 Research
 Open Access
 Published:
Fixed points of mappings satisfying contractive condition of integral type in modular spaces endowed with a graph
Fixed Point Theory and Applications volume 2014, Article number: 220 (2014)
Abstract
Jachymski (Proc. Am. Math. Soc. 136:13591373, 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.
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 BanachCaccioppoli) 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 Acontractions, 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.
Define
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]
A selfmap T on a metric space X is called an Acontraction 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 sconvex 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.
The vector space {X}_{\rho} given by
is called a modular space. In general the modular ρ is not subadditive and therefore does not behave as a norm or a distance. One can associate to a modular an xnorm.
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;
(r_{2}) if {a}_{1},\dots ,{a}_{n} are nonnegative real numbers with {\sum}_{i=1}^{n}{a}_{i}=1, then we have
Define the ρball, {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}) ρconvergent to x\in {X}_{\rho} if \rho ({x}_{n}x)\to 0 as n\to \mathrm{\infty};
(t_{2}) ρCauchy if \rho ({x}_{n}{x}_{m})\to 0 as n and m\to \mathrm{\infty}.
{X}_{\rho} is ρcomplete if any ρCauchy sequence is ρconvergent. Note that ρconvergence does not imply ρCauchy 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}_{n1}))\to 0, then \rho (\alpha l({x}_{n}{x}_{n1}))\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}_{i1},{x}_{i})\in E(G) for i\in \{1,2,\dots ,k\}.
Recall that a graph 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
Since it is more convenient to treat \tilde{G} as a directed graph for which the set of its edges is symmetric, under this convention we have
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]
A mapping T:X\to X is called a Banach Gcontraction 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
In this section, we obtain several fixed point results in the setup of a modular space endowed with a graph. We start with the following definitions. Let {X}_{\rho} be a modular space endowed with a graph 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
and
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;
(A_{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 Tinvariant 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(xTx))}\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
By the definition of α, we obtain
for some k\in (0,1). Thus we have
That is, {\int}_{0}^{\rho (c({T}^{n}x{T}^{n1}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(xTx))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt. □
Theorem 3.6 Let {X}_{\rho} be a ρcomplete 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
Using Lemma 3.5, we have
It follows that \{\frac{c}{mn}{T}^{n}x\} is a ρCauchy sequence in {X}_{\rho}. Since {X}_{\rho} is ρcomplete, there exists a point {x}^{\ast}\in {X}_{\rho} such that \rho (\frac{c}{mn}({T}^{n}x{x}^{\ast}))\to 0. Consequently, \rho (l({T}^{n}x{x}^{\ast}))\to 0.
Now we show that {x}^{\ast} is a fixed point of T. As \rho (\frac{c}{mn}({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
which on taking the limit as p\to \mathrm{\infty} gives
By the definition of function α, we have
From Lemma 2.2, it follows that \rho (c({x}^{\ast}T{x}^{\ast}))=0 and T{x}^{\ast}={x}^{\ast}.
Next, we prove that {x}^{\ast} is a unique fixed point. Suppose that 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 ρcomplete 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}{mn}{T}^{n}x\} is a ρCauchy sequence in {X}_{\rho}. Owing to ρcompleteness of {X}_{\rho}, there exists {x}^{\ast}\in {X}_{\rho} such that \rho (\frac{c}{mn}({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}{mn}(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 ρcomplete modular space endowed with a graph G, where ρ satisfies the {\mathrm{\Delta}}_{2}condition and let T:{X}_{\rho}\to {X}_{\rho} be edgepreserving, 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 HardyRogers 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 HardyRogers type {(G)}_{\rho}contraction if the following conditions hold:
(H_{1}) T preserves edges of G;
(H_{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 HardyRogers 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 HardyRogers type {({G}^{1})}_{\rho}contraction and a HardyRogers type {(\tilde{G})}_{\rho}contraction,

(ii)
{[{x}_{0}]}_{\tilde{G}} is Tinvariant and T{{}_{[{x}_{0}]}}_{\tilde{G}} is a HardyRogers type {({\tilde{G}}_{{x}_{0}})}_{\rho}contraction.
Theorem 3.11 Let {X}_{\rho} be a ρcomplete 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 HardyRogers 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
It follows that
where h=\frac{\eta +\beta +\gamma}{1\beta \gamma}<1. Also,
Taking the limit as n\to \mathrm{\infty}, and using Lemma 2.2, we get
which implies that {lim}_{n}\rho (c({T}^{n}x{T}^{n+1}x))=0.
Let m,n\in \mathbb{N} with m>n. By (4) and Remark 2.5, we get
Thus, \{\frac{c}{mn}{T}^{n}x\} is a ρCauchy sequence in {X}_{\rho}. Since {X}_{\rho} is ρcomplete, there exists a point {x}^{\ast}\in {X}_{\rho} such that \rho (\frac{c}{mn}({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
Taking the limit as n\to \mathrm{\infty}, we have
As (\beta +\gamma )<1, so \rho (c({x}^{\ast}T{x}^{\ast}))=0 and {x}^{\ast}=T{x}^{\ast}.
Next, we prove that {x}^{\ast} is a unique fixed point. Suppose that 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
which further implies that
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 ρcomplete 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 HardyRogers 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 ρcomplete modular space endowed with a graph G, where ρ satisfies the {\mathrm{\Delta}}_{2}condition and T:{X}_{\rho}\to {X}_{\rho} is edgepreserving 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(TxTy))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \eta {\int}_{0}^{\rho (l(xy))}\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(TxTy))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \beta {\int}_{0}^{[\rho ({l}_{1}(xTx))+\rho ({l}_{2}(yTy))]}\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(TxTy))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \gamma {\int}_{0}^{[\rho (\frac{{l}_{1}}{2}(xTy))+\rho (\frac{{l}_{2}}{2}(yTx))]}\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;
(Q_{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 Tinvariant and T{{}_{[{x}_{0}]}}_{\tilde{G}} is a {({\tilde{G}}_{{x}_{0}},\varphi ,\psi )}_{\rho}contraction.
Theorem 3.18 Let {X}_{\rho} be a ρcomplete 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
Thus,
is decreasing and bounded from below and so
converges to a nonnegative number L. If L\ne 0, we obtain
that is, L\le \varphi (L), a contradiction. Hence {\int}_{0}^{\psi (\rho (c({T}^{n}x{T}^{n+1}x)))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\to 0 as n\to \mathrm{\infty}. It follows that \psi (\rho (c({T}^{n}x{T}^{n+1}x)))\to 0. Suppose that
Then there exists a {v}_{\epsilon}\in \mathbb{N} and a sequence {\{{T}^{{n}_{v}}x\}}_{v\ge {v}_{\epsilon}} such that
Hence we get the following:
Assume that there is an \epsilon >0 and there exist {m}_{v},{n}_{v}\in \mathbb{N} such that {m}_{v}>{n}_{v}>v for each v\in \mathbb{N} and
Then we choose the sequence {({m}_{v})}_{v\in \mathbb{N}} and {({n}_{v})}_{v\in \mathbb{N}} such that for each v\in \mathbb{N}, {m}_{v} is minimal in the sense that \psi (\rho (l({T}^{{m}_{v}}x{T}^{{n}_{v}}x)))\ge \epsilon but \psi (\rho (l({T}^{s}x{T}^{{n}_{v}}x)))<\epsilon, for all s\in \{{n}_{v}+1,\dots ,{m}_{v}1\}. Now, let \delta \in {\mathbb{R}}_{+} be such that \frac{l}{c}+\frac{1}{\delta}=1, then we have
Thus, taking the limit as v\to \mathrm{\infty}, and Proposition 2.8, we have
Therefore,
Now,
If v\to \mathrm{\infty}, then we have
a contradiction for \epsilon >0. Hence, \{l{T}^{n}x\} is a ρCauchy sequence. By ρcompleteness 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
Taking the limit as p\to \mathrm{\infty}, we have
Since \rho (l({T}^{{n}_{p}}x{x}^{\ast}))\to 0 (p\to \mathrm{\infty}), we obtain
which implies that \psi (\rho (c({T}^{{n}_{p}+1}xT{x}^{\ast})))\to 0, as p\to \mathrm{\infty}. Thus,
Hence {lim}_{n\to \mathrm{\infty}}\psi (\rho (\frac{c}{2}({x}^{\ast}T{x}^{\ast})))=0 and {x}^{\ast}=T{x}^{\ast}.
Finally, we prove that {x}^{\ast} is a unique fixed point. Suppose that 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 ρcomplete 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 ρcomplete modular space endowed with a graph G, where ρ satisfies the {\mathrm{\Delta}}_{2}condition and let T:{X}_{\rho}\to {X}_{\rho} be edgepreserving 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(TxTy))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \varphi ({\int}_{0}^{\rho (l(xy))}\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(TxTy)))}\phi (t)\phantom{\rule{0.2em}{0ex}}dt\le \eta {\int}_{0}^{\psi (\rho (l(xy)))}\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
Define T:{X}_{\rho}\to {X}_{\rho} as follows:
Then G is weakly connected and {C}_{\rho}graph, {X}_{T} is nonempty, and T is a HardyRogers 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.
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
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/S0002993903072204
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 10.1186/16871812201125
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.2010.12.060
Jachymski J: The contraction principle for mappings on a metric space endowed with a graph. Proc. Am. Math. Soc. 2008, 136: 1359–1373.
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 20
Öztürk M, Girgin E: On some fixedpoint theorems for ψ contraction on metric space involving a graph. J. Inequal. Appl. 2014., 2014: Article ID 39 10.1186/1029242X201439
Samreen M, Kamran T: Fixed point theorems for integral G contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 149
Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 60: 71–76.
Rhaodes BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1977, 26: 257–290.
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/S0161171202007524
Akram M, Zafar AA, Siddiqui AA: A general class of contractions: A contractions. Novi Sad J. Math. 2002, 38(1):25–33.
Bianchini R: Su un problema di S. Reich riguardante la teoria dei punti fissi. Boll. Unione Mat. Ital. 1972, 5: 103–108.
Reich S: Kannan’s fixed point theorem. Boll. Unione Mat. Ital. 1971, 5: 1–11.
Nakano H Tokyo Math. Book Ser. 1. In Modulared SemiOrdered Linear Spaces. Maruzen, Tokyo; 1950.
Musielak T, Orlicz W: On modular spaces. Stud. Math. 1959, 18: 49–65.
Koshi S, Shimogaki T: On F norms of quasimodular spaces. J. Fac. Sci. Hokkaido Univ., Ser. I 1961, 15(3–4):202–218.
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))
Yamamuro S: On conjugate spaces of Nakano spaces. Trans. Am. Math. Soc. 1959, 90: 291–311. 10.1090/S00029947195901323781
Turpin PH: Fubini inequalities and bounded multiplier property in generalized modular spaces. Comment. Math. 1978, 1: 331–353. (Tomus specialis in honorem Ladislai Orlicz)
Luxemburg, WAJ: Banach function spaces. Thesis, Delft Inst. of Techn. Asser., The Netherlands (1955)
Khamsi MA, Kozolowski WK, Reich S: Fixed point theory in modular function spaces. Nonlinear Anal. 1990, 14: 935–953. 10.1016/0362546X(90)90111S
Benavides TD, Khamsi MA, Samadi S: Asymptotically nonexpansive mappings in modular function spaces. J. Math. Anal. Appl. 2002, 265: 249–263. 10.1006/jmaa.2000.7275
Benavides TD, Khamsi MA, Samadi S: Asymptotically regular mappings in modular function spaces. Sci. Math. Jpn. 2001, 53: 295–304.
Benavides TD, Khamsi MA, Samadi S: Uniformly Lipschitzian mappings in modular function spaces. Nonlinear Anal. 2001, 46: 267–278. 10.1016/S0362546X(00)001176
Khamsi MA: A convexity property in modular function spaces. Math. Jpn. 1996, 44: 269–279.
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 705943
Beygmohammadi M, Razani A: Two fixedpoint theorems for mappings satisfying a general contractive condition of integral type in the modular space. Int. J. Math. Math. Sci. 2010., 2010: Article ID 317107
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 64
Mongkolkeha C, Kumam P: Some fixed point results for generalized weak contraction mappings in modular spaces. Int. J. Anal. 2013., 2013: Article ID 247378
Acknowledgements
The authors are thankful to the anonymous referees for their valuable comments and suggestions, which helped to improve the presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Öztürk, M., Abbas, M. & Girgin, E. Fixed points of mappings satisfying contractive condition of integral type in modular spaces endowed with a graph. Fixed Point Theory Appl 2014, 220 (2014). https://doi.org/10.1186/168718122014220
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122014220
Keywords
 connected graph
 modular space
 Banach Gcontraction
 integral type contraction