# Subharmonic Solutions with Prescribed Minimal Periodic for a Class of Second-Order Impulsive Functional Differential Equations

Authors: Qi Wang; Dicheng Lu; Zhaoyue Zheng
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

By using critical point theory and variational methods, we investigate the subharmonic solutions with prescribed minimal period for a class of second-order impulsive functional differential equations. The conditions for the existence of subharmonic solutions are established. In the end, we provide an example to illustrate our main results.


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## Body

## 1. Introduction

During the last 40 years, the theory and applications of impulsive differential equations have been developed, see [1–28]. Recently, some researchers studied the minimal period problem or homoclinic solution for some classes of Hamiltonian systems and classical pendulum equations [29–35]. In [30, 31], using the variational methods and decomposition technique, Yu got some sufficient conditions for the existence of periodic solutions with minimal period pT for the following nonautonomous Hamiltonian systems:
(1.1)x′′(t)+Fx′(t,x)=0,
and a classical forced pendulum equation:
(1.2)x′′(t)+Asin⁡x=f(t),
respectively. In [35], by using critical point theory and variational methods, Luo et al. considered the existence results of subharmonic solutions with prescribed minimal period for a class of second-order impulsive differential equations:
(1.3)u′′(t)+f(t,u(t))=0,a.e.t∈J′,Δu′(tk)=Ik(u(tk)),k∈Z0,
where f∈C(R2,R), Z0=Z+∪Z-, J′=R∖{tk∣k∈Z0}, Ik∈C(R,R+∪{0}), Δu′(tk)=u′(tk+)-u′(tk-), u′(tk±)=limt→tk±u′(t), 0<t1<⋯<tm<T, Ik+m=Ik, T∈R+ and tk=tm+k-T if k∈Z+, while tk=tm+k+1-T if k∈Z-.Motivated by [30, 31, 35], in this paper, we consider the existence results of subharmonic solutions with prescribed minimal period for a class of second-order impulsive functional differential equations:
(1.4)u′′(t-r)+f(t,u(t),u(t-r),u(t-2r))=0,a.e.t∈J′,Δu′(tk)=Ik(u(tk)),k∈Z0,
where r>0,f∈C(R4,R), Z0=Z+∪Z-, J′=R∖{tkk∈Z0}, Ik∈C(R,R+∪{0}), Δu′(tk)=u′(tk+)-u′(tk-), u′(tk±)=limt→tk±u′(t), 0<t1<⋯<tm<r, Ik+m=Ik, r∈R+ and tk=tm+k-r if k∈Z+, while tk=tm+k+1-r if k∈Z-.We make the following assumptions.(A1)f(t,u1,u2,u3)∈C(R4,R) is r-periodic in t for any ui∈C([0,pr],R),i=1,2,3, where p is a positive integer.(A2)F(t,u1,u2)∈C(R3,R) is r-periodic in t and continuously differentiable for any ui∈C([0,pr],R) such that limsup|u1|,|u2|→+∞F(t,u1,u2)/(|u1|2+|u2|2)≤1/2(pr)2=γ and Fu2′(t,u1,u2)+Fu2′(t,u2,u3)=f(t,u1,u2,u3), where Fu2′(t,u1,u2) and Fu2′(t,u2,u3) are r-periodic functions in t.(A3) There are constants α>0, β>0, dj≥0, j=1,2,…,m such that
(1.5)|Ij(u)|≤dj|u|,α2pr-pr(ωp)2-2mpD>0,p2<ps2ω2α,max⁡⁡{0,α(|u1|2+|u2|2)-β(|u1|4+|u2|4)}≤F(t,u1,u2)-Fu2′(t,0,0)u1-Fu2′(t,0,0)u2≤α(|u1|2+|u2|2),whereD=max{dj,j=1,2,…,m}.(A4) Suppose q is rational. If u is a periodic function with minimal period qr, and f(t,u1,u2,u3) is a periodic function with minimal period qr, then q is necessarily an integer.From(A2), we have
(1.6)Fu(t-r)′(t,u(t-r),u(t-2r))+Fu(t-r)′(t,u(t),u(t-r))=f(t,u(t),u(t-r),u(t-2r)).
Therefore, under the assumptions (A1)-(A4), the existence of subharmonic solutions with minimal period for (1.4) has been changed into the existence of subharmonic solutions with minimal period for
(1.4)′u′′(t-r)+Fu(t-r)′(t,u(t-r),u(t-2r))+Fu(t-r)′(t,u(t),u(t-r))=0,t∈(tk-1,tk),Δu′(tk)=Ik(u(tk)),k∈Z0.The outline of the paper is as follows. In Section2, some preliminaries and basic results are established. In Section 3, by using critical point theory, we give sufficient conditions for the existence of of subharmonic solutions with minimal period for the impulsive systems. In Section 4, we give an example to illustrate the application of our main result


## 2. Preliminaries and Basic Results

In the following, we introduce some notations and some necessary definitions.LetT=pr,p≥2. The norm in H1([0,T],R) is denoted by ∥·∥0. Denote the Sobolov space E by
(2.1)E={u∈H1([0,T],R)∣uis absolutely continuous,u(0)=u(T)}
with the inner product
(2.2)(u,v)=∫0T‍[u(t)v(t)+u′(t)v′(t)]dt,u,v∈E,
which induces the norm
(2.3)‖u‖=‖u‖0+‖u′‖0,u∈E.
It is easy to verify that E is a reflexive Banach space.Consider the functionalI defined on E by
(2.4)I(u)=∫0T‍[12|u′(t)|2-F(t,u(t),u(t-r))]dt+∑k∈K‍∫0u(tk)‍Ik(t)dt,
where K={k∈Z0∖tk∈(0,T]}={1,2,…,pm}.We should caution that the solutions minimal periods may not bepr. Define ω=2π/r, and ps as the smallest prime factor of p.DefineE-={u∈E∣u(-t)=-u(t)}, a subspace of the Sobolev space E. For any u∈E,u has a Fourier series expansion u(t)=∑n=0∞‍(ancosnωt/p+bnsinnωt/p). Moreover, u∈E- if and only if u(t)=∑n=0∞‍bnsinnωt/p.We will show that the classicT-solutions of (1.4) or (1.4)′ is equivalent to finding the critical points of I.Similar to the proof [13, 36, 37], we have two lemmas as following.Lemma 2.1.
Suppose thatIk are continuous. Then, the following statements are equivalent:(1)
u∈E is a critical point of I;(2)
u is a classical solution of (1.4) or (1.4)′.Lemma 2.2.
Ifu is a critical point of I on E-, then u is also a critical point of I on X. And the minimal period of u is an integer multiple of r.Now we state some results on nonlinear functional analysis and critical point theory. Suppose thatX is a Banach space and φ:X→R. Say that I is weakly lower semicontinuous if uk⇀u0 means liminfn→∞I(uk)≥I(u0) and I is coercive if lim∥u∥→∞=+∞.Lemma 2.3 (see [38]).
LetE be a real reflexive Banach space and weak sequentially closed. φ∈C1(E,R) is weakly lower semicontinuous and coercive. Then, φ has a critical point u* with minu∈Eφ(u)=φ(u*).
Similar to the proof of [35, Lemma 2.3], we have the following lemma.Lemma 2.4.
Suppose that(A2)-(A3) hold. E- is a weak sequentially closed and φ is coercive and weakly lower semicontinuous on E-.


## 3. Main Results

Theorem 3.1.
Suppose that(A1)-(A4) hold. If
(3.1)‖Fu*(t)′(t,0,0)‖0+‖Fu*(t-r)′(t,0,0)‖0≤qω2p(αT-Tω2p2-2mpD)2(1-αp2/q2ω2)3βT,
then (1.4) has at least one classical periodic solution with the minimal period T=pr.Proof.
It follows from Lemmas 2.3 and 2.4 thatI has a critical point u* with minφu∈E(u)=φ(u*). Next, we show the minimal period of u* is pr. For the sake of a contradiction, let the minimal period of u* be pr/q for some integer q≥2. By Lemma 2.2, we know that q is a factor of p, and so q≥ps.By the Wirtinger inequality and(A1), we have
(3.2)I(u*)=∫0T‍[12|u*′(t)|2-F(t,u*(t),u*(t-r))]dt+∑k∈K‍∫0u*(tk)‍Ik(t)dt≥12‖u*′‖02-∫0T‍[Fu*(t)′(t,0,0)u*(t)+Fu*(t-r)′(t,0,0)u*(t-r)]dt-∫0T‍[F(t,u*(t),u*(t-r))-Fu*(t)′(t,0,0)u*(t)-Fu*(t-r)′(t,0,0)u*(t-r)]dt≥12‖u*′‖02-(‖Fu*(t)′(t,0,0)‖0+‖Fu*(t-r)′(t,0,0)‖0)‖u*‖0-α2‖u*‖02≥12(1-α(pqω)2)‖u*′‖02-pqω(‖Fu*(t)′(t,0,0)‖0+‖Fu*(t-r)′(t,0,0)‖0)‖u*′‖0.
On the other hand, let u-(t)=ρsinωt/p. Then, u-(t) is T-periodic with minimal periodic T. Since Fu(t)′(t,u(t),u(t-r)) and Fu(t-r)′(t,u(t),u(t-r)) are r-periodic, we have
(3.3)∫0T‍Fu-(t)′(t,u-(t),u-(t-r))u-(t)dt=0,∫0T‍Fu-(t-r)′(t,u-(t),u-(t-r))u-(t-r)dt=0.
By the Wirtinger inequality and (A3), we also have
(3.4)I(u-)=∫0T[12|u-′(t)|2-F(t,u-(t),u-(t-r))]dt+∑k∈K‍∫0u-(tk)‍Ik(t)dt‍≤ρT4(ωp)2-∫0T‍[Fu-(t)′(t,0,0)u-(t)+Fu-(t-r)′(t,0,0)u-(t-r)]dt-∫0T‍[F(t,u-(t),u-(t-r))-Fu-(t)′(t,0,0)u-(t)-Fu-(t-r)′(t,0,0)u-(t-r)]dt+mpDρ2≤ρT4(ωp)2-α2∫0T‍|u-(t)|2dt+β2∫0T‍|u-(t)|4dt+mpDρ2≤ρT4(ωp)2-αρT4+3βTρ216+mpDρ2=3βTρ216-14(αT-T(ωp)2-2mpD)ρ.
If I(u-)<I(u*), then this is clearly in contradiction with the assumption for u*. Now, we are going to choose some positive number ρ such that
(3.5)3βTρ216-1/4(αT-T(ωp)2-2mpD)ρ<12(1-α(pqω)2)‖u*‖02.-pqω(‖Fu*(t)′(t,0,0)‖0+‖Fu*(t-r)′(t,0,0)‖0)‖u*′‖0.
Actually, we can choose ρ=4/3βT(αT-T(ω/p)2-2mpD). Then, we need to prove
(3.6)-(1/4(αT-T(ω/p)2-2mpD))23βT/4<-(p/qω(‖Fu*(t)′(t,0,0)‖0+‖Fu*(t-r)′(t,0,0)‖0))22(1-α(p/qω)2).
This is true under the assumption (3.1). Hence, the proof is complete.


## 4. Example

Suppose(4.1)F(t,u1,u2)=120(u12+u22)-120sin⁡2πtr(u12arctanu22+u22arctanu12+u1+u2).
Then,
(4.2)Fu(t-r)′(t,u(t-r),u(t-2r))=110u(t-r)-120sin⁡2πtr(2u(t-r)u(t-2r)1+u4(t-r)2u(t-r)arctanu2(t-r)+2u(t-r)u(t-2r)1+u4(t-r)+1)Fu(t-r)′(t,u(t),u(t-r))=110u(t-r)-120sin⁡2πtr(2u(t-r)arctanu2(t)+2u(t)u(t-r)1+u4(t-r)+1),Fu1′(t,u1,u2)|u1u1=u2=0+Fu2′(t,u1,u2)|u1=u2=0u2=-120sin⁡2πtr(u1+u2).
Let
(4.3)f(t,u(t),u(t-r),u(t-2r))==15u(t-r)-120sin⁡2πtr(2u(t-r)arctanu2(t-r)+2u(t-r)u(t-2r)1+u4(t-r)===============+2u(t-r)arctanu2(t)+2u(t)u(t-r)1+u4(t-r)+2).
Consider the following impulsive system:
(4.4)u′′(t-r)+15u(t-r)-120sin⁡2πtr[2u(t-r)arctanu2(t-r)+2u(t-r)u(t-2r)1+u4(t-r)=+2u(t-r)arctanu2(t)+2u(t)u(t-r)1+u4(t-r)+2]=0,======================∀t∈(tk-1,tk),Δu′(tk)=Ik(u(tk))=0.001|u(tk)|,k∈Z*,
where tk=k-1/2 if k∈Z+, while tk=k+1/2 if k∈Z-.Proof.
Letr=1,T=1, γ=1/20, α=1/20, β=1/20, m=1, D=0.001, ω=2π, ps=2. It is easy to check all the assumptions of Theorem 3.1 are satisfied. Thus, (4.4) has a periodic solution with the minimal period 30.



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*Source: 457163-2012-08-09.xml*