R E S E A R C H
Open Access
Solvability of impulsive partial neutral
second-order functional integro-differential
equations with infinite delay
Shengli Xie
**Correspondence: [email protected]; [email protected] Department of Mathematics and Physics, Anhui University of Architecture, Zipeng Road, Hefei, Anhui 230601, P.R. China
Abstract
Using the Kuratowski measure of noncompactness and progressive estimation method, we obtain the existence results of mild solutions for impulsive partial neutral second-order functional integro-differential equations with infinite delay in Banach spaces. The compactness condition of the impulsive term, some restrictive conditions ona prioriestimation and noncompactness measure estimation have been deleted. Our conditions are simple and our results essentially improve and extend some known results. As applications, some examples are provided to illustrate the obtained results.
MSC: 34K30; 34K40; 35R10; 47D09
Keywords: impulsive partial neutral functional integro-differential equations; mild solutions; fixed point; Banach spaces
1 Introduction
Consider the following impulsive partial neutral second-order functional integro-differ-ential systems with infinite delay in a Banach spaceX:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
d
dt[x(t) +g(t,xt,
t
k(t,s,xs)ds)] =Ax(t) +g(t,xt,
t
k(t,s,xs)ds),
t∈[,b],
x(ti) =Ii(xti), x(ti) =Ii(xti), i= , , . . . ,n,
x=ϕ∈ß, x() =z∈X,
()
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
d
dt[x(t) +g(t,xt,xt)] =Ax(t) +
t
f(s,xs,xs)ds, t∈[,b], x(ti) =Ii(xti,xti), x(ti) =Ii(xti,xti), i= , , . . . ,n,
x=ϕ∈ß, x=ψ∈ß,
()
whereAis the infinitesimal generator of a strongly continuous cosine function of bounded linear operators, (C(t))t∈R, onX. In both cases, the historyxt,xt: (–∞, ]→X,xt(θ) =
x(t+θ) andxt(θ) =x(t+θ) belongs to some abstract phase spaceßdefined axiomatically; g,f,gj, kj,Iij(j= , ) are appropriate functions; =t<t<· · ·<tn<tn+=b are fixed
numbers and the symbolx(ti) represents the jump of the functionxatti, which is defined
byx(ti) =x(t+i) –x(t–i),i= , , . . . ,n.
The study of impulsive functional differential equations is linked to their utility in sim-ulating processes and phenomena subject to short-time perturbations during their evolu-tion. The perturbations are performed discretely and their duration is negligible in com-parison with the total duration of the processes and phenomena. Now impulsive partial neutral functional differential equations have become an important object of investigation in recent years stimulated by their numerous applications to problems arising in mechan-ics, electrical engineering, medicine, biology, ecology,etc.With regard to this matter, we refer the reader to [–] and references therein. However, in order to obtain the existence of solutions in these study papers, the compactness condition on the associated family of operators and the impulsive term, some similar restrictive conditions ona priori estima-tion,
μ=Kb
n
i=
Nci+Nci < , Kb –μ
t
Nmg(s) +Nmf(s) ds<
∞
c
ds
W(s), ()
Kb
NLgb+
n
i=
NLi+NLi +Nlim inf ξ→+
Wf(ξ) ξ
t
mf(s)
ds< , ()
are used. In [–], authors used a strict set contraction mapping fixed point theorem without the compactness assumption on the associated family of operators to obtain the existence results of system () whengi(t,xt) is not an integral operator and the following
system: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
d
dt[x(t) +g(t,xt,x(t))] =Ax(t) +f(t,xt,x(t)), t∈[,b],t=ti, x(ti) =Ii(xti,x(ti)), x(ti) =Ii(xti,x(ti)), i= , , . . . ,n,
x=ϕ∈ß, x() =z,
()
improved and generalized some results in [, ]. However, the compactness condition of the impulsive termsIij(·), some similar restrictive conditions ona prioriestimation (), () and the restrictive condition on measure of noncompactness estimation
Kb
NLgb+
n
i=
NLi+NL i
+ b
η(s)ds< ()
are used in [–]. So far we have not seen the existence results of system ().
In this paper, using the Kuratowski measure of noncompactness and progressive es-timation method, we obtain the existence results of mild solutions of impulsive partial neutral second-order functional integro-differential systems () and (). The compactness condition of impulsive termsIij(·), some restrictive conditions ona prioriestimation and measure of noncompactness estimation (), () and () have been deleted. Our conditions are simple and our results essentially improve and extend some corresponding results in papers [, , , ]. As applications, some examples are provided to illustrate the obtained results.
2 Preliminaries
S(t) is the sine function associated with (C(t))t∈R, which is defined byS(t)x=tC(s)x ds,
x∈X,t∈R. We designate byN,Ncertain constants such thatC(t) ≤NandS(t) ≤ N for everyt∈J= [,b]. We refer the reader to [] for the necessary concepts about cosine functions. Next, we only mention a few results and notations needed to establish our results. As usual we denote by [D(A)] the domain ofAendowed with the graph norm
xA=x+Ax,x∈D(A). Moreover, the notationEstands for the space formed by the
vectorx∈X, for which the functionC(·)xis of classC. It was proved by Kisyński [] that
the spaceEendowed with the norm
xE=x+ sup ≤t≤b
AS(t)x,x∈E,
is a Banach space. The operator-valued functionG(t) =ASC(t(t))CS((tt))is a strongly continuous group of linear operators on the spaceE×Xgenerated by the operatorA=AIdefined onD(A)×E. It follows from this thatAS(t) :E→Xis a bounded linear operator and that AS(t)x→ (t→) for eachx∈E. Furthermore, ifx: [,∞)→Xis a locally integrable function, thenz(t) =tS(t–s)x(s)dsdefines anE-valued continuous function. This is a consequence of the fact that
t
G(t–s)
x(s)
ds=
t
S(t–s)x(s)ds
t
C(t–s)x(s)ds
defines an (E×X)-valued continuous function. Next, we denoteN=supt∈JAS(t)L(E,X),
in whichL(E,X) stands for the Banach space of bounded linear operators fromEintoX, and we abbreviate this notation toL(X) whenE=X.
To describe appropriately our system (), we say that the function u: [σ,τ]→X is a normalized piecewise continuous function on [σ,τ] ifuis piecewise continuous and left continuous on (σ,τ]. We denote byPC([σ,τ],X) the space formed by the normal-ized piecewise continuous functions from [σ,τ] intoX. In particular, we introduce the space PC formed by all functions u: [,b]→X such that u is continuous at t=ti,
u(ti–) =u(ti) andu(ti+) exists for alli= , , . . . ,n. It is clear thatPCendowed with the norm
xpc=supt∈Jx(t)is a Banach space.
Forx∈PC, let
xi(t) =
⎧ ⎨ ⎩
x(t), t∈(ti,ti+],
x(t+i), t=ti,i= , , . . . ,n.
Thenx∈C([ti,ti+],X). Moreover, forV⊆PCandi= , , . . . ,n, we use the notationVifor
Vi={xi:x∈V}. From Lemma . in [], we know that a setV⊆PCis relatively compact
if and only if each setVi={xi:x∈V}is relatively compact inC([ti,ti+],X) (i= , , . . . ,n).
For system (), we give the precise meaning of the derivative in (). We say thatx∈ PCis piecewise smooth ifxis continuously differentiable att=ti,i= , , . . . ,n, and for
t=ti, i= , , . . . ,n, there are the right derivativex(ti+) =lims→+x(ti+s)–sx(ti) and the left
derivativex(t–
i) =lims→–x(ti+ss)–x(ti). Furthermore, we denote the space byPC={x∈PC:
x(t) is continuous att=ti,x(ti–) andx(ti+) exist,i= , , . . . ,n}. ThenPC endowed with
In this work we employ an axiomatic definition of the phase spaceßintroduced by Hale and Kato [] which appropriated to treat retarded impulsive differential equations. For other abstract phase spaces, we can refer to [, ].
Definition .[] The phase spaceßis a linear space of functions mapping (–∞, ] into Xendowed with a seminorm · ß. We assume thatßsatisfies the following axioms.
(A) Ifx: (–∞,σ+b]→X(b> ) is such thatxσ∈ßandx|[σ,σ+b]∈PC([σ,σ+b],X),
then for everyt∈[σ,σ+b)the following conditions hold: (i) xtis inß,
(ii) x(t) ≤Hxtß,
(iii) xtß≤K(t–σ)sup{x(s):σ≤s≤t}+M(t–σ)xσß, whereH> is a
constant;K,M: [,∞)→[,∞),Kis continuous,Mis locally bounded and H,K,Mare independent ofx(·).
(B) The spaceßis complete.
In this paper we denote byα(·) the Kuratowski measure of noncompactness ofX, byαc(·)
the Kuratowski measure of noncompactness ofC([,b],X) and byαpc(·) the Kuratowski
measure of noncompactness ofPC. The following lemma is easy to get.
Lemma . If the cosine function family C(t),t∈R,is equicontinuous andη∈L([,b],R+),
then the set t
S(t–s)u(s)ds,u(s)≤η(s)for a.e. s∈[,b]
is equicontinuous for t∈[,b]. Lemma .[, ]
() IfW⊂PC([,b],X)is bounded,thenα(W(t))≤αpc(W)for anyt∈[,b],where
W(t) ={u(t) :u∈W} ⊆X.
() IfW is piecewise equicontinuous on[,b],thenα(W(t))is piecewise continuous for t∈[,b]andαpc(W) =sup{α(W(t)),t∈[,b]}.
() IfW⊂PC([,b],X)is bounded and piecewise equicontinuous,thenα(W(t))is piecewise continuous fort∈[,b]and
α
t
W(s)ds
≤
t
αW(s) ds, ∀t∈[,b],
wheretW(s)ds={tx(s)ds:x∈W}.
() IfW⊂PC([,b],X)is bounded and the elements ofWare equicontinuous on each
Ji= (ti,ti+](i= , , . . . ,n),then
αpc(W) =max
sup
t∈[,b]
αW(t) , sup
t∈[,b]
αW(t) ,
whereαpc(·)denotes the Kuratowski measure of noncompactness in the space
Lemma .[] Let h: [,b]→E be an integrable function such that h∈PC.Then the function v(t) =tC(t–s)h(s)ds belongs to PC,the function s→AS(t–s)h(s)is integrable
on[,t]for t∈[,b]and
v(t) =h(t) +A t
S(t–s)h(s)ds=h(t) + t
AS(t–s)h(s)ds, t∈[,b].
Lemma .[] Let V ={xn} ⊂L([a,b],X).If there isσ ∈L([a,b],R+) (R+= [, +∞))
such thatxn(t) ≤σ(t)for x∈V and a.e.t∈[a,b],thenα(V(t))∈L([a,b],R+)and
α
t
a
xn(s)ds:n∈N
≤ t
a
αV(s) ds, t∈[a,b].
Lemma .[] (Mónch) Let X be a Banach space,be a bounded open subset in X and∈.Assume that the operator F:→X is continuous and satisfies the following conditions:
() x=λFx,∀λ∈(, ),x∈∂,
() Dis relatively compact ifD⊂co({} ∪F(D))for any countable setD⊂. Then F has a fixed point in.
3 Main results
Firstly, we discuss the existence of mild solutions for the impulsive second-order sys-tem ().
Definition . A functionx: (–∞,b]→Xis said to be a mild solution of system () if x=ϕ,x(·)|J∈PCand
x(t) =C(t)ϕ() +S(t)z+g(,ϕ, ) –
t
C(t–s)g
s,xs,
s
k(s,r,xr)dr
ds
+ t
S(t–s)g
s,xs,
s
k(s,r,xr)dr
ds+
<ti<t
C(t–ti)Ii(xti)
+
<ti<t
S(t–ti)Ii(xti), t∈J. ()
For system (), we make the following hypotheses.
(H) The functionsgj:J×ß×X→X(j= , ) satisfy the following conditions:
() For every(φ,x)∈ß×X,gj(·,φ,x)are strongly measurable andgj(t,·,·)are
continuous for everyt∈J;
() There are integrable functionspj:J→R+(j= , ) such that
gj(t,φ,x)≤pj(t)
φß+x , t∈J,φ∈ß,x∈X;
() For any bounded setV⊂PC, there are integrable functionsγj:J→R+(j= , )
such that
αgj
t,Vt,V(t)
≤γj(t)
α(Vt) +α
(H)kj:×ß→X(j= , ,={(t,s)∈J×J: ≤s≤t≤}) satisfies the following
conditions:
() For everyφ∈ß,kj(·,·,φ)are strongly measurable andkj(t,s,·)are continuous for
every(t,s)∈;
() There are continuous functionsqj:→R+(j= , ) such that
kj(t,s,φ)≤qj(t,s)φß, (t,s)∈,φ∈ß;
() For any bounded setV⊂PC, there are continuous functionsμj:→R+(j= , )
such that
αkj(t,s,Vs) ≤μj(t,s)α(Vs), (t,s)∈.
(H) The functionsIij:ß→X(i= , . . . ,n,j= , ) are continuous and there are constants
cji≥,dij> such that
Iij(φ)≤cjiφß+dji, φ∈ß.
Let the functiony: (–∞,b]→Xbe defined byy=ϕand
y(t) =C(t)ϕ() +S(t)z+g(,ϕ, ) , t∈J.
Theorem . Suppose that the cosine function family C(t),t∈R,is equicontinuous,g,
gsatisfy the condition(H), (H)and(H)are satisfied.Then the impulsive second-order
system()has at least one mild solution.
Proof LetS(b) be the spaceS(b) ={x: (–∞,b]→X,x= ,x|J ∈PC}endowed with the
supremum norm · b. The mapF:S(b)→S(b) is defined by
(Fx)(t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
, t≤,
–tC(t–s)g(s,xs+ys,
s
k(s,r,xr+yr)dr)ds
+tS(t–s)g(s,xs+ys,
s
k(s,r,xr+yr)dr)ds
+<ti<tC(t–ti)Ii(xti+yti)
+<ti<tS(t–ti)Ii(xti+yti), t∈J.
()
Clearly, xt +ytß≤ xtß+supt∈Jytß≤Kbxt+M, whereKb=sup≤t≤bK(t), M=
supt∈Jytß,xt=sup≤s≤tx(s). ThusFis well defined with values inS(b). In addition,
from the axioms of phase space, the Lebesgue dominated convergence theorem and the conditions (H), (H) and (H), we can show thatFis continuous (see []). It is easy to see
that ifxis a fixed point ofF, thenx+yis a mild solution of system (). Firstly, we show that the set
=
x∈S(b) :x=λFxfor someλ∈(, )
Whent∈J= [,t], notice thatxtß≤Kbxt andxt is continuous nondecreasing
onJ. We have, by () and (H),
x(t)≤N t
p(s)
xs+ysß+
s
q(s,r)xr+yrßdr
ds
+N t
p(s)
xs+ysß+
s
q(s,r)xr+yrßdr
ds ≤NM t p(s) +
s
q(s,r)dr
ds
+NKb
t
p(s) +
s
q(s,r)dr
xsds
+NM t
p(s) +
s
q(s,r)dr
ds
+NKb
t
p(s) +
s
q(s,r)dr
xsds
≤(N+N)M t
p(s) +p(s) +
s
q(s,r)dr+
s
q(s,r)dr
ds
+ (N+N)Kb
t
p(s) +p(s) +
s
q(s,r)dr+
s
q(s,r)dr
xsds. ()
Consequently,
xt≤(N+N)M
t
p(s) +p(s) +
s
q(s,r)dr+
s
q(s,r)dr
ds
+ (N+N)Kb
t
p(s) +p(s) +
s
q(s,r)dr+
s
q(s,r)dr
xsds. ()
By well-known Gronwall’s lemma and (), there are constantsG> independent ofx
andλ∈(, ) such thatx(t) ≤Gandxtß≤KbG,t∈J. It follows from this and the
condition (H) that
Ij(xt+yt)≤cj(KbG+M) +dj=δj (j= , ),
xt+ =x(t) +I(xt+yt)≤G+δ.
Nextly, whent∈J= (t,t], let
u(t) = ⎧ ⎨ ⎩
x(t), t∈(t,t],
x(t+
), t=t.
Thenu∈C([t,t],X). Similar to (), we get
u(t)≤(N+N)M t
p(s) +p(s) +
s
q(s,r)dr+
s
q(s,r)dr
ds
+ (N+N)KbG
t
p(s) +p(s) +
s
q(s,r)dr+
s
q(s,r)dr
+ (N+N)Kb
t
t
p(s) +p(s) +
s
q(s,r)dr+
s
q(s,r)dr
usds
+NI(xt+yt)+NI(xt+yt), ()
wherext≤sup≤s≤tx(s)+supt≤s≤tu(s)=:xt+v(t). Equation () implies that
v(t)≤Nδ+Nδ+ (N+N)(M+KbG)
t
a(s)ds
+ (N+N)Kb
t
t
a(s)v(s)ds, t∈[t,t], ()
wherea(s) =p(s) +p(s) +
s
q(s,r)dr+
s
q(s,r)dr. Using Gronwall’s lemma once again
and (), there is a constantG> independent ofvandλ∈(, ) such thatv(t)≤G,
t∈[t,t]. Thencex(t) ≤Gandxtß≤Kb(G+G) fort∈J.
It is similar to the proof above, there is a constantGi> independent ofxandλ∈(, )
such thatx(t) ≤Gi,t∈Ji(i= , , . . . ,n). LetG=max{G,G, . . . ,Gn}, thenx(t) ≤G,
t∈J,i.e.,is bounded.
Lastly, we verify that all the conditions of Lemma . are satisfied. LetR>Gand
R=
x∈S(b) :xb<R
.
Then R is a bounded open set and ∈. SinceR>G, we know thatx=λFxfor any
x∈∂Randλ∈(, ).
Nextly, letV⊂Rbe a countable set andV⊂co({} ∪F(V)). Then
V(t)⊂co{} ∪F(V)(t), t∈[,b]. ()
It follows from (H)-(H) and Lemma . thatF(V) is equicontinuous on every interval
Ji= [ti,ti+] (i= , , . . . ,n), which together with () implies thatV is equicontinuous on
everyJi(i= , , . . . ,n).
Whent∈J= [,t], by the property of noncompactness measure, (H)(), (H)() and
Lemma ., we have
αV(t) ≤αF(V)(t)
≤N t
α
g
s,Vs+ys,
s
k(s,r,Vr+yr)dr
ds
+ N t
α
g
s,Vs+ys,
s
k(s,r,Vr+yr)dr
ds
≤(N+N) t
γ(s) +γ(s) α(Vs) +
s
μ(s,r) +μ(s,r) α(Vr)dr
ds
≤(N+N)Kb
t
k=
γk(s) +
s
i=
μk(s,r)dr
sup
≤τ≤s
whereα(Vt)≤sup≤s≤tα(V(s)). Letm(t) =sup≤s≤tα(V(s)),t∈J. Lemma . implies that
m∈C(J,R+) and
m(t)≤(N+N)Kb
t
γ(s) +γ(s) +
s
μ(s,r) +μ(s,r) dr
m(s)ds, t∈J.
From this and Gronwall’s lemma, we know thatm(t) = andα(V(t)) = ,t∈J. Therefore
V is a relative compact set inC(J,X). Since
≤α(Vt)≤αc(V) = sup ≤t≤tα
V(t) =
andIj(·) is continuous,α(Vt+yt)≤α(Vt) = ,α(Ij(Vt+yt)) = (j= , ).
Whent∈J= [t,t], similar to (), it is easy to get
αV(t) ≤(N+N)Kb
t
γ(s) sup
≤τ≤sα
V(τ) ds
+NαI(Vt+yt) +Nα
I(Vt+yt)
≤(N+N)Kb
t
t
γ(s) sup
t≤τ≤s
αV(τ) ds, t∈J, ()
where γ(s) =γ(s) +γ(s) +
s
(μ(s,r) +μ(s,r))dr. Let q(t) =supt≤s≤tα(V(s)), t∈J.
Equation () implies that
q(t)≤(N+N) t
t
γ(s)q(s)ds, t∈J.
Thereforeα(V(t)) = ,t∈JandVis a relative compact set inC(J,X).
Similarly, we can show thatV is a relative compact set inC(Ji,X) (i= , , . . . ,n), soVis
a relative compact set inS(b). Lemma . concludes thatFhas a fixed point inR. Letx
be a fixed point ofFonS(b). Thenz=x+yis a mild solution of system ().
Nextly, we discuss the existence of mild solutions for the impulsive system ().
Definition . A functionx: (–∞,b]→Xis said to be a mild solution of system () if x=φ,x=ψ,x(·)|J∈PCand
x(t) =C(t)φ() +S(t)ψ() +g(,φ,ψ) – t
C(t–s)gs,xs,xs ds
+ t
S(t–s) s
fr,xr,xr dr ds+
<ti<t
C(t–ti)Ii
xti,xti
+
<ti<t
S(t–ti)Ii
Differentiate () to get
x(t) =AS(t)φ() +C(t)ψ() +g(,φ,ψ) –gt,xt,xt
– t
AS(t–s)gs,xs,xs ds+
t
C(t–s) s
fr,xr,xr dr ds
+
<ti<t
AS(t–ti)Ii
xti,xti +
<ti<t
C(t–ti)Ii
xti,xti , t∈J. ()
Let functionsy,y: (–∞,b]→Xbe defined byy=ϕ,y=ψand
y(t) =C(t)ϕ() +S(t)ψ(), y(t) =AS(t)ϕ() +C(t)ψ(), t∈J. Clearly,
ytß≤Kbyb+Mbϕß=M, ytß≤Kbyb+Mbψß=M,
whereyb=sup≤t≤by(t),yb=sup≤t≤by(t).
LetS(b) be the spaceS(b) ={x: (–∞,b]→X:x
= ,x= ,x(·)|J∈PC}endowed
with the supremum norm · b.
We make the following hypotheses for convenience. (Hf)f :J×ß×ß→Xsatisfies the following conditions:
() For everyx∈S(b), the functiont→f(t,x
t,xt)is strongly measurable andf(t,·,·)is
continuous for everyt∈J;
() There is an integrable functionp:J→R+such that
f(t,u,v)≤p(t)uß+vß , t∈J,u,v∈ß;
() For any bounded setV⊂PC, there is an integrable functionμ:J→R+such that
αft,Vt,Vt
≤μ(t)α(Vt) +α
Vt , t∈J,
whereVt={xt:x∈V},Vt={xt:x∈V} ⊂ß(t∈J),V⊂PC.
(Hg)g:J×ß×ß→Esatisfies the following conditions:
() The functiong(·)is continuous, there are constantsc> ,d≥such thatcKb< and
g(t,u,v)E≤cuß+vß +d, t∈J,u,v∈ß;
() For every bounded setQ⊂S(b), the set of functions{(ω
x)i(t) :x∈Q}is uniformly
equicontinuous onJi= [ti,ti+]for everyi= , , . . . ,n, whereωx(t) =g(t,xt,xt);
() For any bounded setV⊂PC,α(g(t,V
t,Vt))≤c(α(Vt) +α(Vt)),t∈J.
(HI) The functionsIi:ß×ß→E,Ii:ß×ß→X(i= , . . . ,n) are continuous and there
are constantscji≥,dji≥ such that
Ii(u,v)E≤ciuß+vß +di, Ii(u,v)≤ci
uß+vß +di, u,v∈ß.
Theorem . Let the conditions(Hf), (Hg)and(HI)be satisfied,the cosine function family
Proof Let the functionz: (–∞, ]→Xbe defined byz=x,z(t) =x(t),t∈J, the map = (,) :S(b)×S(b)→S(b) be defined by
(x,z)(t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
, t≤,
S(t)g(,ϕ,ψ) –tC(t–s)g(s,xs+ys,zs+ys)ds
+tS(t–s)sf(r,xr,xr)dr ds
+<ti<tC(t–ti)Ii(xti+yti,zti+yti)
+<ti<tS(t–ti)Ii(xti+yti,zti+yti), t∈J,
()
and(x,z)(t) =(x,z)(t),
(x,z)(t) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
, t≤,
C(t)g(,ϕ,ψ) –g(t,xt+yt,zt+yt)
–tAS(t–s)g(s,xs+ys,zs+ys)ds
+tC(t–s)sf(r,xr,xr)dr ds
+<ti<tAS(t–ti)Ii(xti+yti,zti+yti)
+<ti<tC(t–ti)Ii(xti+yti,zti+yti), t∈J.
()
The product spaceS(b)×S(b) is endowed with the norm(x,z)b=xb+zb. Then, are well defined and with values inS(b). In addition, from the axioms of phase space,
the Lebesgue dominated convergence theorem and the conditions (Hf), (Hg) and (HI), we
can show that= (,) is continuous. It is easy to see that if (x,z) is a fixed point of,
thenx+yis a mild solution of system (). Firstly, we show that the set
=
(x,z)∈S(b)×S(b) : (x,z) =λ(x,z) for someλ∈(, )
is bounded. Ifx∈, there exists aλ∈(, ) such thatx=λ(x,z) andz=λ(x,z).
Whent∈J= [,t], it follows from (), () and (Hf)(), (Hg)(), (HI) that
x(t)≤(x,z)(t)≤Ng(,ϕ,ψ)E
+ (N+N) t
gs,xs+ys,zs+ys +
s
fr,xr+yr,zr+yr dr
ds
≤Ng(,ϕ,ψ)E+N t
cxsß+zsß+M+M +d
ds
+N t
s
p(r)xrß+zrß+M+M dr ds
≤Ng(,ϕ,ψ)E+M+M
t
Nc+N
s
p(r)dr
ds+Nbd
+ (Nc+N)Kb
t
+
s
z(t)≤(x,z)(t)≤Ng(,ϕ,ψ)E+c
xt+ytß+zt+ytß +d
+M+M
t
Nc+N
s
p(r)dr
ds+Nbd
+ (Nc+N)Kb
t + s
p(r)drxs+zs ds
≤Ng(,ϕ,ψ)E+M+M c+ t
Nc+N
s p(r)dr ds
+ (Nb+ )d+cKb
xt+zt
+ (Nc+N)Kb
t + s
p(r)drxs+zs ds. ()
Equations () and () imply that
xt+zt≤
–cKb
(N+N)g(,ϕ,ψ)E+ (Nb+Nb+ )d
+M+M c+ (N+N)ct+ (N+N)
t s p(r)dr ds
+(Nc+N+Nc+N)Kb –cKb
t + s
p(r)drxs+zs ds. ()
Sincext+zt∈C(J,X), by Gronwall’s lemma and (), there is a constantG> such
that xt +zt ≤G,t∈J. Thereforex(t)+z(t) ≤G,t∈Jandxtß≤KbG,
ztß≤KbG,t∈J. It follows from this and the condition (HI) that
Ijxt+yt,zt+yt E≤c j
KbG+M+M +dj=:ηj (j= , ),
xt+ =x(t) +I
xt+yt,zt+yt ≤G+η,
zt+ =z(t) +I
xt+yt,zt+yt ≤G+η.
Nextly, whent∈J= (t,t], let
u(t) = ⎧ ⎨ ⎩
x(t), t∈(t,t],
x(t+
), t=t,
v(t) = ⎧ ⎨ ⎩
z(t), t∈(t,t],
z(t+
), t=t.
Thenu,v∈C([t,t],X). Similar to () and (), we get
u(t)≤Ng(,ϕ,ψ)E+ (N+N) t c+ s
p(r)drxs+zs+M+M ds
+Nbd+NIxt+yt,zt+yt E+NI
xt+yt,zt+yt
≤Ng(,ϕ,ψ)E+ (N+N)KbG+M+M
t c+ s p(r)dr ds
+Nbd+KbG+M+M
t t c+ s p(r)dr ds
+Nη+Nη
+ (N+N)Kb
t t c+ s p(r)dr sup
t≤τ≤s
u(τ)+ sup
t≤τ≤s
v(t)≤Ng(,ϕ,ψ)E+cxt+ytß+zt+ytß +d+Nη+Nη
+Nbd+ (N+N)
t
c+
s
p(r)drKbxs+Kbzs+M+M ds
≤Ng(,ϕ,ψ)E+cKbG+M+M +Nη+Nη+ (Nb+ )d
+ (N+N)
KbG+M+M
t
c+
s
p(r)dr
ds
+ (N+N)Kb
t
t
c+
s
p(r)dr
sup
t≤τ≤s
u(τ)+ sup
t≤τ≤s
v(τ)ds
+cKb
sup
t≤s≤tu(s)+t≤s≤tsupv(s). ()
We have, by () and (),
sup
t≤s≤tu(s)+t≤s≤tsup v(s)
≤ e+e
–cKb
+(N+N+N)Kb –cKb
· t
t
c+
s
p(r)dr
sup
t≤τ≤s
u(τ)+ sup
t≤τ≤s
v(τ)ds, t∈[t,t], ()
where
e= (N+N)
g(,ϕ,ψ)E+η+η+
KbG+M+M
t
c+
s
p(r)dr
ds
,
e=
KbG+M+M c+ (N+N)
t
c+
s
p(r)dr
ds
+N(bd+η) +N(bd+η).
Using Gronwall’s lemma once again and (), there is a constantG> such thatu(t)+
v(t) ≤G,t∈[t,t], and sox(t)+z(t) ≤G,t∈J.
It is similar to the proof above, there are constantsGi> such thatx(t)+x(t) ≤Gi,
t∈Ji(i= , , . . . ,n). LetG=max{G,G, . . . ,Gn}, then(x,z)b≤Gandis bounded.
LetR>Gand
R=
(x,z)∈S(b)×S(b) :(x,z)b<R.
ThenRis a bounded open set and (, )∈. SinceR>G, we know that (x,z)=λ(x,z)
for any (x,z)∈∂Randλ∈(, ).
Suppose thatV⊂Ris a countable set andV⊂co({(, )} ∪(V)). Let
V=
x∈S(b) :∃z∈S(b), (x,z)∈V, V=
z∈S(b) :∃x∈S(b), (x,z)∈V. Then we have
V⊂V×V⊂co
{} ∪(V) ×co
{} ∪(V)
⊂co{} ∪(V×V) ×co
It follows from (), () and (Hg)() thatj((V)i×(V)i) (j= , ) are equicontinuous on
every intervalJi(i= , , . . . ,n), which together with () implies that (Vk)i(k= , ) are
equicontinuous on every intervalJi.
In the following, we verify that the setV,Vis relatively compact inPC. Without loss
of generality, we do not distinguishVk|JiandVi, whereVk|Ji(k= , ) is the restriction of
VkonJi= (ti,ti+].
Whent∈J= [,t], by the condition (Hf)(), (Hg)() and Lemma ., we have
αV(t) ≤α
(V×V)(t)
≤N t
αgs,Vs+ys,Vs+ys
ds
+ N t
α
s
fr,Vr+yr,Vr+yr dr
ds
≤(N+N) t
c+
s
μ(r)drα(Vs+ys) +α
Vs+ys
ds
≤(N+N)Kb
t
c+
s
μ(r)dr
×sup
≤τ≤s
αV(τ) + sup
≤τ≤s
αV(τ) ds, ()
αV(t) ≤α
(V×V)(t) ≤α
gt,Vt+yt,Vt+yt
+ (N+N)Kb
t
c+
s
μ(r)dr
×sup
≤τ≤s
αV(τ) + sup
≤τ≤s
αV(τ) ds
≤cKb
sup
≤s≤t
αV(s) + sup
≤s≤t αV(s)
+ (N+N)Kb
t
c+
s
μ(r)dr
×sup
≤τ≤s
αV(τ) + sup
≤τ≤s
αV(τ) ds. ()
Sincemj(t) =:sup≤s≤tα(Vj(s)) (j= , ) are continuous nondecreasing onJ, () and ()
imply that
m(t) +m(t)≤
(N+N+N)Kb
–cKb
t
c+
s
μ(r)drm(s) +m(s) ds. ()
By Gronwall’s lemma and (), we have α(Vk(t)) = (k= , ), t∈J. Lemma .
im-plies thatVk(k= , ) is relatively compact inC(J,X). Note thatα(Vjt+yt)≤α(Vjt)≤
Kbsup≤s≤tα(Vj(s)) = andIj(·,·) (j= , ) is continuous, we have
αIVt+yt,Vt+yt
=αIVt+yt,Vt+yt
Whent∈J= [t,t], it is similar to () and (), we get
αV(t) ≤(N+N)Kb
t
t
c+
s
μ(r)dr
× sup
t≤τ≤s
αV(τ) + sup
t≤τ≤s
αV(τ) ds, ()
αV(t) ≤(N+N)Kb
t
t
c+
s
μ(r)dr
× sup
t≤τ≤s
αV(τ) + sup
t≤τ≤s
αV(τ) ds
+cKb
sup
t≤s≤tα
V(s) + sup t≤s≤tα
V(s)
. ()
Equations () and () imply that
sup
t≤s≤tα
V(s) + sup t≤s≤tα
V(s)
≤(N+N+N)Kb
–cKb
t
t
c+
s
μ(r)dr
sup
t≤τ≤s
αV(τ) + sup
t≤τ≤s
αV(τ) ds.
Consequently,α(Vk(t)) = (k= , ),t∈J. Lemma . implies thatVk(k= , ) are
rela-tively compact inC(J,X).
Similarly, we can show thatVk(k= , ) are relatively compact inC(Ji,X) (i= , , . . . ,n).
SoVk(k= , ) are relatively compact inS(b). In view of Lemma ., we conclude that
has a fixed point inR. Let (x,z) be a fixed point ofonS(b). Thenx+yis a mild solution
of system ().
Theorem . Let the conditions(Hf), (Hg)()and(HI)be satisfied,the cosine function
family C(t),t∈R,be equicontinuous andϕ()∈E.Furthermore,suppose that the following condition is satisfied:
(Hg)()The function g(·)is continuous and g(t,·)satisfies the Lipschitz condition,that
is,there is a constant c> such that g(t,u,v) –g(t,u,v)E≤c
u–uß+v–vß , t∈J,uj,vj∈ß(j= , ),
and cKb< .Then system()has at least one mild solution.
Proof We have, by the condition (Hg)(),
g(t,u,v)E≤cuß+vß +g(t, , )E, t∈J,u,v∈ß(j= , ),
αgt,Vt,Vt
≤cα(Vt) +α
Vt , t∈J.
The rest of the proof is similar to the proof of Theorem ., we omit it.
4 Examples
LetX=L([,π]) and letAbe the operator given byAf =fwith the domain
D(A) =f ∈X:f,fare absolutely continuous,f∈X,f() =f(π) = .
It is well known thatAis the infinitesimal generator of a strongly continuous cosine family C(t),t∈R, onX. Moreover,Ahas discrete spectrum, the eigenvalues are –n,n∈N, with
corresponding normalized eigenvectorszn(ξ) :=
!
πsin(nξ), and the following properties hold:
(a) {zn:n∈N}is an orthonormal basis ofX.
(b) Forf∈X,C(t)f =n∞=cos(nt)(f;zn)zn. Moreover, it follows from this expression
thatS(t)f=∞n=sinn(nt)(f;zn)zn, thatS(t)is compact fort> and thatC(t)= and
S(t)= for everyt∈R. Additionally, we observe that the operatorsC(kπ), k∈N, are not compact.
(c) Ifdenotes the group of translations onXdefined by(t)x(ξ) =x(ξ+t), wherexis the extension ofxwith periodπ, thenC(t) =((t) +(–t));A=B, whereBis
the infinitesimal generator of the groupandE={x∈H(,π) :x() =x(π) = };
see [] for details. In particular, we observe that the inclusionic:E→Xis compact.
In the next application,ßshould be the phase spaceß=C×L(ρ,X) in [], whereρ:
(–∞, ]→Ris a positive Lebesgue integrable function. We can takeH= ,M(t) =γ(–t)
andK(t) = + (–tρ(θ)dθ) fort≥.
Example . Consider the partial neutral functional integro-differential system: ⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ ∂t(
∂
∂tu(t,ξ) +
t
–∞h(s–t)u(s,ξ)ds)
=∂ξ∂u(t,ξ) +
t
–∞F(t,s–t,ξ,u(s,ξ))ds,
u(t, ) =u(t,π) = , t∈[,b],
u(θ,ξ) =ϕ(θ,ξ), θ∈(–∞, ],ξ∈[,π], ∂
∂tu(,ξ) =z(ξ), ξ∈[,π],
u(ti) =
ti
–∞qi(s–ti)u(s,ξ)ds, u(ti) =
ti
–∞qi(s–ti)u(s,ξ)ds, i= , , . . . ,n,
()
whereh(·)∈L∞(R),ϕ∈C×L(ρ,X),z∈X, <t<· · ·<tn<band
(i) The functionF:R→Ris continuous and there is a continuous function
μ:R→Rsuch that|F(t,s,ξ,x)| ≤μ(t,s)|x|,(t,s,ξ,x)∈R.
(ii) The functionqi∈C(R+,R)andci = (
–∞qi(θ)ρ–(θ)dθ)
<∞,i= , , . . . ,n.
(iii) The functionqi∈C(R+,R)andc i = (
–∞q
i(θ)ρ–(θ)dθ)
<∞,i= , , . . . ,n.
(iv) The functionϕdefined byϕ(θ)(ξ) =:φ(θ,ξ)belongs toß.
Assuming that the conditions (i)-(iv) are satisfied, then system () can be modeled as the abstract impulsive Cauchy problem () by defining
g(t,ϕ,x)(ξ) =
–t
–∞h(s)
ϕ(s,ξ)ds+x(ξ),
g(t,ϕ)(ξ) =
–∞F
Ii(ϕ)(ξ) =
–∞
qi(s)ϕ(s,ξ)ds (j= , ),
Ii(ϕ)(ξ) =
–∞
qi(s)ϕ(s,ξ)ds (j= , ),
where
x(ξ) = t
h(s–t)ϕ(,ξ)ds, k(t,s,ϕ)(ξ) =h(s–t)ϕ(,ξ).
Moreover,g(t,·),Iji(·) (i= , , . . . ,n,j= , ) andk(t,s,·) are bounded linear operators,
Iij ≤cji(i= , , . . . ,n,j= , ), g(t,ϕ,x)≤
–t
–∞
a(s)ρ–(s)ds
ϕß+x,
g(t,ϕ)≤d(t)ϕß,
k(t,s,ϕ)≤a(·)∞ϕß, t∈[,b],
whered(t) = (–∞μ(t,s)ρ–(s)ds). If the cosine function familyC(t),t∈R, is
equicon-tinuous, all the conditions of Theorem . are satisfied (see [] for details), so system () has at least one mild solution. However, if we selectρ(s) =e–s,q(s) =sin (iii), we have
c = (
–∞sesds)
=√. But
+ –b ρ(t)dt bmax –t
–∞a
(s)ρ–(s)ds
, + n i= ci+ci
> ,
the restrictive conditions (), () and () do not hold. Thus, our results are different from the corresponding known results.
Example . Consider the partial neutral functional integro-differential system: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ ∂t(
∂
∂tu(t,ξ) +
t –∞
π
b(s–t,η,ξ)u(s,η)dηds)
= ∂
∂ξu(t,ξ) +
t
–∞μ(s,θ)u(s+θ,ξ)dθds,
u(t, ) =u(t,π) = , t∈J= [,b], u(θ,ξ) =ϕ(θ,ξ), θ∈(–∞, ],ξ∈[,π],
∂
∂tu(θ,ξ) =ψ(θ,ξ), θ∈(–∞, ],ξ∈[,π],
u(ti) =
–∞qi(θ)u(ti+θ,ξ)dθ, i= , , . . . ,n, u(ti) =
–∞qi(θ)u(ti+θ,ξ)dθ, i= , , . . . ,n,
()
whereϕ,ψ∈C×L(h,X),ϕ(,·)∈H([,π]), <t<· · ·<tn<band
(v) The functionsb(θ,η,ξ), ∂ξ∂b(θ,η,ξ)are measurable,b(θ,η, ) =b(θ,η,π) = and
c=max
π –∞ π
ρ(θ)
∂ib(θ,η,ξ) ∂ξi
dηdθdξ
:i= ,
<Kb–,
sup t∈J π t π ∂
∂sb(s–t,η,ξ)
dηds dξ<∞.
(vi) The functionμ:R→Ris continuous and
–∞μ(t,θ)ρ–(θ)dθ=d(t) <∞.
Assuming that the conditions (ii)-(vi) are satisfied, system () can be modeled as the abstract Cauchy problem () by defining
g(t,ϕ,ψ)(ξ) =
–∞
π
b(θ,η,ξ)ϕ(θ,η)dηdθ,
f(t,ϕ,ψ)(ξ) =
–∞
μ(t,θ)ψ(θ,ξ)dθ,
Ii(ϕ,ψ)(ξ) =
–∞
qi(θ)ψ(θ,ξ)dθ (j= , ),
Ii(ϕ,ψ)(ξ) =
–∞
qi(θ)ϕ(θ,ξ)dθ (j= , ),
whereϕ,ψ∈ß,Iij(·) (i= , , . . . ,n,j= , ),g(t,·) are bounded linear operators and f(t,ϕ,ψ)≤d(t)ϕß+ψß , d(t) =
–∞
μ(t,s)ρ–(s)ds
, t∈J.
Moreover, for every bounded setQ⊂S(b), it follows from () and the proof in [] that
the set of functions{(ωx)i(t) :x∈Q}is uniformly equicontinuous onJi= [ti,ti+] for every
i= , , . . . ,n. If the cosine function familyC(t),t∈R, is equicontinuous, all the conditions of Theorem . are satisfied, so system () has at least one mild solution.
Remark . From the results of this paper, we know that the compactness condition of the impulsive term, the restrictive conditions on a priori estimation and noncompactness measure estimation can be deleted for the existence results of abstract impulsive func-tional Volterra integro-differential equations Cauchy problems.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author is grateful to the referees for their careful reading and helpful suggestions that have led to the present improved version of the original paper. The work was supported by the Natural Science Foundation of Anhui Province (11040606M01), Anhui Educational Committee (KJ2011A061, KJ2011B052), China.
Received: 5 April 2013 Accepted: 5 August 2013 Published: 9 September 2013 References
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