R E S E A R C H
Open Access
A generalization on weak contractions in
partially ordered
b
-metric spaces and its
application to quadratic integral equations
Reza Allahyari
1, Reza Arab
2*and Ali Shole Haghighi
3*Correspondence: [email protected] 2Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran
Full list of author information is available at the end of the article
Abstract
We introduce the notion of almost generalized (ψ,
ϕ,
L)-contractive mappings, and establish the coincidence and common fixed point results for this class of mappings in partially ordered completeb-metric spaces. Our results extend and improve several known results from the context of ordered metric spaces to the setting of ordered b-metric spaces. As an application, we prove the existence of a unique solution to a class of nonlinear quadratic integral equations.Keywords: fixed point; common fixed point; coincidence point; integral equations; b-metric space; partially ordered set
1 Introduction
Fixed points theorems in partially ordered metric spaces were firstly obtained in by Ran and Reurings [], and then by Nieto and Rodríguez-López []. In this direction sev-eral authors obtained further results under weak contractive conditions (see,e.g., [–]). Berinde initiated in [] the concept of almost contractions and obtained several interesting fixed point theorems. This has been a subject of intense study since then; see,e.g., [–]. Some authors used related notions as ‘condition (B)’ (Babuet al.[]) and ‘almost general-ized contractive condition’ for two maps (Ćirićet al.[]), and for four maps (Aghajaniet al.[]). See also a note by Pacurar []. On the other hand, the concept ofb-metric space was introduced by Czerwik in []. After that, several interesting results of the existence of fixed point for single-valued and multivalued operators inb-metric spaces have been obtained (see [–]). Pacurar [] proved some results on sequences of almost con-tractions and fixed points inb-metric spaces. Recently, Hussain and Shah [] obtained results on KKM mappings in coneb-metric spaces. Using the concepts of partially ordered metric spaces, almost generalized contractive condition, andb-metric spaces, we define a new concept of almost generalized (ψ,ϕ,L)-contractive condition. In this paper, some co-incidence and common fixed point theorems for mappings satisfying almost generalized (ψ,ϕ,L)-contractive condition in the setup of partially ordered completeb-metric spaces are proved. Consistent with [] and [, p.], the following definitions and results will be needed in the sequel.
Definition .[] LetXbe a (nonempty) set ands≥ be a given real number. A function
d:X×X→R+is said to be ab-metric space iff for allx,y,z∈X, the following conditions are satisfied:
(i) d(x,y) = iffx=y, (ii) d(x,y) =d(y,x),
(iii) d(x,y)≤s[d(x,z) +d(z,y)].
The pair (X,d) is called ab-metric space with the parameters.
It should be noted that the class ofb-metric spaces is effectively larger than that of metric spaces, since ab-metric is a metric, whens= .
The following example shows that in general ab-metric does not necessarily need to be a metric (see, also, []).
Example .[] Let (X,d) be a metric space andρ(x,y) = (d(x,y))p, wherep> is a real number. Thenρis ab-metric withs= p–. However, if (X,d) is a metric space, then (X,ρ) is not necessarily a metric space. For example, if X=Ris the set of real numbers and
d(x,y) =|x–y|is the usual Euclidean metric, thenρ(x,y) = (x–y)sis ab-metric onRwith
s= , but it is not a metric onR.
Also, the following example of ab-metric space is given in [].
Example .[] LetXbe the set of Lebesgue measurable functions on [, ] such that
|f(x)|
dx<∞. DefineD:X×X→[,∞) byD(f,g) =
|f(x) –g(x)|
dx. As ( |f(x) –
g(x)|dx) is a metric onX, then, from the previous example,Dis ab-metric onX, with
s= , where theb-metricDis defined withD(x,y) =d(x,y),dis a cone metric (also see [–]).
Khamsi [] also showed that each cone metric space over a normal cone has ab-metric structure.
Definition .[] We shall say that the mappingT isg-nondecreasing if
gx≤gy ⇒ Tx≤Ty.
2 Main results
Throughout the paper, let be the family of all functionsψ: [,∞)→[,∞) satisfying the following conditions:
(a) ψ is continuous, (b) ψ is nondecreasing,
(c) ψ() = <ψ(t)for everyt> .
We denote bythe set of all functionsϕ: [,∞)→[,∞) satisfying the following con-ditions:
(i) ϕis right continuous, (ii) ϕis nondecreasing, (iii) ϕ(t) <tfor everyt> .
Let (X,d,≤) be a partially orderedb-metric space andT:X→Xandg:X→Xbe two mappings. Set
M(x,y) =max
d(gx,gy),d(gx,Tx),d(gy,Ty),d(gx,Ty) +d(gy,Tx) s
and
N(x,y) =mind(gx,Tx),d(gy,Ty),d(gx,Ty),d(gy,Tx).
Now, we introduce the following definition.
Definition . Let (X,d,≤) be a partially orderedb-metric space. We say thatT:X→X
is an almost generalized (ψ,ϕ,L)-contractive mapping with respect tog:X→Xfor some ψ∈,ϕ∈, andL≥ if
ψsd(Tx,Ty)≤ϕψM(x,y)+LψN(x,y) (.)
for allx,y∈Xwithgx≤gy.
Now, we establish some results for the existence of coincidence point and common fixed point of mappings satisfying almost generalized (ψ,ϕ,L)-contractive condition in the setup of partially orderedb-metric spaces. The first result in this paper is the follow-ing coincidence point theorem.
Theorem . Suppose that(X,d,≤)is a partially ordered complete b-metric space.Let T:
X→X be an almost generalized(ψ,ϕ,L)-contractive mapping with respect to g:X→X,
and T and g are continuous such that T is a monotone g-nondecreasing mapping, com-mutative with g and T(X)⊆g(X).If there exists x∈X such that gx≤Tx,then T and g
have a coincidence point in X.
Proof By the given assumptions, there existsx∈Xsuch thatgx≤Tx. SinceT(X)⊆
g(X), we can definex ∈Xsuch that gx=Tx, thengx≤Tx=gx. Also there exists
x∈Xsuch thatgx=Tx. SinceTis a monotoneg-nondecreasing mapping, we have
gx=Tx≤Tx=gx.
Continuing in this way, we construct a sequence{xn}inXsuch that for alln= , , , . . . ,
gxn+=Txn (.)
for which
gx≤gx≤gx≤ · · · ≤gxn≤gxn+≤ · · ·. (.)
If there existsk∈Nsuch that gxk+=gxk, thengxk =Txk. This means thatxk is a coincidence point ofT,g, and the proof is finished. Thus,gxn+=gxnfor alln∈N. From (.) and (.) and the inequality (.) with (x,y) = (xn,xn+), we have
ψd(gxn+,gxn+)
≤ψsd(gxn+,gxn+)
=ψsd(Txn,Txn+)
≤ϕψM(xn,xn+)
+LψN(xn,xn+)
where
M(xn,xn+) =max
d(gxn,gxn+),d(gxn,Txn),d(gxn+,Txn+),
d(gxn,Txn+) +d(gxn+,Txn) s
=max
d(gxn,gxn+),d(gxn,gxn+),d(gxn+,gxn+),
d(gxn,gxn+) s
and
N(xn,xn+) =min
d(gxn,Txn),d(gxn+,Txn+),d(gxn,Txn+),d(gxn+,Txn)
= .
Since
d(gxn,gxn+)
s ≤
d(gxn,gxn+) +d(gxn+,gxn+)
≤max
d(gxn,gxn+),d(gxn+,gxn+)
,
then we get
M(xn,xn+) =max
d(gxn,gxn+),d(gxn+,gxn+)
,
N(xn,xn+) = .
(.)
By (.) and (.), we have
ψd(gxn+,gxn+)
≤ϕψmaxd(gxn,gxn+),d(gxn+,gxn+)
. (.)
Suppose thatmax{d(gxn,gxn+),d(gxn+,gxn+)}=d(gxn+,gxn+) > for somen∈N, then by (.)
ψd(gxn+,gxn+)
≤ϕψd(gxn+,gxn+)
<ψd(gxn+,gxn+)
;
a contradiction. Hence,
maxd(gxn,gxn+),d(gxn+,gxn+)
=d(gxn,gxn+) and thus
ψd(gxn+,gxn+)
≤ϕψd(gxn,gxn+)
<ψd(gxn,gxn+)
.
Thus, we get
ψd(gxn+,gxn+)
<ψd(gxn,gxn+)
for alln∈N. Now, from ψd(gxn,gxn+)
≤ϕψd(gxn–,gxn)
≤ϕψd(gxn–,gxn–)
≤ · · · ≤ϕnψd(gx,gx)
and the property ofϕ, we obtainlimn→∞ψ(d(gxn,gxn+)) = , and consequently
lim
n→∞d(gxn,gxn+) = . (.)
Now, we shall prove that{gxn}is a Cauchy sequence in (X,d). Suppose, on the contrary, that {gxn}is not a Cauchy sequence. Then there exist> and subsequences{gxm(k)},
{gxn(k)}of{gxn}withm(k) >n(k)≥ksuch that
d(gxn(k),gxm(k))≥. (.)
Additionally, corresponding ton(k), we may choosem(k) such that it is the smallest integer satisfying (.) andm(k) >n(k)≥k. Thus,
d(gxn(k),gxm(k)–) <. (.)
Using the triangle inequality inb-metric space and (.) and (.) we obtain
≤d(gxm(k),gxn(k))≤sd(gxm(k),gxm(k)–) +sd(gxm(k)–,gxn(k)) <sd(gxm(k),gxm(k)–) +s.
Taking the upper limit ask→ ∞and using (.) we obtain
≤lim sup
k→∞
d(gxn(k),gxm(k))≤s. (.)
Also
≤d(gxn(k),gxm(k))≤sd(gxn(k),gxm(k)+) +sd(gxm(k)+,gxm(k))
≤sd(gxn(k),gxm(k)) +sd(gxm(k),gxm(k)+) +sd(gxm(k)+,gxm(k))
≤sd(gxn(k),gxm(k)) +
s+sd(gxm(k),gxm(k)+). So from (.) and (.), we have
s ≤lim supk→∞ d(gxn(k),gxm(k)+)≤s
. (.)
Also
≤d(gxm(k),gxn(k))≤sd(gxm(k),gxn(k)+) +sd(gxn(k)+,gxn(k))
≤sd(gxm(k),gxn(k)) +sd(gxn(k),gxn(k)+) +sd(gxn(k)+,gxn(k))
≤sd(gxm(k),gxn(k)) +
s+sd(gxn(k),gxn(k)+).
So from (.) and (.), we have
s ≤lim supk→∞ d(gxm(k),gxn(k)+)≤s
Also
d(gxn(k)+,gxm(k))≤sd(gxn(k)+,gxm(k)+) +sd(gxm(k)+,gxm(k)), so from (.) and (.), we have
s ≤lim sup k→∞
d(gxn(k)+,gxm(k)+). (.)
Linking (.), (.), (.) together with (.) we get
lim sup
k→∞
M(xn(k),xm(k))
=max
lim sup
k→∞ d(gxn(k),gxm(k)), lim sup
k→∞ d(gxn(k),gxn(k)+), lim sup
k→∞ d(gxm(k),gxm(k)+), lim supk→∞d(gxn(k),gxm(k)+) +lim supk→∞d(gxm(k),gxn(k)+)
s
≤max
s, , ,s +s
s
=s.
So,
lim sup
k→∞
M(xn(k),xm(k))≤s. (.)
Similarly, we have
lim sup
k→∞
N(xn(k),xm(k)) = . (.)
Sincem(k) >n(k) from (.), we have
gxn(k)≤gxm(k). Thus,
ψsd(gxn(k)+,gxm(k)+)
=ψsd(Txn(k),Txm(k))
≤ϕψM(xn(k),xm(k))
+LψN(xn(k),xm(k))
.
Passing to the upper limit ask→ ∞, and using (.), (.), and (.), we get ψ(s)≤ψ slim sup
k→∞
d(gxn(k)+,gxm(k)+)
=lim sup
k→∞
ψsd(gxn(k)+,gxm(k)+)
=lim sup
k→∞
ψsd(Txn(k),Txm(k))
≤lim sup
k→∞
ϕψM(xn(k),xm(k))
+lim sup
k→∞
LψN(xn(k),xm(k))
=ϕ ψ lim sup
k→∞
M(xn(k),xm(k))
+Lψ lim sup
k→∞
N(xn(k),xm(k))
which is a contradiction. Thus, we proved that{gxn}is a Cauchy sequence in (X,d). Since
Xis a completeb-metric space, there existsx∈Xsuch that
lim
n→∞gxn+=x. (.)
From the commutativity ofTandg, we have
g(gxn+) =g
T(xn)
=T(gxn). (.)
Lettingn→ ∞in (.) and from the continuity ofT andg, we get
gx= lim
n→∞g(gxn+) =nlim→∞T(gxn) =T nlim→∞gxn
=T(x).
This implies thatxis a coincidence point ofT andg. This completes the proof. Now, we will prove the following result.
Theorem . Suppose that(X,d,≤)is a partially ordered complete b-metric space.Let T:X→X be an almost generalized(ψ,ϕ,L)-contractive mapping with respect to g:X→ X,T is a g-nondecreasing mapping and T(X)⊆g(X).Also suppose
if{gxn} ⊂X is a nondecreasing sequence with gxn→gz in gX,
then gxn≤gz,gz≤g(gz)∀n hold.
(.)
Also suppose gX is closed.If there exists x∈X such that gx≤Tx,then T and g have a
coincidence.Further,if T and g commute at their coincidence points,then T and g have a common fixed point.
Proof As in the proof of Theorem ., we can show that{gxn}is a Cauchy sequence. Since
gXis a closed, there existsx∈Xsuch that
lim
n→∞gxn+=gx. (.)
Now we show thatxis a coincidence point ofTandg. Since from (.) and (.) we have
gxn≤gxfor alln, then by the triangle inequality in ab-metric space and (.), we get
d(gx,Tx)≤sd(gx,gxn+) +sd(gxn+,Tx) =sd(gx,gxn+) +sd(Txn,Tx), ψd(gx,Tx)≤ lim
n→∞ψ
sd(Txn,Tx)
≤ lim
n→∞ψ
sd(Txn,Tx)
≤ lim
n→∞
ϕψM(xn,x)+LψN(xn,x)
≤ϕψd(gx,Tx)<ψd(gx,Tx).
Indeed,
lim
n→∞M(xn,x) =nlim→∞max
d(gxn,gx),d(gxn,Txn),d(gx,Tx),d(gxn,Tx) +d(gx,Txn) s
and
lim
n→∞N(xn,x) =nlim→∞min
d(gxn,Txn),d(gx,Tx),d(gxn,Tx),d(gx,Txn)
= .
Henced(gx,Tx) = , that is,Tx=gx. Thus we proved that T andg have a coincidence. Suppose now thatTandgcommute atx. Sety=Tx=gx. Then
Ty=T(gx) =g(Tx) =gy.
Since from (.) we havegx≤g(gx) =gyand asgx=Txandgy=Ty, from (.) we obtain ψd(Tx,Ty)≤ψsd(Tx,Ty)≤ϕψM(x,y)+LψN(x,y)
=ϕ
ψ
max
d(gx,gy),d(gx,Tx),d(gy,Ty),d(gx,Ty) +d(gy,Tx) s
+Lψmind(gx,Tx),d(gy,Ty),d(gx,Ty),d(gy,Tx) =ϕψd(Tx,Ty)<ψd(Tx,Ty).
Henced(Tx,Ty) = , that is,y=Tx=Ty. Therefore,Ty=gy=y. Thus we proved thatT
andghave a common fixed point.
In the following, we deduce some fixed point theorems from our main results given by Theorems . and ..
Corollary . Let(X,d,≤)be a partially ordered complete b-metric space and T:X→X is a nondecreasing mapping.Suppose there existψ∈,ϕ∈,and L≥such that
ψsd(Tx,Ty)≤ϕψM(x,y)+LψN(x,y),
where
M(x,y) =max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx) s
and
N(x,y) =mind(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)
for all x,y∈X with x≤y.Also suppose either
(a) if{xn} ⊂Xis a nondecreasing sequence withxn→zinX,thenxn≤z,for alln,
holds,or
(b) Tis continuous.
If there exists x∈X such that x≤Tx,then T has a fixed point in X.
Example . Let X be the set of Lebesgue measurable functions on [, ] such that
|x(t)|dt<∞. DefineD:X×X→[,∞) by
D(x,y) =
x(t) –y(t)dt
ThenDis ab-metric onX, withs= . Also, this space can also be equipped with a partial order given by
x,y∈X, xy ⇐⇒ x(t)≤y(t) for anyt∈[a,b].
The operatorT:X→Xdefined by
Tx(t) =tn+et+
√
lnx(t)+
. (.)
Now, we prove thatThas a fixed point. For allx,y∈Xwithx≤y, we have
D(Tx,Ty) =
Tx(t) –Ty(t)dt
≤√
√lnx(t)+ –
√
lny(t)+ dt
≤
lnx(t)+ –lny(t)+ dt
≤
ln |
x(t)|+
|y(t)|+
dt
≤
ln
+|x(t) –y(t)|
|y(t)|+
dt
≤ln
+
x(t) –y(t)dt
≤ln
+
x(t) –y(t)dt
≤ln +D(x,y).
Now, if we defineϕ(t) =ln( +t),ψ(t) =√t, andx= . Thus, by Corollary . we see that
T has a fixed point.
Remark . Corollary . extends and generalizes many existing fixed point theorems in
the literature [, , , ].
The following result is the immediate consequence of Corollary ..
Corollary . Let(X,d,≤)be a partially ordered complete b-metric space and T:X→X is a nondecreasing mapping.Suppose there existsϕ∈such that
sd(Tx,Ty)≤ϕ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx) s
(.)
for all x,y∈X with x≤y.Also suppose either
(a) if{xn} ⊂Xis a nondecreasing sequence withxn→zinX,thenxn≤z,for alln,
(b) Tis continuous.
If there exists x∈X such that x≤Tx,then T has a fixed point in X.
Remark . Corollary . is a generalization to [, Theorem .].
Takingϕ(t) =λt, <λ< , in Corollary . we obtain the following generalization of the results in [, ].
Corollary . Let(X,d,≤)be a partially ordered complete b-metric space and T:X→X is a nondecreasing mapping.Suppose there existsϕ∈such that
sd(Tx,Ty)≤λmax
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx) s
for all x,y∈X with x≤y.Also suppose either
(a) if{xn} ⊂Xis a nondecreasing sequence withxn→zinX,thenxn≤z,for alln,
holds,or
(b) Tis continuous.
If there exists x∈X such that x≤Tx,then T has a fixed point in X.
Corollary . Let(X,d,≤)be a partially ordered complete b-metric space and T:X→X is a nondecreasing mapping.Suppose there existψ∈and≤λ< such that
ψsd(Tx,Ty)≤λψd(x,y)
for all x,y∈X with x≤y.Also suppose either
(a) if{xn} ⊂Xis a nondecreasing sequence withxn→zinX,thenxn≤z,for alln,
holds,or
(b) Tis continuous.
If there exists x∈X such that x≤Tx,then T has a fixed point in X.
3 Application to integral equations
Here, in this section, we wish to study the existence of a unique solution to a nonlinear quadratic integral equation, as an application to the our fixed point theorem. Consider the integral equation
x(t) =h(t) +λ
k(t,s)fs,x(s)ds, t∈I= [, ],λ≥. (.)
Let denote the class of those functionsγ : [, +∞)→[, +∞) for whichγ ∈and (γ(t))p≤γ(tp), for allp≥.
For example,γ(t) =kt, where ≤k< andγ(t) =t+t are in . We will analyze (.) under the following assumptions:
(a) f :I×R→Ris continuous monotone nondecreasing inx,f(t,x)≥and there exist
constant≤L< andγ ∈ such that for allx,y∈Randx≥y
(a) h:I→Ris a continuous function.
(a) k:I×I→Ris continuous int∈Ifor everys∈Iand measurable ins∈Ifor allt∈I
such that
k(t,s)ds≤K
andk(t,s)≥.
(a) There existsα∈C(I)such that
α(t)≤h(t) +λ
k(t,s)fs,α(s)ds.
(a) LpλpKp≤p–.
We consider the space X=C(I) of continuous functions defined on I= [, ] with the standard metric given by
ρ(x,y) =sup
t∈I
x(t) –y(t) forx,y∈C(I).
This space can also be equipped with a partial order given by
x,y∈C(I), x≤y ⇐⇒ x(t)≤y(t) for anyt∈I. Now forp≥, we define
d(x,y) =ρ(x,y)p= sup
t∈I
x(t) –y(t)p=sup
t∈I
x(t) –y(t)p forx,y∈C(I).
It is easy to see that (X,d) is a completeb-metric space withs= p–[].
For anyx,y∈Xand eacht∈I,max{x(t),y(t)}andmin{x(t),y(t)}belong toX and are upper and lower bounds ofx,y, respectively. Therefore, for everyx,y∈X, one can take
max{x,y},min{x,y} ∈Xwhich are comparable tox,y. Now, we formulate the main result of this section.
Theorem . Under assumptions(a)-(a), (.)has a unique solution in C(I).
Proof We consider the operatorT:X→Xdefined by
T(x)(t) =h(t) +λ
k(t,s)fs,x(s)ds fort∈I.
By virtue of our assumptions,T is well defined (this means that ifx∈XthenT(x)∈X). Forx≤y, andt∈Iwe have
T(x)(t) –T(y)(t) =h(t) +λ
k(t,s)fs,x(s)ds–h(t) –λ
k(t,s)fs,y(s)ds
=λ
Therefore,Thas the monotone nondecreasing property. Also, forx≤y, we have
T(x)(t) –T(y)(t)=h(t) +λ
k(t,s)fs,x(s)ds–h(t) –λ
k(t,s)fs,y(s)ds
≤λ
k(t,s)fs,x(s)–fs,y(s)ds
≤λ
k(t,s)Lγy(s) –x(s)ds.
Since the functionγ is nondecreasing andx≤y, we have
γy(s) –x(s)≤γ sup
t∈I
x(s) –y(s)=γρ(x,y),
hence
T(x)(t) –T(y)(t)≤λ
k(t,s)Lγρ(x,y)ds≤λKLγρ(x,y).
Then we obtain
dT(x),T(y)=sup
t∈I
T(x)(t) –T(y)(t)p
≤λKLγρ(x,y)p=λpKpLpγρ(x,y)p
≤λpKpLpγρ(x,y)p=λpKpLpγd(x,y)
≤λpKpLpϕ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx) s
≤
p–ϕ
max
d(x,y),d(x,Tx),d(y,Ty),d(x,Ty) +d(y,Tx) s
.
This proves that the operator T satisfies the contractive condition (.) appearing in Corollary .. Also, letα,βbe the functions appearing in assumption (a); then, by (a),
we getα≤T(α). So, (.) has a solution and the proof is complete.
Example . Consider the following functional integral equation:
x(t) = t
+t+
e–ssint ( +t)
|x(s)|
+|x(s)|ds (.)
fort∈[, ]. Observe that this equation is a special case of (.) with
h(t) = t
+t,
k(t,s) = e –s
+t, f(t,x) = sint
|x|
Indeed, by usingγ(t) =twe see thatγ ∈and (γ(t))p= ( t)p=
ptp≤tp=γ(tp), for
allp≥. Further, for arbitrarily fixedx,y∈Rsuch thatx≥yand fort∈[, ] we obtain
f(t,x) –f(t,y)=sint
|x|
+|x|–
sint
|y|
+|y|
≤
|x–y|=
γ(x–y).
Thus, the functionf satisfies assumption (a) withL=. It is also easily seen thathis a continuous function. Further, notice that the functionkis continuous int∈I for every
s∈Iand measurable ins∈Ifor allt∈Iandk(t,s)≥. Moreover, we have
k(t,s)ds=
e–s +tds=
–e– +t
≤ –e–≤
=K. If we putα(t) =(+tt), we have
α(t) = t
( +t)≤
t +t
≤ t
+t+
e–ssint ( +t)
|α(s)| +|α(s)|ds =h(t) +λ
k(t,s)fs,α(s)ds.
This shows that assumption (a) holds. TakingL=,K= andλ=, then inequality
LpλpKp≤
p– appearing in assumption (a) has the following form:
p ×
p ×
p p ≤
p–.
It is easily seen that each numberp≥ satisfies the above inequality. Consequently, all the conditions of Theorem . are satisfied. Hence the integral equation (.) has a unique solution inC(I).
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
1Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.2Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran.3Department of Mathematics, Karaj Branch, Islamic Azad University, Alborz, Iran.
Received: 23 May 2014 Accepted: 5 September 2014 Published: 24 September 2014 References
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