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NONAUTONOMOUS DYNAMICAL SYSTEMS

MARTIN SCHECHTER

Received 13 March 2006; Revised 10 May 2006; Accepted 15 May 2006

We study the existence of periodic solutions for second-order nonautonomous dynamical systems. We give four sets of hypotheses which guarantee the existence of solutions. We were able to weaken the hypotheses considerably from those used previously for such systems. We employ a new saddle point theorem using linking methods.

Copyright © 2006 Martin Schechter. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

We consider the following problem. One wishes to solve

−x(t)= ∇xVt,x(t), (1.1)

where

x(t)=x1(t),...,xn(t)

(1.2)

is a map fromI=[0,T] toRnsuch that each componentx

j(t) is a periodic function in H1with periodT, and the functionV(t,x)=V(t,x

1,...,xn) is continuous fromRn+1to

Rwith

∇xV(t,x)=

∂V ∂x1,...,

∂V ∂xn

∈CRn+1,Rn. (1.3)

Here H1 represents the Hilbert space of periodic functions in L2(I) with generalized derivatives inL2(I). The scalar product is given by

(u,v)H1=(u,v) + (u,v). (1.4)

For eachx∈Rn, the functionV(t,x) is periodic intwith periodT.

Hindawi Publishing Corporation Boundary Value Problems

(2)

We will study this problem under the following assumptions: (1)

V(t,x)≥0, t∈I,x∈Rn; (1.5)

(2) there are constantsm >0,α≤6m2/T2such that

V(t,x)≤α, |x| ≤m,t∈I,x∈Rn; (1.6)

(3) there is a constantμ >2 such that (t,x)

|x|2 ≤W(t)∈L 1

(I), |x| ≥C,t∈I,x∈Rn, (1.7)

lim sup

|x|→∞

(t,x)

|x|2 0, (1.8)

where

(t,x)=μV(t,x)− ∇xV(t,x)·x; (1.9)

(4) there is a subsete⊂Iof positive measure such that

lim inf

|x|→∞

V(t,x)

|x|2 >0, t∈e. (1.10)

We have the following theorem.

Theorem1.1. Under the above hypotheses, the system (1.1) has a solution. As a variant ofTheorem 1.1, we have the following one.

Theorem1.2. The conclusion inTheorem 1.1is the same if Hypothesis (2) is replaced by (2) there is a constantq >2such that

V(t,x)≤C|x|q+ 1, tI,xRn, (1.11)

and there are constantsm >0,α <2π2/T2such that

V(t,x)≤α|x|2, |x| ≤m,tI,xRn. (1.12)

We also have the following theorem.

Theorem1.3. The conclusions of Theorems1.1and1.2hold if Hypothesis (3) is replaced by (3) there is a constantμ <2such that

(t,x)

|x|2 ≥ −W(t)∈L

1(I), |x| ≥C,tI,xRn,

lim inf

|x|→∞

(t,x) |x|2 0.

(3)

And we have the following theorem.

Theorem1.4. The conclusion ofTheorem 1.1holds if Hypothesis (1) is replaced by (1)

0≤V(t,x)≤C|x|2+ 1, tI,xRn (1.14)

and Hypothesis (3) by (3) the function given by

H(t,x)=2V(t,x)− ∇xV(t,x)·x (1.15)

satisfies

H(t,x)≤W(t)∈L1(I), |x| ≥C,tI,xRn,

H(t,x)−→ −∞, |x| −→ ∞,t∈I,x∈Rn. (1.16)

The periodic nonautonomous problem

x(t)= ∇xVt,x(t) (1.17)

has an extensive history in the case of singular systems (cf., e.g., Ambrosetti-Coti Zelati [1]). The first to consider it for potentials satisfying (1.3) were Berger and Schechter [3]. We proved the existence of solutions to (1.17) under the condition that

V(t,x)−→ ∞ as |x| −→ ∞ (1.18)

uniformly for a.e.t∈I. Subsequently, Willem [16], Mawhin [6], Mawhin and Willem [8], Tang [11,12], Tang and Wu [13–15], Wu and Tang [17] and others proved existence under various conditions (cf. the references given in these publications).

The periodic problem (1.1) was studied by Mawhin and Willem [7,8], Long [5], Tang and Wu [13–15] and others (cf. the refernces quoted in them). Ben-Naoum et al. [2] and Nirenberg (cf. Ekeland and Ghoussoub [4]) proved the existence of nonconstant solutions.

We will prove Theorems1.1–1.4in the next section. We use a linking method of critical point theory (cf. [9,10]). These methods allow us to improve the previous results.

2. Proofs of the theorems

We now give the proof ofTheorem 1.1.

Proof. LetXbe the set of vector functionsx(t) given by (1.2) and described above. It is a Hilbert space with norm satisfying

x2

X= n

j=1 xj2

(4)

We also write

x2=n

j=1 xj2

, (2.2)

where · is theL2(I) norm. Let

N=x(t)∈X:xj(t) constant, 1≤j≤n (2.3)

andM=N⊥. The dimension ofN isn, andX=M⊕N. Proof of the following lemma can be found in [7].

Lemma2.1. Ifx∈M,then

x2

∞≤12Tx2, x ≤T x. (2.4)

We define

G(x)= x22

IV

t,x(t)dt, x∈X. (2.5)

For eachx∈X writex=v+w, wherev∈N,w∈M. For convenience, we will use the following equivalent norm forX:

x2

X= w2+v2. (2.6)

Ifx∈Mand

x2=ρ2=12 Tm

2, (2.7)

thenLemma 2.1implies thatx∞≤m, and we have by Hypothesis (2) thatV(t,x)≤α.

Hence,

G(x)≥ x22

|x(t)|<mαdt≥ρ

2

2αT0. (2.8)

We also note that Hypothesis (1) implies

G(v)≤0, v∈N. (2.9)

Take

A=∂Bρ∩M, ρ2=12 Tm

2,

B=N, (2.10)

where

Bσ=

(5)

By [9, Theorem 1.1],AlinksB. (For background material on linking theory, cf. [10].) Moreover, by (2.8) and (2.9), we have

sup A

[−G]≤0inf

B [−G]. (2.12)

Hence, we may apply [9, Theorem 1.1] to conclude that there is a sequence{x(k)} ⊂X such that

Gx(k)=x(k)22

IV

t,x(k)(t)dt−→c0, (2.13)

Gx(k),z

2 =

x(k),z

I∇xV

t,x(k)(t)·z(t)dt−→0, z∈X, (2.14)

Gx(k),x(k)

2 =

x(k)2

I∇xV

t,x(k)(t)·x(k)(t)dt−→0. (2.15)

If

ρk=x(k)X≤C, (2.16)

then there is a renamed subsequence such thatx(k)converges to a limitxXweakly in Xand uniformly onI. From (2.14) we see that

G(x),z

2 =(x

,z)

I∇xV

t,x(t)·z(t)dt=0, z∈X, (2.17)

from which we conclude easily thatxis a solution of (1.1). If

ρk=x(k)X−→ ∞, (2.18)

letx(k)=x(k)

k. Then,x(k)X=1. Letx(k)=w(k)+v(k), wherew(k)∈Mandv(k)∈N. There is a renamed subsequence such that[x(k)]randx(k)τ, wherer2+τ2=1. From (2.13) and (2.15) we obtain

x(k)22

IV

t,x(k)(t)dt ρ2

k

−→0,

x(k)2

I∇xV

t,x(k)(t)·x(k)(t)dt ρ2k −→0.

(2.19)

Thus,

2IVt,x(k)(t)dt ρ2k −→r

2, (2.20)

I∇xV

t,x(k)(t)·x(k)(t)dt ρ2

k

(6)

Hence, by (1.9),

IHμ

t,x(k)(t)dt ρ2

k

−→

μ 21

r2. (2.22)

Note that

x(k)(t)≤Cx(k)X=C. (2.23)

If

x(k)

(t)−→ ∞, (2.24)

then by (1.8)

lim sup

t,x(k)(t) ρ2

k

=lim sup

t,x(k)(t) x(k)(t)2 x

(k)(t)20. (2.25)

If

x(k)(t)C, (2.26)

then

Hμt,x(k)(t) ρ2

k

−→0. (2.27)

Hence,

lim sup

IHμt,x(k)(t)dt ρ2

k

0. (2.28)

Hence by (2.22)

μ 21

r20. (2.29)

Ifr=0, this contradicts the fact thatμ >2. Ifr=0, then w(k)0 uniformly inI by Lemma 2.1. Moreover,T|v(k)|2= v(k)21. Hence, there is a renamed subsequence such thatv(k)vinN with|v|2=1/T. Hence,x(k)vuniformly inI. Consequently,

|x(k)| → ∞uniformly inI. Thus, by Hypothesis (4),

lim inf

IV

t,x(k)(t)dt ρ2

k

elim inf

Vt,x(k)(t) x(k)(t)2 x

(k)(t)2dt >0. (2.30)

(7)

The proof ofTheorem 1.2is similar to that ofTheorem 1.1with the exception of the inequality (2.8) resulting from Hypothesis (2). In its place we reason as follows: ifx∈M, we have by Hypothesis (2),

G(x)≥ x22

|x|<mα

x(t)2 dt−2C

|x(t)|>m

x(t)q+ 1 dt

≥ x2x22C1 +m−q

|x(t)|>m

x(t)q dt

≥ x212αT2 4π2

−C

|x(t)|>m

x(t)q dt 1 αT2 2π2

x2

X−C

Ix q Xdt 1 αT2 2π2

x2

X−CxqX

= 1 αT2 2π2

−CxqX−2

x2

X

(2.31)

byLemma 2.1. Hence, we have the following lemma. Lemma2.2.

G(x)≥εx2

X, xX≤ρ,x∈M (2.32)

forρ >0sufficiently small, whereε <1[αT2/2π2]. The remainder of the proof is essentially the same.

In provingTheorem 1.3we follow the proof of Theorem 1.1until we reach (2.20). Then we reason as follows. If

x(k)(t)−→ ∞, (2.33)

then

lim inf

t,x(k)(t)

ρ2k =lim inf

t,x(k)(t) x(k)(t)2 x

(k)(t)20. (2.34)

If

x(k)

(t)≤C, (2.35)

then by Hypothesis (3),

Hμt,x(k)(t) ρ2

k

−→0. (2.36)

Hence,

lim inf

IHμt,x(k)(t)dt ρ2

k

(8)

Thus by (2.22)

μ 21

r20. (2.38)

Ifr=0, this contradicts the fact thatμ <2. Ifr=0, then w(k)0 uniformly inI by Lemma 2.1. Moreover,T|v(k)|2= v(k)21. Hence, there is a renamed subsequence such thatv(k)vinN with|v|2=1/T. Hence,x(k)vuniformly inI. Consequently,

|x(k)| → ∞uniformly inI. Thus, by Hypothesis (4),

lim inf

IV

t,x(k)(t)dt ρ2k

elim inf

Vt,x(k)(t) x(k)(t)2 x

(k)(t)2dt >0. (2.39)

This contradicts (2.20). Hence theρkare bounded, and the proof is complete.

In proving Theorem 1.4, we follow the proof ofTheorem 1.1until (2.20). Assume first thatr >0. Note that (2.13) and (2.15) imply that

IH

t,x(k)(t)dt−→ −c. (2.40)

On the other hand, by Hypothesis (1), we have

0←−x(k)22

I

Vt,x(k)(t)dt ρ2

k

≥x(k)22C

Ix

(k)(t)2+ρ2

k

dt

−→r22C

Ix(t)

2 dt.

(2.41)

Hence,x(t)≡0. LetΩ0⊂Ibe the set on whichx(t) =0.The measure ofΩ0is positive. Moreover,|x(k)(t)| → ∞ask→ ∞fortΩ

0. Thus,

IH

t,x(k)(t)dt≤

Ω0

Ht,x(k)(t)dt+

I\Ω0

W(t)dt−→ −∞ (2.42)

by Hypothesis (3). But this contradicts (2.40). Ifr=0, thenw(k)0 uniformly inI byLemma 2.1. Moreover,T|v(k)|2= v(k)21. Thus, there is a renamed subsequence such thatv(k)vinNwith|v|2=1/T. Hence,x(k)(t)vuniformly inI. Consequently,

|x(k)(t)| → ∞uniformly inI. Thus, by Hypothesis (4),

lim inf

IV

t,x(k)(t)dt ρ2k

elim inf

Vt,x(k)(t) x(k)(t)2 x

(k)

(t)2dt >0. (2.43)

(9)

References

[1] A. Ambrosetti and V. Coti Zelati,Periodic Solutions of Singular Lagrangian Systems, Progress in Nonlinear Differential Equations and Their Applications, vol. 10, Birkh¨auser Boston, Mas-sachusetts, 1993.

[2] A. K. Ben-Naoum, C. Troestler, and M. Willem,Existence and multiplicity results for homogeneous second order differential equations, Journal of Differential Equations112(1994), no. 1, 239–249. [3] M. S. Berger and M. Schechter,On the solvability of semilinear gradient operator equations,

Ad-vances in Mathematics25(1977), no. 2, 97–132.

[4] I. Ekeland and N. Ghoussoub,Selected new aspects of the calculus of variations in the large, Bul-letin of the American Mathematical Society39(2002), no. 2, 207–265.

[5] Y. M. Long,Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials, Nonlinear Analysis24(1995), no. 12, 1665–1671.

[6] J. Mawhin,Semicoercive monotone variational problems, Acad´emie Royale de Belgique. Bulletin de la Classe des Sciences73(1987), no. 3-4, 118–130.

[7] J. Mawhin and M. Willem,Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance, Annales de l’Institut Henri Poincar´e. Analyse Non Lin´eaire3(1986), no. 6, 431–453.

[8] ,Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, vol. 74, Springer, New York, 1989.

[9] M. Schechter,New linking theorems, Rendiconti del Seminario Matematico della Universit`a di Padova99(1998), 255–269.

[10] ,Linking Methods in Critical Point Theory, Birkh¨auser Boston, Massachusetts, 1999. [11] C.-L. Tang,Periodic solutions of non-autonomous second order systems withγ-quasisubadditive

potential, Journal of Mathematical Analysis and Applications189(1995), no. 3, 671–675. [12] ,Periodic solutions for nonautonomous second order systems with sublinear nonlinearity,

Proceedings of the American Mathematical Society126(1998), no. 11, 3263–3270.

[13] C.-L. Tang and X.-P. Wu,Periodic solutions for second order systems with not uniformly coercive potential, Journal of Mathematical Analysis and Applications259(2001), no. 2, 386–397. [14] ,Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian

systems, Journal of Mathematical Analysis and Applications275(2002), no. 2, 870–882. [15] ,Notes on periodic solutions of subquadratic second order systems, Journal of Mathematical

Analysis and Applications285(2003), no. 1, 8–16.

[16] W. Willem,Oscillations forc´ees syst`emes hamiltoniens, Public. S´emin. Analyse Non Lin´earie, Uni-versit´e de Franche-Comt´e, Besancon, 1981.

[17] X.-P. Wu and C.-L. Tang,Periodic solutions of a class of non-autonomous second-order systems, Journal of Mathematical Analysis and Applications236(1999), no. 2, 227–235.

Martin Schechter: Department of Mathematics, University of California, Irvine, CA 92697-3875, USA

References

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