# Existence and Multiplicity of Periodic Solutions Generated by Impulses

Authors: Liu Yang; Haibo Chen
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We investigate the
existence and multiplicity of periodic solutions
for a class of second-order differential systems with impulses. By using variational
methods and critical point theory, we obtain such a system possesses
at least one nonzero, two nonzero, or infinitely many periodic solutions generated
by impulses under different conditions, respectively. Recent results in the
literature are generalized and significantly improved.


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## Body

## 1. Introduction

Consider the following second-order impulsive differential equations:(Pλ)ü(t)+Vu(t,u(t))=0,t∈(sk-1,sk),Δu̇(sk)=λgk(u(sk)),u(0)-u(T)=u̇(0)-u̇(T)=0,
where sk,k=1,2,…,m, are instants in which the impulses occur and 0=s0<s1<s2<⋯<sm<sm+1=T, Δu̇(sk)=u̇(sk+)-u̇(sk-) with u̇(sk±)=limt→sk±u̇(t), gk(u)=graduGk(u), Gk∈C1(ℝN,ℝ), V∈C1([0,T]×ℝN,ℝ), Vu(t,u(t))=graduV(t,u), λ>0 is a constant.Impulsive differential equations can be used to describe the dynamics of processes which possess abrupt changes at certain instants. Up to now, impulsive differential systems have been widely applied in many science fields such as control theory, biology, mechanics, see [1–7] and references therein. For general theory of impulsive differential equations, we refer the readers to the monographs as [8–10].The existence of the solutions is one of the most important topics of impulsive differential systems. Many classical methods and tools have been used to study them, such as coincidence degree theory, fixed point theory, and the method of upper and lower solutions. See [11–15] and references therein.Recently, some authors creatively applied variational method to deal with impulsive problems, see [16–21]. The variational method is opening a new approach for dealing with discontinuity problems such as impulses. However, when the problems studied in [16–21] degenerate to the cases without impulses, plenty of the corresponding results can also be obtained. In other words, the effect of impulses was not seen evidently. As pointed out in [22], these results, in some sense, mean that the nonlinear term Vu plays a more important role than the impulsive terms gk do in guaranteeing the existence of solutions. Due to this point, Zhang and Li [22] studied the existence of solutions for impulsive differential systems generated by impulses.Definition 1.1 (see [22]).
A solution is called a solution generated by impulses if this solution is nontrivial when impulsive terms are not zero, but it is trivial when impulsive term is zero.
For example, if problem (Pλ) does not possess non-zero solution when gk≡0 for all 1≤k≤m, then nonzero solution for problem (Pλ) is called solution generated by impulses. In detail, Zhang and Li [22] obtained the following theorem.Theorem A (see [22]).
Assume thatV,W satisfy the following conditions:(V1)
V is continuous differentiable and there exist positive constants b1,b2>0 such that b1|u|2≤-V(t,u)≤b2|u|2 for all (t,u)∈[0,T]×ℝN;(V2)
-V(t,u)≤-Vu(t,u)u≤-2V(t,u) for all (t,u)∈[0,T]×ℝN;(g1)
there exists aθ>2 such that gk(u)u≤θGk(u)<0, for u∈ℝN∖{0} and k=1,2,…,m.Then, problem (P1) possesses at least one non-zero solution generated by impulses.Motivated by the facts mentioned above, in this paper, we will further study the existence of solution for problem (Pλ) generated by impulses under more general conditions. In addition, we will investigate the multiple solutions and infinitely many solutions generated by impulses.Now, we state our results.Theorem 1.2.
Assume that(g1) and the following (V1′), (V2′) hold. (V1′)
V is continuous differentiable and there exist positive constants b>0 and γ∈(1,2] such that(1.1)-V(t,u)≥b|u|γ∀(t,u)∈[0,T]×RN;(V2′)
there exists a constantϱ∈[2,θ) such that(1.2)-V(t,u)≤-Vu(t,u)u≤-ϱV(t,u)∀(t,u)∈[0,T]×RN.
Then, problem(P1) possesses at least one non-zero solution generated by impulses.Remark 1.3.
It is easy to see that(V1′), (V2′) are weaker than (V1), (V2). Therefore, Theorem 1.2 improves Theorem A.
Indeed, taking(1.3)V(t,u)=-|u|γ-|u|ς,1<γ<ς≤2,Gk(u)=-|u|θ,
all conditions in Theorem 1.4 are satisfied, but conditions in Theorem A cannot be satisfied.Theorem 1.4.
Assume that(V1), (V2) hold. Moreover, the following conditions are satisfied: (V3)
Vu(t,u1-u2)=Vu(t,u1)-Vu(t,u2);(H1)
∑k=1mGk(u)≤0 and(1.4)max⁡{limsup⁡u→0⁡(-∑k=1mGk(u)|u|2),limsup⁡|u|→+∞⁡(-∑k=1mGk(u)|u|2)}<A,0<AC12<b3,
where b3=min{1/2,b1} and C1 is a constant which will be defined in Section 2; (H2)
there exists a constant vectorξ=(ξ1,ξ2,…,ξN)∈ℝN∖{0} such that ∑k=1mGk(ξ)<∫0TV(t,ξ)dt.Then, there exists B>0 such that problem (P1) possesses at least two non-zero solutions generated by impulses; moreover, their norms are less than B.Remark 1.5.
Compared with Theorem A, Theorem1.4 deals with the multiple solutions generated by impulses. In Theorem A and Theorem 1.2, impulses are superquadratic. However, some impulses which are subquadratic can satisfy the conditions of Theorem 1.4.Example 1.6.
LetV(t,u)=-|u|2. It is easy to see that conditions (V1), (V2), and (V3) hold. Let T=1, m=1, ξ=(1,0,…,0), and
(1.5)G1(u)={-2|u|3,|u|<1,-6|u|+4,|u|≥1.
Then, we have
(1.6)limsup⁡u→0-G1(u)|u|2=limsup⁡|u|→+∞-G1(u)|u|2=0,-2=G1(ξ)<∫0TV(t,ξ)dt=-1.
Hence, conditions (H1) and (H2) in Theorem 1.4 are satisfied.Theorem 1.7.
Assume that(V1), (V2) hold. Moreover, the following conditions are satisfied: (H3)
let(1.7)α=lim⁡inf⁡ρ→+∞max⁡|u|≤ρ⁡[-∑k=1mGk(u)]ρ2,β=limsup⁡|u|→+∞-∑k=1mGk(u)|u|2,
and Tb2C12α<b3β.Then, for everyλ∈Λ∶=(Tb2/β,b3/C12α), problem (Pλ) possesses an unbounded sequence of solutions generated by impulses.Example 1.8.
LetV(t,u)=(-1/2)|u|2,T=1,m=1,N=1,
(1.8)an=2n!(n+2)!-14(n+1)!,bn=2n!(n+2)!+14(n+1)!,(1.9)g1(u)={π(n+1)![(n+1)!2-n!2]sin(2π(n+1)!(u-an)),|u|∈⋃n∈N[an,bn],0,otherwise,
and g2=max{0,2u}. Let
(1.10)g=g1+g2,-G=∫0ug(t)dt.
Then, -G is a C1 function with -G′=g. From the computation of Example  3.2 in [23], one has C1=2,b2=1/2,b3=1/2 in this case and
(1.11)lim⁡n→+∞-G(an)an2=1,lim⁡n→+∞-G(bn)bn2=5.
Moreover,
(1.12)lim⁡inf⁡|u|→+∞-G(u)u2=1,limsup⁡|u|→+∞-G(u)u2=5.
In addition, -G is nondecreasing. By the monotonicity of -G, one has
(1.13)max⁡|u|≤ρ[-G(u)]=-G(ρ).
Hence, we have
(1.14)α=1,β=5.
Then, for λ∈[1/10,1/4], problem
(1.15)ü(t)-u=0,t≠s1,Δu̇(s1)=λg(u(s1)),u(0)-u(1)=u̇(0)-u̇(1)=0
possesses an unbounded sequence of solutions generated by impulses.Remark 1.9.
In Theorem1.7, substituting ρ→+∞ and |u|→+∞ with ρ→0+ and |u|→0+, applying part (g) of Lemma 2.4 instead of part (f) in the proof, we can obtain a sequence of pairwise distinct solutions generated by impulses.
The paper is organized as follows. In Section2, we present some preliminaries. In Section 3, we give the proof of our main results.


## 2. Preliminaries

In order to prove our main results, we give some definitions and lemmas that will be used in the proof of our main results. Let(2.1)HT1={u:[0,T]→RN∣uisabsoltelycontinuous,u(0)=u(T),u̇∈L2([0,T],Rn)}.
Then, HT1 is a Hilbert space with the inner product(2.2)〈u,v〉=∫0T[(u(t),v(t))+(u̇(t),v̇(t))]dt,∀u,v∈HT1,
where (·,·) denotes the inner product in ℝN. The corresponding norm is defined by (2.3)‖u‖=(∫0T[|u̇(t)|2+|u(t)|2]dt)1/2,∀u∈HT1.
Let ∥·∥2=(∫0T|u(t)|2)1/2 denote the norm of Banach space of Lp([0,T],ℝN). Since (HT1,∥·∥) is compactly embedded in C([0,T],ℝN) (see [24]), we claim that there exists a positive constant C1 such that (2.4)‖u‖∞≤C1‖u‖,
where ∥u∥∞=maxt∈[0,T]|u(t)|.To study the problem (Pλ), we consider the functional I defined by (2.5)I(u)=∫0T[12|u̇|2-V(t,u)]dt+λ∑k=1mGk(u(sk)).
Similar to the proof of Lemma 1 of [22] (see also [20, 21]), we can easily prove the following Lemma 2.1.Lemma 2.1.
SupposeV∈C1[0,T]×ℝN→ℝ, Gk∈C1(ℝN,ℝN), k=1,2,…,m. Then, I is Frechét differentiable with
(2.6)I′(u)v=∫0T[u̇v̇-Vu(t,u)v]dt+λ∑k=1mgk(u(sk))v(sk),
for any u and v in HT1. Furthermore, u is a solution of (Pλ) if and only if u is a critical point of I in HT1.Lemma 2.2 (see [24, 25]).
LetE be a real Banach space and I∈C1(E,ℝ) satisfying the (P.S.) condition. Suppose I(0)=0 and(I1)
there are constantsρ,β>0 such that I|∂Bρ≥β, where Bρ={x∈E∣∥x∥<ρ},(I2)
there is ane∈E∖Bρ such that I(e)≤0.Then, I possesses a critical value c≥β.IfE is a real Banach space, denote by 𝒲X the class of all functionals Φ:E→ℝ possessing the following property: if {un} is a sequence in E converging weakly to u and liminfn→∞Φ(un)≤Φ(u), then {un} has a subsequence converging strongly to u. For example, if E is uniformly convex and g:[0,+∞)→ℝ is a continuous, strictly increasing function, then, by a classical results, the functional u→g(∥u∥) belongs to the class 𝒲X.Lemma 2.3 (see [26]).
LetE be a separable and reflexive real Banach space; let Φ:E→ℝ be a coercive, sequentially weakly lower semicontinuous C1 functional, belonging to 𝒲X, bounded on each bounded subset of E, and whose derivative admits a continuous inverse on E*; J:E→ℝ a C1 functional with compact derivative. Assume that Φ has a strict local minimum u0 with Φ(u0)=J(u0)=0. Finally, setting
(2.7)α′=max⁡{0,limsup⁡‖u‖→+∞J(u)Φ(u),limsup⁡u→u0J(u)Φ(u)},β′=sup⁡u∈Φ-1(0,+∞)J(u)Φ(u),
assume that α′<β′. Then, for each compact interval [a,b]⊂(1/β′,1/α′) (with the conventions 1/0=+∞,1/+∞=0), there exists B>0 with the following property: for every λ∈[a,b] and every C1 functional Ψ:E→ℝ with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation
(2.8)Φ′(x)=λJ′(x)+μΨ′(x)
has at least three solutions in E whose norms are less than B.Now, we recall a result which insures infinitely critical points. For allr>infXΦ, we put (2.9)ψ(r)=inf⁡x∈Φ-1((-∞,r))(sup⁡x∈Φ-1((-∞,r))Ψ(x))-Ψ(x)r-Φ(x),γ∶=lim⁡⁡inf⁡r→+∞ψ(r),δ∶=lim⁡⁡inf⁡r→(inf⁡XΦ)+ψ(r).Lemma 2.4 (see [27]).
LetX be a reflexive real Banach space, let Φ:X→R be a (strongly) continuous, coercive, and sequentially weakly lower semicontinuous and Gâteaux differentiable function, and let Ψ:X→R be a sequentially weakly upper semicontinuous and Gâteaux differentiable function. One has(e)
for everyr>infXΦ and every λ∈(0,1/ψ(r)), the restriction of the functional Φ-λΨ to Φ-1((-∞,r)) admits a global minimum, which is a critical point (local minimum) of Φ-λΨ in X.(f)
ifγ<+∞, then, for each λ∈(0,1/γ), the following alternative holds:
either(f1)
Φ-λΨ possesses a global minimumor(f2)
there is a sequence{xn} of critical points (local minima) of Φ-λΨ such that
(2.10)lim⁡n→+∞⁡Φ(xn)=+∞.(g)
ifδ<+∞, then, for each λ∈(0,1/δ), the following alternative holds:
either(g1)
here is a global minimum ofΦ which is a local minimum of Φ-λΨor(g2)
there is a sequence of pairwise distinct critical points (local minima) ofΦ-λΨ, with limn→+∞Φ(xn)=infXΦ which weakly converges to a global minimum of Φ.


## 3. Proof of the Main Results

In this section, we give the proof of our main results in turn.Ci,i=1,2,… denote different constants.Proof of Theorem1.2.
We firstly show thatI satisfies the P.S. condition. Assume that {uj}j∈ℕ⊂HT1 is a sequence such that {I(uj)}j∈ℕ is bounded and I′(uj)→0 as j→+∞. Then, there exists a constant C2>0 such that
(3.1)|I(uj)|≤C2,|I′(uj)|≤C2,∀j∈N.
By (3.1), (2.4), (2.5), (2.6), (V1′), (V2′), and (g1), we have
(3.2)2C2+C3‖uj‖≥2I(uj)-2θI′(uj)uj=(1-2θ)∫0T|u̇j|2dt+∫0T[2θVu(t,uj)uj-2V(t,uj)]dt+2∑k=1mGk(uj(sk))-2θ∑k=1mgk(uj(sk))uj(sk)≥(1-2θ)∫0T|u̇j|2dt+2(1-ϱθ)∫0Tb|uj|γdt≥(1-2θ)∫0T|u̇j|2dt+2(1-ϱθ)bC1γ-2‖uj‖γ-2∫0T|uj|2dt≥min⁡{(1-2θ),2(1-ρθ)bC1γ-2‖uj‖γ-2}‖uj‖2≥min⁡{(1-2θ)‖uj‖2,2(1-ϱθ)bC1γ-2‖uj‖γ}.
It follows from (3.2) that {uj}j∈ℕ is bounded in HT1. In a similar way to Lemma  2 in [22], we can prove that {uj}j∈ℕ has a convergent subsequence in HT1. Hence, I satisfies the P.S. condition.
Second, we verify(I1) of Lemma 2.2. By (g1) and (2.4), there exists a δ1>0 such that, for any ∥u∥∞≤δ1,
(3.3)|Gk(u)|≤14mC1|u|2.
By (2.4), there exists a 0<δ<1 such that, for any ∥u∥≤δ, the inequality ∥u∥∞≤δ1 holds. Then for u∈HT1 with ∥u∥=δ0, δ0 small enough (0<δ0<min{(2bC1γ-2)1/2-γ,δ}) such that the following inequality holds
(3.4)I(u)≥12∫0T|u̇(t)|2dt+b∫0T|u(t)|γdt-14C1‖u‖∞2≥min⁡{12‖u‖2,bC1γ-2‖u‖γ}-14‖u‖2≥12‖u‖2-14‖u‖2≥14‖δ0‖2.
By (V2′), there exist C3,C4>0 such that
(3.5)-V(t,u)≤C3|u|ϱ+C4,∀(t,u)∈[0,T]×Rn.
By (g1), there exist C5,C6>0 such that
(3.6)Gk(u)≤-C5|u|θ+C6,∀u∈Rn,k∈{1,2,…,m}.
To verify (I2) of Lemma 2.2, choose u∈HT1 such that u(sk)≠0 for some k∈{1,2,…,m}. Hence, we obtain
(3.7)I(tu)≤t22∫0T|u̇(t)|2dt+C3|t|ϱ∫0T|u(t)|ϱdt-C5|t|θ∑k=1m|u(sk)|θ+C7,
where C7 is a positive constant. Since θ>2, ϱ∈[2,θ), (3.7) implies I(tu)→-∞ as t→∞. So, for t large enough, e=tu satisfies condition (I2). By the Mountain Pass Lemma (Lemma 2.2), I possesses at least one non-zero critical point. Then, by Lemma 2.1, problem (P1) has at least one non-zero solution.
Finally, we verify that the solution is generated by impulses. Supposegk≡0 and u is a solution for problem (P1). Then, by (V1′) and (V2′),
(3.8)0=∫0T[|u̇|2-Vu(t,u)u]dt≥∫0T[|u̇|2+b|u|γ]dt≥∫0T|u̇|2dt+b‖u‖∞γ-2∫0T|u|2dt≥∫0T|u̇|2dt+bC1γ-2‖u‖γ-2∫0T|u|2dt≥min⁡{1,bC1γ-2‖u‖γ-2}‖u‖2=min⁡{‖u‖2,bC1γ-2‖u‖γ}.
This implies that u≡0, that is, problem (P1) does not possess any non-zero solutions when impulses are zero. Hence, by Definition 1.1, the non-zero solution obtained above is generated by impulses.Proof of Theorem1.4.
In order to apply Lemma2.3, we let
(3.9)Φ(u)=∫0T[12|u̇(t)|2-V(t,u(t))]dt,J(u)=-∑k=1mGk(u(sk)).
Obviously, E=HT1 is a separable and uniformly convex Banach space. By (V1), we have
(3.10)b3‖u‖2≤Φ(u)≤b4‖u‖2,
where b3=min{1/2,b1}andb4=min{1/2,b2}. Hence, by (3.10) and Lemma 2.1, we can obtain that Φ(u) is a coercive, C1 functional, and bounded on each bounded subset of E. For a sequence {un}⊂HT1, if un⇀u∈HT1 and liminfn→∞Φ(un)≤Φ(u), then un→u∈C[0,T]. This means that ∫0T[(-1/2)|u(t)|2-V(t,u(t))]dt is weakly continuous. Hence, we have that Φ(u)=(1/2)∥u∥2+∫0T[(-1/2)|u(t)|2-V(t,u(t))]dt is sequentially weakly lower semicontinuous and liminfn→∞∥un∥2≤∥u∥2. Therefore, {un} has a subsequence converging strongly to u, that is, Φ(u) belongs to 𝒲X. For any u∈HT1∖{0}, we have
(3.11)〈Φ′(u),u〉=∫0T[(u̇(t),u̇(t))-(Vu(t,u(t)),u(t))]dt≥∫0T[(u̇(t),u̇(t))+b1|u(t)|2]dt≥min⁡⁡{1,b1}‖u‖2.
So, lim∥u∥→∞〈Φ′(u),u〉/∥u∥=+∞, that is, Φ′ is coercive. For any u,v∈HT1, one has
(3.12)〈Φ′(u)-Φ′(v),u-v〉=∫0T(u̇(t)-v̇(t),u̇(t)-v̇(t))dt-∫0T(Vu(t,u(t))-Vu(t,v(t)),u(t)-v(t))dt≥∫0T(u̇(t)-v̇(t),u̇(t)-v̇(t))dt+∫0Tb1|u(t)-v(t)|2dt≥min⁡{1,b1}‖u-v‖2.
So, Φ′ is uniformly monotone. By [28, Theorem  26.A(d)], we have that (Φ′)-1 exists and is continuous. For any u,v∈HT1,
(3.13)〈J′(u),u〉=-∑k=1mgk(u(sk))u(sk).
Let un⇀u∈HT1, then un→u∈C[0,T]. Hence, J′(un)→J′(u), Ψ′(un)→Ψ′(u) as n→∞. Therefore, we have that J′ is a compact operator by [28, Proposition  26.2]. In addition, Φ has a strict local minimum 0 with Φ(0)=J(0)=0. In view of (H1), there exist τ1>τ2>0 such that
(3.14)-∑k=1mGk(u)≤A|u|2
for any |u|∈[0,τ2)∪(τ1,+∞). By the continuity of Gk, ∑k=1mGk(u) is bounded for any |u|∈[τ2,τ1]. We can choose C8>0 and σ>2 such that
(3.15)-∑k=1mGk(u)≤A|u|2+C8|u|σ
for any u∈ℝN. Hence, by (2.4), we have
(3.16)J(u)≤AC12‖u‖2+C8C1σ‖u‖σ
for all u∈HT1. By (3.16) and (H1), we obtain
(3.17)limsup⁡u→0J(u)Φ(u)≤lim⁡⁡sup⁡u→0(AC12b3+C8C1σb3‖u‖σ-2)≤AC12b3<1.
On the other hand, by (H1), -∑k=1mGk(u)≥0. Then, one has
(3.18)-∑k=1mGk(u(sk))Φ(u)=-∑k=1mGk(u(sk))||u(sk)|≤τ1Φ(u)+-∑k=1mGk(u(sk))||u(sk)|>τ1Φ(u)≤C9Φ(u)+-∑k=1mGk(u(sk))||u(sk)|>τ1Φ(u)≤C9b3‖u‖2+A|u(sk)|2b3‖u‖2≤C9b3‖u‖2+AC12‖u‖2b3‖u‖2,
where C9 is a positive constant and f|A denotes the restriction of the functional f to the set A. Hence, we have
(3.19)limsup⁡‖u‖→∞J(u)Φ(u)≤limsup⁡‖u‖→∞(C9b3‖u‖2+AC12b3)≤AC12b3<1.
Combining (3.18) with (3.19), one has α′<1.
By{H2}, we obtain
(3.20)β′=sup⁡u∈Φ-1(0,+∞)J(u)Φ(u)≥-∑k=1mGk(ξ)-∫0TV(t,ξ)dt>1.
Then, for each compact interval [a,b]⊂(1/β′,1/α′), 1∈[a,b], there exists B>0 with the following property: for every f(t), there exists δ>0 such that, for each μ∈[0,δ],
(3.21)Φ′(x)=J′(x)+μΨ′(x).
Let μ=0, then we can obtain that problem (P1) has at least three solutions in E whose norms are less than B.Finally, by Theorem  4 in [22], we can prove that problem (P1) does not possess any non-zero periodic solutions when impulses are zero. By Definition 1.1, problem (P1) has at least two solutions generated by impulses.Proof of Theorem1.7.
Let(3.22)Φ(u)=∫0T[12|u̇(t)|2-V(t,u(t))]dt,Ψ(u)=-∑k=1mGk(u(sk)).
Similar to the proof of Theorems  3.1 and 3.3 in [27], it is easy to see that the functionals Φ,Ψ satisfy the regularity assumptions required in Lemma 2.4. Let us now verify that
(3.23)γ<+∞.
Let {ρn} be a sequence of positive numbers such that ρn→+∞ and
(3.24)lim⁡n→+∞max⁡|u|≤ρn[-∑k=1mGk(u)]ρn2=lim⁡inf⁡ρ→+∞max⁡|u|≤ρn[-∑k=1mGk(u)]ρ2.
If we put, for every n∈ℕ,
(3.25)rn=ρn2b3C12,
then, by (2.4) and (3.10), Φ(u)≤rn implies that ∥u∥∞≤ρn. Since 0∈Φ-1((-∞,rn)), the following inequality holds:
(3.26)ψ(rn)=inf⁡x∈Φ-1((-∞,rn))(sup⁡x∈Φ-1((-∞,rn))Ψ(x))-Ψ(x)rn-Φ(x)≤C12max⁡|u|≤ρn[-∑k=1mGk(u)]ρn2b3.
Hence,
(3.27)0<γ≤C12αb3<+∞.
From λ∈Λ∶=(Tb2/β,b3/C12α), we have
(3.28)Λ⊂(0,1γ).
Fix λ∈Λ and let us verify that the functional I=Φ-λΨ is unbounded from below. In fact, by the choice of λ and the positivity of β, one has that there exists a sequence {yn} in ℝn such that limn→∞|yn|=+∞ and -∑k=1mGk(yn)/|yn|2>Tb2/λ. Let vn(t)=yn. Then,
(3.29)I(vn)=∫0T[-V(t,vn)]dt-λ[-∑k=1mGk(vn)]≤(Tb2-λ-∑k=1mGk(yn)|yn|2)|yn|2.
Note that Tb2-λ(-∑k=1mGk(yn)/|yn|2)<0, I has no global minimum. Hence, by Lemma 2.4, there is a sequence {xn} of critical points (local minima) of Φ-λΨ such that limn→+∞Φ(xn)=+∞. By Theorem  4 in [22], we can prove that problem (Pλ) does not possess any non-zero periodic solutions when impulses are zero. Therefore, by Definition 1.1, the solutions obtained above are all generated by impulses.



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*Source: 310957-2011-06-07.xml*