An application of variational approach to a class of damped vibration problems with impulsive effects on time scales

R E S E A R C H

Open Access

An application of variational approach to a

class of damped vibration problems with

impulsive effects on time scales

Jianwen Zhou, Yanning Wang and Yongkun Li

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

Abstract

In this paper, we present a new approach via variational methods and critical point theory to obtain the existence and multiplicity of solutions to a class of damped vibration problems with impulsive effects on time scales. By establishing a proper variational set, two existence results and two multiplicity results are obtained. Finally, one example is presented to illustrate the feasibility and effectiveness of our results.

Keywords: damped vibration problems; impulse; time scales

1 Introduction

Consider the damped vibration problem with impulsive effects on time scales

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u_{(t) +B(u+u}σ_{)(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(ui)(t), tis right-dense; (ui_{)(σ(tj)), tis right-scattered,}

uit–_j=

limt→t–_j(ui)(t), tis left-dense; (ui_{)(ρ(tj)), tis left-scattered,}

u(t) = (u_(t),u_{(t), . . . ,u}N_{(t)),B= [b}

lm] is an antisymmetryN×Nconstant matrix,A(t) = [alm(t)] is a symmetric N ×N matrix-valued function defined on [,T]T with alm ∈

L∞([,T],R), for alll,m= , , . . . ,N,Iij:R→R(i= , , . . . ,N,j= , , . . . ,p) are continu-ous andF: [,T]T×RN→Rsatisfies the following assumption:

(A) F(t,x)is-measurable intfor everyx∈RN_{and continuously differentiable inxfor}

-a.e.t∈[,T]Tand 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}.

Problem () covers the second-order Hamiltonian system with impulsive effects (when

T=R) ⎧ ⎪ ⎨ ⎪ ⎩ ¨

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

u() –u(T) =u˙() –u˙(T) = , ˙

ui_(t+

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

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

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

_{u(t) +B(u(t) +u(t+ )) +A(t+ )u(t+ )}

+∇F(t+ ,u(t+ )) = , a.e.t∈[,T– ]∩Z,

u() –u(T) = , u() –u(T) = ,

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

WhenT=R,Iij≡,i∈,j∈,BandA(t) are zero matrices, () is the Hamiltonian

system ¨

u(t) +∇F(t,u(t)) = , a.e.t∈[,T];

u() –u(T) =u˙() –u˙(T) = . ()

Mawhin and Willem in [] studied the periodic solutions of () and obtained a series of re-sults. Equation () has also been investigated by several authors using various techniques and different conditions on the nonlinearities, such as the coercive type potential condi-tion (see []) and the even type potential condicondi-tion (see []).

WhenT=R,Iij≡,i∈,j∈,B=  andA(t) is not a zero matrix, He and Wu in

[] researched the existence of solutions for () whenA(t) is negative-definite. Meng and Zhang in [] got some sufficient conditions for the existence of solutions for () by using a minimax theorem. Wuet al.in [] studied the periodic solutions for a class of damped vibration problems.

WhenT=R,Iij≡,i∈,j∈,BandA(t) are not zero matrices, Liet al.in []

researched the existence and multiplicity of solutions for () by variational methods and some critical point theorems.

WhenIij(t)≡,i∈,j∈,BandA(t) are not zero matrices, until now, it is unknown

whether problem () has a variational structure or not.

social sciences [, ]. For example, it can model insect populations that are continuous 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.

Besides, impulsive and periodic boundary value problems on time scales have been stud-ied extensively in the literature. There have been many approaches to study periodic solu-tions of impulsive differential equasolu-tions on time scales, such as method of lower and upper solutions, fixed-point theory, coincidence degree theory and so on. However, the study of solutions for impulsive differential equations on time scales using variational method has received considerably less attention. The variational method is, to the best of our knowl-edge, an effective approach to deal with nonlinear problems on time scales with some type of discontinuities such as impulses (see []).

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

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.

We denote [a,b]κ_T= ([a,b]κ_T)κ_{, therefore [a,b]}κ

T = [a,b]Tifbis left-dense and [a,b]κ_T= [a,ρ(b)]κ

Tifbis left-scattered.

Definition . (Definition ., []) Assume that f : T→ RN is a function, f(t) = (f_(t),f_{(t), . . . ,f}N_{(t)) and lett∈T}κ_{. Then we definef(t) = (f}_(t),f_{(t), . . . ,f}N_(t)) (pro-vided it exists). We callf_{(t) the delta (or Hilger) derivative off att. The functionf is delta} (or Hilger) differentiable 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→RNwe shall refer to 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 letAbe a-measurable subset ofT.f is integrable onAif and} only iffi (i= , , . . . ,N) are integrable onA, and_Af(t)t= (_Af(t)t,_Af(t)t, . . . ,

AfN(t)t).

Definition .([]) LetB⊂T.Bis called a-null set ifμ(B) = . We say that a prop-ertyPholds-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 functionf : [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]Tsatisfying_kn_=(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 will 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∈L

p

([,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

[,T_T,RN),x() =x(T).

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,_,p_T([,T]T,RN) too. That is, we have 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,,pT([,T]T,RN)such that the equalities

are satisfied and

[,T)T

g(t)t= . ()

By identifyingu∈W,_,p_T([,T]T,RN) with its absolutely continuous representativex∈ V,,pT([,T]T,RN), for which () holds, the setW

,p

,T([,T]T,RN) can be endowed with the structure of Banach space. That is, we have 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

u(t) pt+

[,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

u(t),v(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

.

In the sequel, · denotes the norm · _H

,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

byvσ _{and integrate on [,T)}

T, we have

[,T)T

u(t) +Bu+uσ(t) +Aσ(t)uσ(t)+∇Fσ(t),uσ(t)vσ(t)t= . ()

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

[,T)_T

u_(t),vσ_(t)t

= p

j=

[tj,tj+)T

u(t),vσ(t)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_j+vit_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

and

[,T)T

Bu(t) +Buσ(t),vσ(t)t

=

[,T)T

Bu_(t),vσ_(t)t–

[,T)T

Buσ_(t),v(t)t

=

[,T)_T

Bu_(t),vσ_(t)t+

[,T)_T

Bu_(t),v(t)t.

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

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

,T is 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) +∇Fσ(t),uσ(t),vσ(t)t

+

[,T)T

Bu(t),vσ(t)t+

[,T)T

Bu(t),v(t)t

Consider the functionalϕ:H

,T→Rdefined by

ϕ(u) =  

[,T)T

_u(

t) t+ p j= N i=

ui(tj)



Iij(t) dt

+

[,T)T

Buσ_(t),u(t)t– 

[,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

Buσ(t),u(t)t

– 

[,T)T

Aσ(t)uσ(t),uσ(t)t+J(u),

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) +∇Fσ(t),uσ(t),vσ(t)dt

–

[,T)T

Bu(t),vσ(t)t–

[,T)T

Bu(t),v(t)t. ()

Proof SetL(t,x,y) =_|y|₊

(Bx,y) – 

(A(t)x,x) –F(t,x), for allx,y∈RN andt∈[,T)T.

ThenL(t,x,y) satisfies all assumptions of Theorem . in []. Hence, by Theorem . in [], we know that the functionalψ is continuously differentiable onH,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) +∇Fσ(t),u}σ_(t),vσ_(t)dt

–

[,T)T

Bu(t),vσ(t)t+

[,T)T

Buσ(t),v(t)t

–

[,T)T

Bu(t),vσ(t)t–

[,T)T

Bu(t),v(t)t,

On the other hand, by the continuity ofIij,i∈,j∈, one hasφ∈C(HT,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.

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

ui(tj)

≤Cv sup

v∈H_,T,v≤

p

j=

N

i=

Iij

ui k(tj)

–Iij

ui_(t j)

=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.

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

Moreover, we need more preliminaries. We define operatorsG:H,T→(H,T)∗as fol-lows, for anyu∈H,T, which is given by

Gu(v) =

[,T)_T

Bu(t),vσ(t)t,

for allv∈H,T, where (H,T)∗denotes the dual space ofH,T. By the Riesz representation theorem, we can identify (H,T)∗withH,T. Thus,Gucan also be viewed as a function belonging toH,Tsuch thatGu,v=Gu(v) for anyu,v∈H,TandGis a bounded linear self-adjoint operator onH

,T. On the other hand, we can obtained the following lemma in the same way as the proof of Lemma . of [].

Lemma . G is compact on H,T. For anyu∈H

,T, let

q(u) =  

[,T)T

_u˙(t) 

we see that

q(u) =  u

_–



[,T)T

Aσ_{(t) +I} N×N

uσ_{(t) + Bu(t),u}σ_(t)t

=  

(I–K)u,u,

whereK:H

,T→H,Tis the bounded self-adjoint linear operator defined, using Riesz representation theorem, by

Ku,v= Gu,v+

[,T)T

Aσ_{(t) +I} N×N

uσ_(t),uσ_{(t)t, ∀u,v∈H}

,T,

IN×N andIdenoteN×Nidentity matrix and 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 embedding ofH

,TintoC([,T]T,RN) and Lemma . imply thatKis com-pact. By classical spectral theory, we can decomposeH,T into 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. HenceH_{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).I is said to be}

satisfying (PS) condition onXif any sequence{xn} ⊆Xfor whichI(xn) is bounded and

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

Definition .([]) LetXbe a real Banach space andI∈C_{(X,R).Iis said to be}

sat-isfying (C) condition on X if any sequence {xn} ⊆X for which I(xn) is bounded and ( +xn)I(xn)→ asn→ ∞, possesses a convergent subsequence inX.

LetXbe a real Banach space with a direct decompositionX=X_⊕X_{. Consider two}

sequences of subspace

X

⊂X⊂ · · · ⊂X, X⊂X⊂ · · · ⊂X

such that

dimX

n< +∞, dimXn< +∞, 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

m∈Nsuch thatn≥m⇒αn≥α.

Definition .(Definition ., []) LetI∈C(X,R). The functionalIsatisfies condition (C)∗if every sequence (uαn) such thatαnis admissible and

uαn∈Xαn, sup I(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}

as-sumptions:

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

I(u)≥, u∈X,u ≤r,

I(u)≤, u∈X,u ≤r.

(I) Isatisfies condition(C)∗.

(I) Imaps bounded sets into bounded sets.

(I) For everyn∈N,I(u)→–∞asu → ∞,u∈Xn⊕X.

Then I has at least two critical points.

Remark . SinceI∈C_{(X,R), by condition (I}

) of Theorem .,  is the critical point ofI.

Thus, under the conditions of Theorem .,Ihas at least one non-trivial 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),it satisfies(PS)condition,and}

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

(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 an infinite dimensional 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\B_R(E),

then I possesses an unbounded sequence of critical values.

Remark . As shown in [], a deformation theorem can be proved with condition (C) replacing the usual condition (PS), and it turns out that Theorem . and Theorem . hold under condition (C).

In order to state the other critical point theorem we need the following notions. Let

XandY be Banach spaces withXbeing separable and reflexive, and setE=X⊕Y. Let

S⊂X∗be a dense subset. For eachs∈Sthere is a semi-norm onEdefined by

ps:E→R, ps(u) = s(x) +y foru=x+y∈X⊕Y.

We denote byTSthe topology onEinduced by semi-norm family{ps}, and letwandw∗

denote the weak-topology and weak∗-topology, respectively.

For a functional∈C(E,R) we writea={u∈E:(u)≥a}. Recall thatis said to be weak sequentially continuous if for anyuku inE one has limk→∞(uk)v→

(u)vfor eachv∈E,i.e.: (E,w)→(E∗,w∗) is sequentially continuous. Forc∈Rwe say thatsatisfies condition (C)cif 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¯

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

First of all, we give two existence results.

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

(F) lim|x|→∞F_|x(t_|,x)= +∞uniformly for-a.e.t∈[,T]T, (F) lim|x|→F_|(_xt_|,x)= uniformly for-a.e.t∈[,T]T, (F) there existλ> andη>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|η >  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 existβij,γij> ,andξij∈[, )such that

Iij(t) ≤βij+γij|t|ξij for everyt∈R,i∈,j∈,

(F)

t

Iij(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 trivial weak solution.

In order to prove Theorem ., we prove the following lemma.

Lemma . Assume that(A), (F), (F),and (F)are satisfied, then ϕ satisfies

condi-tion(C)∗.

Proof Let{uαn}be a sequence inH



,Tsuch thatαnis admissible and

uαn∈Xαn, sup ϕ(uαn) < +∞,

 +uαn

then there exists a constantC>  such that

ϕ(uαn) ≤C,

 +uαnϕ

_(u

αn)≤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]. 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. Sincealm∈L∞([,T)T,R) 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. ()

Letb=maxl,m=,,...,N{blm}. For∀u∈H,T, we have

[,T)T

Buσ_{(t),u(t)t ≤} 

[,T)T

Buσ_{(t) u(t) t}

≤  

[,T)T

Buσ(t) + u(t) t

≤ bN¯

[,T)T

uσ(t) t+ 

[,T)T

u(t) t. ()

From (F) and (), we have

φ(u) ≤ p

j=

N

i=

|ui(tj)|



βij+γij|t|ξij

dt

≤βpNu∞+γ

p

j=

N

i=

uξ∞ij+

≤βpNCu+γC

ξij+



p

j=

N

i=

uξij+_{, ()}

for allu∈H

,T, whereβ=maxi∈,j∈{βij},γ =maxi∈,j∈{γij}. Combining (), (),

(), (), and Hölder’s inequality, we have

 uαn

 _=ϕ(u

αn) –φ(uαn) +  

[,T)T

uαn(t)



t+ 

[,T)T

Aσ(t)uαn(t),u σ αn(t)

t

–

[,T)T

Buσαn(t),u αn(t)

≤C+βpNCuαn+γC ξij+

 p j= N i=

uαn ξij+_+C



[,T)T

uσαn(t)



t

+ bN¯

[,T)T

uσαn(t)



t+ 



[,T)T

uαn(t)



t

+C

[,T)T

_uσ αn(t)

λ

t+ max

s∈[,ρ] a(s)

[,T)T bσ(t)t

≤C+βpNCuαn+γC ξij  p j= N i=

uαn ξij+₊

uαn



+

C+

 bN¯

Tλ–λ

[,T)T

uσ αn(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|η, ()

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

_∇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|η–Cρη–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),uσ αn(t)

,uσ

αn(t)

– Fσ(t),uσ αn(t)

t

+

[,T)_T

Buαn(t),uαn(t)

t–

[,T)_T

Buαn(t),u σ αn(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)T

Buαn(t),uαn(t)

t–

[,T)T

Buαn(t),uαn(t) +μ(t)u αn(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

–

[,T)T

μ(t)Bu αn(t),u

αn(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 η

t–CρηT–C

[,T)T

bσ(t)t, ()

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

η_{tis bounded. Ifη>λ, by Hölder’s inequality, we} have

[,T)T

uσαn λ

t≤Tη–ηλ

[,T)T

uσαn η

t

λ η

. ()

Sinceξij∈[, ), for alli∈,j∈, by () and (),{uαn}is bounded inH,T. Ifη≤λ, by (), we obtain

[,T)T

uσαn(t) λ

t=

[,T)T

uσαn(t) η

uσαn(t) λ–η

t

≤ uαn λ–η

∞

[,T)T

uσαn(t) η

t

≤Cλ_–ηuαn λ–η

[,T)T

uσ αn(t)

η

t. ()

Sinceξij∈[, ),λ–η< , 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–u

σ_|_{t→. Thus,}

by () and (), we have

[,T)T

u αn–u

_t

=ϕ(uαn) –ϕ(u),uαn–u – p j= N i= Iij

uiαn(tj)

–Iij

ui(tj)uiαn(tj) –u i_(tj)

+

[,T)T

Aσ(t)uσαn–u

,uσαn–u σ

t+ 

[,T)T

Buαn–u

,uαn–u

t

+

[,T)T

∇Fσ(t),uσαn

–∇Fσ(t),uσ,uσαn–u σ

≤ϕ(uαn)uαn–u–

ϕ(u),uαn–u

– p

j=

N

i=

Iij

uiαn(tj)

–Iij

ui(tj)uiαn(tj) –u i_(tj)

+C

[,T)T

uσ αn–u

σ _t+ bN

[,T)T

uσ αn–u

σ _t+ 

[,T)T

u αn–u

_t

+uαn–u∞

[,T)_T

_∇Fσ(t),uσ αn

–∇Fσ(t),uσ _t.

This implies_[,T)

T|˙uαn–u˙|t→, and henceuαn–u →. Therefore,uαn→uin

H

,T. Henceϕsatisfies condition (C)∗.

Now, we prove Theorem ..

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

,T,X=H+with (en)n≥ being its Hilbert

basis,X_=H–_⊕H_{, and define}

X_n=span{e,e, . . . ,en}, n∈N,

X_n=X, n∈N. 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, by Lemma .,ϕsatisfies condition (C)∗.

Secondly, we show thatϕmaps bounded sets into bounded sets. It follows from (), (), (), (), and () that

ϕ(u) = 



[,T)T

u(t) dt+ p

j=

N

i=

ui(tj)



Iij(t) dt+

[,T)T

Buσ(t),u(t)t

– 

[,T)T

Aσ(t)uσ(t),uσ(t)t+J(u)

≤  

[,T)T

u_(t) _t+C 

[,T)T

uσ_(t) _t+βpNC

u

+γCξ_ij+

p

j=

N

i=

uξij+

+ bN

[,T)T

uσ(t) t+ 

[,T)T

u(t) t+C

[,T)T

uσ(t) λt

+ max

s∈[,ρ] a(s)

≤  

CC+bNC+ 

u+βpNCu

+γCξ_ij+

p

j=

N

i=

uξij+_+C

Tuλ∞+C

≤  

CC+bNC+ 

u+βpNCu

+γCξ_ij+

p

j=

N

i=

uξij+_+C

TCλu

λ +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=_Cδ



, 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_(t j)|



|t|dt–

[,T)T

uσ_(t) _t

≥δu–

p

j=

N

i=

u_∞–

[,T)_T

uσ_(t) _t

≥δu–pNCu–C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satisfiesu ≤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.

Letr=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∈Xn⊕X. ()

By (F), there existsρ>  such that

F(t,x)≥C_CC+bNC+  +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C+bNC+  +δ

|x|–C– max

s∈[,ρ]

a(s)b(t), ()

for allx∈RN _{and-a.e.t∈[,T]}

T, whereC=C(C+bN_ +_ +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

Buσ_(t),u(t)t

– 

[,T)_T

Aσ_(t)uσ_(t),uσ_(t)t–

[,T)_T

Fσ(t),uσ_(t)t

≤–δu–+ 

[,T)T

u+(t) t+

[,T)T

Bu+σ(t),u+(t)t

– 

[,T)T

Aσ(t)u+σ(t),u+σ(t)t–

[,T)T

≤–δu–+ 

[,T)T

u+(t) t+bN 

[,T)T

u+σ(t) t

+ 

[,T)T

_u+_(t) _t+C



[,T)T

_u+σ_(t) _t–

[,T)T

Fσ(t),uσ(t)t

≤–δu–+ 

CC+bNC+ u+ 

–C_CC+bNC+  +δ

u_L

+CT+C

≤–δu–+CC+bNC+ u+ 

–CC+bNC+  +δ

u+CT+C

= –δu–+CC+bNC+ u+ 

–CC+bNC+  +δu++u+u– 

+CT+C

≤–δu–+CC+bNC+ 

u+–CC+bNC+  +δu+ 

–δu+u–+CT+C

≤–δu–+CC+bNC+ u+ 

–CC+bNC+  +δu+ 

–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 and one

trivial weak solution.

Example . LetT= ,N= ,t= ,t= . Consider the second order Hamiltonian

sys-tem with impulsive effects

⎧ ⎪ ⎨ ⎪ ⎩ ¨

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

u() –u() =u˙() –u˙() = ,

u˙i_(t

j) =u˙i(t+j) –u˙i(t–j) =Iij(ui(tj)), i= , , , ,j= , ,

()

whereA(t) is the unit matrix,

B= ⎛ ⎜ ⎜ ⎜ ⎝

 – – –

  – –

   –

   

⎞ ⎟ ⎟ ⎟ ⎠

and

Iij(t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

, t≥,

–(t– )_{, ≤t< ,}

t– ,  <t< , –t_{, |t| ≤,}

t+ , – <t< –, –(t+ )_{, – <t≤–,}

, t≤–,

for all i= , , , ,j= , . All conditions of Theorem . hold because ofλ=η=  and

βij=γij= ,ξij= ,ζij= , for alli∈,j∈. According to Theorem ., problem ()

has at least one weak solution.

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

satisfied.

(F) lim sup|x|→F_|(_xt_|,x)≤uniformly for-a.e.t∈[,T]T, (F) F(t,x)≥,for allx∈RNand-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) < +∞.

From the proof of Lemma ., we know thatϕsatisfies condition (C). On the other hand, for any small=__Cδ



, by (F) we know that there is aρ>  (ρ<ρ)

such that

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

By (F), for=_pNCδ 

, there exists

ρ>  such that

Iij(t) ≤|t|, |t| ≤ρ,i∈,j∈. ()

Letρ= _min{ρ,ρ}. Foru∈E 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_(t j)|



|t|dt–

[,T)T

uσ_(t) _t

≥δu–

p

j=

N

i=

u_∞–

[,T)_T

uσ(t) t