R E S E A R C H Open Access
Integral type contractions in modular metric spaces
Bahareh Azadifar1, Ghadir Sadeghi1, Reza Saadati2and Choonkil Park3*
*Correspondence:
3Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea
Full list of author information is available at the end of the article
Abstract
We prove the existence and uniqueness of a common fixed point of compatible mappings of integral type in modular metric spaces.
MSC: Primary 47H09; 47H10; secondary 46A80
Keywords: modular metric space; compatible maps; contractive inequality of integral type; fixed point
1 Introduction and preliminaries
The metric fixed point theory is very important and useful in mathematics. It can be applied in various branches of mathematics, variational inequalities optimization and ap- proximation theory. In , Jungck [] proved a common fixed point theorem for com- muting maps generalizing the Banach contraction mapping principle. This result was fur- ther generalized and extended in various ways by many authors. On the other hand, Sessa [] defined weak commutativity as follows.
Let (X,d) be a metric space, the self-mappingsf,gare said to be weakly commuting if d(fg(x),gf(x))≤d(g(x),f(x)) for allx∈X. Further, Jungck [] introduced more generalized commutativity, the so-called compatibility, which is more general than weak commutativ- ity. Letf,gbe self-mappings of a metric space (X,d). The mappingsf andgare said to be compatible iflimn→∞d(fg(xn),gf(xn)) = , whenever{xn}∞n=is a sequence inXsuch that limn→∞f(xn) =limn→∞g(xn) =zfor somez∈X. Clearly, weakly commuting mappings are compatible, but neither implication is reversible. LetX= [, ) with the usual metric. We define mappingsf andgonXby
f(x) :=
⎧⎨
⎩
if ≤x<,
–x if≤x< and g(x) :=
⎧⎨
⎩
if ≤x<,
–x if≤x< .
Let{xn}∞n= be a sequence inX with limn→∞f(xn) =limn→∞g(xn) =z, thenz= and limn→∞fg(xn) =limn→∞gf(xn) =. Thus the pair (f,g) is compatible onX. In [] Bran- ciari obtained a fixed point theorem for a single mapping satisfying an analogue of the Banach contraction principle for integral type inequality (see also [–]). Vijayarajuet al.
[] proved the existence of the unique common fixed point theorem for a pair of maps sat- isfying a general contractive condition of integral type. Recently, Razani and Moradi []
proved the common fixed point theorem of integral type in modular spaces. The purpose of this paper is to generalize and improve Jungck’s fixed point theorem [] and Branciari’s
©2013Azadifar 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 medium, provided the original work is properly cited.
result [] to compatible maps in metric modular spaces. The notions of a metric modu- lar on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by Chistyakov []. In the sequel, we recall some basic concepts about modular metric spaces.
Definition . A functionω: (,∞)×X×X→[,∞] is said to be a metric modular on Xif it satisfies the following three axioms:
(i) givenx,y∈X,ωλ(x,y) = for allλ> if and only ifx=y;
(ii) ωλ(x,y) =ωλ(y,x)for allλ> andx,y∈X;
(iii) ωλ+μ(x,y)≤ωλ(x,z) +ωμ(z,y)for allλ,μ> andx,y,z∈X.
If, instead of (i), we have only the condition (i)ωλ(x,x) = for allλ> andx∈X, then ωis said to be a (metric) pseudo-modular onX. The main property of a (pseudo)modular ωon a setXis the following: givenx,y∈X, the function <λ→ωλ(x,y)∈[,∞] is non- increasing on (,∞). In fact, if <μ<λ, then (iii), (i)and (ii) imply
ωλ(x,y)≤ωλ–μ(x,x) +ωμ(x,y) =ωμ(x,y)
for allx,y∈X. If follows that at each pointλ> the right limitωλ+(x,y) :=limε→+ωλ+ε(x, y) and the left limitωλ–(x,y) :=limε→+ωλ–ε(x,y) exist in [,∞] and the following two inequalities hold:
ωλ+(x,y)≤ωλ(x,y)≤ωλ–(x,y)
for allx,y∈X. We know that ifx∈X, the setXω={x∈X:limλ→∞ωλ(x,x) = }is a metric space, called a modular space, whose metric is given by
dω(x,y) =inf
λ> :ωλ(x,y)≤λ
for allx,y∈Xω. We know that (see []) ifXis a real linear space,ρ:X→[,∞] and ωλ(x,y) =ρ
x–y λ
for allλ> andx,y∈X, thenρis modular onXif and only ifωis metric modular onX.
Example . The following indexed objectsωare simple examples of (pseudo)modulars on a setX. Letλ> andx,y∈X, we have:
(a) ωaλ(x,y) =∞ifx=y,ωaλ(x,y) = ifx=y;
and if (X,d) is a (pseudo)metric space with (pseudo)metricd, then we also have:
(b) ωbλ(x,y) =d(x,y)ϕ(λ) , whereϕ: (,∞)→(,∞)is a nondecreasing function;
(c) ωcλ(x,y) =∞ifλ≤d(x,y), andωcλ(x,y) = ifλ>d(x,y);
(d) ωdλ(x,y) =∞ifλ<d(x,y), andωdλ(x,y) = ifλ≥d(x,y).
Definition . LetXωbe a modular metric space.
() The sequence{xn}∞n= inXω is said to be convergent tox∈Xωif ωλ(xn,x)→ as n→ ∞for allλ> .
() The sequence{xn}∞n=inXωis said to be Cauchy tox∈Xωifωλ(xn,xm)→ asm,n→
∞for allλ> .
() A subsetCofXωis said to be closed if the limit of a convergent sequence ofCalways belongs toC.
() A subsetCofXωis said to be complete if any Cauchy sequence ofCis a convergent sequence and its limit is inC.
() A subsetCofXωis said to be bounded if for allλ> ,δω(C) =sup{ωλ(x,y);x,y∈ C}<∞.
2 A common fixed point theorem for contractive condition maps
Here, the existence of a common fixed point forω-compatible mappings satisfying a con- tractive condition of integral type in modular metric spaces is studied. We recall the fol- lowing definition.
Definition . LetXωbe a modular metric space induced by metric modularω. Two self- mappingsT,hofXωare calledω-compatible ifωλ(Thxn,hTxn)→, whenever{xn}∞n=is a sequence inXωsuch thathxn→zandTxn→zfor some pointz∈Xωand forλ> .
Theorem . Let Xωbe a complete modular metric space.Suppose that c,k,l∈R+,c>l and T,h:Xω→Xωare twoω-compatible mappings such that T(Xω)⊆h(Xω)and
ωλ c(Tx,Ty)
ϕ(t)dt≤k
ωλ l
(hx,hy)
ϕ(t)dt, (.)
for some k∈(, )and forλ> ,whereϕ:R+→R+is a Lebesgue integrable function which is summable,nonnegative and for allε> ,
ε
ϕ(t)dt> . (.)
If one of T or h is continuous,then there exists a unique common fixed point of T and h.
Proof Letxbe an arbitrary point ofXωand generate inductively the sequence{Txn}∞n=
as follows:Txn=hxn+for eachnandx=x, that is possible asT(Xω)⊆h(Xω). For each integern≥ and for allλ> , (.) shows that
ωλ
c(Txn+,Txn)
ϕ(t)dt≤k
ωλ l
(hxn+,hxn)
ϕ(t)dt
≤k
ωλ
c(Txn,Txn–)
ϕ(t)dt
≤k
ωλ l
(hxn,hxn–)
ϕ(t)dt.
By the principle of mathematical induction, we can easily show that
ωλ
c(Txn+,Txn)
ϕ(t)dt≤kn
ωλ l
(Tx,x)
ϕ(t)dt,
which, upon taking the limit asn→ ∞, yields
n→∞lim
ωλ
c(Txn+,Txn)
ϕ(t)dt≤.
Hence (.) implies that
n→∞lim ωλ
c(Txn+,Txn) = .
We now show that {Txn}∞n= is Cauchy. So, for all ε> , there existsn∈Nsuch that ωλ
c(Txn+,Txn) <εc for alln∈Nwithn≥n andλ> . Without loss of generality, sup- posem,n∈Nandm>n. Observe that forc(m–n)λ > , there existsn λ
m–n∈Nsuch that ω λ
c(m–n)(Txn+,Txn) < ε c(m–n) for alln≥n λ
m–n. We thus obtain ωλ
c(Txn,Txm)
≤ω λ
c(m–n)(Txn,Txn+) +ω λ
c(m–n)(Txn+,Txn+) +· · ·+ω λ
c(m–n)(Txm–,Txm)
< ε
c(m–n)+ ε
c(m–n)+· · ·+ ε c(m–n)
=ε c for alln,m≥n λ
m–n. This implies that{Txn}∞n=is a Cauchy sequence. SinceXωis complete, there existsz∈Xωsuch thatωλ
c(Txn,z)→ asn→ ∞. IfTis continuous, thenTxn→Tz andThxn→Tz. By theω-compatibility ofXω, we haveωλ(hTxn,Thxn)→ asn→ ∞for λ> . Moreover,hTxn→Tzsinceωλ(hTxn,Tz)≤ωλ
(hTxn,Thxn) +ωλ
(Thxn,Tz).
In the sequel, we prove thatzis a common fixed point ofTandh. By (.), we get
ωλ
c(Txn,Txn)
ϕ(t)dt≤k
ωλ l
(hTxn,hxn)
ϕ(t)dt.
Taking the limit asn→ ∞yields
ωλ c(Tz,z)
ϕ(t)dt≤k
ωλ l
(Tz,z)
ϕ(t)dt
≤k
ωλ c(Tz,z)
ϕ(t)dt,
which implies thatωλ
c(Tz,z) = forλ> . HenceTz=z. It follows fromT(Xω)⊆h(Xω) that there exists a pointzsuch thatz=Tz=hz. By (.), we get
ωλ c(Txn,Tz)
ϕ(t)dt≤k
ωλ l
(hTxn,hz)
ϕ(t)dt.
Taking the limit asn→ ∞yields
ωλ c(Tz,Tz)
ϕ(t)dt≤k
ωλ l
(Tz,hz)
ϕ(t)dt,
and so
ωλ c(z,Tz)
ϕ(t)dt≤k
ωλ l
(z,hz)
ϕ(t)dt
≤k
ωλ l
(z,z)
ϕ(t)dt.
Hencez=Tz=hzand alsohz=hTz=Thz=Tz=z(see []). In addition, if one con- sidershto be continuous (instead ofT), then by a similar argument (as above), one can provehz=Tz=z.
Finally, suppose thatzandωare two arbitrary common fixed points ofTandh. Then we have
ωλ c(z,ω)
ϕ(t)dt=
ωλ c(Tz,Tω)
ϕ(t)dt
≤k
ωλ l
(hz,hω)
ϕ(t)dt
≤k
ωλ c(z,ω)
ϕ(t)dt,
which implies thatωλ
c(z,ω) = forλ> and hencez=ω.
The following theorem is another version of Theorem . whenl=c, by adding the restriction thatT,h:B→B, whereBis a closed and bounded subset ofXω.
Theorem . Let Xω be a complete modular metric space, and let B be a closed and bounded subset of Xω.Suppose that T,h:Xω→Xωare twoω-compatible mappings such that T(Xω)⊆h(Xω)and
ωλ c(Tx,Ty)
ϕ(t)dt≤k
ωλ c(hx,hy)
ϕ(t)dt (.)
for all x,y∈B and forλ> ,where c,k∈R+with k∈(, ),andϕ:R+→R+is a Lebesgue integrable function which is summable,nonnegative and for allε> ,ε
ϕ(t)dt> .If one of T or h is continuous,then T and h have a unique common fixed point.
Proof Letx∈Bandm,n∈N. Let{xn}∞n=be the sequence generated in the proof of The- orem .. Then
ωλ
c(Txn+m,Txm)
ϕ(t)dt≤k
ωλ
c(hxn+m,hxm)
ϕ(t)dt
=k
ωλ
c(Txn+m–,Txm–)
ϕ(t)dt
≤k
ωλ
c(Txn+m–,Txm–)
ϕ(t)dt ...
≤km
ωλ c(Txn,x)
ϕ(t)dt
≤km
δω(B)
ϕ(t)dt
forλ> . SinceBis bounded,
n,m→∞lim
ωλ
c(Txn+m,Txm)
ϕ(t)dt≤, which implies thatlimn,m→∞ωλ
c(Txn+m,Txm) = . Therefore,{Txn}∞n=is Cauchy. SinceXω
is complete and Bis closed, there existsz∈Bsuch thatlimn→∞ωλ
c(Txn,z) = . IfT is continuous, thenTxn→TzandThxn→Tz. Then, byω-compatibility ofXω, we have ωλ(hTxn,Thxn)→ asn→ ∞forλ> . Moreover,hTxn→Tz. Next, we prove thatzis a fixed point ofT. It follows from (.) that
ωλ
c(Txn,Txn)
ϕ(t)dt≤k
ωλ c(hTxn,hxn)
ϕ(t)dt.
Taking the limit asn→ ∞yields
ωλ c(Tz,z)
ϕ(t)dt≤k
ωλ c(Tz,z)
ϕ(t)dt.
Soωλ
c(Tz,z) = forλ> and henceTz=z. SinceT(Xω)⊆h(Xω), there exists a pointz such thatz=Tz=hz, and
ωλ c(Txn,Tz)
ϕ(t)dt≤k
ωλ c(hTxn,hz)
ϕ(t)dt.
Taking the limit asn→ ∞yields
ωλ c(z,Tz)
ϕ(t)dt≤k
ωλ c(z,z)
ϕ(t)dt.
Hencez=Tz=hzand alsohz=hTz=Thz=Tz=z(see []). In addition, if one con- sidershto be continuous (instead ofT), then by a similar argument (as above), one can provehz=Tz=z.
Letzandωbe two arbitrary common fixed points ofTandh. Then
ωλ c(z,ω)
ϕ(t)dt=
ωλ c(Tz,Tω)
ϕ(t)dt
≤k
ωλ c(z,ω)
ϕ(t)dt, which implies thatωλ
c(z,ω) = forλ> and hencez=ω.
3 A common fixed point theorem for quasi-contraction maps
In this section, we prove Theorem . for a quasi-contraction map of integral type. To this end, we present the following definition.
Definition . Two self-mappingsT,h:Xω→Xω of a modular metric space Xω are (c,l,q)-generalized contractions of integral type if there exist <q< andc,l∈R+with c>lsuch that
ωλ c(Tx,Ty)
ϕ(t)dt≤q
m(x,y)
ϕ(t)dt, (.)
where m(x,y) =max{ωλ
l(hx,hy),ωλ
l(hx,Tx),ωλ l(hy,Ty),
ωλ l
(hx,Ty)+ωλ l
(hy,Tx)
}, andϕ:R+→ R+is a Lebesgue integrable function which is summable, nonnegative and for all ε> , ε
ϕ(t)dt> ,λ> andx,y∈Xω.
Theorem . Let Xω be a complete modular metric space. Suppose that T and h are (c,l,q)-generalized contractions of integral type self-maps of Xωand T(Xω)⊆h(Xω).If one of T or h is continuous,then T and h have a unique common fixed point.
Proof Choosec> l. Letxbe an arbitrary point ofXωand generate inductively the se- quence{Txn}∞n=as follows:Txn=hxn+andT(Xω)⊆h(Xω). We thus obtain
ωλ
c(Txn+,Txn)
ϕ(t)dt≤q
m(xn+,xn)
ϕ(t)dt
forλ> , where m(xn+,xn) =max
ωλ
l(hxn+,hxn),ωλ
l(Txn,hxn),ωλ
l(hxn+,Txn+), ωλ
l(hxn+,Txn) +ωλ
l(hxn,Txn+)
. It follows fromTxn=hxn+that
m(xn+,xn) =max
ωλ
l(hxn+,hxn),ωλ
l(Txn,Txn+),
+ωλ
l(hxn,Txn+)
.
Moreover, ωλ
l(hxn,Txn+) =ωλ
l(Txn–,Txn+)
≤ωλ
l(Txn–,Txn) +ωλ
l(Txn,Txn+)
≤ωλ
c(Txn–,Txn) +ωλ
c(Txn,Txn+).
Then
m(xn+,xn)≤ωλ
c(Txn,Txn–) and
ωλ
c(Txn+,Txn)
ϕ(t)dt≤q
ωλ
c(Txn,Txn–)
ϕ(t)dt.
Continuing this process, we get
ωλ
c(Txn+,Txn)
ϕ(t)dt≤qn
ωλ c(Tx,x)
ϕ(t)dt.
Solimn→∞ωλ
c(Txn,Txn+) = asntends to infinity. Supposel<c< l. Sinceωλis a de- creasing function, one may writeωλ
c(Txn,Txn+)≤ωλ
c(Txn,Txn+), wheneverc< l≤c.
Taking the limit from both sides of this inequality shows thatlimn→∞ωλ
c(Txn,Txn+) = forl<c< landλ> . Thus we havelimn→∞ωλ
c(Txn,Txn+) = for anyc>l. Now, we show that{Txn}∞n= is Cauchy. Sincelimn→∞ωλ
c(Txn,Txn+) = forλ> , forε> , there existsn∈Nsuch thatωλ
c(Txn+,Txn) <εc for alln∈Nwithn≥nandλ> . Without loss of generality, suppose m,n∈Nandm>n. Observe that for c(m–n)λ > , there exists n λ
m–n∈Nsuch that ω λ
c(m–n)(Txn+,Txn) < ε c(m–n) for alln≥n λ
m–n. Now we have ωλ
c(Txn,Txm)
≤ω λ
c(m–n)(Txn,Txn+) +ω λ
c(m–n)(Txn+,Txn+) +· · ·+ω λ
c(m–n)(Txm–,Txm)
< ε
c(m–n)+ ε
c(m–n)+· · ·+ ε c(m–n)
=ε c for alln,m≥n λ
m–n. This implies{Txn}∞n=is a Cauchy sequence. SinceXωis complete, there existsz∈Xωsuch thatωλ
c(Txn,z)→ asn→ ∞. Next we prove thatzis a fixed point ofT.
If T is continuous, thenTxn→TzandThxn→Tz. By theω-compatibility ofXω, we haveωλ(hTxn,Thxn)→ asn→ ∞forλ> . Moreover,hTxn→Tzsinceωλ(hTxn,Tz)≤ ωλ
(hTxn,Thxn) +ωλ
(Thxn,Tz). Note that
ωλ
c(Txn,Txn)
ϕ(t)dt≤q
m(xn,Txn)
ϕ(t)dt,
where
m(xn,Txn) =max
ωλ
l(hxn,hTxn),ωλ
l(hxn,Txn),ωλ
l(hTxn,TTxn), ωλ
l(hxn,TTxn) +ωλ
l(Txn,hTxn)
.
Taking the limit asn→ ∞, we get
ωλ c(z,Tz)
ϕ(t)dt≤q
ωλ c(z,Tz)
ϕ(t)dt,
and soTz=z. SinceT(Xω)⊆h(Xω), there exists a pointzsuch thatz=Tz=hz. We have
ωλ c(Txn,Tz)
ϕ(t)dt≤q
m(Txn,z)
ϕ(t)dt
and
m(Txn,z) =max
ωλ
l(hTxn,z),ωλ l
hTxn,Txn
,ωλ l(z,Tz), ωλ
l(hTxn,Tz) +ωλ
l(z,Txn)
.
Taking the limit asn→ ∞, we get
ωλ c(z,Tz)
ϕ(t)dt≤q
ωλ c(z,Tz)
ϕ(t)dt.
It follows thatz=Tz=hzand alsohz=hTz=Thz=Tz=z(see []). Moreover, ifhis continuous (instead ofT), then by a similar argument (as above), we can provehz=Tz=z.
Letzandωbe two arbitrary fixed points ofTandh. Then
m(z,ω) =max
ωλ
l(z,ω), , , ωλ
l(z,ω) +ωλ l(z,ω)
=ωλ l(z,ω).
Therefore,
ωλ c(z,ω)
ϕ(t)dt≤q
ωλ l
(z,ω)
ϕ(t)dt,
which implies thatz=ω.
4 Generalization
Here, we extend the results of the last section. We need a general contractive inequality of integral type. LetR+be a set of nonnegative real numbers and consider (∗)φ:R+→R+ as a nondecreasing and right-continuous function such thatφ(t) <tfor anyt> .
To prove the next theorem, we need the following lemma [].
Lemma . Let t> .φ(t) <t if only iflimkφk(t) = ,whereφkdenotes the k-times repeated composition ofφwith itself.
Next, we prove a modified version of Theorem ..
Theorem . Let Xωbe a complete modular metric space.Suppose that c,l∈R+,c>l and T,h:Xω→Xωare twoω-compatible mappings such that T(Xω)⊆h(Xω)and
ωλ c(Tx,Ty)
ϕ(t)dt≤φ
ωλ l
(hx,hy)
ϕ(t)dt
,
whereφis a function satisfying the property(∗),andϕ:R+→R+is a Lebesgue integrable mapping which is summable,nonnegative and for allε> ,ε
ϕ(t)dt> andλ> .If one of T or h is continuous,then T and h have a unique common fixed point.
Proof Letxbe an arbitrary point ofXωand generate inductively the sequence{Txn}∞n=as follows:Txn=hxn+andx=x, that is possible asT(Xω)⊆h(Xω),
ωλ
c(Txn+,Txn)
ϕ(t)dt≤φ
ωλ l
(hxn+,hxn)
ϕ(t)dt
≤φ
ωλ
c(Txn,Txn–)
ϕ(t)dt
≤φ
ωλ l
(hxn,hxn–)
ϕ(t)dt
for each integern≥ andλ> . By the principle of mathematical induction, we can easily see that
ωλ
c(Txn+,Txn)
ϕ(t)dt≤φn
ωλ l
(Tx,x)
ϕ(t)dt
. Taking the limit asn→ ∞, we obtain, by Lemma .,
limn ωλ
c(Txn+,Txn)
ϕ(t)dt= .
Using the same method as in the proof of Theorem ., we show that T andhhave a
unique common fixed point.
Applying the method of proof of Theorem ., we get the following result.
Theorem . Let Xω be a complete modular metric space.Suppose c,l∈R+, c>l and T,h:Xω→Xωsuch that T(Xω)⊆h(Xω)
ωλ c(Tx,Ty)
ϕ(t)dt≤φ
m(x,y)
ϕ(t)dt
,
where m(x,y) =max{ωλ
l(hx,hy),ωλ
l(hx,Tx),ωλ l(hy,Ty),
ωλ l
(hx,Ty)+ωλ l
(hy,Tx)
},φ is a function satisfying the property(∗)andλ> .If one of T or h is continuous, then there exists a unique common fixed point of h and T.
Proof The proof is similar to the proof of Theorem ..
Now we provide examples to validate and illustrate Theorems . and ..
Example . LetXω={n :n∈N} ∪ {}andωλ(x,y) :=|x–y|λ forλ> . Define the mapping T,h:Xω→Xωby
T(x) :=
⎧⎨
⎩
n+ ifx=n,
otherwise and h(x) :=
⎧⎨
⎩
n+ ifx=n,
otherwise.
Then all the hypotheses of Theorem . are satisfied withϕ(t) = tfort> andc= ,l= , k=.
Example . LetXω={n:n∈N} ∪ {}andωλ(x,y) :=|x–y|λ forλ> . Define the mapping T,h:Xω→Xωby
T(x) :=
⎧⎨
⎩
n+ ifx=n,
otherwise and h(x) :=
⎧⎨
⎩
n ifx=n,
otherwise.
Then all the hypotheses of Theorem . are satisfied withϕ(t) =tt– ( –logt) fort> and c=l=λ= ,k=. See [] for details.
5 Application
In the section, we assume thatR= (–∞, +∞),R+= (, +∞),Ndenotes the set of all pos- itive integers, ‘opt’ stands for ‘sup’ or ‘inf’,Y is a Banach space andXωis anω-complete space. Suppose thatS⊆Xω,D⊆YandB(S) denotes the complete space of all bounded real-valued functions onSwith the norm
g=supg(x):x∈S g∈B(S)
and ={ϕ;ϕ:R+→R+}such thatϕis Lebesgue integrable, summable on each compact subset ofR+andε
ϕ(t)dt> for eachε> . We prove the solvability of the functional equations
f(x) =opt
y∈D
u(x,y) +H x,y,f
T(x,y)
(.)
for allx∈SinB(S). First, we recall the following lemma [].
Lemma . ([]) Let E be a set,and let p,q:E→Rbe mappings. Ifopty∈Ep(y)and opty∈Eq(y)are bounded,then
opt
y∈Ep(t) –opt
y∈Eq(t)≤sup
y∈E
p(y) –q(y).
Theorem . Let u:S×D→R,T:S×D→S,H:S×D×R→R,φ∈ .Suppose that u and H are bounded such that
c|H(x,y,g(T(x,y)))–H(x,y,h(T(x,y)))|
λ
ϕ(t)dt≤k
lg–h λ
ϕ(t)dt (.)
for all(x,y,g,h,λ)∈S×D×B(S)×B(S)×R+,and some <k< and c>l> .Then the functional equation(.)has a unique solution w∈B(S)and{Anz}n∈Nconverges to w for each z∈B(S),where the mapping A is defined by
Az(x) =opt
y∈D
u(x,y) +H x,y,z
T(x,y)
(x∈S). (.)
Proof By boundedness ofuandH, there existsM> such that supu(x,y)+H
x,y,z
T(x,y): (x,y,t)∈S×D×R
≤M. (.)
It is easy to show thatAis a self-mapping inB(S) by (.), (.) and Lemma .. Using [, Theorem .] andφ∈ , we conclude that for eachε> , there existsδ> such that
C
ϕ(t)dt<ε, ∀C⊆[, M] withm(C)≤δ, (.)
wherem(C) denotes the Lebesgue measure ofC.
Letx∈S,h,g∈B(S). Suppose thatopty∈D=infy∈D. Clearly, forc>l, (.) implies that there existy,z∈Dsatisfying
Ag(x) >u(x,y) +H x,y,g
T(x,y) –δ
c; (.)
Ah(x) >u(x,z) +H x,z,h
T(x,z) –δ
c; (.)
Ag(x)≤u(x,z) +H x,z,g
T(x,z)
; (.)
Ah(x)≤u(x,y) +H x,y,h
T(x,y)
. (.)
Put H =H(x,y,g(T(x,y))), H=H(x,y,h(T(x,y))), H=H(x,z,g(T(x,z))),H =H(x,z, h(T(x,z))).
From (.) and (.), it follows that c
Ag(x) –Ah(x)
>c H
x,y,g
T(x,y) –H
x,y,h
T(x,y) –δ
>l H
x,y,g
T(x,y) –H
x,y,h
T(x,y) –δ
≥–max lH
x,y,g
T(x,y) –H
x,y,h
T(x,y), lH
x,z,g
T(x,z) –H
x,z,h
T(x,z)–δ
= –max
l|H–H|,l|H–H| –δ.
Similarly, from (.) and (.), we get c
Ah(x) –Ag(x)
>c H
x,z,h
T(x,z) –H
x,z,g
T(x,z) –δ
>l H
x,z,h
T(x,z) –H
x,z,g
T(x,z) –δ
≥–max lH
x,y,g
T(x,y) –H
x,y,h
T(x,y), lH
x,z,g
T(x,z) –H
x,z,h
T(x,z)–δ
= –max
l|H–H|,l|H–H| –δ.
So c
Ag(x) –Ah(x)
<max
l|H–H|,l|H–H| +δ.
Then
c|Ag(x) –Ah(x)|
λ <max
l|H–H|
λ ,l|H–H| λ
+δ
λ (.)
for eachλ> .
Similarly, we infer that (.) holds also foropty∈D=supy∈D. Combining (.), (.) and (.) yields
c|Ag(x)–Ah(x)|
λ
ϕ(t)dt≤ max{
l|H–H|
λ ,l|H–λH|}+δλ
ϕ(t)dt
=max
l|H–H| λ +δλ
ϕ(t)dt,
l|H–H|
λ +λδ
ϕ(t)dt
=max
l|H–H| λ
ϕ(t)dt+
l|H–H| λ +δλ l|H–H|
λ
ϕ(t)dt,
l|H–H| λ
ϕ(t)dt+
l|H–H| λ +δλ l|H–H|
λ
ϕ(t)dt
≤max
l|H–H| λ
ϕ(t)dt,
l|H–H|
λ
ϕ(t)dt
+max
l|H–H| λ +δλ l|H–H|
λ
ϕ(t)dt,
l|H–H| λ +δλ l|H–H|
λ
ϕ(t)dt
≤k
lg–h λ
ϕ(t)dt+ε, which means that
cAg–Ah λ
ϕ(t)dt≤k
lg–h λ
ϕ(t)dt+ε
for eachλ> . Lettingε→+in the above inequality, we deduce that
cAg–Ah λ
ϕ(t)dt≤k
lg–h λ
ϕ(t)dt.
Thus Theorem . follows from Theorem .. This completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Author details
1Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran.
2Department of Mathematics, Iran University of Science and Technology, Tehran, Iran.3Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea.
Acknowledgements
The authors are grateful to the two anonymous reviewers for their valuable comments and suggestions.
Received: 1 May 2013 Accepted: 17 September 2013 Published:07 Nov 2013
References
1. Jungck, G: Commuting mappings and fixed point. Am. Math. Mon.83, 261-263 (1976)
2. Sessa, S: On a weak commutativity conditions of mappings in fixed point consideration. Publ. Inst. Math. (Belgr.)32, 146-153 (1982)
3. Jungck, G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci.11, 771-779 (1986) 4. Branciari, A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int.
J. Math. Math. Sci.29, 531-536 (2002)
5. Samet, B, Vetro, C: An integral version of Ciric’s fixed point theorem. Mediterr. J. Math.9, 225-238 (2012) 6. Samet, B, Vetro, C: Berinde mappings in orbitally complete metric spaces. Chaos Solitons Fractals44, 1075-1079
(2011)
7. Vetro, C: On Branciari’s theorem for weakly compatible mappings. Appl. Math. Lett.23, 700-705 (2010) 8. Vijayaraju, P, Rhoades, BE, Mohanra, R: A fixed point theorem for a pair of maps satisfying a general contractive
condition of integral type. Int. J. Math. Math. Sci.15, 2359-2364 (2005)
9. Razani, A, Moradi, R: Common fixed point theorems of integral type in modular spaces. Bull. Iran. Math. Soc.35, 11-24 (2009)
10. Chistyakov, VV: Modular metric spaces I: basic concepts. Nonlinear Anal. TMA72, 1-14 (2010)
11. Kaneko, H, Sessa, S: Fixed point theorems for compatible multi-valued and single-valued mappings. Int. J. Math.
Math. Sci.12, 257-262 (1989)
12. Singh, SP, Meade, BA: On common fixed point theorem. Bull. Aust. Math. Soc.16, 49-53 (1977)
13. 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, 64 (2011)
14. Hewitt, E, Stromberg, K: Real and Abstract Analysis. Springer, Berlin (1978)
10.1186/1029-242X-2013-483
Cite this article as:Azadifar et al.:Integral type contractions in modular metric spaces.Journal of Inequalities and Applications2013, 2013:483
