, 1: 331–353. (Tomus specialis in honorem Ladislai Orlicz)MathSciNetGoogle Scholar - Luxemburg, WAJ: Banach function spaces. Thesis, Delft Inst. of Techn. Asser., The Netherlands (1955)Google Scholar
- Khamsi MA, Kozolowski WK, Reich S: Fixed point theory in modular function spaces.
*Nonlinear Anal.*1990, 14: 935–953. 10.1016/0362-546X(90)90111-SView ArticleMathSciNetGoogle Scholar - Benavides TD, Khamsi MA, Samadi S: Asymptotically non-expansive mappings in modular function spaces.
*J. Math. Anal. Appl.*2002, 265: 249–263. 10.1006/jmaa.2000.7275View ArticleMathSciNetGoogle Scholar - Benavides TD, Khamsi MA, Samadi S: Asymptotically regular mappings in modular function spaces.
*Sci. Math. Jpn.*2001, 53: 295–304.MathSciNetGoogle Scholar - Benavides TD, Khamsi MA, Samadi S: Uniformly Lipschitzian mappings in modular function spaces.
*Nonlinear Anal.*2001, 46: 267–278. 10.1016/S0362-546X(00)00117-6View ArticleMathSciNetGoogle Scholar - Khamsi MA: A convexity property in modular function spaces.
*Math. Jpn.*1996, 44: 269–279.MathSciNetGoogle Scholar - Mongkolkeha C, Kumam P: Fixed point and common fixed point theorems for generalized weak contraction mappings of integral type in modular spaces.
*Int. J. Math. Math. Sci.*2011., 2011: Article ID 705943Google Scholar - Beygmohammadi M, Razani A: Two fixed-point theorems for mappings satisfying a general contractive condition of integral type in the modular space.
*Int. J. Math. Math. Sci.*2010., 2010: Article ID 317107Google Scholar - Liu Z, Li X, Kang SM, Cho SY: Fixed point theorems for mappings satisfying contractive conditions of integral type and applications.
*Fixed Point Theory Appl.*2011., 2011: Article ID 64Google Scholar - Mongkolkeha C, Kumam P: Some fixed point results for generalized weak contraction mappings in modular spaces.
*Int. J. Anal.*2013., 2013: Article ID 247378Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.