R E S E A R C H
Open Access
Existence of solutions of a second-order
impulsive differential equation
Zhiguo Luo
1, Jingli Xie
2*and Guoping Chen
2*Correspondence:
[email protected] 2College of Mathematics and
Statistics, Jishou University, Jishou, Hunan 416000, P.R. China Full list of author information is available at the end of the article
Abstract
This paper is concerned with the existence of solutions of a second-order impulsive differential equation with mixed boundary condition. We obtain sufficient conditions for the existence of a unique solution, at least one solution, at least two solutions and infinitely many solutions, respectively, by using critical point theorems. The main results are also demonstrated with examples.
MSC: 34B15; 34B18; 34B37; 58E30
Keywords: impulsive differential equation; critical point theory; existence of solutions
1 Introduction
Nowadays, with the rapid development of science and technology, many people have realized that the theory of impulsive differential equations is not only richer than the corre-sponding theory of differential equations but it also represents a more natural framework for mathematical modeling of real world phenomena. Hence, it has become an effective tool to study some problems of biology, medicine, physics and so on [, ]. Significant progress has been made in the theory of systems of impulsive differential equations in recent twenty years (see [–] and the references cited therein). We generally consider impulses in the positionuandufor the second-order differential equationu=f(t,u,u). However, it is well known that in the motion of spacecraft instantaneous impulses depend on the position, which results in jump discontinuities in velocity, with no change in the position. This motivates us to consider the following second-order impulsive differential equation:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–u(t) +g(t)u(t) =f(t,u(t)), t=tj,t∈J= [,T],T> ,
–u(tj) =Ij(u(tj)), j= , , . . . ,m,
u() = , αu(T) +βu(T) = ,
(.)
wheref∈C(J×R,R),g∈L∞[,T],g(t) > ,Ij∈C(R,R), =t<t<t<· · ·<tm<tm+= T,α,βare constants withα≥,β> , and the operatoris defined asu(tj) =u(tj+) –
u(t–
j), whereu(tj+)(u(tj–)) denotes the right-hand (left-hand) limit ofuattj.
In recent years, some classical tools such as some fixed point theorems in cones, topo-logical degree theory and the upper and lower solutions method combined with the mono-tone iterative technique [–] have been widely used to get solutions of impulsive differ-ential equations. On the other hand, in the last few years, some researchers have studied
the existence of solutions for impulsive differential equations with boundary conditions via variational methods [–]. In this paper, we consider (.) by using critical point theory and variational methods.
The rest of this paper is organized as follows. In Section we present several important lemmas. In Section , we present existence results of equation (.) by using critical point theory and variational methods.
2 Preliminaries
In the following, we first introduce some notations and some necessary definitions.
Definition .[] LetXbe a real reflexive Banach space. For any sequence{uk} ⊂X, if
{ϕ(uk)}is bounded andϕ(uk)→ ask→ ∞possesses a convergent subsequence, then
we say thatϕsatisfies the Palais-Smale condition (PS condition).
Definition .[] Letϕ:X→Rbe differentiable andc∈R. We say thatϕsatisfies the (PS)ccondition if the existence of a sequence{uk}inX, such thatϕ(uk)→c,ϕ(uk)→
ask→ ∞, implies thatcis a critical value ofϕ.
It is clear that the PS condition implies the (PS)ccondition for eachc∈R.
Lemma .[] Let H be a Hilbert space and a:H×H→R be a bounded bilinear form. If a is coercive,i.e.,there existsα> such that a(u;u)≥α u for every u∈H,then for any
σ∈H(the conjugate space of H),there exists a unique u∈H such that
a(u,v) = (σ,v) for every v∈H.
Moreover,if a is also symmetric,then the functionalϕ:H→R defined byϕ=a(v,v) – (σ,v)attains its minimum at u.
Lemma .[] Ifϕis weakly lower semi-continuous on a reflexive Banach space X and has a bounded minimizing sequence, thenϕ has a minimum on X. The existence of a bounded minimizing sequence will be in particular insured whenϕis coercive,i.e.,such thatϕ(u)→+∞if u → ∞.
Lemma .[] For the functional F:M⊆X→R with M not empty,minu∈MF(u) =a has a solution in case the following hold:
(i) Xis a real reflexive Banach space;
(ii) Mis bounded and weakly sequentially closed;
(iii) Fis weakly sequentially lower semi-continuous onM,i.e.,by definition,for each sequence{uk}inMsuch thatukuask→ ∞,we haveF(u)≤limk→∞F(uk).
Lemma .[] Let X be a Banach space andϕ∈C(X,R).Assume that there exist u∈ X,u∈X and a bounded open neighborhoodof usuch that u∈X\and
inf
∂ϕ>max
ϕ(u),ϕ(u).
Let
and
c=inf
h∈smax∈[,]ϕ
h(s).
Ifϕsatisfies the(PS)c,then c is a critical value ofϕand c>max{ϕ(u),ϕ(u)}.
Lemma .[] Let X be an infinite dimensional Banach space and letϕ∈C(X,R)be even,satisfying the(PS),andϕ() = .If X=V⊕W,where V is finite dimensional,andϕ
satisfies the following conditions:
(i) There exist constantsρ,σ> such thatϕ|∂Bρ∩W≥σ;
(ii) For each finite dimensional subspaceV⊂X,there is anR=R(V)such that ϕ(u)≤for everyu∈Vwith u >R;
thenϕpossesses an unbounded sequence of critical values.
Let
H[,T] =u∈L[,T] :u∈L[,T]
and
H[,T] =u∈L[,T] :u,u∈L[,T] .
TakeH={u∈H([,T]) :u() = }. ThenHis a Hilbert space, and the inner product
(u,v) = T
u(t)v(t)dt+ T
g(t)u(t)v(t)dt
induces the norm
u =
T
u(t) dt+ T
g(t)u(t)dt
.
For u∈H([,T]), we have that u andu are both absolutely continuous andu∈ L([,T]), henceu(t
j) =u(tj+) –u(t–j) = for anyt∈J. Ifu∈H, thenuis absolutely
continuous andu∈L([,T]). In this case, the one-side derivativesu(t+
j) andu(tj–) may
not exist. So, by a classical solution of (.), we mean a functionu∈C([,T]) satisfying the following conditions: For everyj= , , . . . ,m,uj=u|(tj,tj+)∈H(tj,tj+);usatisfies the
boundary condition of (.) and the first equation of (.);u(tj+) andu(tj–),j= , , . . . ,m, exist and the impulsive conditions of (.) hold.
Takingv∈Hand multiplying (.) byvand integrating from toT, we have
– T
u(t)v(t)dt+ T
g(t)u(t)v(t)dt= T
ft,u(t)v(t)dt.
This leads to T
u(t)v(t)dt+ T
g(t)u(t)v(t)dt–
m
j= Ij
u(tj) v(tj) + α
βu(T)v(T)
= T
Thus, a weak solution of (.) is a functionu∈Hsuch that (.) holds for anyv∈H. By the regularity theory, the weak solution is a classical solution. Now, we defineϕ:H→Rby
ϕ(u) =
for anyv∈H. Obviously,ϕis continuous, and a critical point ofϕgives a weak solution of (.).
Lemma . If the function u∈H is a critical point of the functionalϕ,then u is a solution of system(.).
Combining (.) and (.), one has
We get
–u(t) +g(t)u(t) =ft,u(t) a.e.t∈(tj,tj+).
Thus,usatisfies the equation in (.). Therefore, by (.) we have
–
m
j=
u(tj) +Ij
u(tj)
v(tj) +
u(T) +α
βu(T)
v(T) = , ∀v∈H. (.)
Next we prove thatusatisfies the impulsive and the boundary condition in (.). If the impulsive condition in (.) does not hold, then there exist somej∈ {, , . . . ,m}such that
u(tj) +Ij
u(tj) = . (.)
Pickv(t) =mi=,+i=j(t–ti), then
–
m
j=
u(tj) +Ij
u(tj)
v(tj) +
u(T) +α
βu(T)
v(T)
= –
m
j=
u(tj) +Ij
u(tj)
m+
i=,i=j
(tj–ti) +
u(T) +α
βu(T)
m+
i=,i=j
(tm+–ti)
= –u(tj) +Ij
u(tj)
m+
i=,i=j
(tj–ti)= . (.)
This is a contradiction. Sousatisfies the impulsive condition in (.) and (.) implies
u(T) +α
βu(T)
v(T) = , ∀v∈H. (.)
Ifαu(T) +βu(T)= , then αβu(T) +u(T)= . Pickv(t) =mi=(t–ti). One has
u(T) +α
βu(T)
m
i=
(tm+–ti)= . (.)
This contradicts (.), sousatisfies the boundary condition. Therefore,uis a solution of
system (.).
Lemma . If u∈H,then u ∞≤T u ,where u ∞=maxt∈[,T]|u(t)|.
Proof The proof follows easily from the Hölder inequality. The detailed argument is sim-ilar to the proof of Lemma . in [], and we thus omit it here.
3 Main results
3.1 Existence of a unique solution
Theorem . Assume that dj(j= , , . . . ,m)are fixed constants,f(t,u) =σ(t)∈L(,T)
and Ij(t) =dj(j= , , . . . ,m),then system(.)has a unique solution u,and u minimizes the
functional(.).
Proof We define the bilinear form
a:H×H→R, a(u,v) = T
u(t)v(t) +g(t)u(t)v(t) dt+α
βu(T)v(T)
and the linear operator
l:H→R, l(v) = T
σ(t)v(t)dt–
m
j= djv(tj).
It is evident thatais continuous and symmetric andlis bounded. Moreover,ais coercive. By Lemma ., system (.) has a unique solutionu, anduminimizes the functional (.).
Example . Consider the following boundary value problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–u(t) +u(t) = , t=t,t∈J= [, ], –u(t) = , t=,
u() = , u() +u() = .
(.)
Hereg(t) = ,f(t,u) = ,Ij(u) = ,T= ,j= ,α= ,β= . Applying Theorem ., problem
(.) has a unique solution. By simple calculations, we obtainu(t) =e–(et–e–t),t∈[,
], u(t) =(e –e–)e–t,t∈[
, ].
3.2 Existence of at least one solution
In this section we derive conditions under which system (.) admits at least one solution. For this purpose, we introduce the following assumption.
(H) There exista,b,aj,bj> ,γ,γj∈[, ),j= , , . . . ,m, such that
f(t,u)≤a+b|u|γ, Ij(u)≤aj+bj|u|γj
fort∈J.
Theorem . Assume that(H)is satisfied,then system(.)has at least one solution u, and u minimizes the functional(.).
Proof According to (H), we have
ϕ(u) = u
– T
Ft,u(t) dt–
m
j= u(tj)
Ij(t)dt+ α
βu
(T)
≥ u
–T
a u ∞+ b
γ+ u
γ+ ∞
–
m
j=
aj u ∞+
bj γj+ u
γj+
≥ u
–T
aT u + b
γ + T
γ+ u γ+
–
m
j=
ajT
u + bj
γj+
T
γj+ u γj+
for allu∈H. This implies thatϕis coercive.
Let{un}be a weakly convergent sequence touinH, then{un}converges uniformly tou
inC[,T]. Set
ϕ(u) = – T
Ft,u(t) dt–
m
j= u(tj)
Ij(t)dt+ α
βu
(T),
ϕ(u) =
T
u(t) +g(t)u(t)dt,
thenϕ(u) =ϕ(u) +ϕ(u). Soϕis weakly sequentially continuous. Clearly,ϕis continuous and convex, which implies thatϕis weakly sequentially lower semi-continuous. There-fore,ϕis weakly sequentially lower semi-continuous onH.
By Lemma ., the functionalϕhas a minimum which is a critical point ofϕ. Hence, system (.) has at least one solution.
Example . Consider the following boundary value problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–u(t) +u(t) = +u(t), t=t,t∈J= [, ],
–u(t) = +u(t), t=
, u() = , u() +u() = .
(.)
Hereg(t) = ,f(t,u) = +u(t),Ij(u) = +u(t),T= ,j= ,α= ,β= . Clearly, (H) is
satisfied. Applying Theorem ., problem (.) has at least one solution.
3.3 Existence of at least two distinct solutions
In this section, we derive some sufficient conditions under which the functionalϕadmits at least two distinct critical points; consequently, (.) admits at least two distinct solu-tions. We first introduce some assumpsolu-tions.
(H) limu→f(tu,u)= uniformly fort∈J,limu→ Ij(u)
u = .
(H) There exist constantsμ> andr≥such that for everyt∈Jandu∈Rwith |u| ≥r,
<μF(t,u)≤uf(t,u), <μ
u
Ij(s)ds≤uIj(u), j= , , . . . ,m,
whereF(t,u) =uf(t,s)ds.
Theorem . Assume that(H)and(H)are satisfied.Then system(.)has at least two
Proof The proof will be given in three steps. Step . The functionalϕsatisfies the PS condition.
Let{un} ⊂Hsuch that{ϕ(un)}is a bounded sequence andlimn→∞ϕ(un) = . We
Byϕ(un)→ andunuasn→+∞, we have
ϕ(un) –ϕ(u)(un–u)→ asn→ ∞. (.)
So (.), (.) and (.) yield un–u → inH,i.e.,{un}strongly converges touinH.
Therefore, the functionalϕsatisfies the PS condition.
Step . We show that there existsρ> such that the functionalϕhas a local minimum u∈Bρ={u∈H: u <ρ}.
Firstly, we claim thatBρis bounded and weakly sequentially closed.
In fact, let{un} ⊆Bρand{un}uasn→ ∞. By the Mazur theorem [], there exists
a sequence of convex combinations
vn= n
j=
αnjuj,
n
j=
αnj= , αnj≥,j∈N
such thatvn→uinH.{vn} ⊂Bρandu∈Bρ, sinceBρis a closed convex set.
Secondly, we claim that the functionalϕis weakly sequentially lower semi-continuous onBρ.
Let
ϕ(u) = – T
Ft,u(t) dt–
m
j= u(tj)
Ij(t)dt+ α
βu
(T),
ϕ(u) =
T
u(t) +g(t)u(t)dt,
thenϕ(u) =ϕ(u) +ϕ(u). By{un}uonH, we see that{un}uniformly converges tou
inC[,T]. Soϕis weakly sequentially continuous. Clearly,ϕis continuous and convex, which implies thatϕis weakly sequentially lower semi-continuous. Therefore,ϕis weakly sequentially lower semi-continuous onBρ.
Thirdly, we claim thatϕhas a minimumu∈Bρ.
In fact, H is a reflexive Banach space,Bρ is bounded and weakly sequentially closed
andϕis weakly sequentially lower semi-continuous onBρ. So, by Lemma ., there exists
u∈Bρsuch thatϕ(u) =min{ϕ(u) :u∈Bρ}.
Finally, we claim thatϕ(u) <infu∈∂Bρϕ(u).
By (H), letε=mT > , there existsδ> such that|u|<δimplies
F(t,u)≤ε |u|
,
u
Ij(t)dt≤ ε
|u|
fort∈J. (.)
Consequently, by Lemma ., for u ≤ √δ
T, we have
ϕ(u) = u
– T
Ft,u(t) dt–
m
j= u(tj)
Ij(t)dt+ α
βu
(T)
≥ u
– T
ε
u(t)
dt–
m
j=
ε
u(tj)
+ α βu
≥ u
–εT u
–εmT u
≥ u
– u
– u
≥ u
.
ChooseC=δT,ρ=√δ
T, thenϕ(u)≥C> for anyu∈∂Bρ. Besides,ϕ(u)≤ϕ() = <
C≤ϕ(u) for anyu∈∂Bρ. Soϕ(u) <infu∈∂Bρϕ(u). Hence,ϕ has a local minimumu∈
Bρ={u∈H: u <ρ}.
Step . We prove that there existsuwith u >ρsuch thatϕ(u) <infu∈∂Bρϕ(u).
Condition (H) implies that there existb,b,cj,dj> ,j= , , . . . ,m, such that
F(t,u)≥b|u|μ–b,
u
Ij(t)dt≥cj|u|μ–dj (.)
fort∈J,u∈R(see []). Then we have
ϕ(u) = u
– T
Ft,u(t) dt–
m
j= u(tj)
Ij(t)dt+ α
βu
(T)
≤ u
–bTμ u μ
–b T–
m
j=
cjT μ
T μ–dj + α
βu
(T). (.)
Sinceμ> , (.) implieslimu →∞ϕ(u) = –∞. Therefore, we can chooseuwith u >ρ sufficiently large such thatϕ(u) <infu∈∂Bρϕ(u).
Let
c=inf
h∈tmax∈[,]ϕ
h(t) ,
where
=h∈C[, ],H :h() =u,h() =u.
By Lemma .,cis a critical value ofϕ, that is, there exists a critical pointu∗. Therefore, u,u∗are two critical points ofϕ, and they are solutions of (.).
Example . Consider the following boundary value problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–u(t) +u(t) = (u(t))
sint, t=tj,t∈J= [, ],
–u(tj) = u(tj), j= ,
u() = , u() +u() = .
(.)
Hereg(t) = ,f(t,u) =(u(t))sint,Ij(u) =
u
,T= ,j= ,α= ,β= . Letμ= ,r= .
3.4 Existence of infinitely many solutions
In this section, we derive some conditions under which system (.) admits infinitely many distinct solutions. To this end, we need the following assumption.
(H) f(t,u)andIj,j= , , . . . ,m, are odd aboutu.
Theorem . Assume that(H), (H)and(H)are satisfied.Then system(.)has in-finitely many solutions.
Proof We apply Lemma . to finish the proof. Clearly,ϕ∈C(H,R) is even sincef(t,u) andIj(u) are odd aboutu, andϕ() = . The arguments of Theorem . show that the
functionalϕsatisfies the PS condition. In the same way as in Theorem ., we can easily verify that conditions (i) and (ii) of Lemma . are satisfied. According to Lemma .,ϕ
possesses infinitely many critical points,i.e., system (.) has infinitely many solutions.
Example . Consider the following boundary value problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–u(t) +u(t) =tu(t), t=t
j,t∈J= [, ],
–u(tj) = u(tj), j= ,
u() = , u() +u() = .
(.)
Hereg(t) = ,f(t,u) =tu,I
j(u) =u,T= ,j= ,α= ,β= . Obviously,f(t,u),Ij(u) are
odd aboutu. Letμ= ,r= . Clearly, (H) and (H) are satisfied. Applying Theorem ., problem (.) has infinitely many solutions.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ZL and JX conceived of the study and drafted the manuscript. GC participated in the discussion. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, P.R. China.2College of Mathematics
and Statistics, Jishou University, Jishou, Hunan 416000, P.R. China.
Acknowledgements
The authors are very grateful to the referees for their very helpful comments and suggestions, which greatly improved the presentation of this paper. This work is supported by the Scientific Research Fund of Hunan Provincial Education Department (No: 13K029).
Received: 10 November 2013 Accepted: 4 April 2014 Published:06 May 2014
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