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Common fixed point results of a pair of generalized \((\psi,\varphi )\)contraction mappings in modular spaces
 Mahpeyker Öztürk^{1}Email author,
 Mujahid Abbas^{2, 3} and
 Ekber Girgin^{1}
https://doi.org/10.1186/s136630160503x
© Öztürk et al. 2016
 Received: 30 May 2015
 Accepted: 15 January 2016
 Published: 17 February 2016
Abstract
In this paper, we establish the existence of a common fixed point of almost generalized contractions on modular spaces. As an application, we present some fixed and common fixed point results for such mappings on modular spaces endowed with a graph. The existence of fixed and common points of mappings satisfying generalized contractive conditions of integral type is also obtained in such spaces. Some examples are presented to support the results obtained herein. Our results generalize and extend various comparable results in the existing literature.
Keywords
 modular space
 Banach Gcontraction
 almost contraction
MSC
 47H10
 54H25
 54E50
1 Introduction
Over the past two decades the development of fixed point theory in metric spaces has attracted considerable attention due to numerous applications in areas such as variational and linear inequalities, optimization, and approximation theory. The classical Banach contraction principle is one of the most useful results in nonlinear analysis. It ensures the existence and uniqueness of the fixed point of nonlinear operators satisfying the strict contraction condition. It also shows that the fixed point can be approximated by means of a Picard iteration. Due to its applications in mathematics and other related disciplines, the Banach contraction principle has been generalized in many directions. Extensions of the Banach contraction principle have been obtained either by generalizing the domain of the mapping (see, e.g., [1, 2]) or by extending the contractive condition on the mappings [3, 4]. The existence of fixed points in ordered metric spaces has been studied by Ran and Reurings [5], Theorem 2.1. Subsequently, Nieto and RodríguezLópez [6] extended the results in [5], Theorem 2.1 for nondecreasing mappings and applied them to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. Since then, a number of results have been proved in the framework of ordered metric spaces. In 2008, Jachymski [7] investigated a new approach in metric fixed point theory by replacing the order structure with a graph structure on a metric space. In this way, the results proved in ordered metric spaces are generalized (see for details [7] and the references therein). Abbas and Nazir [8] obtained some fixed point results for a power graphic contraction pair endowed with a graph. Beg and Butt [9] proved fixed point theorems for setvalued mappings on a metric space with a graph. In this direction, we refer to [10–12] and the references mentioned therein.
The concept of a modular space was initiated by Nakano [13] and was redefined and generalized by Musielak and Orlicz [14]. In addition to it, the most important development of this theory is due to Mazur and Musielak, Luxemburg and Turpin (see [15–17]). The fixed point theory in modular function spaces has recently got a great deal of attention of researchers, for example, Khamsi [18] (see also [10, 11, 15, 17, 19–26]). Kuaket and Kumam [27] and Mongkolkeha and Kumam [28–30] proved some fixed and common fixed point results for generalized contraction mappings in modular spaces. Also, Kumam [22] obtained some fixed point theorems for nonexpansive mappings in arbitrary modular spaces.
The aim of this paper is to prove common fixed point results of a pair of mappings satisfying an almost generalized \({ ( {\psi,\varphi } ) }\)contraction condition in the setting of modular spaces. We provide an example to show that our results are a substantial generalization of comparable results in the existing literature. As an application of the results obtained herein, we obtain fixed and common fixed point results in the framework of modular space endowed with a directed as well as undirected graph. Some examples are presented to support the results proved herein. The existence of common fixed points of mappings satisfying a contractive condition of integral type is also obtained in such spaces.
2 Preliminaries
 (1)
\(F(T)=\{x\in X:Tx=x\}=\{z\}\),
 (2)
for any \(x_{0}\in X\), the Picard iteration \(x_{n}=T^{n}x_{0} \) converges to z.
A sequence as in the above definition is called a sequence of successive approximations of T starting from \(x_{0}\).
The Banach contractive condition forces the mappings to be continuous. It is natural to ask if there do or do not exist weaker contractive conditions that ensure the existence and uniqueness of a fixed point but do not imply the continuity of mappings. Kannan [23], by considering a weaker contractive conditions, proved the existence of a fixed point for a mapping that can have a discontinuity. Following Kannan’s result, a lot of papers were devoted to obtaining fixed point or common fixed point theorems for various classes of contractive type conditions that do not require the continuity of the mappings; see, for example, [24] and [31].
The following definition is more suitable in this context.
Definition 2.1
This concept was introduced by Berinde as a ‘weak contraction’ in [3]. But in [4], Berinde renamed the ‘weak contraction’ as an ‘almost contraction’, which is more appropriate.
Berinde [3] proved some fixed point results for an almost contraction in the setting of a complete metric space and generalized the results in [23, 24], and [31].
Recently Babu et al. [32] considered the class of mappings that satisfy ‘condition (B)’ as follows.
They proved the following fixed point theorem.
Theorem 2.2
([32], Theorem 2.3)
Let \((X,d)\) be a complete metric space and \(T:X\rightarrow X\) be a map satisfying condition (B). Then T has a unique fixed point.
Afterwards Berinde [33] introduced the concept of a generalized almost contraction as follows.
Theorem 2.3
Let \((X,d)\) be a complete metric space and \(T:X\rightarrow X\) a generalized almost contraction. Then T has a unique fixed point.
A point \(y\in X\) is called a point of coincidence of two selfmappings f and T on X if there exists a point \(x\in X\) such that \(y=Tx=fx\). The point x is called coincidence point of a pair \(( f,T ) \).
Abbas et al. [34] introduced a generalization of ‘condition (B)’ for a pair of selfmaps and obtained a unique point of coincidence. Ciric et al. [35] extended the concept of the generalized almost contraction to two mappings and obtained some common fixed point results in a complete metric space.
Consistent with [14], some basic facts and notations needed in this paper are recalled as follows.
Definition 2.4
 (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}):

\(\rho(\alpha x+\beta y)\leq\rho(x)+\rho(y)\), whenever \(\alpha+\beta=1\), and \(\alpha,\beta\geq0\).
If (m_{3}) is replaced with \(\rho(\alpha x+\beta y)\leq \alpha^{s}\rho(x)+\beta^{s}\rho(y)\) where \(\alpha^{s}+\beta^{s}=1\), \(\alpha,\beta\geq0\), and \(s\in ( {0,1} ] \), then ρ is called sconvex modular. If \(s=1\), then we say that ρ is convex modular.
The following are some consequences of condition (m_{3}).
Remark 2.5
[20]
 (r_{1}):

For \(a, b\in \mathbb{R}\) with \(\vert a\vert <\vert b\vert \) we have \(\rho ( {ax} ) <\rho ( {bx} ) \) for all \(x\in X\).
 (r_{2}):

For \({a_{1}},\ldots,{a_{n}}\in \mathbb{R}_{+}\) with \(\sum_{i=1}^{n}{a_{i}}=1\), we have$$\rho \Biggl( {\sum_{i=1}^{n}{a_{i}} {x_{i}}} \Biggr) \leq \sum_{i=1}^{n}{ \rho ( {x_{i}} ) }\quad \text{for any } {{x_{1}}, \ldots,{x_{n}}\in X}. $$
Proposition 2.6
[30]
Let \(X_{\rho}\) be a modular space. If \(a,b\in{{\mathbb{R}_{+}}}\) with \(b\geq a\), then \(\rho ( {ax} ) \leq\rho ( {bx} ) \).
A mapping \(\rho:{\mathbb{R}}\rightarrow{}[0,\infty]\) defined by \(\rho(x)=\sqrt{\vert x\vert }\) is a trivial example of a modular functional.
 (a)
the \(\Delta_{2}\)type condition if there exists \(K>0\) such that for any \(x\in X_{\rho}\), we have \(\rho(2x)\leq K\rho(x)\);
 (b)
the \({\Delta_{2}}\)condition if \(\rho ( {2{x_{n}}} ) \rightarrow0\) as \(n\rightarrow\infty\), whenever \(\rho ( {x_{n}} ) \rightarrow0\) as \(n\rightarrow\infty\).
Definition 2.7
 (t_{1}):

ρconvergent to \(x\in X_{\rho}\) if \(\rho (x_{n}x)\rightarrow0\) as \(n \rightarrow\infty\);
 (t_{2}):

ρCauchy if \(\rho(x_{n}x_{m})\rightarrow0\) as \(n, m\rightarrow\infty\).
\(X_{\rho}\) is called 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 \(\Delta_{2}\)condition. We know that [25] the norm and modular convergence are also the same when we deal with the \(\Delta_{2}\)type condition.
In the sequel, suppose the modular function ρ is convex and satisfies the \(\Delta_{2}\)type condition.
Mongkolkeha and Kumam [30] proved the existence of a fixed point generalized weak contractive mapping in modular space as follows.
Theorem 2.8
Definition 2.9
Let \({X_{\rho}}\) be a modular space and \(T:{X_{\rho}}\rightarrow{X_{\rho}}\) be a selfmap. We say that T is ρcontinuous when if \(\rho ( {x_{n}  x} ) \to0 \), then \(\rho ( {Tx_{n}  Tx} ) \to0 \) as \(n \to\infty\).
3 Common fixed point of almost generalized \((\psi,\varphi )\)contraction
We set Ψ = {\(\psi: [ {0,\infty} ) \rightarrow [ {0,\infty} ) : \psi\) a continuous nondecreasing function and \(\psi ( t ) =0\) if and only if \(t=0\)} and Φ = {\(\varphi : [ {0,\infty} ) \rightarrow [ {0,\infty} ) : \varphi\) a lowersemi continuous function and \(\varphi ( t ) =0\) if and only if \(t=0\)}.
In this section, we obtain common fixed point results for a pair of mappings satisfying the generalized \(( {\psi,\varphi} ) \)contractive condition in the framework of a modular space.
Theorem 3.1
Proof
 Step 1.:

Prove that \(\rho ( {{x_{n}}{x_{n+1}}} ) \rightarrow0\) as \(n\rightarrow\infty\).
 Step 2.:

Now we show that \(\{ {x_{n}} \} \) is a ρCauchy sequence.
 Step 3.:

We prove the existence of a fixed point of one mapping.
 Step 4.:

We prove that z is a fixed point of a mapping T.
 Step 5.:

To prove the uniqueness of a common fixed point of two mappings.
The following results are obtained directly from Theorem 3.1.
Corollary 3.2
Corollary 3.3
Corollary 3.4
Define Ϝ = {\(\xi:{\mathbb{R}_{+}}\rightarrow {\mathbb{R}_{+}}:\varphi\) is a Lebesgue integral mapping which is summable, nonnegative and satisfies \(\int _{0}^{\varepsilon}\xi(t)\, dt>0\), for each \(\varepsilon>0\)}.
Corollary 3.5
Proof
Take \(\psi ( t ) =\int _{0}^{t}{ \xi ( t ) }\, dt\) and \(\varphi ( t ) = ( {1k} ) t\) for all \(t\in [ {0,\infty} ) \). The result then follows from Theorem 3.1. □
Corollary 3.6
4 Common fixed points on modular spaces with a directed graph
Let \({X_{\rho}}\) be a ρmodular space and \(\Delta =\{(x,x):x\in X\}\) denote the diagonal of \({X_{\rho}}\times{X_{\rho}}\). 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 \(\Delta\subseteq E(G)\). Also assume that G has no parallel edges and G is a weighted graph in the sense that each edge \((x,y)\) is assigned the weight \(\rho(xy)\). Since ρ is a modular functional on \(X_{\rho}\), the weight assigned to each vertex x to vertex y does not need to be zero and whenever a zero weight is assigned to some edge \((x,y)\), it reduces to a \((x,x)\) having weight 0. The graph G is identified with 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 \(k\in \mathbb{N}\) is a finite sequence \(\{x_{n}\}\) of vertices such that \(x=x_{0},\ldots,x_{k}=y\) and \((x_{i1},x_{i})\in E(G)\) for \(i\in\{ 1,2,\ldots,k\}\).
For \(x,y\in V(G)\), we have \(x\mathbin{R}y\) if and only if there is a path in G from x to y. If G is such that \(E(G)\) is symmetric, then for \(x\in V(G)\), the equivalence class \([x]_{G}\) in \(V(G)\) defined by the relation R is \(V(G_{x})\).
Theorem 4.1
 (i)If \(\{ {x_{n} } \} \) is a sequence in \(X_{\rho}\) such that \(\rho ( {x_{n}  x} ) \to0 \) and \(( {{x_{2n}},{x_{2n+1}}} ) \in E ( G ) \) for all \(n\geq0\), then there exists a subsequence \({ \{ {{x_{{2{n_{p}}}}}} \} }\) of \({ \{ {{x_{{2n}}}} \} }\) such that
 (i_{a}):

T is ρcontinuous and \(( {x,{x_{{2{n_{p}}+1}}}} ) \in E ( G ) \) for all \(p\geq0\) or
 (i_{b}):

S is ρcontinuous and \(( { {x_{{2{n_{p}}}}}},x ) \in E ( G ) \) for all \(p\geq0\).
 (ii)There is a sequence \({ \{ {x_{n}} \} }\) in \(X_{\rho}\) such that$$\begin{aligned}& ( {{x_{2n}},S{x_{2n}}} ) \in E ( G )\quad \textit{implies that} \quad ( {{x_{2n+2}},S{x_{2n+2}}} ) \in E ( G ) \quad \textit{and} \\& ( {{x_{2n+1}},T{x_{2n+1}}} ) \in E ( G ) \quad \textit{implies that}\quad ( {{x_{2n+3}},T{x_{2n+3}}} ) \in E ( G ) . \end{aligned}$$
 (iii)
 (iv)
\({X_{ST}}\) is nonempty.
Then S and T have a common fixed point.
Proof
We note that Theorem 4.1 does not guarantee the uniqueness of a common fixed point.
To obtain the uniqueness, an additional assumption as given in the following theorem is required.
Theorem 4.2
In addition to the conditions of Theorem 4.1, assume that for any two common fixed point \({x^{\ast}}\), \({y^{\ast}}\) of S and T, there exists \(z\in X_{p}\) such that \(( {{x^{\ast }},z} ) \in E ( G ) \) and \(( {z,{y^{\ast}}} ) \in E ( G ) \). Then \({x^{\ast}}={y^{\ast}}\).
Proof
Example 4.3
The following results are obtained directly from Theorem 4.1.
Corollary 4.4
 (i)If \(\{ {x_{n} } \} \) is a sequence in \(X_{\rho}\) such that \(\rho ( {x_{n}  x} ) \to0 \) and \(( {{x_{2n}},{x_{2n+1}}} ) \in E ( G ) \) for all \(n\geq0\), then there exists a subsequence \({ \{ {{x_{{2{n_{p}}}}}} \} }\) of \({ \{ {{x_{{2n}}}} \} }\) such that
 (i_{a}):

T is ρcontinuous and \(( {x,{x_{{2{n_{p}}+1}}}} ) \in E ( G ) \) for all \(p\geq0\) or
 (i_{b}):

S is ρcontinuous and \(( { {x_{{2{n_{p}}}}}},x ) \in E ( G ) \) for all \(p\geq0\).
 (ii)There is a sequence \({ \{ {x_{n}} \} }\) in \(X_{\rho}\) such that$$\begin{aligned}& ( {{x_{2n}},S{x_{2n}}} ) \in E ( G )\quad \textit{implies that} \quad ( {{x_{2n+2}},S{x_{2n+2}}} ) \in E ( G )\quad \textit{and} \\& ( {{x_{2n+1}},T{x_{2n+1}}} ) \in E ( G )\quad \textit{implies that}\quad ( {{x_{2n+3}},T{x_{2n+3}}} ) \in E ( G ) . \end{aligned}$$
 (iii)
\(\psi ( {\rho ( {SxTy} ) } ) \leq\psi ( {M ( {x,y} ) } ) \varphi ( {M ( {x,y} ) } ) \), for any \(( {x,y} ) \in E ( G ) \).
 (iv)
\({X_{ST}}\) is nonempty.
Then S and T have a common fixed point.
Corollary 4.5
 (i)If \(\{ {x_{n} } \} \) is a sequence in \(X_{\rho}\) such that \(\rho ( {x_{n}  x} ) \to0 \) and \(( {{x_{2n}},{x_{2n+1}}} ) \in E ( G ) \) for all \(n\geq0\), then there exists a subsequence \({ \{ {{x_{{2{n_{p}}}}}} \} }\) of \({ \{ {{x_{{2n}}}} \} }\) such that
 (i_{a}):

T is ρcontinuous and \(( {x,{x_{{2{n_{p}}+1}}}} ) \in E ( G ) \) for all \(p\geq0\) or
 (i_{b}):

S is ρcontinuous and \(( { {x_{{2{n_{p}}}}}},x ) \in E ( G ) \) for all \(p\geq0\).
 (ii)There is a sequence \({ \{ {x_{n}} \} }\) in \(X_{\rho}\) such that$$\begin{aligned}& ( {{x_{2n}},S{x_{2n}}} ) \in E ( G ) \quad \textit{implies that}\quad ( {{x_{2n+2}},S{x_{2n+2}}} ) \in E ( G )\quad \textit{and} \\& ( {{x_{2n+1}},T{x_{2n+1}}} ) \in E ( G )\quad \textit{implies that} \quad ( {{x_{2n+3}},T{x_{2n+3}}} ) \in E ( G ) . \end{aligned}$$
 (iii)
\(\rho ( {SxTy} ) \leq M ( {x,y} ) \varphi ( {M ( {x,y} ) } ) \) for any \(( {x,y} ) \in E ( G ) \).
 (iv)
\({X_{ST}}\) is nonempty.
Then S and T have a common fixed point.
Corollary 4.6
 (i)If \(\{ {x_{n} } \} \) is a sequence in \(X_{\rho}\) such that \(\rho ( {x_{n}  x} ) \to0 \) and \(( {{x_{2n}},{x_{2n+1}}} ) \in E ( G ) \) for all \(n\geq0\), then there exists a subsequence \({ \{ {{x_{{2{n_{p}}}}}} \} }\) of \({ \{ {{x_{{2n}}}} \} }\) such that
 (i_{a}):

T is ρcontinuous and \(( {x,{x_{{2{n_{p}}+1}}}} ) \in E ( G ) \) for all \(p\geq0\) or
 (i_{b}):

S is ρcontinuous and \(( { {x_{{2{n_{p}}}}}},x ) \in E ( G ) \) for all \(p\geq0\).
 (ii)There is a sequence \({ \{ {x_{n}} \} }\) in \(X_{\rho}\) such that$$\begin{aligned}& ( {{x_{2n}},S{x_{2n}}} ) \in E ( G ) \quad \textit{implies that}\quad ( {{x_{2n+2}},S{x_{2n+2}}} ) \in E ( G ) \quad \textit{and} \\& ( {{x_{2n+1}},T{x_{2n+1}}} ) \in E ( G ) \quad \textit{implies that}\quad ( {{x_{2n+3}},T{x_{2n+3}}} ) \in E ( G ) . \end{aligned}$$
 (iii)There exist \(k\in [ {0,1} ) \) and \(L\geq0\) such thatfor any \(( {x,y} ) \in E ( G ) \).$$\rho ( {SxTy} ) \leq k M ( {x,y} ) +L N ( {x,y} ) $$
 (iv)
\({X_{ST}}\) is nonempty.
Then S and T have a common fixed point.
Corollary 4.7
 (i)If \(\{ {x_{n} } \} \) is a sequence in \(X_{\rho}\) such that \(\rho ( {x_{n}  x} ) \to0 \) and \(( {{x_{2n}},{x_{2n+1}}} ) \in E ( G ) \) for all \(n\geq0\), then there exists a subsequence \({ \{ {{x_{{2{n_{p}}}}}} \} }\) of \({ \{ {{x_{{2n}}}} \} }\) such that
 (i_{a}):

T is ρcontinuous and \(( {x,{x_{{2{n_{p}}+1}}}} ) \in E ( G ) \) for all \(p\geq0\) or
 (i_{b}):

S is ρcontinuous and \(( { {x_{{2{n_{p}}}}}},x ) \in E ( G ) \) for all \(p\geq0\).
 (ii)There is a sequence \({ \{ {x_{n}} \} }\) in \(X_{\rho}\) such that$$\begin{aligned}& ( {{x_{2n}},S{x_{2n}}} ) \in E ( G )\quad \textit{implies that} \quad ( {{x_{2n+2}},S{x_{2n+2}}} ) \in E ( G ) \quad \textit{and} \\& ( {{x_{2n+1}},T{x_{2n+1}}} ) \in E ( G ) \quad \textit{implies that}\quad ( {{x_{2n+3}},T{x_{2n+3}}} ) \in E ( G ) . \end{aligned}$$
 (iii)There exists \(k\in [ {0,1} ) \) and \(L\geq0\) such thatfor each \(( {x,y} ) \in E ( G ) \).$$\int _{0}^{\rho ( {SxTy} ) }{\xi ( t ) \,dt}\leq k \int _{0}^{M ( {x,y} ) }{\xi ( t ) \,dt}+L \int _{0}^{N ( {x,y} ) }{\xi ( t ) \,dt} $$
 (iv)
\({X_{ST}}\) is nonempty.
Then S and T have a common fixed point.
Proof
The proof follows from Theorem 4.1 by taking \(\psi ( t ) =\int _{0}^{t}{\xi ( t ) }\,dt\) and \(\varphi ( t ) = ( {1k} ) t\) for all \(t\in [ {0,\infty} ) \). □
5 Fixed point results on modular spaces involving undirected graph
Recently, Öztürk et al. [10] obtained some fixed point results for mappings satisfying a contractive condition of integral type in modular spaces endowed with a graph using the \(C_{\rho}\)graph and being orbitally \(G_{\rho}\)continuous. To apply this property, we modify for \(C=1\) as follows.
Definition 5.1
Let \(\{ {{T^{n}}x} \} \) be a sequence in \(X_{\rho}\) for some \(x\in X_{\rho}\) such that \(\rho ( {T^{n} x  x^{*} } ) \to0 \) for \({x^{\ast}}\in {X_{\rho}}\) and \(( {{T^{n}}x,{T^{n+1}x}} ) \in E ( G ) \) for all \(n\in{\mathbb{N}}\). Then a graph G is called a modified \(C_{\rho}\)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 5.2
A mapping \(T:X_{\rho}\rightarrow X_{\rho}\) is called orbitally \(G_{\rho}\)continuous if for all \(x,y\in X_{\rho }\) and any sequence \((n_{p})_{p\in{\mathbb{N}}}\) of positive integers, \(\rho ( {{T}^{{n}_{{p}} } x  y} ) \to0 \) and \(\rho ( {T ( {{T}^{{n}_{{p}} } x} )  T ( y )} ) \to0 \) as \(p\rightarrow\infty\).
Next, we define an almost generalized \(( {\psi,\varphi} ) \)G̃contraction mapping.
Definition 5.3
 (i)
T preserves edges of G.
 (ii)There exists \(L\geq0\) such thatholds for all \((x,y)\in E(G)\).$$\psi \bigl( {\rho ( {TxTy} ) } \bigr) \leq\psi \bigl( {{m} ( {x,y} ) } \bigr)  \varphi \bigl( {{m} ( {x,y} ) } \bigr) +L \psi \bigl( {{n} ( {x,y} ) } \bigr) $$
Remark 5.4
 (i)
T is both an almost generalized \(( {\psi,\varphi} )\)\(G^{1}\)contraction mapping and an almost generalized \(( {\psi ,\varphi} )\)G̃contraction mapping.
 (ii)
\([ {x_{0} } ]_{\widetilde {G}}\) is Tinvariant and \({T_{{ { [ {x_{0}} ]} }_{\widetilde {G}}}}\) is an almost generalized \(( {\psi,\varphi} )\)\({\widetilde {G}_{x_{0} }}\)contraction mapping.
Theorem 5.5
 (i)
G is weakly connected and modified \(C_{\rho}\)graph;
 (ii)
T is an almost generalized \(( {\psi,\varphi} )\)G̃contraction;
 (iii)
\(X_{T}= \{ {x\in X: ( {x,Tx} ) \in E ( G ) } \}\) is nonempty,
In Theorem 5.5, if we replace the condition that G is a modified \({C_{\rho}}\)graph with modified orbitally \(G_{\rho}\)continuity of T, then we have the following theorem.
Theorem 5.6
Let \({X_{\rho}}\) be a ρcomplete modular space endowed with a graph G, and \(T:{X_{\rho}}\rightarrow{X_{\rho }}\) an almost generalized \(( {\psi,\varphi} ) \)G̃contraction mapping and modified orbitally \(G_{\rho}\)continuous. If \({X_{T}}\) is nonempty and the graph G is weakly connected, then T is a PO.
Example 5.7
Declarations
Acknowledgements
The authors express their gratitude to the referees for valuable comments, critical remarks, and suggestions, which helped them in improving the presentation of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided 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.
Authors’ Affiliations
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