R E S E A R C H
Open Access
F
(
ψ
,
ϕ
)
-Contraction in terms of measure
of noncompactness with application for
nonlinear integral equations
Farzaneh Nikbakhtsarvestani
1, S Mansour Vaezpour
2and Mehdi Asadi
3**Correspondence: [email protected]; [email protected] 3Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran Full list of author information is available at the end of the article
Abstract
In this paper, some new generalization of Darbo’s fixed point theorem is proved by using aF(ψ,
ϕ)-contraction in terms of a measure of noncompactness. Our result
extends to obtaining a common fixed point for a pair of compatible mappings. The paper contains an application for nonlinear integral equations as well.MSC: 47H10; 34A12; 54H25
Keywords: fixed point; measure of noncompactness;F(ψ,
ϕ)-contraction
1 Introduction and preliminaries
A contractive condition in terms of a measure of noncompactness, which was first used by Darbo, is one of the fruitful tools to obtain fixed point and common fixed point theo-rems. The extensions of these contractions in linear and integral type which are known as generalizations of Darbo’s fixed point theorem, are considered by many authors; see, for example, [–] and the references therein.
Recently, Khodabakhshi [] obtained some new common fixed point results with the technique associated with a measure of noncompactness for two commuting operators.
Inspired by the class ofα-ψcontractive type mappings which was introduced by Samet
et al.[], Ansari [] presented the weaker class of this contraction named F(ψ,ϕ )-contraction and used it to obtain fixed point and common fixed point results.
This paper mainly aims at employing theF(ψ,ϕ)-contraction and its property in terms of a measure of noncompactness to investigate a fixed point and a common fixed point for a pair of compatible mappings.
Now we present some definitions, notations and results which will be needed later. Throughout this paper we assume thatEis an infinite dimensional Banach space. If C
is a subset ofEthen the symbolsco(C) andMEandNEdenote the closure of convex hull
ofCand the family of nonempty bounded subsets ofEand the subfamily consisting of all relatively compact subsets ofE, respectively.
The measure of noncompactness was introduced by Kuratowski [],
μ(S) :=inf
δ> :S=
n
i=
Sifor someSiwithdiam(Si)≤δfor ≤i≤n<∞
,
for a bounded subsetSof a metric spaceX.
From now on we will use the following definition for the measure of noncompactness.
Definition .([]) A mappingμ:ME→[,∞) is said to be a measure of
noncompact-ness inEif it satisfies the following conditions:
(A) ∅ =Kerμ={X∈ME:μ(X) = } ⊆NE;
(A) X⊆Y⇒μ(X)≤μ(Y); (A) μ(X) =μ(coX) =μ(X);
(A) μ(λX+ ( –λ)Y)≤λμ(X) + ( –λ)μ(Y)forλ∈[, ];
(A) If{Xn}is a sequence of closed sets fromMEsuch thatXn+⊆Xn, (n≥) and if
limn→∞μ(Xn) = then the intersection setX∞=
∞
n=Xnis nonempty.
The familyKerμdescribed in (A) is said to be the kernel of the measure of noncom-pactnessμ. Observe thatX∞∈Kerμ, sinceμ(X∞)≤μ(Xn) for anyn.
Definition .([]) A functionψ: [,∞)→[,∞) is called an altering distance func-tion if the following properties are satisfied:
() ψis nondecreasing and continuous; () ψ(t) = if and only ift= .
We denote bythe class of altering distance functions.
Definition .([]) An ultra altering distance function is a continuous, nondecreasing mappingϕ: [,∞)→[,∞) such thatϕ(t) > fort> andϕ()≥.
We denote bythe class of ultra altering distance functions.
Definition .([]) A mappingF: [,∞)→Ris called aC-class function if it is
con-tinuous and satisfies the following axioms:
() F(s,t)≤s;
() F(s,t) =simplies that eithers= ort= ; for alls,t∈[,∞).
Note for someFwe haveF(, ) = . We denoteC-class functions byC.
Definition .([]) A pair of self-mappingsFandGonXis weakly compatible if there exists a pointx∈Xsuch thatF(x) =G(x) impliesFGx=GFxi.e., they commute at their coincidence point.
Proposition .([, Proposition .]) Let f and g be weakly compatible self-mappings of a set X.If f and g have a unique point of coincidence,w=f(x) =g(x).Then w is the unique common fixed point of f and g.
Lemma .([]) Let X be a nonempty set and f :X→X be a function.Then there exists a subset E⊆X such that f(E) =f(X)and f :E→X is one to one.
Theorem .(Schauder []) Let C be a closed,convex subset of a Banach space E.Then every compact,continuous map F:C→C has at least one fixed point.
As a significant generalization of Schauder’s fixed point theorem, we have the following fixed point theorem.
Theorem .(Darbo []) Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T :C→C be a continuous mapping.Assume that there exists a constant k∈[, )such that
μT(X)≤kμ(X)
for any subset X of C.Then T has a fixed point.
2 Main results
This section starts by some of the theorems and corollaries related to fixed point are ob-tained by usingF(ψ,ϕ)-contraction in terms of a measure of noncompactness. Next, for a pair of compatible mappings a common fixed point theorem is considered. In the sequel, theorems are proved in integral type to obtain a fixed point and a common fixed point. Our results generalized Darbo’s fixed point theorem and a fixed point theorem which was recently proved.
Theorem . Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T:C→C be a continuous mapping,such that
ψμT(M)≤Fψμ(M),ϕμ(M),
for any subset M of C and whereψ∈,ϕ∈and F∈C.Then T has a fixed point.
Proof Define a sequence{Cn}∞n=setting
C:=C, Cn:=coT(Cn–),
wheren= , , . . . . Now let us prove that
Cn+⊆Cn, T(Cn)⊆Cn, ()
for everyn= , , . . . . The first inclusion will be proved via mathematical induction. Let
n= . SinceC=C,Cis convex and closed,T(·) :C→C, we haveC=co(T(C))⊂C.
Now assume thatCn⊂Cn–. Thenco(T(Cn))⊂co(T(Cn–)). So we obtainCn+⊂Cn. The
second inclusion follows immediately from the first one,T(Cn)⊂co(T(Cn)) =Cn+⊂Cn.
If there existsN∈Nsuch thatμ(N) = thenCNis compact and Schauder’s fixed point
Taking into consideration thatF∈C, the property (A) from Definition (.) and using the inequality given in the theorem, we have
ψμ(Cn+)
=ψμcoT(Cn)
=ψμT(Cn)
≤Fψμ(Cn)
,ϕμ(Cn)
≤ψμ(Cn)
, ()
for everyn= , , . . . . From () and condition (A) of Definition (.) we conclude that
μ(Cn+)≤μ(Cn) for everyn= , , . . . . This means that the sequence{μ(Cn)}∞n=is not
increasing, and consequently there existsr≥ such thatlimn→∞μ(Cn) =r. Sinceψ is
continuous, according to () we get
ψ(r)≤Fψ(r),φ(r)≤ψ(r), and hence
Fψ(r),ϕ(r)=ψ(r).
The last inequality and the inclusionF∈Cyieldψ(r) = orϕ(r) = . These equalities and the inclusionsψ∈,ϕ∈imply thatr= . So, we obtain
lim
n→∞μ(Cn) = . ()
Let C∞=∞n=Cn. Since Cn+⊂Cn, Cn is bounded, closed, and convex for every n=
, , . . . , we see thatC∞is also bounded, closed, and convex, so the equality () and prop-erty (A) of Definition (.) imply thatC∞is nonempty and compact. From the inclusion () it follows that
T(C∞) =T
∞
n=
Cn
⊂
∞
n=
T(Cn)
⊂
∞
n=
Cn
=C∞.
Finally, by virtue of Schauder’s fixed point theorem we see that the mapT:C∞→C∞has
a fixed point inC∞. SinceC∞⊂C, we conclude that the mapT has a fixed point inC.
The proof is completed. If we letF(s,t) =ksin Theorem . we get the following result.
Corollary . Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T:C→C be a continuous mapping.Assume that
ψμ(TM)≤kψμ(M),
for M⊆C.Then T has a fixed point.
Corollary . Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T:C→C be a continuous mapping.Assume that
μ(TM)≤Fμ(M),ϕμ(M),
for M⊆C.Then T has a fixed point.
Theorem . Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T:C→C be continuous mapping.Assume that there existψ∈,ϕ∈and F∈Csuch that the inequality
ψμT(C)≤FψM(X,Y),ϕM(X,Y),
is satisfied for every noncompact subset X,Y of C,where M(X,Y) =maxμ(X),μT(X),μT(Y),μT(X)∪T(Y).
Then T has a fixed point.
Proof Define a sequence{Cn}∞n=setting
C:=C, Cn:=co
T(Cn–)
,
wheren= , , . . . . Analogously to Theorem . it is possible to show that
Cn+⊆Cn, T(Cn)⊆Cn, ()
for everyn= , , . . . .
Ifμ(CN) = for someN∈NthenCNis a compact set and by virtue of Schauder’s
theo-rem the continuous mapT:CN→CN has a fixed point inCN⊂C.
Now suppose thatμ(CN) > for everyn∈N. From () and condition (A) of
Defi-nition . we conclude that μ(Cn+)≤μ(Cn) for everyn= , , . . . . This means that the
sequence {μ(Cn)}∞n= is not increasing, and consequently there exists r≥ such that
limn→∞μ(Cn) =r.
Taking into consideration thatF∈C, condition (A), we have
ψμ(Cn+)
=ψμcoT(Cn)
=ψT(Cn)
≤FψM(Cn,Cn+)
,ϕM(Cn,Cn+)
≤ψM(Cn,Cn+)
, ()
where
M(Cn,Cn+) =max
μ(Cn),μ
T(Cn)
,μT(Cn+)
,μT(Cn)∪T(Cn+)
. SinceT(Cn)⊂CnandCn+⊂Cnwe have
μT(Cn)
≤μ(Cn), μ
T(Cn+)
≤μT(Cn)
,
μT(Cn)∪T(Cn+)
for everyn≥. Thus we obtain
M(Cn,Cn+) =μ(Cn),
and consequently
ψM(Cn,Cn+)
=ψμ(Cn)
, ϕM(Cn,Cn+)
=ϕμ(Cn)
, for everyn≥. The last equalities and () yield
ψμ(Cn+)
≤FψM(Cn,Cn+)
,ϕM(Cn,Cn+)
≤ψμ(Cn)
, and hence
ψμ(Cn+)
≤Fψμ(Cn)
,ϕμ(Cn)
≤ψμ(Cn)
,
for everyn≥. Sincelimn→∞μ(Cn) =r;ψ,ϕandFare continuous functions, we get Fψ(r),ϕ(r)=ψ(r).
The inclusionF∈C yieldsψ(r) = orϕ(r) = . These equalities and inclusionsψ∈,
ϕ∈imply thatr= . So, we obtain
lim
n→∞μ(Cn) = .
From now on the proof repeats the proof of Theorem .. The theorem is proved. By takingψ(t) =twe have the following.
Corollary . Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T:C→C be a continuous mapping.Assume that there existψ∈,ϕ∈and F∈Csuch that the inequality
μT(C)≤FψM(X,Y),ϕM(X,Y),
is satisfied for every noncompact subset X,Y of C,where M(X,Y) =maxμ(X),μT(X),μT(Y),μT(X)∪T(Y).
Then the map T has a fixed point.
Theorem . Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T,S:C→C be continuous mappings.Assume that:
(a). The range ofTcontains the range ofS. (b). For anyM⊂C:
ψ(μT(M)≤FψμS(M),ϕμS(M), ()
Then:
(i). The setsA={x∈C:S(x) =x}andB={x∈C:T(x) =x}are nonempty and closed. (ii). TandShave a coincidence point.
(iii). IfT andSare weakly compatible.ThenSandThave a unique common fixed point.
Proof LetC=C, chooseC⊂Esuch thatS(C)⊆T(C) andC:=coS(C). This can be
done since the range ofTcontains the range ofS. We have
S(C)⊆coS(C)⊆RS⊆RT,
so there existsCsuch thatS(C)⊆T(C).
Continuing this process having chosenCninEwe obtainCn+ inEsuch thatS(Cn)⊆ T(Cn+) andCn+:=coSCn.
If we putCn+:=coS(Cn), then
S(Cn)⊆Cn+=coS(Cn)⊆C and T(Cn+)⊆C, ∀n∈N∪ {},
so
S(Cn)∩T(Cn+)⊆C,
thereforeS(Cn)⊆T(Cn+) for everyn∈N∪ {}because the cases
S(Cn)∩T(Cn+) =∅ and T(Cn+)⊆S(Cn)
are impossible, sinceRS⊆RT.
We observe thatCn+⊆CnandSCn⊆Cnforn∈N∪ {}, because C=co
S(C)
⊆C.
LetCn⊆Cn–so
Cn+=co
S(Cn)
⊆coS(Cn–)
=Cn.
And also
S(Cn)⊂S(Cn–)⊂co
S(Cn–)
=Cn.
Ifμ(CN) = , for someN∈N, thenThas a fixed point inC, because Schauder’s fixed point
theorem guarantees this. Supposeμ(Cn) > for eachn∈N. Therefore we get
ψμ(Cn+)
=ψμco(SCn)
=ψμS(Cn)
≤ψμT(Cn+)
≤FψμS(Cn+)
,ϕμS(Cn+)
≤Fψμ(Cn+)
,ϕμ(Cn+)
≤ψμ(Cn+)
. ()
SinceCn+⊂Cn for everyn= , , . . . , the condition (A) of Definition . implies that
μ(Cn+)≤μ(Cn) for everyn= , , . . . . This means that the sequence{μ(Cn)}∞n=is not
increasing, and consequently there existsr≥ such thatlimn→∞μ(Cn) =r. By () we find
that
ψ(r)≤Fψ(r),ϕ(r)≤ψ(r), so
Fψ(r),ϕ(r)=ψ(r),
according to the property ofFwe haveψ(r) = orϕ(r) = . Hence
r= lim
n→∞μ(Cn) = .
Also sinceCn+⊆Cnby property (A) of Definition .C∞=∞n=Cnis nonempty and
compact.
Moreover, sinceCnandCare convex, andS(Cn)⊂Cn,S:Cn→Cnforn= , , , . . . and
soS:C∞→C∞, now Schauder’s fixed point theorem ensuresShas a fixed point and the
setA={x∈C:S(x) =x}is nonempty and closed.
Similarly to S; T has a fixed point and by continuity of T, B={x∈C:T(x) =x} is nonempty and closed. By Lemma ., take
D:=x∈C:S(x) =xorT(x) =x
and define a mapg:=S(D)→S(D) byg(Sx) =Tx. Clearlygis well defined. Now if we put
X:S(D) andE:=Bin Lemma ., theng(E) =g(X), sogis one to one. Now by using () we have
ψμgS(M)=ψμT(M)≤FψμS(M),ϕμS(M),
so according to Theorem . there existsz∈Eand it is unique, sincegis one to one, such thatg(Sz) =Sz, which impliesTz=Sz.
HenceTandShave a unique coincidence point thus from Proposition . and it follows thatTandShave a unique common fixed point.
Example . LetC= [, ] be a subset ofR. TakeS,T: [, ]→[, ] defined by
T(x) =
x and S(x) = x, also let
M= [, ], ϕ(t) = t
, ψ(t) =t, F(s,t) =
and
μ(X) =diam(X).
It is clear thatψ∈andϕ∈. To verify the hypotheses of Theorem .:
(a). RS= [,]⊂RT= [, ].
By takingC= [, ]andC= [,]we have
S[, ]=
,
=T
,
,
the algorithm ofCnfollows by
C=
,
, C=
,
, C=
,
, · · ·,
such thatS(Cn)⊆T(Cn+).
(b). ψ(μ(T[, ])) =,ψ(μ(S[, ])) =,ϕ(μ(S[, ])) =, and also
F
,
= ,
therefore
ψ
μT(M)= ≤F
ψμS(M),ϕμS(M)=
, () thus the setsF={x∈C:S(x) =x}andK={x∈C:T(x) =x}are nonempty and closed. AlsoT() =S(), so is a coincidence point. Finally sinceT andScommute at , that is,ST() =TS(), soT andSare weakly compatible and is a common fixed point ofS
andT.
Theorem . Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T:C→C be a continuous mapping such that
ϕ(μ(T(M)))
f(t)dt≤F
ϕ(μ(M))
f(t)dt,
ψ(μ(M))
f(t)
, ()
for any subset M of C and where f : [,∞)→[,∞)be a Lebesgue integrable function,
which is summable on each compact of[,∞)andεf(t)dt> for eachε> andψ∈,
ϕ∈and F∈C.Then T has a fixed point.
Proof Define a sequence{Cn}as follows:
C:=C, Cn:=coTCn–, forn= , , . . . .
Ifμ(CN) = for someN∈N. ThenT has a fixed pointCby the proof of previous
the-orems. Supposeμ(Cn) > for alln∈N. SinceCn+⊂Cnfor everyn= , , . . . , the
the sequence{μ(Cn)}∞n=is not increasing, and consequently there existsr≥ such that
limn→∞μ(Cn) =r. From the inclusionsψ∈,ϕ∈we obtainlimn→∞ψ(μ(Cn)) =ψ(r)
andlimn→∞ϕ(μ(Cn)) =ϕ(r). Sincef(·) is Lebesgue integrable on each compact subset of
[,∞), we see that
lim
n→∞
ψ(μ(Cn))
f(t)dt=
ψ(r)
f(t)dt, lim
n→∞
ϕ(μ(Cn))
f(t)dt=
ϕ(r)
f(t)dt.
The last equalities, inclusionF∈Cand () imply that
ϕ(r)
f(t)dt≤F
ϕ(r)
f(t)dt,
ψ(r)
f(t)dt
≤
ϕ(r)
f(t)dt,
and hence
F
ϕ(r)
f(t)dt,
ψ(r)
f(t)dt
=
ϕ(r)
f(t)dt.
Thus we see that
ϕ(r)
f(t)dt= , or
ψ(r)
f(t)dt= ,
and consequently
ϕ(r) = or ψ(r) = .
From the last equalities it follows thatr= ,i.e.limn→∞μ(Cn) =r. Also sinceCn+⊆Cn
by property (A) of Definition .C∞=∞n=Cnis a nonempty, closed, and convex subset
ofC. Moreover, we know thatC∞ belongs toker(μ). SoC∞is compact and invariant by
the mappingT. Consequently, Schauder’s fixed point theorem implies thatT has a fixed point inC∞. SinceC∞⊂Cthe proof is complete. Remark . Putf(t) = ,ϕ(t) =tandF(s,t) =ksfort∈[,∞) in Theorem .. Then
μT(X)=
ϕ(μ(T(X)))
f(t)dt
≤F
ϕ(μ(X))
f(t)dt,
ψ(μ(X))
f(t)dt
=k
ϕ(μ(X))
f(t)dt
=kμ(X),
thus we get Darbo’s fixed point theorem.
Corollary . Let C be a nonempty,bounded, closed,and convex subset of a Banach space E and let T:C→C be a continuous mapping such that
ϕ(Tx–Ty)
f(t)dt≤F
ϕ(x–y)
f(t)dt,
ψ(x–y)
f(t)dt
for any x,y∈C where f : [,∞)→[,∞)be a Lebesgue integrable function which is summable on each compact set of[,∞)andεf(t)dt> for eachε> andψ∈,ϕ∈
and F∈C.Then T has a fixed point.
Proof Defineμ:mE→R+withμ(X) =diam(X) for anyX∈mE, wherediam(X) is a
diam-eter of the setX. It is easy to verify thatμis a measure of noncompactness on spaceE. By assumption, we have
ϕ(supx,y∈XTx–Ty)
f(t)dt≤F
ϕ(supx,y∈Xx–y)
f(t)dt,
ψ(supx,y∈Xx–y)
f(t)dt
,
thus we get
ϕ(μ(TX))
f(t)dt≤F
ϕ(μ(X))
f(t)dt,
ψ(μ(X))
f(t)dt
,
so according to Theorem . we get the result.
Corollary . Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T:C→C be continuous mapping such that
ϕ(μ(T(X)))
f(t)dt≤F
ϕ(μ(X))
f(t)dt,
ψ(μ(X))
f(t)dt
–ψ∗
ψ(μ(X))
f(t)dt
,
for any subset X of C and where f : [,∞)→[,∞)be a Lebesgue integrable function,
which is summable on each compact subset of[,∞)andεf(t)dt> for eachε> and
ψ,ψ∗∈,ϕ∈and F∈C.Then T has a fixed point.
Proof Define a sequence{Cn}∞n=setting
C:=C, Cn:=coT(Cn–),
wheren= , , . . . . Analogously to Theorem . it is possible to show that
Cn+⊆Cn, T(Cn)⊆Cn, ()
for everyn= , , . . . . From condition (A) of Definition . we conclude thatμ(Cn+)≤
μ(Cn) for everyn= , , . . . . This means that the sequence{μ(Cn)}∞n=is not increasing,
and consequently there existsr≥ such thatlimn→∞μ(Cn) =r.
Ifμ(CN) = for someN∈NthenCN is a compact set and using Schauder’s theorem we
conclude that the continuous mapT:CN→CN has a fixed point inCN⊂C.
Now suppose thatμ(CN) > for everyn∈N. Taking into consideration thatF∈C,ψ∗∈
ψ and condition (A), we have
ϕ(μ(Cn+))
f(t)dt=
ϕ(μ(co(TCn)))
f(t)dt
=
ϕ(μ(TCn))
≤F
ϕ(μ(Cn))
f(t)dt,
ψ(μ(Cn))
f(t)dt
–ψ∗
ψ(μ(Cn))
f(t)dt
≤
ϕ(μ(Cn))
f(t)dt–ψ∗
ψ(μ(Cn))
f(t)dt
. ()
Sincelimn→∞μ(Cn) =r,F∈C,ϕ∈,ψ∗∈, the functionf(·) is Lebesgue summable on
the compact subset of [,∞), and from () we obtain
ϕ(r)
f(t)dt≤
ϕ(r)
f(t)dt–ψ∗ ψ(r)
f(t)dt
,
and hence
ψ∗ ψ(r)
f(t)dt
≤.
From the last inequality it follows thatψ(r) = and consequentlyr= ,i.e.limn→∞μ(Cn) =
.
Continuing the proof analogously to the proof of the theorem . we obtain the result.
Theorem . Let C be a nonempty,bounded,closed,and convex subset of a Banach space E and let T,S:C→C be continuous mappings.Assume that:
(a). The range ofTcontains the range ofS. (b). For anyM⊆C:
ϕ(μ(T(M))
f(t)dt≤F
ϕ(μ(S(M)))
f(t)dt,
ψ(μ(S(M)))
f(t)dt
, ()
whereψ∈,ϕ∈and F∈C.
Then:
(i). The setsA={x∈C:S(x) =x}andB={x∈C:T(x) =x}are nonempty and closed. (ii). TandShave a coincidence point.
(iii). IfTandSare weakly compatible,thenSandT have a common fixed point.
Proof LetC=C,Cn:=coSCn–forn= , , . . . . As is pointed out in Theorem .,Cn+in
Esuch thatT(Cn+) =S(Cn). We have
T(Cn)⊆Cn and also S(Cn)⊂S(Cn–)⊂co
S(Cn–)
=Cn,
for alln∈N. Thus
ϕ(μ(Cn+))
f(t)dt=
ϕ(μ(co(S(Cn))))
f(t)dt
=
ϕ(μ(S(Cn)))
f(t)dt
=
ϕ(μ(T(Cn+)))
f(t)dt
≤F
ϕ(μ(S(Cn+)))
f(t)dt,
ψ(μ(S(Cn+)))
f(t)dt
≤F
ϕ(μ(Cn+))
f(t)dt,
ψ(μ(Cn+))
f(t)dt
≤
ϕ(μ(Cn+))
f(t)dt
≤
ϕ(μ(Cn))
f(t)dt. ()
The remaining part is similar to the previous proofs.
3 Application
In this section, since integral equations arise in different problems of theory and applica-tions (see,e.g., [–] and the references therein). We consider the existence of solutions for the following nonlinear integral equation inBC(R+), the space of bounded and
con-tinuous functionsx(·) :R+→R+:
x(t) =λf
t,
t
gt,s,xα(s)ds
+ ( –λ)
t
gt,s,x(s)ds, t≥,λ∈(, ) ()
for any nonempty bounded subsetXofBC(R+),x∈XandT> and> , let
ωT(x,) =supx(t) –x(s):s,t∈[,T],|t–s| ≤,
ωT(X,) =supωT(x,) :x∈X, ωT(X) =lim
→ω
T(X,),
ω(X) = lim
T→∞ω T
(X), X(t) =
x(t) :x∈X,
diamX(t) =supx(t) –y(t):x,y∈X,
and
μ(X) =ω(X) +t→∞limsup diamX(t). ()
Banaś has shown in [] that the functionμis a measure of noncompactness in the space
BC(R+).
Theorem . The nonlinear integral equation()has at least one solution in the space
BC(R+),if the following conditions are satisfied:
(A). The functionα:R+→R+is continuous,α(t)→ ∞ast→ ∞.
(A). The functionf :R+×R→Ris continuous and
f(t,x) –f(t,y)≤ψ|x–y|,
moreover,ψ andϕ are an altering distance function and an ultra altering distance function,respectively whichψ satisfies for allt,s∈R+,ψ(t) +ψ(s)≤ψ(s+t)and
ψ(t) <t.
(A).
(A). The functiong:R+×R+×R→Ris continuous and there exists a continuous function
b:R+×R+→R+which is increasing with respect to the first component,satisfying
g(t,s,x)≤b(t,s),
for allt,s∈R+andx∈Rwhere
lim
t→∞
t
b(t,s)ds= .
(A).
ft,Kx(t)=Kft,x(t),
where
Kx(t) =
t
gt,s,x(s)ds.
For the following remark, by definition, commuting mappings means for a pair of self-mappingsK,L:X→Xthat there exists a pointx∈Xsuch thatKL(x) =LK(x).
Remark . Note that by hypothesis (A) there exists constantV> such that
V=sup
t≥
v(t) =sup
t≥
t
b(t,s)ds
.
Proof Put
Q=x∈BCR+:x ≤r=L+V, (Fx)(t) =ft,x(t), (Kx)(t) =
t
gt,s,x(s)ds.
Thus equation () becomes
x(t) = (Hx)(t) =λFKxα(t)+ ( –λ)(Kx)(t)
we define the operationG:C(R+)→C(R+) by
G(x) =Hx– ( –λ)(Kx)(t)
λ =FKx
α(t),
whereC(R+) is the space of continuous functions onR+.
Khodabakhshi and Vaezpour in [] have shown thatG,Kare continuous onQ, bounded, commuting mappings,RG⊆RKand alsoG,Khave a common fixed point. The conditions
(a) and (b) of Theorem . hold. By referring to [] we get
On the other hand
g(t,s,x)≤b(t,s),
for allt,s∈R+. And forT> such thatt,t∈[,T] andx∈Xwe have
Kx(t) –Kx(t)= t
gt,s,x(s)
ds–
t
gt,s,x(s)
ds
≤ t
gt,s,x(s)
ds
≤ t
b(t,s)ds
≤
T
b(T,s)ds, ()
so
ωT(KX)≤
T
b(T,s)ds; ()
by takingT→ ∞
ω(KX)≤ lim
T→∞
T
b(T,s)ds= . ()
Also
Kx(t) –Ky(t)=
t
gt,s,x(s)ds–
t
gt,s,y(s)ds
≤ t
gt,s,x(s)ds
≤ t
b(t,s)ds
≤
T
b(T,s)ds, ()
so we have
lim sup
T→∞ diam
KX(T)≤lim sup
T→∞
T
b(T,s)ds= ; ()
by () and () we have
μ(KX) =μ
ω(KX) +lim sup
T→∞ diam(KX)
=μ() + = ,
therefore
thus for theC-class functionFand the ultra altering distance functionϕ, defined in Defi-nitions . and ., respectively, we can get
ψμG(X)≤ψμK(X)=Fψμ(KX),ϕμ(KX),
sinceK andGcommutes, so they are weakly compatible; therefore according to Theo-rem .,GandKhave a common fixed point soHhas a fixed point and thus the functional integral equation () has at least one solution.
4 Conclusion
In Theorem . of [] the conditionTS=ST for two self-mappingsT,Sis used. But in this article for achieving a common fixed point from two self-maps the hypothesis of the common range which is weaker than the hypotheses of a commuting map is utilized.
In this article the measure of noncompactness is used instead of the metricd, which is used in [] and []. Also, contractions associated with a measure of noncompactness in two linear and integral types and the application in solving integral equations are consid-ered, while in the two mentioned article just the linear contractions in terms of the metric
dis used for achieving a fixed point.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have read and approved the final manuscript.
Author details
1Young Researchers and Elite club, Shiraz Branch, Islamic Azad University, Shiraz, Iran.2Department of Mathematics and Computer Sciences, Amirkabir University of Technology, Tehran, Iran.3Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran.
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Received: 25 August 2017 Accepted: 13 October 2017
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