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Öztürk et al. Fixed Point Theory and Applications 2014, 2014:220
http://www.fixedpointtheoryandapplications.com/content/2014/1/220
RESEARCH
Open Access
Fixed points of mappings satisfying
contractive condition of integral type in
modular spaces endowed with a graph
Mahpeyker Öztürk1* , Mujahid Abbas2,3 and Ekber Girgin1
*
Correspondence:
[email protected]
1
Department of Mathematics,
Sakarya University, Sakarya, 54187,
Turkey
Full list of author information is
available at the end of the article
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: connected graph; modular space; Banach G-contraction; integral 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 []. This
Banach contraction principle [] 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 []. 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.,
[, ]). Jachymski [] 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 [] and the references therein).
For further work in this direction, we refer to, e.g., [–].
In , Kannan [] 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 [] and
the references therein. On the other hand, Branciari [] obtained a ﬁxed point theorem
©2014 Öztürk et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
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for a single valued mapping satisfying an analog of Banach’s contraction principle for an
integral type inequality. Recently, Akram et al. [] introduced a new class of contraction
maps, called A-contractions, which is a proper generalization of Kannan’s mappings [],
Bianchini’s mappings [], and Reich type mappings [].
The theory of modular spaces was initiated by Nakano [] in connection with the theory
of ordered spaces which was further generalized by Musielak and Orlicz [] (see also
[–]). The study of ﬁxed point theory in the context of modular function spaces was
initiated by Khamsi et al. [] (see also [–]). Also, some ﬁxed point theorems have
been proved for mappings satisfying contractive conditions of integral type in modular
space [, ].
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 [, , ].
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 ∈ X and limn→∞ T n x = z for all x ∈ X.
Deﬁne = {ϕ : R+ → R+ : ϕ is a Lebesgue integral mapping which is summable, non
negative and satisﬁes  ϕ(t) dt > , for each > }.
Let A = {α : R+ → R+ : α is continuous and a ≤ kb for some k ∈ [, ) whenever a ≤
α(a, b, b) or a ≤ α(b, a, b) or a ≤ α(b, b, a) for all a, b}.
Let ψ : R+ → R+ be a nondecreasing mapping which satisﬁes the following conditions:
(ψ ) ψ(x) =  if and only if x = ;
(ψ ) for a sequence {xn } in R+ , we have ψ(xn ) →  if and only if xn →  as n → ∞;
(ψ ) for every x, y ∈ R+ , we have ψ(x + y) ≤ ψ(x) + ψ(y).
The collection of all such mappings will be denoted by .
Deﬁne
 = φ : R+ → R+ : φ is increasing, upper semi-continuous and φ(t) < t, ∀t >  .
Theorem . [] Let (X, d) be a complete metric space, η ∈ [, ), and T : X → X a mapping. Suppose that

d(Tx,Ty)
d(x,y)
ϕ(t) dt ≤ η
ϕ(t) dt

is satisﬁed for every x, y ∈ X, where ϕ ∈ . Then T has a unique ﬁxed point z ∈ X and for
each x ∈ X, we have limn→∞ T n x = z.
Lemma . [] Let (X, d) be a metric space, ϕ ∈ , and {xn } a nonnegative sequence.
Then
x
x
(a) limn→∞ xn = x implies that limn→∞  n ϕ(t) dt =  ϕ(t) dt;
x
(b) limn→∞  n ϕ(t) dt = if and only if limn→∞ xn = .
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Deﬁnition . [] A selfmap T on a metric space X is called an A-contraction if for any
x, y ∈ X and for some α ∈ A, the following condition holds:
d(Tx, Ty) ≤ α d(x, y), d(x, Tx), d(y, Ty) .
Now, we recall some basic facts and notations as regards modular spaces. For more details the reader may consult [].
Deﬁnition . Let X be an arbitrary vector space. A functional ρ : X → [, ∞] is called
modular if for any arbitrary x, y in X:
(m ) ρ(x) =  if and only if x = ;
(m ) ρ(αx) = ρ(x) for every scalar α with |α| = ;
(m ) ρ(αx + βy) ≤ ρ(x) + ρ(y) if α + β = , α ≥ , β ≥ .
If (m ) is replaced by ρ(αx + βy) ≤ α s ρ(x) + β s ρ(y) if α s + β s = , where s ∈ (, ], α ≥ ,
β ≥  then we say that ρ is s-convex modular. If s = , then ρ is called convex modular.
√
ρ : R → [, ∞] deﬁned by ρ(x) = |x| is a simple example of a modular functional.
The vector space Xρ given by
Xρ = x ∈ X; ρ(λx) →  as λ → 
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 . [] The following are immediate consequences of condition (m ):
(r ) if a, b ∈ R (set of all real numbers) with |a| < |b|, then ρ(ax) < ρ(bx) for all x ∈ X;
(r ) if a , . . . , an are nonnegative real numbers with ni= ai = , then we have
ρ
n
i=
ai xi ≤
n
ρ(xi )
(x , . . . , xn ∈ X).
i=
Deﬁne the ρ-ball, Bρ (x, r), centered at x ∈ Xρ with radius r as
Bρ (x, r) = h ∈ Xρ; ρ(x – h) ≤ r .
A point x ∈ Xρ is called a ﬁxed point of T : Xρ → Xρ if T(x) = x.
A function modular is said to satisfy the  -type condition if there exists K >  such
that for any x ∈ Xρ we have ρ(x) ≤ Kρ(x). A modular ρ is said to satisfy the  -condition
if ρ(xn ) →  as n → ∞, whenever ρ(xn ) →  as n → ∞.
Deﬁnition . Let Xρ be a modular space. The sequence {xn } ⊂ Xρ is said to be:
(t ) ρ-convergent to x ∈ Xρ if ρ(xn – x) →  as n → ∞;
(t ) ρ-Cauchy if ρ(xn – xm ) →  as n and m → ∞.
Xρ 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  -condition.
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Proposition . [] Let Xρ be a modular space. If a, b ∈ R+ with b ≥ a, then ρ(ax) ≤
ρ(bx).
Proposition . [] Suppose that Xρ is a modular space, ρ satisﬁes the  -condition and
{xn }n∈N is a sequence in Xρ . If ρ(c(xn – xn– )) → , then ρ(αl(xn – xn– )) → , as n → ∞,
where c, l, α ∈ R+ with c > l and cl + α = . Now we give some basic deﬁnitions from graph
theory needed in the sequel.
Throughout this paper, = {(x, x) : x ∈ X} denotes the diagonal of X × 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 ⊆ 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 ∈ N is a ﬁnite
sequence {xn }, n ∈ {, , , . . . , k} of vertices such that x = x , . . . , xk = y and (xi– , xi ) ∈ E(G)
for i ∈ {, , . . . , k}.
Recall that a graph G is connected if there is a path between any two vertices and it
˜ is connected, where G
˜ denotes the undirected graph obtained
is weakly connected if G
from G by ignoring the direction of edges. Denote by G– the graph obtained from G by
reversing the direction of the edges. Thus
E G– = (x, y) ∈ X × X : (y, x) ∈ E(G) .
()
˜ as a directed graph for which the set of its edges is
Since it is more convenient to treat G
symmetric, under this convention we have
˜ = E(G) ∪ E G– .
E(G)
()
Let Gx 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 ∈ 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 (Gx ) = [x]G .
Deﬁnition . [, Deﬁnition .] A mapping T : X → X is called a Banach G-contraction
if and only if:
(a) for each x, y in X with (x, y) ∈ E(G), we have (T(x), T(y)) ∈ E(G), that is, T preserves
edges of G;
(b) there exists α in (, ) such that for each x, y ∈ X with (x, y) ∈ E(G) implies
d T(x), T(y) ≤ αd(x, y).
()
That is, T decreases weights of edges of G.
For any x, y ∈ V , (x, y) ∈ E such that V ⊆ V (G), E ⊆ E(G), (V , E ) 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 Xρ be a modular space
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endowed with a graph G and let T : Xρ → Xρ be a mapping. Denote
XT = x ∈ X : (x, Tx) ∈ E(G) .
Deﬁnition . Let {T n x} be a sequence, there exists C >  such that
ρ C T n x – x∗ →  for x∗ ∈ Xρ
and
T n x, T n+ x ∈ E(G) for all n ∈ N.
Then a graph G is called a Cρ -graph if there exists a subsequence {T np x} of {T n x} such
that (T np x, x∗ ) ∈ E(G) for p ∈ N.
Deﬁnition . A mapping T : Xρ → Xρ is called orbitally Gρ -continuous for all x, y ∈ Xρ
and any sequence (np )p∈N of positive integers, if there exists C >  such that
np
ρ C T np x – y → ,
T x, T np + x ∈ E(G) imply
ρ C T T np x – T(y) → ,
as p → ∞.
Deﬁnition . A mapping T is called a (G, A)ρ -contraction if it satisﬁes the following
conditions:
(A ) T preserves edges of G;
(A ) there exist nonnegative numbers l, c with l < c such that
ρ(c(Tx–Ty))
ϕ(t) dt

ρ(l(x–y))
≤α

ρ(l(x–Tx))
ϕ(t) dt,

ρ(l(y–Ty))
ϕ(t) dt,
ϕ(t) dt

holds for each (x, y) ∈ E(G), and some α ∈ A and ϕ ∈ .
Remark . Let T : Xρ → Xρ be a (G, A)ρ -contraction. If there exists x ∈ Xρ such that
Tx ∈ [x ]G˜ , then
˜ A)ρ -contraction,
(i) T is both a (G– , A)ρ -contraction and a (G,
˜
(ii) [x ]G˜ is T-invariant and T|[x ] G˜ is a (Gx , A)ρ -contraction.
Lemma . Let T : Xρ → Xρ be a (G, A)ρ -contraction. If x ∈ XT , then there exists r(x, Tx) ≥
 such that
ρ(c(T n x–T n+ x))
ϕ(t) dt ≤ k n r(x, Tx)

holds for all n ∈ N, where r(x, Tx) =
ρ(c(x–Tx))

ϕ(t) dt.
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Proof Let x ∈ XT , that is, (x, Tx) ∈ E(G). Then by induction, we have (T n x, T n+ x) ∈ E(G)
for all n ∈ N. Now, we have
ρ(c(T n x–T n+ x))
ϕ(t) dt

ρ(l(T n– x–T n x))
≤α
ϕ(t) dt

ρ(l(T n– x–T n x))
ρ(l(T n x–T n+ x))
ϕ(t) dt,

≤α
ρ(l(T n– x–T n x))
ϕ(t) dt,

ρ(l(T n– x–T n x))
ϕ(t) dt,
ϕ(t) dt .
ϕ(t) dt,


ρ(c(T n x–T n+ x))

By the deﬁnition of α, we obtain
ρ(c(T n x–T n+ x))
ρ(l(T n– x–T n x))
ϕ(t) dt ≤ k

ρ(c(T n– x–T n x))
ϕ(t) dt ≤ k

ϕ(t) dt

for some k ∈ (, ). Thus we have
ρ(c(T n x–T n+ x))
ρ(l(T n– x–T n x))
ϕ(t) dt ≤ k

ϕ(t) dt

ρ(c(T n– x–T n x))
≤k
ϕ(t) dt

≤k·k
ρ(l(T n– x–T n– x))
ϕ(t) dt

ρ(c(T n– x–T n– x))
≤ k
ϕ(t) dt

..
.
≤k
ϕ(t)
≤k
ρ(l(x–Tx))
n

ρ(c(x–Tx))
n
ϕ(t).

That is,
ρ(c(T n x–T n– x))

ϕ(t) dt ≤ k n r(x, Tx) for all n ∈ N, where r(x, Tx) =
ρ(c(x–Tx))

ϕ(t) dt.
Theorem . Let Xρ be a ρ-complete modular space endowed with a graph G, where ρ
˜ A)ρ -contraction. If the set XT is
satisﬁes the  -condition and let T : Xρ → Xρ be a (G,
nonempty, the graph G is weakly connected and a Cρ -graph, then T is a PO.
Proof If x ∈ XT , then Tx ∈ [x]G˜ and (T n x, T n+ x) ∈ E(G) for all n ∈ N. Let m, n ∈ N with
m > n. Note that
c n
T x – T mx
ρ
m–n
c n+
c m–
c n
T x – T n+ x +
T x – T n+ x + · · · +
T x – T mx
=ρ
m–n
m–n
m–n
n
≤ ρ c T x – T n+ x + ρ c T n+ x – T n+ x + · · · + ρ c T m– x – T m x .
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Using Lemma ., we have
c (T n x–T m x))
ρ( m–n

ρ(c(T n x–T n+ x))
ϕ(t) dt ≤
ρ(c(T n+ x–T n+ x))
ϕ(t) dt +

ϕ(t) dt

ρ(c(T m– x–T m x))
+ ··· +
ϕ(t) dt

≤ k n r(x, Tx) + k n+ r(x, Tx) + · · · + k m– r(x, Tx)
≤
kn
r(x, Tx) → ,
–k
as n → ∞.
c
It follows that { m–n
T n x} is a ρ-Cauchy sequence in Xρ . Since Xρ is ρ-complete, there exists
c
(T n x – x∗ )) → . Consequently, ρ(l(T n x – x∗ )) → .
a point x∗ ∈ Xρ such that ρ( m–n
c
∗
Now we show that x is a ﬁxed point of T. As ρ( m–n
(T n x – x∗ )) → , (T n x, T n+ x) ∈ E(G)
for all n ∈ N and G is a Cρ -graph, there exists a subsequence {T np x} of {T n x} such that
˜ and T is a (G,
˜ A)ρ -contraction, it
(T np x, x∗ ) ∈ E(G) for each p ∈ N. Since (T np x, x∗ ) ∈ E(G)
follows that
ρ(c(T np + x–Tx∗ ))
ϕ(t) dt

ρ(l(T np x–x∗ ))
≤α

ρ(l(T np x–T np + x))
ϕ(t) dt,
ρ(l(x∗ –Tx∗ ))
ϕ(t) dt,

ϕ(t) dt ,

which on taking the limit as p → ∞ gives
ρ(c(x∗ –Tx∗ ))
ϕ(t) dt ≤ α , ,

ρ(l(x∗ –Tx∗ ))
ϕ(t) dt ≤ α , ,

ρ(c(x∗ –Tx∗ ))
ϕ(t) dt .

By the deﬁnition of function α, we have
ρ(c(x∗ –Tx∗ ))
ϕ(t) dt ≤ k ·  = .

From Lemma ., it follows that ρ(c(x∗ – Tx∗ )) =  and Tx∗ = x∗ .
Next, we prove that x∗ is a unique ﬁxed point. Suppose that T has another ﬁxed point
y∗ ∈ Xρ – {x∗ }. Since G is a Cρ -graph, there exists a subsequence {T np x} of {T n x} such
that (T np x, x∗ ) ∈ E(G) and (T np x, y∗ ) ∈ E(G) for each p ∈ N. Furthermore, G is weakly con˜ and we have
nected, (x∗ , y∗ ) ∈ E(G),
ρ(c(x∗ –y∗ ))
ρ(c(Tx∗ –Ty∗ ))
ϕ(t) dt =

ϕ(t) dt

ρ(l(x∗ –y∗ ))
≤α

ϕ(t) dt,

ρ(l(x∗ –y∗ ))
≤α
ϕ(t) dt, , 

ρ(c(x∗ –y∗ ))
≤α

ρ(l(x∗ –Tx∗ ))
ϕ(t) dt,
ϕ(t) dt, ,  .
ρ(l(y∗ –Ty∗ ))
ϕ(t) dt

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By the deﬁnition of α and Lemma ., we have
, and x∗ = y∗ .
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ρ(c(x∗ –y∗ ))

ϕ(t) dt ≤ k ·  = , ρ(c(x∗ – y∗ )) =
In Theorem ., if we replace the condition that G is a Cρ -graph with orbitally Gρ -continuity of T, then we have the following theorem.
Theorem . Let Xρ be a ρ-complete modular space endowed with a graph G, where ρ
˜ A)ρ -contraction and orbitally Gρ satisﬁes the  -condition and let T : Xρ → Xρ be a (G,
continuous. If the set XT is nonempty and the graph G is weakly connected, then T is a
PO.
c
T n x} is a ρ-Cauchy sequence
Proof If x ∈ XT , then Theorem . implies that { m–n
c
in Xρ . Owing to ρ-completeness of Xρ , there exists x∗ ∈ Xρ such that ρ( m–n
(T n x –
∗
n
n+
x )) → . As (T x, T x) ∈ E(G) for all n ∈ N and T is orbitally Gρ -continuous, we have
c
ρ( m–n
(T(T n x) – T(x∗ ))) → , as n → ∞. That is, Tx∗ = x∗ . Assume that y∗ is another ﬁxed
point of T. Following arguments similar to those in the proof of Theorem ., we obtain
y∗ = x∗ .
Corollary . Let Xρ be a ρ-complete modular space endowed with a graph G, where ρ
satisﬁes the  -condition and let T : Xρ → Xρ be edge-preserving, the set XT nonempty
and graph G be weakly connected and a Cρ -graph. If there exist nonnegative numbers l, c
with l < c such that
ρ c(Tx – Ty) ≤ α ρ l(x – y) , ρ l(x – Tx) , ρ l(y – Ty)
˜ and some α ∈ A, then T is a PO.
holds for all (x, y) ∈ E(G)
Now, we introduce Hardy-Rogers type (G)ρ -contraction and obtain related ﬁxed point
results.
Deﬁnition . Let Xρ be a modular space. A mapping T : Xρ → Xρ is called a HardyRogers type (G)ρ -contraction if the following conditions hold:
(H ) T preserves edges of G;
(H ) there exist nonnegative numbers li , c with li < c for i = , . . . ,  such that
ρ(c(Tx–Ty))

ρ(l (x–y))
ϕ(t) dt ≤ η
[ρ(l (x–Tx))+ρ(l (y–Ty))]
ϕ(t) dt + β

ϕ(t) dt

l
l
[ρ(  (x–Ty))+ρ(  (y–Tx))]
+γ
ϕ(t) dt

holds for each (x, y) ∈ E(G) with nonnegative numbers η, β, γ such that η +β +γ < 
and ϕ ∈ .
Remark . Let Xρ be a modular space endowed with a graph G and let T : Xρ → Xρ be
a Hardy-Rogers type (G)ρ -contraction. If there exists x ∈ Xρ such that Tx ∈ [x ]G˜ , then
(i) T is both a Hardy-Rogers type (G– )ρ -contraction and a Hardy-Rogers type
˜ ρ -contraction,
(G)
˜ x )ρ -contraction.
(ii) [x ]G˜ is T-invariant and T|[x ] G˜ is a Hardy-Rogers type (G
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Theorem . Let Xρ be a ρ-complete modular space endowed with a graph G, where ρ
˜ ρ -contraction.
satisﬁes the  -condition and let T : Xρ → Xρ be a Hardy-Rogers type (G)
Assume that the set XT is nonempty and the Cρ -graph G is weakly connected. Then T is a
PO.
Proof If x ∈ XT , then Tx ∈ [x]G˜ and (T n x, T n+ x) ∈ E(G) for all n ∈ N. Note that
ρ(c(T n x–T n+ x))
ϕ(t) dt

ρ(l (T n– x–T n x))
≤η
[ρ(l (T n– x–T n x))+ρ(l (T n x–T n+ x))]
ϕ(t) dt + β
ϕ(t) dt


l
l
[ρ(  (T n– x–T n+ x))+ρ(  (T n x–T n x))]
+γ

ρ(c(T n– x–T n x))
≤η
ϕ(t) dt
[ρ(c(T n– x–T n x))+ρ(c(T n x–T n+ x))]
ϕ(t) dt + β

ϕ(t) dt

[ρ(c(T n– x–T n x))+ρ(c(T n x–T n+ x))]
+γ
ϕ(t) dt.

It follows that
ρ(c(T n x–T n+ x))
ϕ(t) dt,


where h =
ρ(c(T n– x–T n x))
ϕ(t) dt ≤ h
η+β+γ
–β–γ
< . Also,
ρ(c(T n x–T n+ x))
ρ(c(x–Tx))
ϕ(t) dt ≤ hn
()
ϕ(t) dt.


Taking the limit as n → ∞, and using Lemma ., we get
ρ(c(T n x–T n+ x))
ϕ(t) dt = ,
lim
n

which implies that limn ρ(c(T n x – T n+ x)) = .
Let m, n ∈ N with m > n. By () and Remark ., we get
c (T n x–T m x))
ρ( m–n
ϕ(t) dt ≤

ρ(c(T n+ x–T n+ x))
ϕ(t) dt +

ρ(c(T m– x–T m x))
ϕ(t) dt

≤h
ϕ(t) dt

+ ··· +
ρ(c(x–Tx))
n
ϕ(t) dt + h

hn
–h
ρ(c(x–Tx))
n+
ϕ(t) dt

+ ··· + h
≤
ρ(c(T n x–T n+ x))
ρ(c(x–Tx))
m–
ϕ(t) dt

ρ(c(x–Tx))

ϕ(t) dt → ,
as n → ∞.
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c
Thus, { m–n
T n x} is a ρ-Cauchy sequence in Xρ . Since Xρ is ρ-complete, there exists a point
c
∗
x ∈ Xρ such that ρ( m–n
(T n x – x∗ )) → . As G is a Cρ -graph, there exists a subsequence
{T np x} such that (T np x, x∗ ) ∈ E(G) for all p ∈ N. Also, (T np x, x∗ ) ∈ E(G) for all p ∈ N. Now
we have
ρ(c(T np + x–Tx∗ ))
ρ(l (T np x–x∗ ))
ϕ(t) dt ≤ η

ϕ(t) dt

[ρ(l (T np x–T np + x))+ρ(l (x∗ –Tx∗ ))]
+β
ϕ(t) dt

l
l
[ρ(  (T np x–Tx∗ ))+ρ(  (x∗ –T np + x))]
+γ
ϕ(t) dt.

Taking the limit as n → ∞, we have
ρ(c(x∗ –Tx∗ ))
ρ(c(x∗ –Tx∗ ))
ϕ(t) dt ≤ (β + γ )
ϕ(t) dt.


As (β + γ ) < , so ρ(c(x∗ – Tx∗ )) =  and x∗ = Tx∗ .
Next, we prove that x∗ is a unique ﬁxed point. Suppose that T has another ﬁxed point
∗
y ∈ Xρ – {x∗ }. Since G is a Cρ -graph, there exists a subsequence {T np x} of {T n x} such that
(T np x, x∗ ) ∈ E(G) and (T np x, y∗ ) ∈ E(G) for each p ∈ N. As G is weakly connected, we have
˜ and
(x∗ , y∗ ) ∈ E(G),
ρ(c(x∗ –y∗ ))
ρ(c(Tx∗ –Ty∗ ))
ϕ(t) dt =
ϕ(t) dt


ρ(l (x∗ –y∗ ))
≤η
[ρ(l (x∗ –Tx∗ ))+ρ(l (y∗ –Ty∗ ))]
ϕ(t) dt + β

ϕ(t) dt

l
l
[ρ(  (x∗ –Ty∗ ))+ρ(  (y∗ –Tx∗ ))]
+γ
ϕ(t) dt,

which further implies that
ρ(c(x∗ –y∗ ))
ϕ(t) dt ≤ (η + γ )

Since (η + γ ) < ,
ρ(c(x∗ –y∗ ))
ϕ(t) dt.

ρ(c(x∗ –y∗ ))

ϕ(t) dt = . The result follows.
In Theorem ., if we replace the condition that G is a Cρ -graph with orbitally Gρ continuity of T, then we have the following theorem.
Theorem . Let Xρ be a ρ-complete modular space endowed with a graph G, where ρ
˜ ρ -contraction,
satisﬁes the  -condition and let T : Xρ → Xρ be a Hardy-Rogers type (G)
which is orbitally Gρ -continuous. Assume that the set XT is nonempty and the graph G is
weakly connected. Then T is a PO.
In the following suppose that Xρ is a ρ-complete modular space endowed with a graph
G, where ρ satisﬁes the  -condition and T : Xρ → Xρ is edge-preserving such that the
set XT is nonempty.
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Corollary . Assume
(i) the Cρ -graph G is weakly connected and
(ii) there exist nonnegative numbers l, c with l < c such that
ρ(c(Tx–Ty))
ρ(l(x–y))
ϕ(t) dt ≤ η

ϕ(t) dt

˜ with η ∈ (, ) and ϕ ∈ . Then T is a PO.
holds for each (x, y) ∈ E(G)
Corollary . Assume
(i) the Cρ -graph G is weakly connected and
(ii) there exist nonnegative numbers l , l , c with l , l < c such that
ρ(c(Tx–Ty))
[ρ(l (x–Tx))+ρ(l (y–Ty))]
ϕ(t) dt ≤ β

ϕ(t) dt

˜ with β ∈ (,  ) and ϕ ∈ . Then T is a PO.
holds for each (x, y) ∈ E(G)

Corollary . Assume
(i) the Cρ -graph G is weakly connected and
(ii) there exist nonnegative numbers l , l , c with l , l < c such that
ρ(c(Tx–Ty))
l
l
[ρ(  (x–Ty))+ρ(  (y–Tx))]
ϕ(t) dt ≤ γ

ϕ(t) dt

˜ with γ ∈ (,  ) and ϕ ∈ . Then T is a PO.
for each (x, y) ∈ E(G)

Now we introduce the (G, φ, ψ)ρ -contraction and obtain some ﬁxed point results.
Deﬁnition . A mapping T : Xρ → Xρ is called a (G, φ, ψ)ρ -contraction if the following
conditions hold:
(Q ) T preserves edges of G;
(Q ) there exist nonnegative numbers l, c with l < c such that
ψ(ρ(c(Tx–Ty)))

ψ(ρ(l(x–y)))
ϕ(t) dt ≤ φ
ϕ(t) dt

holds for each (x, y) ∈ E(G), where ψ ∈ , φ ∈  , and ϕ ∈ .
Remark . Let T : Xρ → Xρ be a (G, φ, ψ)ρ -contraction. If there exists x ∈ Xρ such
that Tx ∈ [x ]G˜ , then
˜ φ, ψ)ρ -contraction,
(i) T is both a (G– , φ, ψ)ρ -contraction and a (G,
˜ x , φ, ψ)ρ -contraction.
(ii) [x ]G˜ is T-invariant and T|[x ] G˜ is a (G
Theorem . Let Xρ be a ρ-complete modular space endowed with a graph G, where
˜ φ, ψ)ρ -contraction. If XT is
ρ satisﬁes the  -condition and let T : Xρ → Xρ be a (G,
nonempty and Cρ -graph G is weakly connected, then T is a PO.
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Proof If x ∈ XT , then (T n x, T n+ x) ∈ E(G) for all n ∈ N. First, we show that the sequence
{ψ(ρ(c(T n x – T n+ x)))} converges to . From Deﬁnition ., we have
ψ(ρ(c(T n x–T n+ x)))
ϕ(t) dt ≤ φ
ψ(ρ(l(T n– x–T n x)))
ϕ(t) dt


ψ(ρ(l(T n– x–T n x)))
<
ϕ(t) dt

ψ(ρ(c(T n– x–T n x)))
<
ϕ(t) dt.

Thus,
ψ(ρ(c(T n x–T n+ x)))
ϕ(t) dt

is decreasing and bounded from below and so
ψ(ρ(c(T n x–T n+ x)))
ϕ(t) dt

converges to a nonnegative number L. If L = , we obtain
ψ(ρ(c(T n x–T n+ x)))
L = lim
n→∞ 
n→∞
ψ(ρ(l(T n– x–T n x)))
ϕ(t) dt

ψ(ρ(c(T n– x–T n x)))
≤ lim φ
n→∞
ϕ(t) dt ≤ lim φ
ϕ(t) dt ,

ψ(ρ(c(T n x–T n+ x)))
that is, L ≤ φ(L), a contradiction. Hence 
ϕ(t) dt →  as n → ∞. It follows
n
n+
that ψ(ρ(c(T x – T x))) → . Suppose that
lim sup ψ ρ c T n x – T n+ x = ε > .
n→∞
Then there exists a vε ∈ N and a sequence {T nv x}v≥vε such that
ψ ρ c T nv x – T nv + x → ε > , v → ∞,
ε
ψ ρ c T nv x – T nv + x ≥ , ∀v ≥ vε .

()
()
Hence we get the following:
ψ(ρ(c(T nv x–T nv + x)))
 = lim
v→∞ 
ε

ϕ(t) dt ≥
ϕ(t) dt > .

Assume that there is an ε >  and there exist mv , nv ∈ N such that mv > nv > v for each
v ∈ N and
ψ ρ l T mv x – T nv x ≥ ε.
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Then we choose the sequence (mv )v∈N and (nv )v∈N such that for each v ∈ N, mv is minimal in the sense that ψ(ρ(l(T mv x – T nv x))) ≥ ε but ψ(ρ(l(T s x – T nv x))) < ε, for all s ∈
{nv + , . . . , mv – }. Now, let δ ∈ R+ be such that cl + δ = , then we have
ε
ψ(ρ(l(T mv x–T nv x)))
ϕ(t) dt ≤

ϕ(t) dt

ψ(ρ(c(T mv x–T nv + x)))
≤
ψ(ρ(δl(T nv + x–T nv x)))
ϕ(t) dt +

ϕ(t) dt

ψ(ρ(l(T mv – x–T nv x)))
≤φ
ϕ(t) dt +

ϕ(t) dt

ψ(ρ(l(T mv – x–T nv x)))
≤
ψ(ρ(δl(T nv + x–T nv x)))
ψ(ρ(δl(T nv + x–T nv x)))
ϕ(t) dt +

ε
≤
ψ(ρ(δl(T nv + x–T nv x)))
ϕ(t) dt +

ϕ(t) dt

ϕ(t) dt.

Thus, taking the limit as v → ∞, and Proposition ., we have
ψ(ρ(δl(T nv + x–T nv x)))
ϕ(t) dt → .

Therefore,
ψ(ρ(l(T mv x–T nv x)))
ϕ(t) dt → ε,
v → ∞.

Now,
ψ(ρ(l(T mv x–T nv x)))
ϕ(t) dt

ψ(ρ(c(T mv + x–T nv + x)))
≤
ψ(ρ(δl(T nv + x–T nv x)))
ϕ(t) dt +

ϕ(t) dt

ψ(ρ(δl(T mv + x–T mv x)))
+
ϕ(t) dt

ψ(ρ(l(T mv x–T nv x)))
≤φ
ϕ(t) dt +

ψ(ρ(δl(T nv + x–T nv x)))
ϕ(t) dt

ψ(ρ(δl(T mv + x–T mv x)))
+
ϕ(t) dt.

If v → ∞, then we have

ε
ϕ(t) dt ≤ φ
ε
ϕ(t) dt ,

a contradiction for ε > . Hence, {lT n x} is a ρ-Cauchy sequence. By ρ-completeness of Xρ ,
there exists x∗ ∈ Xρ such that ρ(l(T n x – x∗ )) →  as n → ∞ and (T n x, T n+ x) ∈ E(G) for all
n ∈ N and G is a Cρ -graph, then there exists a subsequence {T np x} such that (T np x, x∗ ) ∈
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E(G) for each p ∈ N. Also, (T np x, x∗ ) ∈ E(G) for each p ∈ N. From Remark . and (ψ ), it
follows that
c
c ∗
c ∗
x – Tx∗
x – T np + x + T np + x – Tx∗
=ψ ρ
ψ ρ



np +
∗
np +
x +ρ c T
x – Tx∗
≤ψ ρ c x –T
≤ ψ ρ c x∗ – T np + x + ψ ρ c T np + x – Tx∗ .
Taking the limit as p → ∞, we have
ψ(ρ(c(T np + x–Tx∗ )))
ϕ(t) dt ≤ φ

≤
ψ(ρ(l(T np x–x∗ )))
ϕ(t) dt

ψ(ρ(l(T np x–x∗ )))
ϕ(t) dt.

Since ρ(l(T np x – x∗ )) →  (p → ∞), we obtain
ψ(ρ(l(T np x–x∗ )))
lim
p→∞ 
ϕ(t) dt ≤ ,
which implies that ψ(ρ(c(T np + x – Tx∗ ))) → , as p → ∞. Thus,
ψ ρ c x∗ – T np + x + ψ ρ c T np + x – Tx∗ → ,
as p → ∞.
Hence limn→∞ ψ(ρ( c (x∗ – Tx∗ ))) =  and x∗ = Tx∗ .
Finally, we prove that x∗ is a unique ﬁxed point. Suppose that T has another ﬁxed point
y∗ ∈ Xρ – {x∗ }. Since G is a Cρ -graph, there exists a subsequence {T np x} of {T n x} such
that (T np x, x∗ ) ∈ E(G) and (T np x, y∗ ) ∈ E(G) for each p ∈ N. Furthermore, as G is weakly
˜ We have
connected, (x∗ , y∗ ) ∈ E(G).
ψ(ρ(c(x∗ –y∗ )))
ψ(ρ(c(Tx∗ –Ty∗ )))
ϕ(t) dt =

ϕ(t) dt

ψ(ρ(l(x∗ –y∗ )))
≤φ
ϕ(t) dt

ψ(ρ(l(x∗ –y∗ )))
<
ϕ(t) dt

ψ(ρ(c(x∗ –y∗ )))
<
ϕ(t) dt,

a contradiction. Hence, x∗ = y∗ .
In Theorem ., if we replace the condition that G is a Cρ -graph with orbitally Gρ continuity of T, then we have the following theorem.
Theorem . Let Xρ be a ρ-complete modular space endowed with a graph G, where
˜ φ, ψ)ρ -contraction, which is orρ satisﬁes the  -condition and let T : Xρ → Xρ be a (G,
bitally Gρ -continuous. Assume that the set XT is nonempty and the graph G is weakly connected. Then T is a PO.
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In the following corollaries, suppose that Xρ is a ρ-complete modular space endowed
with a graph G, where ρ satisﬁes the  -condition and let T : Xρ → Xρ be edge-preserving
and the set XT be nonempty.
Corollary . Assume
(i) the Cρ -graph G is weakly connected and
(ii) there exist nonnegative numbers l, c with l < c such that
ρ(c(Tx–Ty))

ρ(l(x–y))
ϕ(t) dt ≤ φ
ϕ(t) dt

˜ with φ ∈  and ϕ ∈ . Then T is a PO.
hold for each (x, y) ∈ E(G)
Corollary . Assume
(i) the Cρ -graph G is weakly connected and
(ii) there exist nonnegative numbers l, c with l < c such that
ψ(ρ(c(Tx–Ty)))
ψ(ρ(l(x–y)))
ϕ(t) dt ≤ η

ϕ(t) dt

˜ with η ∈ (, ) and ϕ ∈ . Then T is a PO.
for each (x, y) ∈ E(G)
Now we provide examples in support of our results.
Example . Let Xρ = {, , , , , } and ρ(x) = |x|, for all x ∈ Xρ . Consider
˜ = (, ), (, ), (, ), (, ), (, ) ∪ (, x) : x ∈ Xρ ,
E(G)
and Tx = , x ∈ Xρ . Then G is weakly connected and Cρ -graph, XT is nonempty and T is
˜ A)ρ -contraction where c =  , l =  , ϕ(t) = . Hence, T is a PO.
a (G,


Example . Let Xρ = {, , , } and ρ(x) = |x|, for all x ∈ Xρ . Consider
˜ = (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ) .
E(G)
Deﬁne T : Xρ → Xρ as follows:
Tx =
, x ∈ {, },
, x ∈ {, }.
Then G is weakly connected and Cρ -graph, XT is nonempty, and T is a Hardy-Rogers type
˜ ρ -contraction where c = , l = l = l = , l = l =  , η =  , β =  , γ = , and ϕ(t) = .
(G)



Moreover,  is a unique ﬁxed point of T.
Example . Let Xρ = [, ] and ρ(x) = |x|, for all x ∈ Xρ . Consider
E(G) = (, ) ∪ (, x) : x ≥ / ∪ (x, y) : x, y ∈ (, ] ,
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and Tx = x , x ∈ Xρ . Then G is weakly connected and a Cρ -graph, XT is nonempty and T
˜ φ, ψ)ρ -contraction where c =  , l =  , ϕ(t) = , φ(ξ ) = ξ , and ψ(ω) = ω . Thus all
is a (G,


+ξ

conditions of Theorem . are satisﬁed. Moreover, T is a PO.
Remark . In the above examples, if we use ρ(x) = x , the conclusions remain the same.
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.
Author details
1
Department of Mathematics, Sakarya University, Sakarya, 54187, Turkey. 2 Department of Mathematics and Applied
Mathematics, University of Pretoria, Lynnwood Road, Pretoria, 0002, South Africa. 3 Department of Mathematics, Lahore
University of Management Sciences (LUMS), Phase-II, Opposite sector U, DHA Lahore Cantt., Lahore, 54792, Pakistan.
Acknowledgements
The authors are thankful to the anonymous referees for their valuable comments and suggestions, which helped to
improve the presentation of this paper.
Received: 30 May 2014 Accepted: 14 October 2014 Published: 29 Oct 2014
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Cite this article as: Öztürk et al.: Fixed points of mappings satisfying contractive condition of integral type in
modular spaces endowed with a graph. Fixed Point Theory and Applications 2014, 2014:220
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