# Variational Approach to Impulsive Problems: A Survey of Recent Results

Authors: Fang-fang Liao; Juntao Sun
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We present a survey on the existence of nontrivial solutions to impulsive differential equations by using variational methods, including solutions to boundary value problems, periodic solutions, and homoclinic solutions.


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## Body

## 1. Introduction

There are many processes and phenomena in the real world, which are subjected during their development to the short-term external influences. Their duration is negligible compared with the total duration of the studied phenomena and processes. Therefore, it can be assumed that these external effects are “instantaneous”; that is, they are in the form of impulses. The investigation of such “leaps and bounds” developing dynamical states is a subject of different sciences: mechanics, control theory, pharmacokinetics, epidemiology, population dynamics, economics, ecology, and so forth. We refer the reader to [1–6].During the last two decades, the theory of the existence of solutions for impulsive differential equations has attracted much attention because of its important applications; see [3, 7–12]. Classical approaches to such problems include fixed point theory and the method of upper and lower solutions. In 2008, Tian and Ge first applied variational methods in the study of boundary value problems (BVPs) of impulsive differential equations. From then on, variational methods have been widely used to study impulsive problems, such as boundary value problems, periodic solutions, and homoclinic solutions. We refer the reader to [3, 13–22].The main aim of this survey is to present some recent existence results for impulsive differential equations via variational methods. More precisely, we will explore the variational framework of impulsive differential equations and study the existence and multiplicity of solutions of boundary value problems, periodic solutions, and homoclinic solutions.The rest of this paper is organized as follows. In Section2, we will state some important results for the second-order impulsive differential equations with two kinds of boundary conditions. Periodic solutions to impulsive problems will be considered in Section 3. In Section 4, homoclinic solutions to impulsive differential equations will be studied.Finally in this section, we must note that besides the results presented in this survey many interesting and important results on impulsive differential equations via variational methods have been obtained by other researchers; see, for example, [23–31] and the references cited therein.


## 2. Boundary Value Problems

In this section, we recall some results of boundary value problems for impulsive differential equations.Firstly, we consider a class of impulsive problems with Dirichlet boundary condition(1)-u′′(t)+λu(t)=f(t,u(t))+g(t),a.e.t∈[0,T],Δu′(tj)=u′(tj+)-u′(tj-)=Ij(u(tj)),j=1,2,…,p,u(0)=u(T)=0,
where λ is a constant, f:[0,T]×R→R is continuous, g∈L2[0,T] and Ij:R→R, j=1,2,...,p, are continuous.From now on, we assume thatλ>-(π2/T2) and define
(2)L(u,u)=12∫0T‍|u′(t)|2dt+λ2∫0T‍|u(t)|2dt,000000000000000000foranyu∈H01(0,T).
Then we have the following.Proposition 1 (see [21, Proposition 1.1]).
Ifλ>-(π2/T2), then there exist constants b>a>0 such that
(3)a∥u∥H012≤L(u,u)≤b∥u∥H012,foranyu∈H01(0,T).Definition 2.
One says that a functionu∈H01(0,T) is a weak solution to problem (1) if
(4)∫0T(u′(t),v′(t))+λ(u(t),v(t))dt+∑j=1pIj(u(tj))v(tj)=∫0Tf(t,u)vdt+∫0Tg(t)vdt
holds for any v∈H01(0,T).Theorem 3 (see [15, Theorem 4.2]).
Suppose thatg≡0, f is bounded, and the impulsive functions Ij are bounded. Then problem (1) has at least one weak solution to λ>-(π2/T2).Iff is sublinear at infinity on u and the impulsive functions Ij have sublinear growth, then we have the following.Theorem 4 (see [15, Theorem 4.3]).
Assume thatg≡0 and the following conditions are satisfied. (F1)
There exista,b>0 and γ∈(0,1) such that
(5)|f(t,u)|≤a+b|u|γ,forevery(t,u)∈[0,T]×R;(I1)
there existaj,bj>0 and γj∈[0,1)(j=1,2,…,p) such that
(6)|Ij(u)|≤aj+bj|u|γj,foreveryu∈R(j=1,2,…,p).Then problem (1) has at least one weak solution to λ>-(π2/T2).Iff is superlinear at infinity on u, then we have the following.Theorem 5 (see [21, Theorem 1.2]).
Assume that(I1) holds and the following conditions are satisfied. (F2)
There is a constantμ>2 such that 0<μF(t,u)≤uf(t,u) for every t∈[0,T] and u∈R∖{0};(G)
g∈L2[0,T] satisfies the following inequality
(7)a-Mλ1-∑j=1pT(aj+bj)-T∥g∥L2λ1>0,where M≔sup⁡⁡{F(t,u):t∈[0,T],|u|=1} and a is defined in Proposition 1.
Then problem (1) has at least one weak solution to λ>-(π2/T2).Furthermore, we have the following result of multiple solutions.Theorem 6 (see [21, Theorem 1.3]).
Letg(t)≡0. Assume that the conditions (F2), (I1), and (G) hold. Moreover, f and the impulsive functions Ij are odd about u; then problem (1) has infinitely many solutions.Next, we consider another Dirichlet impulsive problem(8)-u′′(t)+q(t)u(t)=f(t,u(t)),a.e.t∈[0,T],Δu′(tj)=u′(tj+)-u′(tj-)=Ij(u(tj)),j=1,2,…,p,u(0)=u(T)=0,
where q∈L∞[0,T], f:[0,T]×R→R is continuous, and Ij:R→R, j=1,2,…,p, are continuous.Theorem 7 (see [20, Theorem 3.1]).
Letdj(j=1,2…,p) be fixed constants. If f(t,u)=σ(t)∈L2(0,T) and Ij(u)=dj(j=1,2,…,p); then problem (8) has a unique solution u if ess inf⁡t∈[0,T]q(t)>-(π2/T2).Similar to Theorem4, we have the following.Theorem 8 (see [20, Theorem 3.2]).
Assume that conditions(F1) and (I1) hold. Then problem (8) has at least one weak solution to ess inf⁡t∈[0,T]q(t)>-(π2/T2).If condition(I1) is replaced by the following condition:(I3)
Ij(j=1,2…,p) satisfy ∫0u‍Ij(s)ds≥0 for all u∈R,then we have the following.Theorem 9 (see [30, Theorem 3.1]).
Assume that conditions(F1) and (I3) hold. Then problem (8) has at least one weak solution to ess inf⁡t∈[0,T]q(t)>-(π2/T2).Now we study a class of impulsive Hamiltonian systems with periodic boundary conditions(9)-u¨+A(t)u=λ∇F(t,u)+μ∇G(t,u),a.e.t∈[0,T],Δ(u˙i(tj))=u˙i(tj+)-u˙i(tj-)=Iij(ui(tj)),000000000000i=1,2,…,N,j=1,2,…,l,u(0)-u(T)=u˙(0)-u˙(T)=0,
where A:[0,T]→RN×N is a continuous map from the interval [0,T] to the set of N-order symmetric matrices, λ, μ are two positive parameters, T is a real positive number, u(t)=(u1(t),u2(t),…,uN(t)), tj,j=1,2,…,l, are the instants where the impulses occur and 0=t0<t1<t2<⋯<tl<tl+1=T, Iij:R→R(i=1,2,…,N,j=1,2,…,l) are continuous, and F,G:[0,T]×RN→R are measurable with respect to t, for every u∈RN, continuously differentiable in u, for almost every t∈[0,T], and satisfy the following standard summability condition:
(10)sup⁡|u|≤bmax⁡⁡{|F(·,u)|,|G(·,u)|,|∇F(·,u)|,000000000|∇G(·,u)|}∈L1(0,T),∀b>0.IfX is a real Banach space, denote by WX the class of all functionals Φ:X→R possessing the following property: if {un} is a sequence in X converging weakly to u∈X and liminf⁡n→∞Φ(un)≤Φ(u), then {un} has a subsequence converging strongly to u.For example, ifX is uniformly convex and g:[0,+∞)→R is a continuous, strictly increasing function, then, by a classical result, the functional u→g(∥u∥) belongs to the class WX.Theorem 10 (see [32]).
LetX be a separable and reflexive real Banach space; let Φ:X→R be a coercive, sequentially weakly lower semicontinuous C1 functional, belonging to WX, bounded on each bounded subset of X, whose derivative admits a continuous inverse on X*; J:X→R  is a C1 functional with compact derivative. Assume that Φ has a strict local minimum u0 with Φ(u0)=J(u0)=0. Finally, setting
(11)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 R>0 with the following property: for every λ∈[a,b] and every C1 functional Ψ:X→R with compact derivative, there exists δ>0 such that, for each μ∈[0,δ],
(12)Φ′(x)=λJ′(x)+μΨ′(x)
has at least three solutions in X whose norms are less than R.The following two results of Ricceri guarantee the existence of three solutions for a given equation.Theorem 11 (see [33]).
LetX be a reflexive real Banach space; I⊆R is an interval; let Φ:X→R be a sequentially weakly lower semicontinuous C1 functional, bounded on each bounded subset of X, whose derivative admits a continuous inverse on X*; -J:X→R is a C1 functional with compact derivative. Assume that
(13)lim⁡∥x∥→+∞(Φ(x)-λJ(x))=+∞
for all λ∈I and that there exists ρ∈R such that
(14)sup⁡λ∈Iinf⁡x∈X(Φ(x)+λ(ρ-J(x)))<inf⁡x∈Xsup⁡λ∈I(Φ(x)+λ(ρ-J(x))).
Then there exist a nonempty open set A⊆I and a positive number R with the following property: for every λ∈A and every C1 functional -Ψ:X→R with compact derivative, there exists δ>0 such that, for each μ∈[0,δ],
(15)Φ′(x)-λJ′(x)-μΨ′(x)=0
has at least three solutions in X whose norms are less than R.Proposition 12 (see [34]).
LetX be a nonempty set and Φ,J two real functions on X. Assume that there are r>0 and x0,x1∈X such that
(16)Φ(x0)=J(x0)=0,Φ(x1)>r,sup⁡x∈Φ-1((-∞,r])J(x)<rJ(x1)Φ(x1).
Then, for each ρ, satisfying
(17)sup⁡x∈Φ-1((-∞,r])J(x)<ρ<rJ(x1)Φ(x1),
one has
(18)sup⁡λ≥0inf⁡x∈X(Φ(x)+λ(ρ-J(x)))<inf⁡x∈Xsup⁡λ≥0(Φ(x)+λ(ρ-J(x))).We always assume that the following conditions hold:(A1)
A(t)=(aij(t)) is a symmetric matrix with aij∈L∞([0,T]) for every t∈[0,T];(A2)
there exists a positive constantν such that
(19)A(t)u·u≥ν|u|2,foreveryu∈RN,a.e.t∈[0,T];(I1)
for anyi=1,2,…,N, and j=1,2,…,l, Iij(t) are nondecreasing in t∈R and
(20)Iij(t)t≥0,∀t∈R.Define the Sobolev space by
(21)HT1≔{L2u:[0,T]⟶RN:uisabsolutelycontinuous,0000u(0)=u(T),andu˙∈L2(0,T)}
with the inner product
(22)〈u,v〉=∫0T(u(t),v(t))dt+∫0T(u˙(t),v˙(t))dt,hhhhhhhhhhhhhhhhhhhhhhhhhh∀u,v∈HT1,
where (·,·) denotes the inner product in RN. The corresponding norm is defined by
(23)∥u∥HT1=(∫0T|u(t)|2dt+∫0T|u˙(t)|2dt)1/2,∀u∈HT1.
For every u,v∈HT1, we define
(24)〈u,v〉1=∫0T(A(t)u(t),v(t))dt+∫0T(u˙(t),v˙(t))dt,∀u,v∈HT1
and observe that, using assumptions (A1) and (A2), (24) defines an inner product in HT1, whose corresponding norm is
(25)∥u∥≔〈u,u〉11/2.
A simple computation shows that
(26)(A(t)u,u)=∑i,j=1Nai,j(t)uiuj≤∑i,j=1N∥ai,j∥∞|u|2,
for every t∈[0,T] and u∈RN. So, putting together (A2) and (26), one has
(27)m∥u∥HT1≤∥u∥≤M∥u∥HT1,
where m=min⁡⁡{1,ν} and M=max⁡⁡{1,∑i,j=1N‍∥ai,j∥∞}; that is, the norm ∥·∥ is equivalent to (23). At this point, since (HT1,∥·∥HT1) is compactly embedded in C(0,T) (see [35]), due to (27) we claim that there exists a positive number k such that
(28)∥u∥C≤k∥u∥,
where k=(1/m)max⁡⁡{T,1/T} and ∥u∥C=max⁡t∈[0,T]|u(t)|.Ifu∈HT1, then u is absolutely continuous and u˙∈L2(0,T;RN). In this case, Δu˙(t)=u˙(t+)-u˙(t-)=0 is not necessarily valid for every t∈(0,T) and the derivative u˙ may present some discontinuities. This leads to the impulsive effects.Following the ideas of [15], take v∈HT1 and multiply the two sides of the equality
(29)-u¨+A(t)u-λ∇F(t,u)-μ∇G(t,u)=0
by v and integrate from 0 to T; we have
(30)∫0T[-u¨+A(t)u-λ∇F(t,u)-μ∇G(t,u)]vdt=0.
Moreover, combining u˙(0)-u˙(T)=0, one has
(31)∫0T(u¨(t),v(t))dt=∑j=0l∫tjtj+1(u¨(t),v(t))dt=∑j=0l((u˙(tj+1-),v(tj+1-))00000-(u˙(tj+),v(tj+)))-∑j=0l∫tjtj+1(u˙(t),v˙(t))dt=∑j=0l∑i=1N(u˙i(tj+1-)vi(tj+1-)00000000-u˙i(tj+)vi(tj+))-∑j=0l∫tjtj+1(u˙(t),v˙(t))dt=u˙(T)v(T)-u˙(0)v(0)-∑j=1l∑i=1NΔu˙i(tj)vi(tj)-∫0T‍(u˙(t),v˙(t))dt=-∑j=1l∑i=1N‍Iij(ui(tj))vi(tj)-∫0T‍(u˙(t),v˙(t))dt.
Combining (30), we get
(32)∫0T(u˙(t),v˙(t))+(A(t)u(t),v(t))dt+∑j=1l∑i=1NIij(ui(tj))vi(tj)-λ∫0T∇F(t,u)vdt-μ∫0T∇G(t,u)vdt=0.
Considering the above equality, we introduce the following concept of a weak solution to problem (9).Definition 13.
One says that a functionu∈HT1 is a weak solution to problem (9) if
(33)∫0T(u˙(t),v˙(t))+(A(t)u(t),v(t))dt+∑j=1l∑i=1NIij(ui(tj))vi(tj)=λ∫0T∇F(t,u)vdt+μ∫0T∇G(t,u)vdt
holds for any v∈HT1.In order to study problem (9), we will use the functionals Φ,J,Ψ:HT1→R defined by putting
(34)Φ(u)=12∥u∥2+∑j=1l∑i=1N∫0ui(tj)Iij(t)dt=ϕ(u)+ψ(u),J(u)≔∫0TF(t,u)dt,Ψ(u)≔∫0TG(t,u)dt,
respectively, for every u∈HT1; here,
(35)ϕ(u)=12∥u∥2,ψ(u)=∑j=1l∑i=1N∫0ui(tj)Iij(t)dt.
By the continuity of Iij, i=1,2,…,N, j=1,2,…,l, we get that functional Φ is a continuous Gâteaux differential functional whose Gâteaux derivative is the functional Φ′(u), given by
(36)〈Φ′(u),v〉=∫0T(A(t)u(t),v(t))+(u˙(t),v˙(t))dt+∑j=1l∑i=1NIij(ui(tj))vi(tj),
for any u,v∈HT1. On the other hand, condition (10) implies that J and Ψ are well defined, continuously Gâteaux differentiable in HT1. More precisely, their Gâteaux derivatives are
(37)〈J′(u),v〉=∫0T∇F(t,u)vdt,〈Ψ′(u),v〉=∫0T∇G(t,u)vdt,
respectively, for every u,v∈HT1. Put
(38)q=∑i,j=1N∥ai,j∥∞,λ1=inf⁡⁡{∥u∥2+2∑j=1l∑i=1N∫0ui(tj)Iij(t)dt2∫0TF(t,u(t))dt:u∈HT1,000000000∫0TF(t,u(t))dt>0∥u∥2+2∑j=1l∑i=1N∫0ui(tj)Iij(t)dt2∫0TF(t,u(t))dt},λ2=(max⁡⁡{0,limsup⁡∥u∥→+∞2∫0T‍F(t,u(t))dt∥u∥2+2∑j=1l∑i=1N∫0ui(tj)Iij(t)dt,0000000000limsup⁡|u|→02∫0TF(t,u(t))dt∥u∥2+2∑j=1l∑i=1N∫0ui(tj)Iij(t)dt})-1D=(T-tl)2t1tl2+t13tl2(tl2+tlT+T2)+(tl-t1)+T-tltl2+13tl2(T3-tl3)>0.
Then we have the following.Theorem 14 (see [16, Theorem 3.1]).
Suppose that(A1)-(A2) and (I1) hold. Moreover, there exist a constant p>0 and constant vector ξ¯=(ξ¯1,…,ξ¯N)∈RN∖{0} such that (R1)
max⁡{limsup⁡u→0(max⁡t∈[0,T]F(t,u)/|u|2),  limsup⁡|u|→+∞(max⁡t∈[0,T]F(t,u)/|u|2)}<p;(R2)
pTk2<(∫0T‍F(t,ξ¯)dt/(qT|ξ¯|2+2∑j=1l∑i=1N∫0ξ¯i‍Iij(t)dt)).
Then, for each compact interval[a,b]⊂(λ1,λ2), there exists R>0 with the following property: for every λ∈[a,b] and G∈C there exists δ>0 such that, for each μ∈[0,δ], problem (9) has at least three weak solutions whose norms are less than R.Theorem 15 (see [16, Theorem 3.2]).
Suppose that(A1)-(A2) and (I1) hold and there exist h,q∈L1(0,T;R+), three positive constants c,d,α with c>kdDm/2, and α∈[0,2) such that (H1)
max⁡t∈[0,T],|u|≤cF(t,u)<(Dmd2/T)(∫t1tl‍F(t,dE)dt/(2DMd2+4∑j=1l∫0d‍I1j(t)dt)); here, E=(1,0,…,0)∈RN;(H2)
F(t,u)≥0 for each t∈[0,t1]∪[tl,T], |u|≤dT/tl;(H3)
|F(t,u)|≤h(t)|u|α+q(t), for all u∈RN and a.e. t∈[0,T].Then, there exist a nonempty open set A⊂[0,+∞) and a positive number R with the following property: for every λ∈A and every G∈C there exists δ>0 such that, for each μ∈[0,δ], problem (9) has at least three weak solutions whose norms are less than R.As a special case ofF(t,u), let F(t,u)=b(t)f(u), where b(t)∈L1(0,T;R)∖{0}, f(u)∈C1(RN;R). Then we get the following two corollaries.Corollary 16 (see [16, Corollary 3.1]).
Suppose that(A1)-(A2) and (I1) hold. Moreover, there exist a constant p>0 and constant vector ξ¯=(ξ¯1,…,ξ¯N)∈RN∖{0} such that (R1)′
max⁡t∈[0,T]b(t)·max⁡{limsup⁡u→0(f(u)/|u|2),  limsup⁡|u|→+∞(f(u)/|u|2)}<p;(R2)′
pTk2<(f(ξ¯)∫0T‍b(t)dt/(qT|ξ¯|2+2∑j=1l∑i=1N∫0ξ¯i‍Iij(t)dt)).Then, for each compact interval [a,b]⊂(λ1,λ2), there exists R>0 with the following property: for every λ∈[a,b] and G∈C there exists δ>0 such that, for each μ∈[0,δ], problem (9) has at least three weak solutions whose norms are less than R.Corollary 17 (see [16, Corollary 3.2]).
Suppose that(A1)-(A2) and (I1) hold. Furthermore, there exist five positive constants c, d, α, β, and γ with c>kdDm/2 and α∈[0,2) such that (H1)′
max⁡|u|≤cf(u)<(Dmd2/∥b∥L1)  (f(dE)∫t1tl‍b(t)dt/(DMd2+2∑j=1l∫0d‍I1j(t)dt));(H2)′
b(t)f(u)≥0 for any t∈[0,t1]∪[tl,T], |u|≤dT/tl;(H3)′
|f(u)|≤γ+β|u|α for every u∈RN.Then, there exist a nonempty open set A⊂[0,+∞) and a positive number R with the following property: for every λ∈A and every G∈C there exists δ>0 such that, for each μ∈[0,δ], problem (9) has at least three weak solutions whose norms are less than R.Finally, we consider the existence of infinitely many solutions for a class of impulsive systems with periodic boundary conditions(39)-u¨+A(t)u=∇W(t,u),a.e.t∈[0,T],Δ(u˙i(tj))=u˙i(tj+)-u˙i(tj-)=Iij(ui(tj)),hhhhhhhhhhhhi=1,2,…,N,j=1,2,…,l,u(0)-u(T)=u˙(0)-u˙(T)=0,
where A:[0,T]→RN×N is a continuous map from the interval [0,T] to the set of N-order symmetric matrices, T is a real positive number, u(t)=(u1(t),u2(t),…,uN(t)), tj, j=1,2,…,l, are the instants where the impulses occur and 0=t0<t1<t2<⋯<tl<tl+1=T, Iij:R→R(i=1,2,…,N,j=1,2,…,l) are continuous, and W:[0,T]×RN→R are measurable with respect to t, for every u∈RN, continuously differentiable in u, for almost every t∈[0,T], and satisfy the following standard summability condition:
(40)sup⁡|u|≤bmax⁡⁡{|W(·,u)|,|∇W(·,u)|}∈L1(0,T),∀b>0.Now we state our main results.Theorem 18 (see [36, Theorem 1.1]).
Assume thatW is even in u and the following conditions hold: (H1)
Iij(i=1,…,N,j=1,2,…,l) are odd and satisfy
(41)2∫0sIij(t)dt-Iij(s)s≥0,∫0sIij(t)dt≥0∀s∈R;(H2)
there exist constantsaij,bij>0 and γij∈[0,1) such that
(42)|Iij(s)|≤aij+bij|s|γijfors∈R;(H3)
lim⁡|u|→+∞(W(t,u)/u2)=+∞ uniformly for t∈[0,T];(H4)
W(t,0)≡0, 0≤W(t,u)=o(|u|2) as |u|→0 uniformly for t∈[0,T];(H5)
there exist constantsα>1, 1<β<1+((α-1)/α), c1,c2>0, and L>0 such that for every t∈[0,T] and u∈RN with |u|≥L,
(43)(∇W(t,u),u)-2W(t,u)≥c1|u|α,|∇W(t,u)|≤c2|u|β.Then problem (39) has infinitely many weak solutions.Theorem 19 (see [36, Theorem 1.2]).
Assume that the following conditions hold:(S1)
Iij(i=1,2,…,N,j=1,2,…,l) are odd and satisfy Iij(s)s≥0 for any s∈R;(S2)
there exist constantsbij>0 and γij∈[0,1) such that
(44)|Iij(s)|≤bij|s|γijfors∈R;(S3)
W(t,u)=(ρ/2)|u|2+F(t,u), where ρ is a positive constant and not a spectrum point of -(d2/dt2)+A, F is even in u, and F(t,0)≡0;(S4)
there existδ1,δ2∈[1,2) with δ1<min⁡1≤i≤N,1≤j≤l{γij}+1, and d1,d2>0 such that
(45)d1|u|δ1≤F(t,u),|∇F(t,u)|≤d2|u|δ2-1,hhhhhhhhhhhhhhh∀(t,u)∈[0,T]×RN.Then problem (39) has infinitely many weak solutions.Theorem 20 (see [36, Theorem 1.3]).
Assume thatW is even in u, (S1)-(S2) and the following conditions hold: (J1)
W(t,0)≡0 for any t∈[0,T];(J2)
there are constantsk1>0 and ζ1∈[1,2) with ζ1<min⁡1≤i≤N,1≤j≤l{γij}+1 such that
(46)W(t,u)≥k1|u|ζ1,forany(t,u)∈[0,T]×RN;(J3)
there exist constantsk2>0 and ζ2∈[1,2) such that
(47)|∇W(t,u)|≤k2|u|ζ2-1,forany(t,u)∈[0,T]×RN.Then problem (39) has infinitely many weak solutions.


## 3. Periodic Solutions

Firstly, we consider periodic solutions for second-order regular impulsive problems with a perturbation:(48)u′′(t)-Vu(t,u)=f(t),fort∈(sk-1,sk),Δu′(sk)=gk(u(sk)),
where k∈Z, u∈Rn, Δu′(sk)=u′(sk+)-u′(sk-) with u′(sk±)=lim⁡t→sk±, Vu(t,u)=graduV(t,u), f∈C(R,Rn), gk(u)=graduGk(u), Gk∈C1(Rn,Rn) for each k∈Z, and there exist m∈N and T∈R+ such that 0=s0<s1<⋯<sm=T, sk+m=sk+T, and gk+m≡gk for all k∈Z (i.e., gk is m-periodic in k).Theorem 21 (see [20, Theorem 1]).
Assume thatf is continuous and T-periodic and gk is continuous, m-periodic in k. Furthermore, assume that V and gk satisfy the following conditions: (V1)
V:R×Rn→R is continuous differentiable and T-periodic, and there exist positive constants b1,b2>0 such that
(49)b1|u|2≤-V(t,u)≤|u|2,∀(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 existsμ>2 such that
(50)gk(u)u≤μGk(u)<0,foru∈Rn∖{0},k=1,2,…,m.Then there exists b-f>0 such that, for any positive integer l, if bf,1≤b-fl-1/2, then problem (48) possesses at least one nonzero lT-periodic solution.Theorem 22 (see [20, Theorem 1]).
Suppose thatV is even about u and satisfies (V1)-(V2), f≡0, and gk is continuous, odd about u, m-periodic in k, and satisfies the following: (g2)
for allk=1,2,…,m,
(51)liminf⁡|u|→∞Gk(u)|u|2≥0.Then there exists Cg>0 such that if
(52)limsup⁡|u|→0Gk(u)|u|2≤-Cg,
then problem (48) possesses at least lmn distinct pairs of lT-periodic solutions generated by impulses.Theorem 23 (see [14, Theorem 1]).
Assume thatf≡0 and gk is continuous, m-periodic in k satisfying (g1). Furthermore, assume that V satisfies the following condition: (V3)
V:R×Rn→R is continuous differentiable and T-periodic, and V(t,u)≥(1/2)Vu(t,u)u>0 for all t∈[0,T] and u∈Rn∖{0}.Then problem (48) possesses one nonzero periodic solution generated by impulses.Corollary 24 (see [14, Corollary 1]).
Assume thatf≡0 and gk is continuous, m-periodic in k satisfying (g1). Furthermore, assume that V satisfies the following condition: (V4)
Vu(t,u)=αu+Wu(t,u) for all t∈[0,T] and u∈Rn, where α>2/μ for some μ>2 and W(t,u)≥(1/2)Wu(t,u)u>0 for all t∈[0,T] and u∈Rn∖{0}.Then problem (48) possesses one nonzero periodic solution generated by impulses.Theorem 25 (see [14, Theorem 2]).
Assume thatf≡0 and V and gk satisfy the following conditions: (V5)
V(t,u)≥(1/2)Vu(t,u)u≥0 for all (t,u)∈[0,T]×Rn;(g3)
gk(u)=2u+wk(u), where wk=graduWk(u) and satisfies the fact that there exists μ>2 such that
(53)wk(u)u≤μWk(u)<0,foru∈Rn∖{0},hhhhhhhhhhhhhhhhhhhhhhk=1,2,…,m.Then problem (48) possesses one nonzero periodic solution generated by impulses.Now we discussT-periodic solution to second-order impulsive problems with singularity as follows:
(54)u′′(t)-1uα(t)=e(t),a.e.t∈(0,T),Δu′(tj)=Ij(u(tj)),j=1,2,…,p-1,
where α≥1, e∈L1([0,T],R) is T-periodic, Δu′(tj)=u′(tj+)-u′(tj-) with u′(tj±)=lim⁡t→tj±u′(t); 0=t0<t1<t2<⋯<tp-1<tp=T and tj+p=tj+T; Ij∈C(R,R)(j=1,2,…,p-1) and Ij+p≡Ij.Theorem 26 (see [18, Theorem 1.2]).
Assume that(S1)
e∈L1([0,T],R) is T-periodic and ∫0T‍e(t)dt≥0;(S2)
there exist two constantsm,M such that, for any s∈R,
(55)m≤Ij(s)≤M,j=1,2,…,p-1,where
(56)m≤M<-1p-1∫0Te(t)dt≤0.
Then problem (54) has at least a positive T-periodic solution.Set(57)HT1={u′L2u:R⟶R∣uisabsolutelycontinuous,000u′∈L2((0,T),R),andu(t)=u(t+T)fort∈R}
with the inner product
(58)(u,v)=∫0Tu(t)v(t)dt+∫0Tu′(t)v′(t)dt,∀u,v∈HT1.
The corresponding norm is defined by
(59)∥u∥HT1=(∫0T|u(t)|2dt+∫0T|u′(t)|2dt)1/2,∀u∈HT1.
Then HT1 is a Banach space (in fact it is a Hilbert space).In order to study problem (54), for any λ∈(0,1), we consider the following modified problem:
(60)u′′(t)+fλ(u(t))=e(t),a.e.t∈(0,T),Δu′(tj)=Ij(u(tj)),j=1,2,…,p-1.
Following the ideas of [15], we introduce the following concept of a weak solution to problem (60).Definition 27.
One says that a functionu∈HT1 is a weak solution to problem (60) if
(61)∫0Tu′(t)v′(t)dt+∑j=1p-1Ij(u(tj))v(tj)-∫0Tfλ(u(t))v(t)dt+∫0Te(t)v(t)dt=0
holds for any v∈HT1.LetFλ∈C1(R,R) be defined by
(62)Fλ(s)=∫1sfλ(t)dt,
and consider the functional
(63)Φλ:HT1⟶R,
defined by
(64)Φλ(u)≔12∫0T|u′(t)|2dt+∑j=1p-1∫0u(tj)Ij(s)ds-∫0TFλ(u(t))dt+∫0Te(t)u(t)dt.
Clearly, Φλ is well defined on HT1, continuously Gâteaux differentiable functional whose Gâteaux derivative is the functional Φλ′(u), given by
(65)Φλ′(u)v=∫0Tu′(t)v′(t)dt+∑j=1p-1Ij(u(tj))v(tj)-∫0Tfλ(u(t))v(t)dt+∫0Te(t)v(t)dt,
for any v∈HT1. Moreover, it is easy to verify that Φλ is weakly lower semicontinuous. Indeed, if {un}⊂HT1, u∈HT1, and un⇀u, then {un} converges uniformly to u on [0,T] and un→u on L2([0,T]), and combining the fact that liminf⁡n→∞∥un∥HT1≥∥un∥HT1, we have
(66)liminf⁡n→∞Φλ(un)=liminf⁡n→∞(12∥un∥HT12-12∫0T|un(t)|2dt000000000+∑j=1p-1∫0un(tj)Ij(s)ds000000000-∫0TFλ(un(t))dt000000000+∫0Te(t)un(t)dt)≥12∫0T|u′(t)|2dt+∑j=1p-1∫0u(tj)Ij(s)ds-∫0TFλ(u(t))dt+∫0Te(t)u(t)dt=Φλ(u).
By the standard discussion, the critical points of Φλ are the weak solutions of problem (1); see [15, 16].The following version of the mountain pass theorem will be used in our argument.Theorem 28 (see Theorem 4.10, [35]).
LetX be a Banach space and let φ∈C1(X,R). Assume that there exist x0,x1∈X and a bounded open neighborhood Ω of x0 such that x1∈X∖Ω¯ and
(67)max⁡⁡{φ(x0),φ(x1)}<inf⁡x∈∂Ωφ(x).
Let
(68)Γ={h∈C([0,1],X):h(0)=x0,h(1)=x1},c=inf⁡h∈Γmax⁡s∈[0,1]φ(h(s)).
If φ satisfies the (PS) condition, that is, a sequence {un} in X satisfying φ(un) is bounded and φ′(un)→0 as n→∞ has a convergent subsequence, then c is a critical value of φ and c>max⁡⁡{φ(x0),φ(x1)}.IfIj(u(tj))≡bj (j=1,2,…,p-1), then we have the following result of problem (54).Theorem 29 (see [17, Theorem 1.2]).
Assume thate∈L1([0,T],R) is T-periodic. Then problem (54) has a positive T-periodic weak solution u∈HT1 if and only if ∑j=1p-1bj+∫0T‍e(t)dt<0.Remark 30.
From Theorem29 we can see that if ∫0T‍e(t)dt≥0 but ∑j=1p-1bj+∫0T‍e(t)dt<0, then problem (54) still admits a positive T-periodic solution. This shows that the existence of positive T-periodic solution to problem (54) depends on the forced term e and impulsive functions bj together, not single one.


## 4. Homoclinic Solutions

In this section, we are interested in the existence of homoclinic solutions for problem (48).First we consider the case off≡0. When Vu(t,u) is asymptotically linear, we show that there exists a nonzero homoclinic solution as the limit of a sequence 2lT-periodic solutions as l goes to ∞, and when Vu(t,u) is sublinear, we attain a nonzero homoclinic solution in the same way by strengthening the effect of impulses. The results are as follows.Theorem 31 (see [14, Theorem 3]).
Supposing thatf≡0, V is T-periodic in t and satisfies condition (V4), and gk satisfies condition (g1), then problem (48) possesses at least one nonzero homoclinic solution generated by impulses.Theorem 32 (see [14, Theorem 4]).
Supposing thatf≡0, V is T-periodic in t and satisfies condition (V5), and gk satisfies condition (g3), then problem (48) possesses at least one nonzero homoclinic solution generated by impulses.Ifn=1 and gk(u)≡I(u) for all k=1,2,…,m, then we have the following result for problem (48) when Vu satisfies the superlinear condition at infinity on u.Theorem 33 (see [13, Theorem 1.1]).
Assume thatf≡0 and the following conditions hold: (H1)
there exists a positive numberT such that
(69)Vu(t+T,u)=Vu(t,u),V(t+T,u)=V(t,u)hhhhhhhhhhhhhhhhhhhhhhhhhllhh∀(t,u)∈R2;(H2)
lim⁡u→0Vu(t,u)u=0 uniformly for t∈R;(H3)
there exists aμ>2 such that
(70)uVu(t,u)≥μV(t,u)>0,∀(t,u)∈R×R∖{0};(H4)
there exist constantsa0>0 and a1>0 such that
(71)V(t,u)≥a0|u|μ,∀|u|≥1,t∈R;V(t,u)≤a0|u|μ,∀|u|≤1,t∈R;(H5)
there exists a constantb, with 0<b<((μ-2)/(μ+2)Tm), such that
(72)|I(u)|≤b|u|,2∫0uI(t)dt-I(u)u≤0.Then there exists a nontrivial weak homoclinic solution to problem (48).The proof of Theorem33 is based on the following version of Mountain pass theorem.Theorem 34 (see [37]).
LetE be a Banach space and let φ∈C1(E,R), e∈E, r>0 be such that ∥e∥>r and b:=inf⁡∥y∥=rφ(y)>φ(0)≥φ(e). Let
(73)Γ={h∈C([0,1],X):h(0)=x0,h(1)=x1},c≔inf⁡h∈Γmax⁡s∈[0,1]φ(h(s)).
Then, for each ϵ>0, δ>0, there exists y∈E such that (M1)
c-2ϵ≤φ(y)≤c+2ϵ;(M2)
dist(y,E)≤2δ;(M3)
∥φ′(y)∥≤8ϵ/δ.Next we consider the case off≢0 for problem (48).Theorem 35 (see [20, Theorem 3]).
Assume thatV satisfies conditions (V1)-(V2) and gk satisfies (g1). Furthermore, gk satisfies the following condition: (g4)
gk∈C(R) is m-periodic in k, and lim⁡|u|→0(gk(u)/|u|)=0 for k=1,2,…,m.Then there exists b-f>0 such that if ∫R‍|f(t)|2dt≤b-f2, then problem (48) possesses at least one nonzero homoclinic solution.IfV satisfies some other conditions, then we still obtain the existence result for problem (48).Theorem 36 (see [19, Theorem 1]).
Assume that the following conditions hold:(V6)
V(t,u)=-K(t,u)+W(t,u), where K, W∈C1(R×Rn,R), and is T-periodic in its first variable;(K1)
there exist constantsa1>0 and γ∈(1,2] such that for all (t,u)∈[0,T]×Rn(74)K(t,0)≡0,K(t,u)≥a1|u|γ;(K2)
there existsθ∈(1,2] such that
(75)K(t,u)≤Ku(t,u)u≤θK(t,u),∀(t,u)∈[0,T]×Rn;(W1)
W(t,0)=0, Wu(t,u)=o(|u|), as |u|→0 uniformly for t∈[0,T];(W2)
there existsr>2 and b1>0 such that
(76)W(t,u)≤b1|u|r,∀(t,u)∈[0,T]×Rn;(W3)
there existμ>1, a2>0, and β(t)∈L1(R,R+) such that μ>r-γ and
(77)Wu(t,u)u-2W(t,u)≥a2|u|μ-β(t),hhhhhhhhhhhhhl∀(t,u)∈[0,T]×Rn;(W4)
W(t,u)/|u|2→+∞ as |u|→+∞ uniformly in t∈[0,T];(G1)
Gk(0)=0 and lim⁡|u|→0gk(u)|u|=0 for k=1,2,…,m;(G2)
there existsb2>0 such that Gk(u)≤b2|u|r, u∈Rn, k=1,2,…,m;(G3)
gk(u)u-2Gk(u)≥0, u∈Rn∖{0}, k=1,2,…,m;(F1)
f∈C(R,Rn)∩L2(R,Rn)∩Lμ/(μ-1) satisfying
(78)(∫R‍|f(t)|2dt)1/2<1Cmin⁡⁡{12,a1-b1-M},where M≔sup⁡⁡{Gk(u):k=1,2,…,m,|u|=1} and a1>b1+M.
Then problem (48) possesses at least one nonzero homoclinic solution.



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*Source: 382970-2014-05-20.xml*