Existence and multiplicity of solutions for some second-order systems on time scales with impulsive effects

R E S E A R C H

Open Access

Existence and multiplicity of solutions for

some second-order systems on time scales

with impulsive effects

Jianwen Zhou

_{, Yanning Wang}

_{and Yongkun Li}

*_{Correspondence: [email protected]} 1_{Department of Mathematics,} Yunnan University, Kunming, Yunnan 650091, People’s Republic of China

Full list of author information is available at the end of the article

Abstract

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effects

⎧ ⎪ ⎨ ⎪ ⎩

u2(t) +A(σ(t))u(σ(t)) +∇F(σ(t),u(σ(t))) = 0,

-a.e.

u(0) –u(T) =u(0) –u(T) = 0,

(ui)(t+_j) – (ui)(t–_j) =Iij(ui(tj)), i= 1, 2,. . .,N,j= 1, 2,. . .,p,

wheret0= 0 <t1<t2<· · ·<tp<tp+1=T,tj∈[0,T]T(j= 0, 1, 2,. . .,p+ 1),

u(t) = (u1_(t),u2_{(t),. . .,u}N_(t))∈RN_{,A(t) = [dlm(t)] is a symmetricN×Nmatrix-valued} function defined on [0,T]_Twithdlm∈L∞([0,T]T,R) for alll,m= 1, 2,. . .,N,Iij:R→R (i= 1, 2,. . .,N,j= 1, 2,. . .,p) are continuous andF: [0,T]_T×RN_{→R. Finally, two} examples are presented to illustrate the feasibility and effectiveness of our results.

MSC: 34B37; 34N05

Keywords: nonautonomous second-order systems; time scales; impulse; variational approach

1 Introduction

Consider the nonautonomous second-order system on time scales with impulsive effects

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

u_{(t) +A(σ(t))u(σ(t)) +∇F(σ(t),u(σ(t))) = , -a.e.t∈[,T]}κ T; u() –u(T) =u_{() –u(T) = ,}

(ui_)(t+

j) – (ui)(t–j) =Iij(ui(tj)), i= , , . . . ,N,j= , , . . . ,p,

(.)

wheret=  <t<t<· · ·<tp<tp+=T,tj∈[,T]T(j= , , , . . . ,p+ ),

uit+_j=

⎧ ⎨ ⎩

limt→t+

j(u

i_{)(t), tis right-dense;}

(ui_)(σ(t

j)), tis right-scattered,

uit–_j=

⎧ ⎨ ⎩

limt→t–

j (u

i_{)(t), tis left-dense;}

(ui_)(ρ(t

j)), tis left-scattered,

u(t) = (u_(t),u_{(t), . . . ,u}N_{(t)),A(t) = [d}

lm(t)] is a symmetricN×N matrix-valued

func-tion defined on [,T]T with dlm ∈L∞([,T]T,R) for all l,m= , , . . . ,N, Iij :R→R

(i= , , . . . ,N,j= , , . . . ,p) are continuous andF: [,T]T×RN →Rsatisfies the fol-lowing assumption:

(A) F(t,x)is-measurable intfor everyx∈RN _{and continuously differentiable inxfor} -a.e.t∈[,T]T, and there exista∈C(R+,R+),bσ ∈L(,T;R+)such that

F(t,x)≤a|x|b(t), ∇F(t,x)≤a|x|b(t)

for allx∈RN _{and-a.e.t∈[,T]}

T, where∇F(t,x)denotes the gradient ofF(t,x) inx.

For the sake of convenience, in the sequel, we denote ={, , , . . . ,N},={, , , . . . ,p}.

WhenIij≡,i∈A,j∈BandA(t) is a zero matrix, (.) is the Hamiltonian system on

time scales

⎧ ⎨ ⎩

u_{(t) +∇F(σ(t),u(σ(t))) = , a.e.t∈[,T],}

u() –u(T) =u_{() –u(T) = .} (.)

In [], the authors study the Sobolev’s spaces on time scales and their properties. As ap-plications, they present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for (.).

WhenIij(t)≡,i∈A, j∈BandA(t) is not a zero matrix, until now the variational

structure of (.) has not been studied.

Problem (.) covers the second-order Hamiltonian system with impulsive effects (when T=R)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨

u(t) +A(t)u(t) +∇F(t,u(t)) = , a.e.t∈[,T];

u() –u(T) =u() –˙ u(T˙ ) = ,

u˙i_(t

j) =u˙i(t+j) –u˙i(t–j) =Iij(ui(tj)), i∈,j∈,

(.)

as well as the second-order discrete Hamiltonian system (whenT=Z,T∈N,T≥)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u(t+ ) – u(t+ ) +u(t) +A(t+ )u(t+ )

+∇F(t+ ,u(t+ )) = , t∈[,T– ]∩Z,

u() –u(T) = ,u(T) –u() =u(T+ ) –u(),

ui_(t

j+ ) –ui(tj+ ) –ui(tj) +ui(tj– ) =Iij(ui(tj)), i∈,j∈.

In [], the authors establish some sufficient conditions on the existence of solutions of (.) by means of some critical point theorems whenA(t)≡. WhenT=R, until now, it is unknown whether problem (.) has a variational structure or not.

differential equations with or without delays occur in biology, medicine, mechanics, engi-neering, chaos theory and so on (see [–]).

For a second-order differential equationu=f(t,u,u), one usually considers impulses in the position uand the velocityu. However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that result in jump discon-tinuities in velocity, but with no change in position (see []). The impulses only on the velocity occur also in impulsive mechanics (see []). An impulsive problem with impulses in the derivative only is considered in [].

The study of dynamical systems on time scales is now an active area of research. One of the reasons for this is the fact that the study on time scales unifies the study of both discrete and continuous processes, besides many others. The pioneering works in this direction are Refs. [–]. The theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis in , providing a rich theory that unifies and extends discrete and continuous analy-sis [, ]. The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical tech-nology, population dynamics, biotechnology and economics, neural networks and social sciences (see []). For example, it can model insect populations that are continuous while in season (and may follow a difference scheme with variable step-size), die out in winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population.

There have been many approaches to study solutions of differential equations on time scales, such as the method of lower and upper solutions, fixed-point theory, coincidence degree theory and so on (see [, –]). In [], authors used the fixed point theorem of strict-set-contraction to study the existence of positive periodic solutions for functional differential equations with impulse effects on time scales. However, the study of the ex-istence and multiplicity of solutions for differential equations on time scales using the variational method has received considerably less attention (see, for example, [, ]). The variational method is, to the best of our knowledge, novel and it may open a new approach to deal with nonlinear problems, with some type of discontinuities such as impulses.

Motivated by the above, we research the existence of variational construction for prob-lem (.) in an appropriate space of functions and study the existence of solutions for (.) by some critical point theorems in this paper. All these results are new.

2 Preliminaries and statements

In this section, we present some fundamental definitions and results from the calculus on time scales and Sobolev’s spaces on time scales that will be required below. These are a generalization toRnof definitions and results found in [].

Definition .([, Definition .]) LetTbe a time scale. Fort∈T, the forward jump operatorσ:T→Tis defined by

σ(t) =inf{s∈T,s>t} for allt∈T,

while the backward jump operatorρ:T→Tis defined by

(supplemented byinf∅=supTandsup∅=infT, where∅denotes the empty set). A point t∈Tis called right-scattered, left-scattered, ifσ(t) >t,ρ(t) <thold, respectively. Points that are right-scattered and left-scattered at the same time are called isolated. Also, ift<

supTandσ(t) =t, thentis called right-dense, and ift>infTandρ(t) =t, thentis called left-dense. Points that are right-dense and left-dense at the same time are called dense. The setTκ _{which is derived from the time scaleTas follows. IfThas a left-scattered} maximumm, thenTκ_{=T–{m}; otherwise,T}κ_=T.

Whena,b∈T,a<b, we denote the intervals [a,b]T, [a,b)Tand (a,b]TinTby

[a,b]T= [a,b]∩T, [a,b)T= [a,b)∩T, (a,b]T= (a,b]∩T,

respectively. Note that [a,b]κ

T= [a,b]Tifbis left-dense and [a,b]κT= a,b)T= [a,ρ(b)]Tifb is left-scattered. We denote [a,b]κ

T = ([a,b]κT)κ, therefore [a,b]κ



T = [a,b]Tifbis left-dense and [a,b]κ

T = [a,ρ(b)]κTifbis left-scattered.

Definition .([, Definition .]) Assume thatf :T→Ris a function and lett∈Tκ_.

Then we definef_{(t) to be the number (provided it exists) with the property that given} any> , there is a neighborhoodUoft(i.e.,U= (t–δ,t+δ)∩Tfor someδ> ) such that

fσ(t)–f(s)–f(t)σ(t) –s≤σ(t) –s for alls∈U.

We callf_{(t) the delta (or Hilger) derivative off att. The functionf is delta (or Hilger)} differentiable onTκ_{providedf(t) exists for allt∈T}κ_{. The functionf:T}κ_{→Ris then} called the delta derivative off onTκ_.

Definition .([, Definition .]) Assume thatf:T→RN _{is a function,}

f(t) =f_(t),f_{(t), . . . ,f}N_(t),

and lett∈Tκ_{. Then we definef(t) = (f}

(t),f(t), . . . ,fN(t)) (provided it exists). We callf_{(t) the delta (or Hilger) derivative off att. The functionf is delta (or Hilger)} differ-entiable providedf_{(t) exists for allt∈T}κ_{. The functionf:T}κ_→RN _{is then called the}

delta derivative off onTκ_.

Definition . ([, Definition .]) For a functionf :T→R, we will talk about the second derivativef _{providedf is differentiable onT}κ_{= (T}κ₎κ _{with derivativef}₌ (f_):Tκ_→R.

Definition .([, Definition .]) For a function f :T→RN, we will talk about the second derivativef _{providedf is differentiable onT}κ_{= (T}κ₎κ _{with derivativef}₌ (f_):Tκ_→RN_.

Definition . ([, Definition .]) Assume that f : T → RN _{is a function, f(t) =}

(f_(t),f_{(t), . . . ,f}N_{(t)) and let A be a-measurable subset ofT.f is integrable onA if}

and only iffi_{(i= , , . . . ,N) are integrable onA, and}

Af(t)t= (

Af(t)t,

Af(t)t,

. . . ,_AfN_(t)t).

Definition .([, Definition .]) LetB⊂T.Bis called a-null set ifμ(B) = . Say that a propertyPholds-almost everywhere (-a.e.) onB, or for-almost all (-a.a.) t∈Bif there is a-null setE⊂Bsuch thatPholds for allt∈B\E.

Forp∈R,p≥, we set the space

Lp

[,T)_T,RN=

u: [,T)_T→RN:

[,T)T

f(t)pt< +∞

with the norm

f_Lp

=

[,T)T

f(t)pt



p

.

We have the following theorem.

Theorem .([, Theorem .]) Let p∈Rbe such that p≥.Then the space Lp( [,T)_T, RN_{) is a Banach space together with the norm ·}

Lp. Moreover, L( [a,b)T,RN) is a Hilbert space together with the inner product given for every(f,g)∈Lp( [a,b)T,RN)× Lp( a,b)_T,RN)by

f,g_L

=

[a,b)T

f(t),g(t)t,

where(·,·)denotes the inner product inRN_.

Definition . ([, Definition .]) A function f : [a,b]T → RN,f(t) = (f(t),f(t), . . . ,fN_{(t)). We say thatf is absolutely continuous on [a,b]}

T (i.e.,f ∈AC([a,b]T,RN)) if for every > , there existsδ>  such that if{[ak,bk)T}nk= is a finite pairwise disjoint family of subintervals of [a,b]Tsatisfyingnk=(bk–ak) <δ, then

n

k=|f(bk) –f(ak)|<.

Now, we recall the Sobolev spaceW,_,p_T([,T]T,RN) on [,T]Tdefined in []. For the sake of convenience, in the sequel we letuσ_{(t) =u(σ(t)).}

Definition .([, Definition .]) Letp∈Rbe such thatp>  andu: [,T]T→RN. We say thatu∈W,p_,T([,T]T,RN) if and only ifu∈Lp( [,T)T,RN) and there existsg: [,T]κ_T→RN suchg∈Lp( [,T)T,RN) and

[,T)T

u(t),φ(t)t= –

[,T)T

g(t),φσ(t)t, ∀φ∈C_T_,rd[,T]T,RN. (.)

Forp∈R,p> , we denote

V,,pT

[,T]T,RN=x∈AC[,T]T,RN:x∈Lp

It follows from Remark . in [] that

V,p

[,T]T,RN

⊂W,p

[,T]T,RN

is true for every p∈Rwithp> . These two sets are, as a class of functions, equiva-lent. It is the characterization of functions inW,p_,T([,T]T,RN) in terms of functions in V,,pT([,T]T,RN) too. That is the following theorem.

Theorem .([, Theorem .]) Suppose that u∈W,p_,T([,T]T,RN)for some p∈Rwith p> ,and that(.)holds for g∈Lp( [,T)T,RN).Then there exists a unique function x∈V,p_,T([,T]T,RN)such that the equalities

x=u, x=g -a.e. on [,T)_T (.)

are satisfied and

[,T)T

g(t)t= . (.)

By identifyingu∈W,,pT([,T]T,RN) with its absolutely continuous representativex∈

V,_,p_T([,T]T,RN) for which (.) holds, the setW,p,T([,T]T,RN) can be endowed with

the structure of a Banach space. That is the following theorem.

Theorem .([, Theorem .]) Assume p∈Rand p> .The set W,p_,T([,T]T,RN)is

a Banach space together with the norm defined as

u_W,p

,T=

[,T)T

uσ_(t)p_t+

[,T)T

u_(t)p_t



p

∀u∈W,p,T

[,T]T,RN. (.)

Moreover,the set H,T=W

,

,T([,T]T,RN)is a Hilbert space together with the inner

prod-uct

u,v_H

,T=

[,T)T

uσ(t),vσ(t)t+

[,T)T

u(t),v(t)t ∀u,v∈H,T.

The Banach spaceW,p,T([,T]T,RN) has some important properties.

Theorem .([, Theorem .]) There exists C> such that the inequality

u∞≤CuH_,T (.)

holds for all u∈H

,T,whereu∞=maxt∈[,T]T|u(t)|. Moreover,if_[,T)Tu(t)t= ,then

u∞≤Cu_L

Theorem .([, Theorem .]) If the sequence{uk}k∈N⊂W,p,T([,T]T,RN)converges

weakly to u in W,,pT([,T]T,RN),then{uk}k∈Nconverges strongly in C([,T]T,RN)to u.

Theorem .([, Theorem .]) Let L: [,T]T×RN ×RN →R, (t,x,y)→L(t,x,y) be-measurable in t for each(x,y)∈RN ×RN and continuously differentiable in(x,y) for-almost every t∈[,T]T. If there exist a∈C(R+,R+),b∈L([,T]T,R+)and c∈ Lq([,T]T,R+) ( <q< +∞)such that for-almost t∈[,T]Tand every(x,y)∈RN×RN, one has

L(t,x,y)≤a|x|b(t) +|y|p,

Lx(t,x,y)≤a

|x|b(t) +|y|p, (.)

Ly(t,x,y)≤a

|x|c(t) +|y|p–,

where_p+_q= ,then the functional:W,p_,T([,T]T,RN)→Rdefined as

(u) =

[,T)_T

Lσ(t),uσ_(t),u(t)t

is continuously differentiable on W,p,T([,T]T,RN)and

(u),v=

[,T)T Lx

σ(t),uσ(t),u(t),vσ(t)

+Ly

σ(t),uσ_{(t),u(t),v(t)t. (.)}

3 Variational setting

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we make a variational structure. From this variational structure, we can reduce the problem of finding solutions of (.) to the one of seeking the critical points of a corresponding functional.

If u ∈H

,T, by identifying u ∈H,T with its absolutely continuous representative

x∈ V,_,T([,T]T,RN) for which (.) holds, then u is absolutely continuous and u˙ ∈ L_{( [,T)}

T;RN). In this case,u(t+) –u(t–) =  may not hold for somet∈(,T)T. This leads to impulsive effects.

Takev∈H

,Tand multiply the two sides of the equality

u_{(t) +Aσ(t)uσ(t)+∇Fσ(t),u}σ_{(t)= }

byvσ _{and integrate on [,T)}

T, then we have

[,T)T

u(t) +Aσ(t)uσ(t)+∇Fσ(t),uσ(t)vσ(t)t= . (.)

Moreover, combiningu_{() –u(T) = , one has}

[,T)T

u(t),vσ(t)t

=

p

j=

[tj,tj+)_T

=

p

j=

ut_j–_+,vt_j–_+–ut+_j,vt_j+–

[tj,tj+)

u(t),v(t)t

= p j= _N i=

uit_j–_+vit_j–_+–uit+_jvit_j+–

[tj,tj+)

u(t),v(t)t

=u_{(T)v(T) –u()v() –}

p j= N i= Iij

ui(tj)

vi(tj) –

[,T)T

u_(t),v(t)t

= – p j= N i= Iij

ui(tj)

vi(tj) –

[,T)_T

u_(t),v(t)t.

Combining (.), we have

[,T)T

u(t),v(t)t+

p j= N i= Iij

ui(tj)

vi(tj)

–

[,T)T

Aσ(t)uσ(t),vσ(t)t–

[,T)T

∇Fσ(t),uσ(t),vσ(t)t= .

Considering the above, we introduce the following concept solution for problem (.).

Definition . We say that a functionu∈H

,Tis a weak solution of problem (.) if the

identity

[,T)T

u(t),v(t)t+

p j= N i= Iij

ui(tj)

vi(tj)

=

[,T)T

Aσ(t)uσ(t),vσ(t)t+

[,T)T

∇Fσ(t),uσ(t),vσ(t)t

holds for anyv∈H,T.

Consider the functionalϕ:H

,T→Rdefined by

ϕ(u) =  

[,T)_T

u(t)t+

p j= N i=

ui_(t j)



Iij(t) dt

– 

[,T)T

Aσ(t)uσ(t),uσ(t)t+J(u)

=ψ(u) +φ(u), (.)

where

J(u) = –

[,T)T

Fσ(t),uσ(t)t,

ψ(u) =  

[,T)T

u(t)t– 

[,T)T

and

φ(u) =

p

j=

N

i=

ui(tj)



Iij(t) dt.

Lemma . The functionalϕis continuously differentiable on H ,Tand

ϕ(u),v=

[,T)T

u(t),v(t)t+

p

j=

N

i= Iij

ui(tj)

vi(tj)

–

[,T)T

Aσ(t)uσ(t),vσ(t)–∇Fσ(t),uσ(t),vσ(t)t. (.)

Proof Set L(t,x,y) =  |y|

_– 

(A(t)x,x) –F(t,x) for all x,y∈R

N _{andt∈[,T]}

T. Then L(t,x,y) satisfies all assumptions of Theorem .. Hence, by Theorem ., we know that the functionalψ is continuously differentiable onH,Tand

ψ(u),v=

[,T)T

u(t),v(t)–Aσ(t)uσ(t),vσ(t)–∇Fσ(t),uσ(t),vσ(t)t

for allu,v∈H ,T.

On the other hand, by the continuity ofIij,i∈,j∈, one has thatφ∈C(H,T,R) and

φ(u),v=

p

j=

N

i= Iij

ui(tj)

vi(tj)

for allu,v∈H,T. Thus,ϕis continuously differentiable onH,Tand (.) holds.

By Definition . and Lemma ., the weak solutions of problem (.) correspond to the critical points ofϕ.

Moreover, we need more preliminaries. For anyu∈H,T, let

q(u) =  

[,T)T

uσ(t)–Aσ(t)uσ(t),uσ(t)t.

We see that

q(u) =  u

_– 

[,T)T

Aσ(t) +IN×N

uσ(t),uσ(t)t

=  

(I–K)u,u,

whereK:H,T→H,Tis the bounded self-adjoint linear operator defined, using the Riesz

representation theorem, by

Ku,v=

[,T)T

Aσ(t) +IN×N

IN×NandIdenote anN×Nidentity matrix and an identity operator, respectively. By (.), ϕ(u) can be rewritten as

ϕ(u) =q(u) +φ(u) +J(u)

= 

(I–K)u,u+φ(u) +J(u). (.)

The compact imbedding ofH,TintoC([,T]T,RN) implies thatKis compact. By classical

spectral theory, we can decomposeH,Tinto the orthogonal sum of invariant subspaces

forI–K

H,T=H–⊕H⊕H+,

whereH_{=ker(I–K) andH}–_,H+_{are such that, for someδ> ,}

q(u)≤–δu ifu∈H–, (.)

q(u)≥δu ifu∈H+. (.)

Remark . K has only finitely many eigenvaluesλiwithλi>  sinceKis compact on

H_T. HenceH– is finite dimensional. Notice thatI–Kis a compact perturbation of the self-adjoint operatorI. By a well-known theorem, we know that  is not in the essential spectrum ofI–K. Hence,H_{is a finite dimensional space too.}

To prove our main results, we need the following definitions and theorems.

Definition .([,P]) LetXbe a real Banach space andI∈C_{(X,R).Iis said to be} satisfying (PS) condition onXif any sequence{xn} ⊆Xfor whichI(xn) is bounded and

I(xn)→ asn→ ∞, possesses a convergent subsequence inX.

Firstly, we state the local linking theorem.

LetXbe a real Banach space with a direct decompositionX=X⊕X. Consider two sequences of a subspace

X_⊂X_⊂ · · · ⊂X, X_⊂X_⊂ · · · ⊂X

such that

dimX_n< +∞, dimX_n< +∞, n∈N

and

X=

n∈N X

n, X=

n∈N X

n.

For every multi-indexα= (α,α)∈N, we denote byXα the spaceXα⊕Xα. We say

α≤β⇔α≤β,α≤β. A sequence (αn)⊂Nis admissible if, for everyα∈N, there is

Definition .([, Definition .]) LetI∈C_{(X,R). The functionalIsatisfies the (C)}* condition if every sequence (uαn) such thatαnis admissible and

uαn∈Xαn,supI(uαn)<∞,

 +uαn

I(uαn)→

contains a subsequence which converges to a critical point ofI.

Theorem .[, Theorem .] Suppose that I∈C_{(X,R)satisfies the following} assump-tions:

(I) X={}andIhas a local linking atwith respect to(X,X);that is,for someM> ,

I(u)≥, u∈X,u ≤M,

I(u)≤, u∈X,u ≤M.

(I) Isatisfies(C)*_condition.

(I) Imaps bounded sets into bounded sets.

(I) For everyn∈N,I(u)→–∞asu → ∞,u∈X

n⊕X.

Then I has at least two critical points.

Remark . SinceI∈C_{(X,R), by the condition (I) of Theorem .,  is the critical point} ofI. Thus, under the conditions of Theorem .,Ihas at least one nontrivial critical point.

Secondly, we state another three critical point theorems.

Theorem .([, Theorem .]) Let E be a Hilbert space with E=E⊕Eand E=E_⊥. Suppose I∈C_{(E,R),satisfies(PS)condition,and}

(I) I(u) = _Lu,u+b(u),whereLu=LPu+LPu andLκ:Eκ →Eκ is bounded and self-adjoint,κ= , ,

(I) bis compact,and

(I) there exist a subspaceE⊂Eand setsS⊂E,Q⊂Eand constantsα>ωsuch that

(i) S⊂EandI|S≥α, (ii) Qis bounded andI|∂Q≤ω,

(iii) Sand∂Qlink.

Then I possesses a critical value c≥α.

Theorem .([, Theorem .]) Let E be a Banach space.Let I∈C_{(E,R)be an even}

functional which satisfies the(PS)condition and I() = .If E=V⊕W,where V is finite dimensional,and I satisfies

(I) there are constantsρ,α> such thatI|∂Bρ∩W≥α,whereBρ={x∈E:x<ρ},

(I) for each finite dimensional subspaceE⊂E,there is an R=R(E)such thatI≤on

E\BR(E),

In order to state another critical point theorem, we need the following notions. LetXand Y be Banach spaces withXbeing separable and reflexive, and setE=X⊕Y. LetS⊂X* be a dense subset. For eachs∈S, there is a semi-norm onEdefined by

ps:E→R, ps(u) =s(x)+y foru=x+y∈X⊕Y.

We denote byT_Sthe topology onEinduced by a semi-norm family{ps}, and letwandw* denote the weak-topology and weak*-topology, respectively.

For a functional∈C_{(E,R), we write}

a={u∈E:(u)≥a}. Recall that is said

to be weak sequentially continuous if, for anyuku inE, one haslimk→∞(uk)v→ (u)vfor eachv∈E,i.e.,: (E,w)→(E*_,w*_{) is sequentially continuous. Forc∈R, we} say thatsatisfies the (C)ccondition if any sequence{uk} ⊂Esuch that(uk)→cand

( +uk)(uk)→ ask→ ∞contains a convergent subsequence.

Suppose that

() for anyc∈R,cisTS-closed, and: (c,TS)→(E*,w*)is continuous; () there existsρ> such thatκ:=inf(∂Bρ∩Y) > , where

Bρ=

u∈E:u<ρ;

() there exist a finite dimensional subspaceY⊂Y andR>ρsuch thatc¯:=sup(E) <

∞andsup(E\S) <inf(Bρ∩Y), where

E:=X⊕Y, and S=

u∈E:u ≤R

.

Theorem .([]) Assume thatis even and()-()are satisfied.Thenhas at least m=dimYpairs of critical points with critical values less than or equal toc provided¯

satisfies the(C)ccondition for all c∈[κ,¯c].

Remark . In our applications, we take S=X* so thatTS is the product topology on E=X⊕Ygiven by the weak topology onXand the strong topology onY.

4 Main results

Lemma . φis compact on H ,T.

Proof Let{uk} ⊂H,T be any bounded sequence. SinceH,T is a Hilbert space, we can

assume thatuku. Theorem . implies thatuk–u∞→. By (.), we have

φ(uk) –φ(u)

= sup

v∈H

,T,v≤

φ(uk) –φ(u),v

= sup

v∈H,T,v≤

p

j=

N

i= Iij

ui_k(tj)

–Iij

ui(tj)

vi(tj)

≤ v∞ sup

v∈H_,T,v≤ p

j=

N

i=

Iij

ui_k(tj)

–Iij

≤Cv sup v∈H_,T,v≤

p

j=

N

i=

Iij

ui_k(tj)

–Iij

ui(tj)

=C sup

v∈H

,T,v≤

p

j=

N

i=

Iij

ui_k(tj)

–Iij

ui(tj).

The continuity ofIijand this imply thatφ(uk)→φ(u) inH,T. The proof is complete.

First of all, we give two existence results.

Theorem . Suppose that(A)and the following conditions are satisfied.

(F) lim|x|→∞F_|(_xt_|,x)= +∞uniformly for-a.e.t∈[,T]T,

(F) lim|x|→F_|(_xt_|,x)= uniformly for-a.e.t∈[,T]T, (F) there existλ> andβ>λ– such that

lim sup

|x|→∞ F(t,x)

|x|λ <∞ uniformly for-a.e.t∈[,T]T

and

lim inf

|x|→∞

(∇F(t,x),x) – F(t,x)

|x|β >  uniformly for-a.e.t∈[,T]T,

(F) there existsr> such that

F(t,x)≥, ∀|x| ≤r,and-a.e.t∈[,T]T,

(F) there existaij,bij> andξij∈[, )such that

Iij(t)≤aij+bij|t|ξij for everyt∈R,i∈,j∈,

(F) _tIij(s) ds≤for everyt∈R,i∈,j∈,

(F) there existsζij> such that



t



Iij(s) ds–Iij(t)t≥ for alli∈,j∈and|t| ≥ζij,

and

lim t→

Iij(t)

t =  for alli∈,j∈.

Then problem(.)has at least two weak solutions.The one is a nontrivial weak solution, the other is a trivial weak solution.

Proof By Lemma .,ϕ∈C_{(X,R). SetX=H}

,T,X=H+with (en)n≥being its Hilbertian basis,X_=H–_⊕H_{and define}

Then we have

X_⊂X_⊂ · · · ⊂X, X_⊂X_⊂ · · · ⊂X, X=

n∈N X

n, X=

n∈N X

n

and

dimX_n< +∞, dimX_n< +∞, n∈N.

We divide our proof into four parts in order to show Theorem .. Firstly, we show thatϕsatisfies the (C)*_condition.

Let{uαn}be a sequence inH,Tsuch thatαnis admissible and

uαn∈Xαn,supϕ(uαn)< +∞,

 +uαn

ϕ(uαn)→,

then there exists a constantC>  such that

ϕ(uαn)≤C,

 +uαn

ϕ(uαn)≤C (.)

for all largen. On the other hand, by (F), there are constantsC>  andρ>  such that

F(t,x)≤C|x|λ (.)

for all|x| ≥ρand-a.e.t∈[,T]T. By (A) one has

F(t,x)≤ max s∈[,ρ]

a(s)b(t) (.)

for all|x| ≤ρand-a.e.t∈[,T]T. It follows from (.) and (.) that

F(t,x)≤ max s∈[,ρ]

a(s)b(t) +C|x|λ (.)

for allx∈RN and-a.e.t∈[,T]T. Sincedlm∈L∞([,T]) for alll,m= , , . . . ,N, there

exists a constantC≥ such that

_[,T)

T

Aσ(t)uσ(t),uσ(t)t≤C

[,T)_T

uσ(t)t, ∀u∈H,T. (.)

From (F) and (.), we have that

φ(u)≤

p

j=

N

i=

_|ui(tj)|



aij+bij|t|ξij

dt

≤apNu∞+b

p

j=

N

i=

uξ∞ij+

≤apNCu+bC

p

j=

N

i=

for allu∈H

,T, wherea=maxi∈,j∈{aij},b=maxi∈,j∈{bij}. Combining (.), (.), (.) and Hölder’s inequality, we have

 uαn

 _=ϕ(u

αn) –φ(uαn) +

 

[,T)T

uσ αn(t)



t

+ 

[,T)_T

Aσ(t)uσαn(t),u

σ αn(t)

t–J(uαn)

≤C+apNCuαn+bC

p j= N i=

uαn

ξij+_+C

[,T)T

uσαn(t)



t

+C

[,T)T

uσαn(t)

λ

t+ max s∈[,ρ]

a(s)

[,T)T bσ(t)t

≤C+apNCuαn+bC

p j= N i=

uαn

ξij+

+CTλ–λ

[,T)T

uσαn(t)

λ t  λ +C

[,T)T

uσαn(t)

λ

t+C (.)

for all largen, whereC=maxs∈[,ρ]a(s)

[,T)Tb

σ_{(t)t. On the other hand, by (F), there}

existC>  andρ>  such that

∇F(t,x),x– F(t,x)≥C|x|β (.)

for all|x| ≥ρand-a.e.t∈[,T]T. By (A),

_∇F(t,x),x– F(t,x)≤Cb(t) (.)

for all|x| ≤ρand-a.e.t∈[,T]T, whereC= ( +ρ)maxs∈[,ρ]a(s). Combining (.) and (.), one has

∇F(t,x),x– F(t,x)≥C|x|β–Cρ_β–Cb(t) (.)

for allx∈RN _{and-a.e.t∈[,T]}

T. According to (F), there existsC>  such that



t



Iij(s) ds–Iij(t)t≥–C for alli∈,j∈andt∈R. (.)

Thus, by (.), (.) and (.), we obtain

C≥ϕ(uαn) –

ϕ(uαn),uαn

= φ(uαn) –

φ(uαn),uαn

+

[,T)T

∇Fσ(t),uσαn(t)

,uσαn(t)

– Fσ(t),uσαn(t)

t = p j= N i= 

uiαn(tj)



Iij(t) dt–Iij

uiαn(tj)

uiαn(tj)

+

[,T)T

∇Fσ(t),uσαn(t)

,uσαn(t)

– Fσ(t),uσαn(t)

t

≥–pNC+C

[,T)T

uσαn

β

t–Cρ_βT–C

[,T)T

bσ(t)t (.)

for all largen. From (.),_[,T)T|uσ αn|

β_{tis bounded. Ifβ>λ, by Hölder’s inequality, we}

have

[,T)T

uσαn(t)

λ

t≤Tβ–βλ

[,T)T

uσαn(t)

β

t

λ β

. (.)

Sinceξij∈[, ) for alli∈,j∈, by (.) and (.),{uαn}is bounded inH,T. Ifβ≤λ,

by (.), we obtain

[,T)T

uσαn(t)

λ

t=

[,T)T

uσαn(t)

β uσαn(t)

λ–β

t

≤ uαn

λ–β ∞

[,T)T

uσαn(t)

β

t

≤C_λ–βuαn

λ–β

[,T)T

uσαn(t)

β

t. (.)

Sinceξij∈[, ),λ–β< , by (.) and (.),{uαn}is also bounded inH,T. Hence,{uαn}

is also bounded inH

,T. Going if necessary to a subsequence, we can assume thatuαnu

inH,T. From Theorem ., we haveuαn–u∞→ and

[,T)T|u σ αn(t) –u

σ_(t)|_t→.

Since

[,T)T

uαn(t) –u

(t)t

=ϕ(uαn) –ϕ(u),uαn–u

+

[,T)T

Aσ(t)uσαn(t) –u

σ_(t),

uσαn(t) –u

σ_(t) t – p j= N i= Iij

ui_αn(tj)

–Iij

ui(tj)

ui_αn(tj) –ui(tj)

+

[,T)_T

∇Fσ(t),uσ αn(t)

–∇Fσ(t),uσ_(t),uσ αn(t) –u

σ_(t)t.

This implies_[,T)T|u αn(t) –u

_(t)|_{t→, and henceu}

αn–u →. Therefore,uαn→u

inH

,T. Henceϕsatisfies the (C)*condition.

Secondly, we show thatϕmaps bounded sets into bounded sets. It follows from (.), (.), (.) and (.) that

ϕ(u)=  

[,T)T

u(t)t+

p j= N i=

ui(tj)



Iij(t) dt

– 

[,T)T

Aσ(t)uσ(t),uσ(t)t+J(u)

≤ 



[,T)T

u(t)t+C 

[,T)T

+bC

p

j=

N

i=

uξij+

+C

[,T)T

uσ(t)λt+ max s∈[,ρ]

a(s)

[,T)T bσ(t)t

≤ 

Cu

_{+apNCu+bC}

p

j=

N

i=

uξij+_+CTuλ

∞+C

≤ 

Cu

_+apNC

u+bC

p

j=

N

i=

uξij+_+C

TCλuλ+C

for allu∈H

,T. Thus,ϕmaps bounded sets into bounded sets.

Thirdly, we claim thatϕhas a local linking at  with respect to (X_,X_). Applying (F), for=δ_, there existsρ>  such that

F(t,x)≤|x| (.)

for all|x| ≤ρand-a.e.t∈[,T]T. By (F), for=_pNCδ _, there existsρ>  such that

Iij(t)≤|t|, |t| ≤ρ,i∈,j∈. (.)

Letρ=min{ρ,ρ}. Foru∈Xwithu ≤rρ_C_, by (.), (.), (.), (.) and (.), we have

ϕ(u) =q(u) +

p

j=

N

i=

ui(tj)



Iij(t) dt–

[,T)T

Fσ(t),uσ(t)t

≥δu–

p

j=

N

i=

_|ui(tj)|



Iij(t)dt–

[,T)T

uσ(t)t

≥δu–

p

j=

N

i=

|ui(tj)|



|t|dt–

[,T)T

uσ(t)t

≥δu–

p

j=

N

i=

u_∞–

[,T)T

uσ(t)t

≥δu–pNCu–u

≥δu–δ u

_–δ u



= δ u

_.

This implies that

ϕ(u)≥, ∀u∈Xwithu ≤r.

On the other hand, it follows from (F) that

for allu∈H

,T. Letu=u–+u∈X satisfyu ≤r Cr. Using (F), (.), (.), (.) and (.), we obtain

ϕ(u) =q(u) +φ(u) –

[,T)T

Fσ(t),uσ(t)t

≤–δu––

[,T)T

Fσ(t),uσ(t)t

≤–δu–.

This implies that

ϕ(u)≤, ∀u∈Xwithu ≤r.

LetM=min{r,r}. Thenϕsatisfies the condition (I) of Theorem .. Finally, we claim that for everyn∈N,

ϕ(u)→–∞ asu → ∞,u∈X_n⊕X.

For givenn∈N, sinceX

n⊕Xis a finite dimensional space, there existsC>  such that

u ≤CuL, ∀u∈X_n⊕X. (.)

By (F), there existsρ>  such that

F(t,x)≥C_(C+δ)|x| (.)

for all|x| ≥ρand-a.e.t∈[,T]T. From (A), we get

_F(t,x)≤ max s∈[,ρ]

a(s)b(t) (.)

for all|x| ≤ρand-a.e.t∈[,T]T. Equations (.) and (.) imply that

F(t,x)≥C_(C+δ)|x|–C– max

s∈[,ρ]

a(s)b(t) (.)

for allx∈RN and-a.e.t∈[,T]T, whereC=C_(C+δ)ρ_. Using (.), (.), (.), (.), (.) and (.), we have, foru=u+_+u_+u–_∈X

n⊕X=Xn⊕H⊕H–,

ϕ(u) =  

[,T)_T

u_(t)_t+

p

j=

N

i=

ui_(t j)



Iij(t) dt

– 

[,T)_T

Aσ(t)uσ(t),uσ(t)t–

[,T)_T

Fσ(t),uσ(t)t

≤–δu–+ 

[,T)T

u+(t)t

– 

[,T)T

Aσ(t)u+σ(t),u+σ(t)t–

[,T)T

≤–δu–+ 

[,T)T

u+(t)t+C 

[,T)T

u+σ(t)t

–

[,T)T

Fσ(t),uσ(t)t

≤–δu–+C  u

+_–C

(C+δ)u σ

L+CT+ max

s∈[,ρ] a(s)

[,T)T bσ(t)t

≤–δu–+Cu+– (C+δ)u+CT+C

= –δu–+Cu+ 

– (C+δ)u++u+u– 

+CT+C

≤–δu–+Cu+ 

– (C+δ)u+ 

–δu+u–+CT+C

≤–δu–+Cu+ 

– (C+δ)u+ 

–δu+CT+C

= –δu+CT+C,

whereC=maxs∈[,ρ]a(s)

[,T)_Tb

σ_{(t)t. Hence, for everyn∈N,ϕ(u)→–∞asu →}

∞andX

n⊕X.

Thus, by Theorem ., problem (.) has at least one nontrivial weak solution. The proof

is complete.

Example . LetT=R,T =π

, N= , t= π

. Consider the second-order Hamiltonian system with impulsive effects

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨

u(t) +A(t)u(t) +∇F(t,x) = , a.e.t∈[,π ];

u() –u(π_) =u() –˙ u(˙ π_) = ,

u(t) =˙ u(t˙ +

) –u(t˙ –) =I(u(t)),

(.)

whereA(t) = ,

F(t,x) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

|x|_{, |x| ≥,} 

–√x– √

–√, 

√

 <x< ,

, |x| ≤√,

 √–x+

√

√–, –≤x< –

√



for allx∈Randt∈[,π ], I(t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

, t≥,

–(t– )_{, ≤t< ,}

t– ,  <t< ,

–t_{, |t| ≤,}

t+ , – <t< –,

–(t+ )_{, – <t≤–,}

then all conditions of Theorem . hold. According to Theorem ., problem (.) has at least one nontrivial weak solution. In fact,

u(t) =

⎧ ⎨ ⎩

√cost, t∈[,π_];

√sint, t∈[π ,

π ]

is the solution of problem (.).

Theorem . Assume that(A), (F), (F), (F)and the following conditions are satisfied.

(F) lim sup_|x|→F_|(t,x)

x| ≤uniformly for-a.e.t∈[,T]T,

(F) there exist constantsμ> andr≥such that(∇F(t,x),x)≥μF(t,x) > for allt∈ [,T]Tand|x| ≥r,

(F) F(t,x)≥for allx∈RN _{and-a.e.t∈[,T]}

T.

Then problem(.)has at least one nontrivial weak solution.

Proof SetE=H+,E=H–⊕HandE=H,T. ThenEis a real Hilbert space,E=E⊕E, E=E⊥_ anddim(E) < +∞.

Firstly, we prove thatϕsatisfies the (PS) condition. Indeed, let{uk} ⊂H,Tbe a sequence

such that|ϕ(uk)| ≤Candϕ(uk)→ ask→ ∞. As the proof of Theorem ., it suffices

to show that{uk}is bounded inH

,T. By (F) there exist positive constantsC,Csuch

that

F(t,x)≥C|x|μ_{–C, ∀t∈[,T]}

T,∀x∈Rn (.)

(see []). By (F), (.) and (.), we have

C+uk

≥ϕ(uk) –

ϕ(uk),uk

= φ(uk) –

φ(uk),uk

+

[,T)T

∇Fσ(t),uσ_k(t),u_kσ(t)– Fσ(t),uσ_k(t)t

=

p

j=

N

i=



ui_k(tj)



Iij(t) dt–Iij

ui_k(tj)

ui_k(tj)

+

[,T)T

∇Fσ(t),uσ

k(t)

,uσ

k(t)

– Fσ(t),uσ

k(t)

t

= –pNC+

[,T)T

∇Fσ(t),uσ

k(t)

,uσ

k(t)

– Fσ(t),uσ

k(t)

t

= –pNC+ (μ– )

[,T)_T

Fσ(t),uσ_k(t)t

+

[,T)T

∇Fσ(t),uσ_k(t),uσ_k(t)–μFσ(t),uσ_k(t)t

≥–pNC+ (μ– )

[,T)T

+

[,T)T

∇Fσ(t),uσ_k(t),uσ_k(t)–μFσ(t),uσ_k(t)t

≥–pNC+ (μ– )C

[,T)T

|uσ_k|(t)μt– (μ– )CT–C (.)

for largek, whereC= (r+μ)maxs∈[,r]a(s)

[,T)Tb

σ_{(t)t. Equation (.) implies that}

there existsC>  such that

[,T)T

uσ_k(t)μt≤C +uk. (.)

Combining (.), (.), (.) and (.), we obtain

μc+uk

≥μϕ(uk) –

ϕ(uk),uk

=

μ

 –  [,T)T

u_k(t)–Aσ(t)uσ_k(t),uσ_k(t)t

+

[,T)T

∇Fσ(t),uσ

k(t)

,uσ

k(t)

–μFσ(t),uσ

k(t)

t+μφ(uk) –

φ(uk),uk

=

μ

 –  [,T)T

u_k(t)–Aσ(t)uσ_k(t),u_kσ(t)t+ (μ– )φ(uk) + φ(uk)

+

[,T)T

∇Fσ(t),uσ

k(t)

,uσ

k(t)

–μFσ(t),uσ

k(t)

t–φ(uk),uk ≥ μ  –  uk–

μ  –  – μ  –  C [,T)T

_uσ

k(t)



t–C–pNC

– (μ– )apNCuk– (μ– )bC

p j= N i=

ukξij+

≥

μ

 – 

uk–

μ  –  – μ  –  C

Tμμ–

[,T)T

uσ k(t) μ t  μ

–C–pNC– (μ– )apNCuk– (μ– )bC

p j= N i=

ukξij+

≥

μ

 – 

uk–

μ  –  – μ  –  C

Tμμ– _C



 +ukμ

–C–pNC– (μ– )apNCuk– (μ– )bC

p j= N i=

ukξij+ _(.)

for largek. Sinceμ> ,ξij∈[, ), by (.),{uk}is bounded inH,T.

For any small=δ_, by (F) we know that there is aρ>  such that

F(t,x)≤|x|, for|x|<ρ-a.e.t∈[,T]T. (.)

By (F), for=_pNCδ _, there existsρ>  such that