R E S E A R C H
Open Access
A fixed point theorem for nonautonomous
type superposition operators and integrable
solutions of a general nonlinear functional
integral equation
Fuli Wang
*Dedicated to Professor Shih-sen Chang on his 80th birthday
*Correspondence:
School of Mathematics and Physics, Changzhou University, Gehu Road, Changzhou, 213164, China
Abstract
We first establish a new fixed point theorem for nonautonomous type superposition operators. After that, we prove the existence of integrable solutions for a general nonlinear functional integral equation in anL1space on an unbounded interval by using our theorem. Our main tool is the measure of weak noncompactness. MSC: Primary 47H30; 47H08
Keywords: superposition operators; fixed points; measure of weak noncompactness; nonlinear functional integral equations
1 Introduction
It is well known that the class of nonlinear operator equations of various types has many useful applications in describing numerous problems of the real world. A number of equa-tions which include several given operators have arisen in many branches of science such as the theory of optimal control, economics, biological, mathematical physics and engi-neering. The present paper is concerned with the solvability of the following quite general nonlinear functional integral equation:
x(t) =f
t,x(t), t
k(t,s)us,x(s)ds
, t∈R+:= [,∞), (.)
inL:=L(R+), the space of Lebesgue integrable functions onR+. Here,u:R+×R→R
andf :R+×R→Rare two given functions, whilekis a given real function defined on
R+×R+.
Among nonlinear operators, there is a distinguished class called superposition opera-tors. The solvability of Eq. (.) is closely related to the fixed points of the nonautonomous type superposition operator, which is asked to prove that there existsx∈Dsatisfying the following operator equation:
x=F(x,Ax) (.)
for two given operatorsF:X×Y→XandA:D⊂X→X, whereXandYare two Banach spaces. Our goal in this paper is to study under what conditions Eq. (.) is solvable in an Lspace. To this end, we establish a fixed point theorem for the solvability of Eq. (.) in
advance.
The organization of this paper is as follows. In Section , we gather some notions and preliminary facts, including the concepts and properties of the measure of weak noncom-pactness, which will be needed in our further considerations. In Section , we establish a new fixed point theorem for Eq. (.). In Section , we prove the existence of integrable solutions for Eq. (.) by virtue of the measure of weak noncompactness.
2 Preliminaries
Definition . Let I be an interval in R. A function f : I× R→R is said to be a Carathéodory function if:
(a) for each fixedx∈R, the functionf(·,x)is Lebesgue measurable inI;
(b) for almost everywhere (a.e., for short) fixedt∈I, the functionf(t,·) :R→Ris continuous.
Letm(I) be a set of all measurable functionsψ:I→R. Iff is a Carathéodory function, thenf defines a mappingNf:m(I)→m(I) by (Nfψ)(t) =f(t,ψ(t)). This mapping is called the superposition operator (or Nemytskii operator) associated tof. The theory concerning superposition operators is presented in [].
For a given measurable functionψ:I→R, the composite operatorNf◦ψ(·) :=f(·,ψ(·))
which mapsIintoRis said to be a nonautonomous type superposition operator. By gener-alizing this concept, the solvability of Eq. (.) may be thought the existence of fixed points of the nonautonomous type superposition operatorNF◦AonD.
The following theorem was proved by Krasnosel’skii [] (see also []) in the case when I is a bounded interval and has been extended to an unbounded interval by Appell and Zabrejko [].
Theorem .(see [, Theorem ., pp.]) Let I be a(bounded or unbounded)interval
inR.The superposition operatorNfmaps L(I)into L(I)if and only if there exist a function L
+(I)and a constant b> such that
f(t,x)≤a(t) +b|x|,
where L
+(I)denotes a positive cone of the space L(I).
In this case, the operator Nf is continuous and bounded in the sense that it maps bounded sets in bounded sets.
The following Scorza-Dragoni theorem explains the structure of the Carathéodory func-tions on a bounded interval.
Theorem .(see [, Theorem ]) Let I be a bounded interval ofR,and let f :I×R→R be a Carathéodory function. Then, for eachε> ,there exists a closed subset Dε of the
Next, we gather together some notations and preliminary facts of some weak topology feature which will be needed in our further considerations. LetB(X) be a collection of all nonempty bounded subsets of a Banach spaceX, and letW(X) be a subset ofB(X) consisting of all weakly compact subsets ofX. Also, letUrdenote a closed ball inXcentered in and with radiusr.
In what follows, we accept the following definition [].
Definition . LetXbe a Banach space; letM,M andMbe inB(X). A functionμ:
B(X)→R+is said to be a measure of weak noncompactness if it satisfies the following
conditions:
() the familyker(μ) :={M∈B(X) :μ(M) = }is nonempty andker(μ)is contained in the set of relatively weakly compact sets ofX;
() M⊂M⇒μ(M)≤μ(M);
() μ(conv(M)) =μ(M), whereconv(M)refers to the closed convex hull ofM; () μ(λM+ ( –λ)M)≤λμ(M) + ( –λ)μ(M)forλ∈[, ];
() if(Mn)∞n=is a decreasing sequence of nonempty, bounded and weakly closed subsets
ofXwithlimn→∞μ(Mn) = , thenM∞:=∞n=Mnis nonempty.
The familyker(μ) described in () is called the kernel of the measure of weak noncom-pactnessμ. Note that the intersection setM∞from () belongs toker(μ) sinceμ(M∞)≤
μ(Mn) for everyn∈Nandlimn→∞Mn= .
The first important example of a measure of weak noncompactness has been defined by De Blasi [] as follows:
ω(M) =inf r> :∃W∈W(X) such thatM⊂W+Ur.
The De Blasi measure of weak noncompactness has some interesting properties. It plays a significant role in nonlinear analysis and has many applications.
Nevertheless, it is rather difficult to express the De Blasi measure of weak noncompact-ness with the help of a convenient formula in a concrete Banach space. Such a formula is only known in the case of the space ofL(I), whereIis a bounded subinterval ofR. In [], Appell and De Pascale gave toωthe following simple form in spaces:
ω(M) =lim
ε→ψsup∈M
sup
D
ψ(t)dt:D⊂I,meas(D)≤ε
for all bounded subsetsMofL(I), wheremeas(·) denotes the Lebesgue measure.
For a nonempty and bounded subsetMof the spaceL(R+), Banas and Knap []
con-structed a useful measure of weak noncompactness as follows:
c(M) :=lim
ε→ψsup∈M
sup
D
ψ(t)dt:D⊂R+,meas(D)≤ε
,
d(M) := lim
T→∞ψsup∈M
∞
T
ψ(t)dt
.
Finally, let us put
Based on the following criterion for weak noncompactness due to Dieudonné [], it was shown that the functionμis a measure of weak noncompactness in the spaceL(R+).
Theorem . A bounded set N is relatively weakly compact in L(R+)if and only if the
following two conditions are satisfied:
() for anyε> ,there existsδ> such that ifmeas(D)≤δthenD|ψ(t)|dt≤εfor all ψ∈N,
() for anyε> ,there existsT> such thatT∞|ψ(t)|dt≤εfor allψ∈N.
The nonlinear contractive property of the operators plays some important roles in our subsequent considerations.
Definition . LetDbe a subset of the Banach spaceX. An operatorT:D→Xis said to be nonlinear contractive (or aϕ-nonlinear contraction) if there exists a continuous and nondecreasing functionϕ:R+→R+withϕ(r) <rforr> such that
Tx–Tx ≤ϕ
x–x
for allx,x∈D.
Remark . If we takeϕ(r) =λrwith ≤λ< , then such aϕ-nonlinear contraction is
also said to beλ-contraction.
Lemma . Let X and Y be two Banach spaces,and letDbe a subset of Y.If F:X×Y→X is continuous,and for any y∈Dthe operator F(·,y)is aϕ-nonlinear contraction,then there exists a continuous map J:D→X such that Jy=F(Jy,y)for any y∈D.
Proof For arbitrary fixedy∈D, the mappingF(·,y) defined byx→F(x,y) is aϕ-nonlinear contraction and mapsXintoX, so it has a unique fixed point by [, Theorem ]. Let us denote byJ:D→Xthe map which assigns to eachy∈Dthe unique point inXsuch that Jy=F(Jy,y). Thus, the mapJis well defined.
For anyy∈Dand a sequence (yn)n∈N⊂Dwhich converges toy, we have
Jyn–Jy=F(Jyn,yn) –F(Jy,y)
≤F(Jyn,yn) –F(Jy,yn)+F(Jy,yn) –F(Jy,y)
≤ϕJyn–Jy
+F(Jy,yn) –F(Jy,y),
which implies
Jyn–Jy–ϕ
Jyn–Jy
≤F(Jy,yn) –F(Jy,y).
Letrn:=Jyn–Jy. Since the operatorFis continuous, we obtainrn–ϕ(rn)→ asn→ ∞.
The properties of the functionϕshow thatrn→, that is,Jyn→Jy. The continuity ofJ
3 Fixed point theorem of nonautonomous type superposition operators
Theorem . Let X and Y be two Banach spaces,and letDbe a nonempty subset of X. Suppose that the operators A:D→Y and F:X×Y→X satisfy the following:
() AandFis continuous,
() for anyy∈A(D),the operatorF(·,y)is aϕ-nonlinear contraction, () there exists a nonempty,compact and convex subsetPofDsuch that
x=F(x,Az) ⇒ x∈P for allz∈P.
Then there is a point x inDsuch that x=F(x,Ax).
Proof Let us denote byJ:A(D)→Xthe map which assigns to eachy∈A(D) the unique point inXsuch thatJy=F(Jy,y). From Lemma ., the mapJis well defined and continuous onA(D).
By assumption (), for anyz∈P, we infer that there isx= (J◦A)z∈P such thatx= F(x,Az). This shows that (J◦A)(P)⊂P. SinceAandJare all continuous, the composite operatorJ◦Ais continuous onP. Now applying the Schauder fixed point theorem, we conclude thatJ◦Ahas at least one fixed pointx∈P ⊂Dsuch that (J◦A)x=x, which implies that
F(x,Ax) =F(J◦A)x,Ax= (J◦A)x=x.
This completes the proof.
Remark . There are some fixed point theorems, which involve several operators such as the operatorsTx:=Ax+Bxin a Banach space, orTx:=AxBx+Cxin Banach algebras etc., and they may be formulated by Theorem . in a coincident form.
For example, letF(x,Ax) :=Ax+BxandDbe a nonempty, convex and closed set of a Banach spaceXin Theorem ., whereA:D→Xis compact and continuous,B:D→X is a contraction mapping andAx+By∈Dfor allx,y∈D. Then we immediately obtain the celebrated Krasnosel’skii fixed point theorem (see [, Theorem .., pp.]), which implies that Theorem . is a generalization of the Krasnosel’skii fixed point theorem. In fact, if we takeP=conv((I–B)–A(M)), thenPsatisfies the assumption () of Theorem .
(see the proof of [, Lemma .., pp.]).
As another example, letF(x,Ax) :=AxBx+CxandDbe a closed convex and bounded subset of the Banach algebraXin Theorem ., whereB:D→Xis completely continuous, andA,C:X→Xsatisfy
Ax–Ay ≤φA
x–y, Cx–Cy ≤φC
x–y, ∀x,y∈X,
whereφA,φC:R+→R+are two continuous nondecreasing functions satisfyingφA() = φC() = and
MφA(r) +φC(r) <r, ∀r>
It is obvious thatF(·,y) is aϕ-nonlinear contraction for anyy∈A(D) withϕ(r) =MφA(r) + φC(r), and if we takeP=conv((I–AC)–B(D)), it is also easily proved thatP satisfies the
assumption () of Theorem .. Thus, we obtain the fixed point theorem for the operator AB+C in Banach algebras, which implies that Theorem . is a generalization of [, Theorem .].
4 The solvability of general nonlinear integral equations inL1space
In this section, we study the existence of integrable solutions for Eq. (.). A number of functional integral equations, such as the following:
x(t) =f
t,
t
k(t,s)f
s,x(s)ds
, t∈R; (.)
x(t) =gt,x(t)+f
t,
t
k(t,s)f
s,x(s)ds
, t∈R; (.)
x(t) =f
t,x(t)+f
t,x(t)
t
k(t,s)us,x(s)ds, t∈R, (.)
may all be illustrated as special examples of Eq. (.).
Solutions to Eq. (.) will be sought inL:=L(R+), the space of Lebesgue integrable
functions onR+with values inR, endowed with the standard normx:=∞
|x(t)|dt.
Here are some hypotheses on the nonlinear functions involved in Eq. (.).
Assumption . Assume that
(H) u:R+×R→Ris a Carathéodory function, and there exist a functiona∈L +and a
constantb> such that|u(t,x)| ≤a(t) +b|x|;
(H) k:R+×R+→Ris a Carathéodory function, andess sup
s∈R+
∞
s |k(t,s)|dt<∞; (H) f :R+×R→Ris a Carathéodory function; there exist two positive numbersα,β
and a functiong∈L+such that|f(t,x(t),y(t))| ≤g(t) +α|x(t)|+β|y(t)|for a.e.t∈R+; (H) α+bβK+g ≤ifg= , otherwiseα+bβK< , whereKdenotes the norm
of the linear Volterra integral operatorKgenerated by the functionk; (H) for an arbitrary fixedy(t) =tk(t,s)u(s,z(s))dswithz∈Ur, wherersatisfies
r≥g
+βka
–α–bβK,
there exists a continuous and nondecreasing functionϕ:R+→R+withϕ(r) <rfor
r> such that
∞
ft,x(t),y(t)
–ft,x(t),y(t)
dt≤ϕx–x
, ∀x,x∈L.
Remark . First notice that Eq. (.) may be written in an abstract form by Eq. (.), whereFis the superposition operator associated to the functionf (F:=Nf, the superpo-sition operator of double variables type was proposed by []):
F:L×L→L,
andA:=K◦Nuappears as the composition of the superposition operator associated tou with the linear Volterra integral operator defined by
K:L×L→L,
ψ→Kψ:R+→R; Kψ(t) = t
k(t,s)ψ(s)ds.
Our aim is now to prove that the nonautonomous type superposition operatorNF◦A has a fixed point inL(R+). Before starting to study this problem, we give some remarks
to illustrate that the operatorsAandFare well defined as follows.
() It should be noted that assumption (H) leads to the estimate
tk(t,s)ψ(s)ds= ∞
s
k(t,s)dt ∞
ψ(s)ds
≤
ess sup
s∈R+
∞
s
k(t,s)dt
ψ, ψ∈L,
which shows that the linear Volterra integral operatorKis continuous from anL
space into itself, andK ≤ess sups∈R+
∞
s |k(t,s)|dt.
() Assumption (H) shows that the superposition operatorNuis continuous and maps bounded sets ofLinto bounded sets ofLby Theorem ..
() Note thatα|x|+β|y|being an equivalent norm of(x,y)inR, according to the
Lucchetti-Patrone theorem (see [] or [, Theorem ]), assumption (H) shows that the superposition operatorNf is continuous and maps bounded sets ofL×L
into bounded sets ofL.
Theorem . If Assumption.is verified,then the equation
x(t) =f
t,x(t), t
k(t,s)us,x(s)ds
, t∈R,
that is,Eq. (.)has at least a solution x∈L.
Proof It is clear that the solutions of the operator equationx=F(x,Ax) satisfy Eq. (.). We will use Theorem . to prove the present theorem, thus the assumptions of Theorem . have to be checked. Our proving is divided into several steps.
() By Remark ., the operators A:L→L,F:L×L→L are well defined and
continuous, and then the assumption () of Theorem . is fulfilled.
() By (H), for arbitrary fixedy=Azwithz∈Ur, there exists a continuous and non-decreasing functionϕ:R+→R+such that
F(x,y) –F(x,y)=
∞
ft,x(t),y(t)
–ft,x(t),y(t)dt
≤ϕx–x
() If there isx∈Lsuch thatx(t) =f(t,x(t),Az(t)) forz∈Ur, then by (H) we have
ft,x(t),Az(t)≤g(t) +αx(t)+βAz(t)=g(t) +αx(t)+β(K◦Nu◦z)(t).
It follows that
x= ∞
ft,x(t),Az(t)dt≤ g+αx+βKa+bz,
that is,
x ≤( –α)–g+βKa+bz ≤( –α)–g+βKa+bβKr
≤r
sinceg+βKa ≤r( –α–bβK) by (H). This shows that the nonautonomous
type superposition operatorNF◦AmapsUrinto itself. () LetP:=Ur, and let
Pn:=conv
x∈L:x(t) =f
t,x(t), t
k(t,s)us,z(s)ds
,z∈Pn–
, n∈N.
ThenPn(n= , , , . . .) are all nonempty closed convex, and then they are weakly closed. Moreover, we haveP⊂Ur=Pfrom step (), and by the induction we may infer that
Pn⊂Pn–for alln∈N.
On the other hand, for eachε> and a nonempty measurable subsetDofR+such that
meas(D)≤ε, we know that if there existz∈Pn–andx∈Pnsuch that
x(t) =f
t,x(t), t
k(t,s)us,z(s)ds
,
then
D
x(t)dt= D f t,x(t), t
k(t,s)us,z(s)dsdt
≤
D
g(t)dt+α
D
x(t)dt+βK
D
a(t)dt+b
D
z(t)dt,
which implies that
D
x(t)dt≤( –α)–
D
g(t)dt+βK
D
a(t)dt+bβK
D
z(t)dt
.
Taking into account the fact that the set consisting of one element is weakly compact, the use of Theorem . leads to
lim
ε→sup
D
g(t)dt:D⊂R+,meas(D)≤ε
= ,
lim
ε→sup
D
a(t)dt:D⊂R+,meas(D)≤ε
As a result,
c(Pn) = lim
ε→xsup∈P
n
sup
D
x(t)dt:D⊂R+,meas(D)≤ε
≤( –α)–bβKlim
ε→z∈Psup
n–
sup
D
z(t)dt:D⊂R+,meas(D)≤ε
=λc(Pn–), (.)
whereλ:= ( –α)–bβK< by (H).
In the sequel let us fix arbitrarily the numberT> . Then, forz∈Pn–andx∈Pnwith
x(t) =f
t,x(t), t
k(t,s)us,z(s)ds
,
we have ∞
T
x(t)dt= ∞ T f t,x(t), t
k(t,s)us,z(s)dsdt
≤ ∞
T
g(t)dt+α
∞
T
x(t)dt+βK
∞
T
a(t)dt+b ∞
T
z(t)dt
,
which implies that
∞
T
x(t)dt≤( –α)–
∞
T
g(t)dt+βK
∞
T
a(t)dt+bβK
∞
T
z(t)dt
,
and the use of Theorem . leads to
lim
T→∞
∞
T
g(t)dt= , and lim
T→∞
∞
T
a(t)dt= .
As a result,
d(Pn) = lim
ε→xsup∈P
n
∞
T
x(t)dt≤ε
≤λlim
ε→z∈Psup
n–
∞
T
z(t)dt≤ε
=λd(Pn–). (.)
Thus, combining estimates (.) and (.), we obtain that
μ(Pn)≤λμ(Pn–).
Further, fromμ(Pn)≤λμ(Pn–)≤ · · · ≤λnμ(P) forn∈N, we obtain that
lim
n→∞μ(Pn) = .
() In this final step, let us prove that P is compact. To this end, for any sequence (xn)n∈N⊂Pwith
xn(t) =f
t,xn(t),
t
k(t,s)us,zn(s)
ds
, (zn)n∈N⊂P, (.)
we shall show that (xn)n∈Npossesses a convergent subsequence inL.
SincePis weakly compact, for an arbitrary fixedε> , by Theorem . there existsT> such that
∞
T
xm(t) –xn(t)dt≤ ε
(.)
for allm,n∈N.
Moreover, for the sequence (zn)n∈Nin (.), let
yn(t) :=
t
k(t,s)us,zn(s)
ds, n∈N.
According to Theorem ., there exists a closed subsetDεof [,T] such that the functions
kanduare continuous onDε×[,T] andDε×R, respectively, wheremeas([,T]\Dε)≤ε.
By takingt,t∈Dεwitht<t, we have
yn(t) –yn(t)=
tk(t,s)u
s,zn(s)
ds–
t
k(t,s)u
s,zn(s)
ds
≤ t
k(t,s) –k(t,s)u
s,zn(s)ds+
t
t
k(t,s)u
s,zn(s)
ds
≤ωTk,|t–t|
T
a(s) +bzn(s)ds+k
t
t
a(s) +bzn(s)ds
≤ωTk,|t–t|
a+br+k t
t
a(s)ds+bk t
t
zn(s)ds, (.)
wherek:=max{|k(t,s)|: (t,s)∈Dε×[,T]}, and ωT(k,|t–t|) denotes the modulus of
continuity of the functionkon the setDε×[,T].
Since (zn)n∈N⊂P is relatively weakly compact and the set consisting of one element is
also weakly compact, by Theorem . we infer that the termstt|zn(s)|dsand
t t a(s)ds
in (.) may all be arbitrarily small provided that the number |t–t|is small enough.
Thus, we obtain that the sequence (yn(t))n∈Nis equicontinuous onC(Dε) (the space of all
continuous functions defined onDε).
On the other hand, we have
yn(t)= t
k(t,s)us,zn(s)
ds≤
t
k(t,s)a(s) +bzn(s)ds
≤ka+bzn
≤ka+br:=Y,
which implies the sequence (yn(t))n∈Nis uniformly bounded onC(Dε).
infer that the sequence (xn(t))n∈N withxn(t) =f(t,xn(t),yn(t)) is uniformly bounded and
equicontinuous onC(Dε). Hence, by applying the Arzéla-Ascoli theorem, we obtain that
the set{xn:n∈N}forms a relatively compact set inC(Dε).
Note that our reasoning does not depend on the choice ofε. Thus we can construct a sequence (D/k)k∈Nof closed subsets of the interval [,T] such thatmeas([,T]\D/k)→
ask→ ∞, and the sequence (xn)n∈Nis relatively compact in every spaceC(D/k). Passing
to subsequences if necessary, we can assume that (xn)n∈N is a Cauchy sequence in each
spaceC(D/k) fork∈N.
In what follows, by virtue of the fact thatP is weakly compact, let us choose a number
δ> such that for each closed subsetDδ of the interval [,T] withmeas([,T]\Dδ)≤δ
satisfies
[,T]\Dδ
xm(t) –xn(t)dt≤ ε
for allm,n∈N. (.)
Since (xn)n∈N is a Cauchy sequence in each space C(D/k), there is k ∈N such that
meas([,T]\D/k)≤δand form,n≥k
xm–xnC(D/k)≤
ε
( +meas(D/k))
. (.)
Consequently, (.) and (.) imply that form,n≥kwe have
T
xm(t) –xn(t)dt
=
D/k
xm(t) –xn(t)dt+
[,T]\D/k
xm(t) –xn(t)dt< ε
. (.)
Now, combining (.) and (.) form,n≥k, we obtain that
xm–xn=
T
xm(t) –xn(t)dt+
∞
T
xm(t) –xn(t)dt<ε,
which shows that (xn)n∈N is a Cauchy sequence in an L space. Thus, the sequence
(xn)n∈N⊂Phas a convergent subsequence, which implies that the closed setPis compact.
This shows that the assumptions of Theorem . are all fulfilled, which completes the
proof.
Remark . The techniques of the proof of Theorem . based on Carathéodory condi-tions and the Scorza-Dragoni theorem were already used in [–] for proving the solv-ability of Eq. (.), (.),etc.
Finally, we provide an example, which is not included in Eq. (.)-(.), and which may be treated by our Theorem ..
Example . Consider the following nonlinear integral equation:
ψ(t) = t et –
arctanψ(t) +t +
sint +|ψ(t)|sin
t
( +t)–+ψ(s)
et+s ds
fort∈R+. In order to show that such an equation admits a solution inL, we are going to
check that the conditions of Theorem . are satisfied. Define the functions as follows:
k:R+×R+→R, k(t,s) =e–(t+s);
u:R+×R→R, u(t,x) =( +t)–+x;
f :R+×R→R, f(t,x,y) =te–t–arctanx +t +
sintsiny +|x| .
It obvious thatu,kandf are all Carathéodory functions. Takinga(t) = ( +t)–andb= ,
we have
u(t,x)=
( +t)–+x≤( +t)–+|x|=a(t) +b|x|.
So,usatisfies (H). Takingg(t) =te–t,α= / andβ= /, we have
f(t,x,y)=te–t–arctanx +t +
sintsiny +|x|
≤te–t+ |x|+
|y|=g(t) +α|x|+β|y|.
It follows thatf satisfies (H). By a simple calculation, we obtain that
g= ∞
te–tdt=
, K ≤sups≥
∞
s
k(t,s)dt=sup
s≥
∞
s
e–(t+s)dt= .
It follows that
α+bβK+g ≤
+ +
= ,
which shows that (H) and (H) are satisfied.
From the inequality
ft,x(t),y(t)
–ft,x(t),y(t)
≤
+tarctanx(t) –arctanx(t)+
sin +tsin|x(t)|y(t)–sintsiny(t) +|x(t)|
,
≤
+tx(t) –x(t)+
|x(t) –x(t)|
+|x(t)|+|x(t)|+|x(t)x(t)|
≤
x(t) –x(t), it follows that
∞
ft,x(t),y(t)
–ft,x(t),y(t)dt≤
x–x, ∀x,x∈L
for ally∈L. So (H) is satisfied forϕ(r) :=r.
Since the assumptions (H)-(H) are all satisfied, we apply Theorem . to derive the
Competing interests
The author declares that they have no competing interests.
Author’s contributions
This paper is completed by the sole author.
Acknowledgements
The author is grateful to the referees for the careful reading of the manuscript. The remarks motivated the author to make some valuable improvements.
Received: 24 April 2014 Accepted: 25 November 2014 Published:08 Dec 2014
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10.1186/1029-242X-2014-487
