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

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.

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: 𝑥^′′(𝑡)+𝐹^′_𝑥(𝑡,𝑥)=0, (1.1) and a classical forced pendulum equation: 𝑥^′′(𝑡)+𝐴sin𝑥=𝑓(𝑡), (1.2) 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: 𝑢^′′(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,a.e.𝑡∈𝐽′,Δ𝑢^′𝑡_𝑘=𝐼_𝑘𝑢𝑡_𝑘,𝑘∈𝑍₀, (1.3) where 𝑓∈𝐶(𝑅²,𝑅), 𝑍₀=𝑍⁺∪𝑍⁻, 𝐽′=𝑅⧵{𝑡_𝑘∣𝑘∈𝑍₀}, 𝐼_𝑘∈𝐶(𝑅,𝑅⁺∪{0}), Δ𝑢^′(𝑡_𝑘)=𝑢^′(𝑡⁺_𝑘)−𝑢^′(𝑡⁻_𝑘), 𝑢^′(𝑡^±_𝑘)=lim_𝑡→𝑡^±_𝑘𝑢^′(𝑡), 0<𝑡₁<⋯<𝑡_𝑚<𝑇, 𝐼_𝑘+𝑚=𝐼_𝑘, 𝑇∈𝑅⁺ and 𝑡_𝑘=𝑡_𝑚+𝑘−𝑇 if 𝑘∈𝑍⁺, while 𝑡_𝑘=𝑡_𝑚+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: 𝑢^′′(𝑡−𝑟)+𝑓(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟),𝑢(𝑡−2𝑟))=0,𝑎.𝑒.𝑡∈𝐽′,Δ𝑢^′𝑡_𝑘=𝐼_𝑘𝑢𝑡_𝑘,𝑘∈𝑍₀, (1.4) where 𝑟>0,𝑓∈𝐶(𝑅⁴,𝑅), 𝑍₀=𝑍⁺∪𝑍⁻, 𝐽^′=𝑅⧵{𝑡_𝑘𝑘∈𝑍₀}, 𝐼_𝑘∈𝐶(𝑅,𝑅⁺∪{0}), Δ𝑢^′(𝑡_𝑘)=𝑢^′(𝑡⁺_𝑘)−𝑢^′(𝑡⁻_𝑘), 𝑢^′(𝑡^±_𝑘)=lim_𝑡→𝑡^±_𝑘𝑢^′(𝑡), 0<𝑡₁<⋯<𝑡_𝑚<𝑟, 𝐼_𝑘+𝑚=𝐼_𝑘, 𝑟∈𝑅⁺ and 𝑡_𝑘=𝑡_𝑚+𝑘−𝑟 if 𝑘∈𝑍⁺, while 𝑡_𝑘=𝑡_𝑚+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 𝑢_𝑖∈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 𝑢_𝑖∈𝐶([0,𝑝𝑟],𝑅) such that limsup_{|𝑢1|,|𝑢2|→+∞}𝐹(𝑡,𝑢₁,𝑢₂)/(|𝑢₁|²+|𝑢₂|²)≤1/2(𝑝𝑟)²=𝛾 and 𝐹^′_𝑢2(𝑡,𝑢₁,𝑢₂)+𝐹^′_𝑢2(𝑡,𝑢₂,𝑢₃)=𝑓(𝑡,𝑢₁,𝑢₂,𝑢₃), where 𝐹^′_𝑢2(𝑡,𝑢₁,𝑢₂) and 𝐹^′_𝑢2(𝑡,𝑢₂,𝑢₃) are 𝑟-periodic functions in 𝑡. (𝐴₃) There are constants 𝛼>0, 𝛽>0, 𝑑_𝑗≥0, 𝑗=1,2,…,𝑚 such that 𝐼_𝑗(𝑢)≤𝑑_𝑗|𝑢|,𝛼₂𝜔𝑝𝑟−𝑝𝑟𝑝²−2𝑚𝑝𝐷>0,𝑝²<𝑝²_𝑠𝜔²𝛼,max0,𝛼𝑢₁²+𝑢₂²−𝛽𝑢₁⁴+𝑢₂⁴≤𝐹𝑡,𝑢₁,𝑢₂−𝐹^′_𝑢2(𝑡,0,0)𝑢₁−𝐹^′_𝑢2(𝑡,0,0)𝑢₂≤𝛼𝑢₁²+𝑢₂², (1.5) where 𝐷=max{𝑑_𝑗,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 𝐹^′_{𝑢()𝑡−𝑟}(𝑡,𝑢(𝑡−𝑟),𝑢(𝑡−2𝑟))+𝐹^′_{𝑢()𝑡−𝑟}(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟))=𝑓(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟),𝑢(𝑡−2𝑟)). (1.6) 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 𝑢^′′(𝑡−𝑟)+𝐹^′_{𝑢()𝑡−𝑟}(𝑡,𝑢(𝑡−𝑟),𝑢(𝑡−2𝑟))+𝐹^′_{𝑢()𝑡−𝑟}(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟))=0,𝑡∈𝑡_𝑘−1,𝑡_𝑘,Δ𝑢^′𝑡_𝑘=𝐼_𝑘𝑢𝑡_𝑘,𝑘∈𝑍₀. (1.4)^′

The outline of the paper is as follows. In Section 2, 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.

Let 𝑇=𝑝𝑟,𝑝≥2. The norm in 𝐻¹([0,𝑇],𝑅) is denoted by ‖⋅‖₀. Denote the Sobolov space 𝐸 by 𝐸=𝑢∈𝐻¹([0,𝑇],𝑅)∣𝑢isabsolutelycontinuous,𝑢(0)=𝑢(𝑇) (2.1) with the inner product (𝑢,𝑣)=^𝑇₀𝑢(𝑡)𝑣(𝑡)+𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡,𝑢,𝑣∈𝐸, (2.2) which induces the norm ‖𝑢‖=‖𝑢‖₀+𝑢^′₀,𝑢∈𝐸. (2.3) It is easy to verify that 𝐸 is a reflexive Banach space.

Consider the functional 𝐼 defined on 𝐸 by 𝐼(𝑢)=^𝑇₀12𝑢′(𝑡)²−𝐹(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟))𝑑𝑡+_𝑘∈𝐾^{𝑢(𝑡𝑘)}₀𝐼_𝑘(𝑡)𝑑𝑡, (2.4) where 𝐾={𝑘∈𝑍₀⧵𝑡_𝑘∈(0,𝑇]}={1,2,…,𝑝𝑚}.

We should caution that the solutions minimal periods may not be 𝑝𝑟. Define 𝜔=2𝜋/𝑟, and 𝑝_𝑠 as the smallest prime factor of 𝑝.

Define 𝐸={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(𝑎_𝑛cos𝑛𝜔𝑡/𝑝+𝑏_𝑛sin𝑛𝜔𝑡/𝑝). Moreover, 𝑢∈𝐸 if and only if 𝑢(𝑡)=∑^∞_𝑛=0𝑏_𝑛sin𝑛𝜔𝑡/𝑝.

We will show that the classic 𝑇-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 that 𝐼_𝑘 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. If 𝑢 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 that 𝑋 is a Banach space and 𝜑∶𝑋→𝑅. Say that 𝐼 is weakly lower semicontinuous if 𝑢_𝑘⇀𝑢₀ means liminf_𝑛→∞𝐼(𝑢_𝑘)≥𝐼(𝑢₀) and 𝐼 is coercive if lim_{‖𝑢‖→∞}=+∞.

Lemma 2.3 (see [38]). Let 𝐸 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 min_𝑢∈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 𝐹^′_{𝑢∗(𝑡)}(𝑡,0,0)₀+𝐹^′_{𝑢∗()𝑡−𝑟}(𝑡,0,0)₀≤𝑞𝜔𝜔2𝑝𝛼𝑇−𝑇²𝑝²−2𝑚𝑝𝐷21−𝛼𝑝²/𝑞²𝜔²,3𝛽𝑇 (3.1) 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 that 𝐼 has a critical point 𝑢_∗ with min𝜑_𝑢∈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≥𝑝_𝑠.

By the Wirtinger inequality and (𝐴₁), we have 𝐼𝑢_∗=^𝑇₀12𝑢^′_∗(𝑡)²−𝐹𝑡,𝑢_∗(𝑡),𝑢_∗(𝑡−𝑟)𝑑𝑡+_𝑘∈𝐾^{𝑢∗(𝑡𝑘)}₀𝐼_𝑘(𝑡)≥1𝑑𝑡2𝑢^′_∗²₀−^𝑇₀𝐹^′_{𝑢∗(𝑡)}(𝑡,0,0)𝑢_∗(𝑡)+𝐹^′_{𝑢∗()𝑡−𝑟}(𝑡,0,0)𝑢_∗(−𝑡−𝑟)𝑑𝑡_𝑇0𝐹𝑡,𝑢_∗(𝑡),𝑢_∗(𝑡−𝑟)−𝐹^′_{𝑢∗(𝑡)}(𝑡,0,0)𝑢_∗(𝑡)−𝐹^′_{𝑢∗()𝑡−𝑟}(𝑡,0,0)𝑢_∗(≥1𝑡−𝑟)𝑑𝑡2𝑢^′_∗²₀−𝐹^′_{𝑢∗(𝑡)}(𝑡,0,0)₀+𝐹^′_{𝑢∗()𝑡−𝑟}(𝑡,0,0)₀𝑢_∗0−𝛼2𝑢_∗²₀≥12𝑝1−𝛼𝑞𝜔²𝑢^′_∗²₀−𝑝𝐹𝑞𝜔^{′𝑢∗(𝑡)}(𝑡,0,0)₀+𝐹^′_{𝑢∗()𝑡−𝑟}(𝑡,0,0)₀𝑢^′_∗0. (3.2) On the other hand, let 𝑢(𝑡)=√𝜌sin𝜔𝑡/𝑝. Then, 𝑢(𝑡) is 𝑇-periodic with minimal periodic 𝑇. Since 𝐹^′_𝑢(𝑡)(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟)) and 𝐹^′_{𝑢(𝑡−𝑟)}(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟)) are 𝑟-periodic, we have ^𝑇₀𝐹^′_𝑢(𝑡)(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟))𝑢(𝑡)𝑑𝑡=0,^𝑇₀𝐹^′_{𝑢()𝑡−𝑟}(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟))𝑢(𝑡−𝑟)𝑑𝑡=0. (3.3) By the Wirtinger inequality and (𝐴₃), we also have 𝐼(𝑢)=^𝑇₀12𝑢^′(𝑡)²−𝐹(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟))𝑑𝑡+_𝑘∈𝐾^{𝑢(𝑡𝑘)}₀𝐼_𝑘(𝑡)≤𝑑𝑡𝜌𝑇4𝜔𝑝²−^𝑇₀𝐹^′_𝑢(𝑡)(𝑡,0,0)𝑢(𝑡)+𝐹^′_{𝑢()𝑡−𝑟}(𝑡,0,0)𝑢(𝑡−𝑟)𝑑𝑡−^𝑇₀𝐹(𝑡,𝑢(𝑡),𝑢(𝑡−𝑟))−𝐹^′_𝑢(𝑡)(𝑡,0,0)𝑢(𝑡)−𝐹^′_{𝑢()𝑡−𝑟}(𝑡,0,0)𝑢(𝑡−𝑟)𝑑𝑡+𝑚𝑝𝐷𝜌2≤𝜌𝑇4𝜔𝑝²−𝛼2^𝑇₀|𝑢(𝑡)|²𝛽𝑑𝑡+2^𝑇₀|𝑢(𝑡)|⁴𝑑𝑡+𝑚𝑝𝐷𝜌2≤𝜌𝑇4𝜔𝑝²−𝛼𝜌𝑇4+3𝛽𝑇𝜌²+16𝑚𝑝𝐷𝜌2=3𝛽𝑇𝜌²−1164𝜔𝛼𝑇−𝑇𝑝²−2𝑚𝑝𝐷𝜌. (3.4) 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 3𝛽𝑇𝜌²𝜔16−1/4𝛼𝑇−𝑇𝑝²1−2𝑚𝑝𝐷𝜌<2𝑝1−𝛼𝑞𝜔²𝑢_∗²₀.−𝑝𝐹𝑞𝜔^{′𝑢∗(𝑡)}(𝑡,0,0)₀+𝐹^′_{𝑢∗()𝑡−𝑟}(𝑡,0,0)₀𝑢^′_∗0. (3.5) Actually, we can choose 𝜌=4/3𝛽𝑇(𝛼𝑇−𝑇(𝜔/𝑝)²−2𝑚𝑝𝐷). Then, we need to prove −1/4𝛼𝑇−𝑇𝜔/𝑝²−2𝑚𝑝𝐷²<−𝐹3𝛽𝑇/4𝑝/𝑞𝜔^′_{𝑢∗(𝑡)}(𝑡,0,0)₀+𝐹^′_{𝑢∗()𝑡−𝑟}(𝑡,0,0)₀²21−𝛼𝑝/𝑞𝜔². (3.6) This is true under the assumption (3.1). Hence, the proof is complete.

4. Example

Suppose 𝐹𝑡,𝑢₁,𝑢₂=1𝑢20²¹+𝑢²₂−120sin2𝜋𝑡𝑟𝑢²₁arctan𝑢²₂+𝑢²₂arctan𝑢²₁+𝑢₁+𝑢₂. (4.1) Then, 𝐹^′_{𝑢()𝑡−𝑟}(()(=1𝑡,𝑢𝑡−𝑟,𝑢𝑡−2𝑟))𝑢()−110𝑡−𝑟20sin2𝜋𝑡𝑟2𝑢(𝑡−𝑟)arctan𝑢²()+()𝑢()𝑡−𝑟2𝑢𝑡−𝑟𝑡−2𝑟1+𝑢⁴()𝐹𝑡−𝑟+1_{′𝑢()𝑡−𝑟}((𝑡)(=1𝑡,𝑢,𝑢𝑡−𝑟))𝑢()−110𝑡−𝑟20sin2𝜋𝑡𝑟2𝑢(𝑡−𝑟)arctan𝑢²(𝑡)+2𝑢(𝑡)𝑢(𝑡−𝑟)1+𝑢⁴(),𝐹𝑡−𝑟+1_′𝑢1𝑡,𝑢₁,𝑢_{2𝑢1=𝑢2=0}𝑢₁+𝐹^′_𝑢2𝑡,𝑢₁,𝑢_{2𝑢1=𝑢2=0}𝑢₂1=−20sin2𝜋𝑡𝑟𝑢₁+𝑢₂. (4.2) Let 𝑓((𝑡)()(=1𝑡,𝑢,𝑢𝑡−𝑟,𝑢𝑡−2𝑟))5𝑢()−1𝑡−𝑟20sin2𝜋𝑡𝑟2𝑢(𝑡−𝑟)arctan𝑢²()+()𝑢()𝑡−𝑟2𝑢𝑡−𝑟𝑡−2𝑟1+𝑢⁴()()𝑡−𝑟+2𝑢𝑡−𝑟arctan𝑢²(𝑡)+2𝑢(𝑡)𝑢(𝑡−𝑟)1+𝑢⁴().𝑡−𝑟+2 (4.3) Consider the following impulsive system: 𝑢^′′()+1𝑡−𝑟5𝑢()−1𝑡−𝑟20sin2𝜋𝑡𝑟2𝑢(𝑡−𝑟)arctan𝑢²()+()𝑢()𝑡−𝑟2𝑢𝑡−𝑟𝑡−2𝑟1+𝑢⁴()()𝑡−𝑟+2𝑢𝑡−𝑟arctan𝑢²(𝑡)+2𝑢(𝑡)𝑢(𝑡−𝑟)1+𝑢⁴()𝑡𝑡−𝑟+2=0,∀𝑡∈_𝑘−1,𝑡_𝑘,Δ𝑢^′𝑡_𝑘=𝐼_𝑘𝑢𝑡_𝑘=0.001𝑢𝑡_𝑘,𝑘∈𝑍^∗, (4.4) where 𝑡_𝑘=𝑘−1/2 if 𝑘∈𝑍⁺, while 𝑡_𝑘=𝑘+1/2 if 𝑘∈𝑍⁻.

Proof. Let 𝑟=1,𝑇=1, 𝛾=1/20, 𝛼=1/20, 𝛽=1/20, 𝑚=1, 𝐷=0.001, 𝜔=2𝜋, 𝑝_𝑠=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.

Acknowlegdment

Research was supported by Anhui Provincial Nature Science Foundation (090416237, 1208085MA13), Research Fund for Doctoral Station of Ministry of Education of China (20103401120002, 20113401110001), 211 Project of Anhui University (02303129, 02303303-33030011, 02303902-39020011, KJTD002B), and Foundation of Anhui Education Bureau (KJ2012A019).