doi:10.1155/2009/137084
Research Article
The Existence of Periodic Solutions for
Non-Autonomous Differential Delay Equations
via Minimax Methods
Rong Cheng
1, 21College of Mathematics and Physics, Nanjing University of Information Science and Technology,
Nanjing 210044, China
2Department of Mathematics, Southeast University, Nanjing 210096, China
Correspondence should be addressed to Rong Cheng,[email protected]
Received 9 April 2009; Accepted 19 October 2009
Recommended by Ulrich Krause
By using variational methods directly, we establish the existence of periodic solutions for a class of nonautonomous differential delay equations which are superlinear both at zero and at infinity.
Copyrightq2009 Rong Cheng. 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 and Main Result
Many equations arising in nonlinear population growth models1, communication systems
2, and even in ecology3can be written as the following differential delay equation:
xt −αfxt−1, 1.1
where f ∈ CR,R is odd and αis parameter. Since Jone’s work in 4, there has been a great deal of research on problems of existence, multiplicity, stability, bifurcation, uniqueness, density of periodic solutions to 1.1 by applying various approaches. See 2, 4–23. But most of those results concern scalar equations1.1and generally slowly oscillating periodic solutions. A periodic solutionxtof1.1is called a “slowly oscillating periodic solution” if there exist numbersp >1 andq > p1 such thatxt>0 for 0< t < p,xt<0 forp < t < q,
andxtq xtfor allt.
In a recent paper17, Guo and Yu applied variational methods directly to study the following vector equation:
wheref ∈CRn,Rnis odd andr >0 is a given constant. By using the pseudo index theory in24, they established the existence and multiplicity of periodic solutions of1.2withf
satisfying the following asymptotically linear conditions both at zero and at infinity:
fx B0xo|x|, as|x| −→0,
fx B∞xo|x|, as|x| −→ ∞, 1.3
whereB0 andB∞are symmetricn×nconstant matrices. Before Guo and Yu’s work, many
authors generally first use the reduction technique introduced by Kaplan and Yorke in7to reduce the search for periodic solutions of1.2withn1 and its similar ones to the problem of finding periodic solutions for a related system of ordinary differential equations. Then variational method was applied to study the related systems and the existence of periodic solutions of the equations is obtained.
The previous papers concern mainly autonomous differential delay equations. In this paper, we use minimax methods directly to study the following nonautonomous diff erential-delay equation:
xt −ft, xt−r, 1.4
where f ∈ CR×Rn,Rn is odd with respect tox and satisfies the following superlinear conditions both at zero and at infinity
lim
|x| →0
ft, x
|x| 0, uniformly int,
lim
|x| → ∞
ft, x
|x| ∞, uniformly int.
1.5
When1.2satisfies1.3, we can apply the twist condition between the zero and at infinity forfto establish the existence of periodic solutions of1.2. Under the superlinear conditions
1.5, there is no twist condition forf, which brings difficulty to the study of the existence of periodic solutions of1.4. But we can use minimax methods to consider the problem without twist condition forf.
Throughout this paper, we assume that the following conditions hold.
H1ft, x∈CR×Rn,Rnis odd with respect toxand 2r-periodic with respect tot.
H2writef f1, f2, . . . , fn. There exist constantsμ >2 andR1>0 such that
0< μ xi
0 fi
t, x1, . . . , xi−1, yi, xi1, . . . , xn
dyi≤xifit, x 1.6
for allx∈Rnwith|x
i|> R1, for allt∈0,2randi1,2, . . . , n. H3there exist constantsc1 >0,R2>0 and 1< λ <2 such that
fit, x< c1|xi|λ 1.7
for allx∈Rnwith|x
Then our main result can be read as follows.
Theorem 1.1. Suppose thatft, x∈CR×Rn,Rnsatisfies1.5and the conditionsH1–H3
hold. Then1.4possesses a nontrivial4r-periodic solution.
Remark 1.2. We shall use a minimax theorem in critical point theory in25to prove our main result. The ideas come from25–27.Theorem 1.1will be proved inSection 2.
2. Proof of the Main Result
First of all in this section, we introduce a minimax theorem which will be used in our discussion. LetEbe a Hilbert space withEE1⊕E2. LetP1, P2 be the projections ofEonto E1andE2, respectively.
Write
Λ ϕ∈C0,2r×E|ϕ0, u uandP2ϕt, u P2u−Φt, u
, 2.1
whereΦ:0,2r×E → E2is compact.
Definition 2.1. LetS, Q⊂E,andQbe boundary. One callsSand∂Qlink if wheneverϕ∈Λ
andϕt, ∂Q∩S∅for allt, thenϕt, Q∩S /∅.
Definition 2.2. A functional φ ∈ C1E,R satisfies P S condition, if every sequence that {xm} ⊂E,φxm → 0 andφxmbeing bounded, possesses a convergent subsequence.
Then25, Theorem 5.29can be stated as follows.
Theorem A. LetEbe a real Hilbert space withEE1⊕E2,E2E⊥1and inner product·,·. Suppose
φ∈C1E,RsatisfiesP Scondition,
I1φx 1/2Ax, xψx, whereAz A1P1xA2P2xandAi:Ei → Eiis bounded
and selfadjoint,i1,2,
I2ψis compact, and
I3there exists a subspaceE⊂Eand setsS⊂E,Q⊂Eand constantsα > ωsuch that iS⊂E1andφ|S≥α,
iiQis bounded andφ|∂Q≤ω,
iiiSand∂Qlink.
Thenφpossesses a critical valuec≥α.
Let
Ft, x x1
0 f1
t, y1, x2, . . . , xn
dy1· · · xn
0 fn
t, x1, . . . , xn−1, yn
dyn. 2.2
Lemma 2.3. Under the conditions ofTheorem 1.1, the functionFsatisfies the following.
iFt, x∈C10,2r×Rn,Ris 2r-periodic with respect totandFt, x≥0for allt, x∈
0,2r×Rn,
ii
lim
|x| →0 Ft, x
|x|2 0, uniformly int, 2.3
lim
|x| → ∞ Ft, x
|x|2 ∞, uniformly int. 2.4
iiiThere exist constantsc2,L > 0,andR > 0such that for allx x1, . . . , xn∈ Rnwith
|x|> Land|xi| ≥R,i1,2, . . . , n, andt∈0,2r
0< μFt, x≤x, Ft, x, 2.5
Ft, x≤c
2|x|λ, 2.6
where·,·denotes the inner product inRn.
Proof. The definition ofFimpliesidirectly. We prove caseiiand caseiii.
Caseii. Let
x1rsinθ1, x2 rsinθ1conθ2, x3rsinθ1sinθ2conθ3,
· · ·
xn−1rsinθ1sinθ2sinθ3· · ·sinθn−2cosθn−1, xnrsinθ1sinθ2sinθ3· · ·sinθn−2sinθn−1.
2.7
Then|x|2r2and|x| → 0 or|x| → ∞is equivalent tor → 0 orr → ∞, respectively.
From1.5and L’Hospital rules, we have2.3by a direct computation.
Caseiii. ByH2, we have a constantL1 √
nR1such that 0< μFt, x≤x, Ft, x
for|x|> L1with|xi| ≥R1.
Now we prove|Ft, x| ≤c2|x|λfor|x|> L2√nR2with|xi| ≥R2, that is,
f2
1t, x1 · · ·fn2t, xn≤c22 x21· · ·xn2
λ
. 2.8
Firstly, it follows from|f1t, x1| ≤c1|x1|λthatf12t, x1≤c12|x1|2λ.
Now we showf2
1t, x1 f22t, x2≤c21|x1|2λ|x2|2λ. Let|x1|λτcosθ,|x2|λτsinθ.
By 1< λ <2, 1−sin2θ≥1−sin2/λλ, that is,cos2/λsin2/λλ≥1. Then
x2 1x22
λ
τ2 cos2/λθsin2/λθ≥τ2|x
By reducing method, we have
x12λ· · ·x2nλ≤ x21· · ·x2nλ|x|2λ. 2.10
Thus, the inequality|Ft, x| ≤c2|x|λfor|xi| ≥R2holds.
TakeLmax{L1, L2}andRmax{R1, R2}. Then2.5and2.6hold with|x|> Land |xi|> R.
Below we will construct a variational functional of1.4defined on a suitable Hilbert space such that finding 4r-periodic solutions of 1.4is equivalent to seeking critical points of the functional.
Firstly, we make the change of variable
t−→ π
2rtν
−1t. 2.11
Then1.4can be changed to
xt −νf t, x t−π
2
, 2.12
wherefisπ-periodic with respect tot. Therefore we only seek 2π-periodic solution of2.12 which corresponds to the 4r-periodic solution of1.4.
We work in the Sobolev spaceHW1/2,2S1,R2N. The simplest way to introduce this space seems as follows. Every functionx∈L2S1,Rnhas a Fourier expansion:
xt a0
∞
m1
amcosmtbmsinmt, 2.13
wheream, bmaren-vectors.His the set of such functions that
x2|a 0|2
∞
m1
m |am|2|bm|2
<∞. 2.14
With this norm · ,His a Hilbert space induced by the inner product·,·defined by
x, y2πa0, a0
π ∞
m1
mam, am
bm, bm
, 2.15
whereya0m∞1amcosmtbmsinmt. We define a functionalφ:H → Rby
φx 2π
0
1 2 x
t π
2
, xtdtν 2π
0
By Riesz representation theorem, H identifies with its dual space H∗. Then we define an operator A:H→H∗H by extending the bilinear form:
Ax, y 2π
0
x tπ
2
, ytdt, ∀x, y∈H. 2.17
It is not difficult to see thatAis a bounded linear operator onHand kerARn. Define a mappingψ:H → Ras
ψx ν
2π
0
Ft, xtdt, 2.18
Then the functionalφcan be rewritten as
φx 1
2Ax, xψx, ∀x∈H. 2.19
According to a standard argument in24, one has for anyx, y∈H,
φx, y 2π
0
1 2 x
tπ
2
−x t−π
2
, ytdtν 2π
0
ft, xt, ytdt. 2.20
Moreover according to28,ψ:H → His a compact operator defined by
ψx, yν 2π
0
ft, xt, ytdt. 2.21
Our aim is to reduce the existence of periodic solutions of2.12to the existence of critical points ofφ. For this we introduce a shift operatorΓ:H → Hdefined by
Γxt x tπ
2
. 2.22
It is easy to compute thatΓis bounded and linear. MoreoverΓis isometric, that is,Γxx andΓ4id, whereiddenotes the identity mapping onH.
Write
Ex∈H:Γ2xt −xt. 2.23
Lemma 2.4. Critical points ofφ|EoverEare critical points ofφonH, whereφ|Eis the restriction of
φoverE.
For anyy∈H, we have
Remark 2.5. ByLemma 2.4, we only need to find critical points ofφ|EoverE. Therefore in the followingφwill be assumed onE.
Forx∈E,xtπ −xtyields thata0 0, wherea0is in the Fourier expansion of
HenceAis self-adjoint onE.
LetE andE−denote the positive definite and negative definite subspace ofAinE, respectively. ThenEE⊕E−. LettingE1E,E2E−, we see thatI1of Theorem A holds.
Sinceψis compact, I2of Theorem A holds. Now we establishI3of Theorem A by the
following three lemmas.
Lemma 2.6. Under the assumptions ofTheorem 1.1,iofI3holds forφ.
Proof. From the assumptions ofTheorem 1.1andLemma 2.3, one has
By2.3, for anyε >0, there is aδ >0 such that
Ft, x≤ε|x|2, ∀t∈0, π, |x| ≤δ. 2.28
Therefore, there is anMMε>0 such that
Ft, x≤ε|x|2M|x|λ1, ∀t, x∈0, π×Rn. 2.29
SinceEis compactly embedded inLsS1,Rnfor alls≥1 and by2.29, we have
2π
0
Ft, xdt≤εx2
L2MxλLλ11 ≤ εc5Mc6xλ−1
x2. 2.30
Consequently, forx∈E1E,
φx≥ x2−ν εc
5Mc6xλ−1
x2. 2.31
Chooseε 3νc5−1andρso that 3νMc6ρλ−11. Then for anyx∈∂Bρ∩E1,
φx≥ 1
3ρ
2. 2.32
ThusφsatisfiesiofI3withS∂Bρ∩E1andα 1/3ρ2.
Lemma 2.7. Under the assumptions ofTheorem 1.1,φsatisfiesiiofI3.
Proof. Sete∈S∂Bρ∩E1and let
Q{se: 0≤s≤2s1} ⊕B2s1∩E2, 2.33
wheres1is free for the moment.
LetEE−⊕span{e}. Write
Kx∈E:x1, λ− inf x∈E−,x1
Ax−, x−, λ sup x∈E,x1
|Ax, x|.
2.34
Case1. Ifx−> γxwithγ λ/λ−, one has
φsx 1
2Asx
, sx1
2
Asx−, sx−−ν 2π
0
Ft, sxdt
≤ −1
2λ
−s2x−21
2λ
s2x2≤0.
Case2. Ifx− ≤γx, we have
1x2x2x−2 ≤ 1γ2x2. 2.36
That is
x2≥ 1
1γ2 >0. 2.37
DenoteK {x∈K:x− ≤γx}. By appendix, there existsε1>0 such that∀u∈K,
meas{t∈0, π:|ut| ≥ε1} ≥ε1. 2.38
Now forx x x− ∈K, setΩx {t ∈ 0, π :|xt| ≥ ε1}. By2.4, for a constantM0 A/νε3
1>0, there is anL3>0 such that
Ft, z≥M0|x|2, ∀|x| ≥L3 uniformly int. 2.39
Choosings1≥L3/ε1, fors≥s1,
Ft, sxt≥M0|sxt|2≥M0s2ε21, ∀t∈Ωx. 2.40
Fors≥s1, we have
φsx 1
2s
2Ax, x1
2s
2Ax−, x−−ν2π 0
Ft, sxdt
≤ 1
2As
2−ν
Ωx
Ft, sxdt
≤ 1
2As
2−M
0s2ε21measΩx
≤ 1
2As
2−M
0s2ε31−1
2As
2<0.
2.41
Henceforth,φsx≤0 for anyx∈Kands≥s1, that is,φ|∂Q≤0. TheniiofI3holds.
Lemma 2.8. Sand∂Qlink.
Proof. Supposeϕ∈Λandϕ∂Q∩S∅for allt∈0, π. Then we claim that for eacht∈0, π, there is awt∈Qsuch thatφt, wt∈S, that is,
P ϕwt 0, wtρ, 2.42
whereP:E → E−is a projection. Define
as follows:
Gt, use 1−tuP ϕuse1−tstI−Pϕuse−ρe. 2.44
It is easy to see that
Gt, use us−ρe /0, asuse∈∂Q. 2.45
However,
G1, use P ϕuse I−Pϕuse−ρe
G0, use us−ρe. 2.46
According to topological degree theory in29, we have
degG1,·;Q,0 degG0,·;Q,0
degidE−;E−∩B2s1,0degs−ρ,0,2s1,01 2.47
sinceρ∈0,2s1. ThereforeSand∂Qlink.
Now it remains to verify thatφsatisfiesP S-condition.
Lemma 2.9. Under the assumptions ofTheorem 1.1,φsatisfiesP S-condition.
Proof. Suppose that
φxm≤M, φxm−→0, asm−→ ∞. 2.48
We first show that{xm}is bounded. If{xm}is not bounded, then by passing to a subsequence if necessary, letxm → ∞asm → ∞.
By2.4, there exists a constantM >0 such thatFt, x> c7|x|2as|x|> M. By2.5,
one has
2φxm−
φxm, xm
2π
0
xm, νFt, xm
−2νFt, xm
dt
≥ 2π
0
νμ−2Ft, xmdt
≥c7ν
μ−2
2π
0
|xm|2dt.
2.49
This yields
2π 0 |xm|
2dt
xm −→
Writeκ1/2λ−1. By2.6, there is a constantc9 >0 such that
This inequality and2.50imply that
⎛ By the above inequality, one has
Similarly, we have
x
m
xm −→
0 asm−→ ∞. 2.57
Thus it follows from2.56and2.57that
1 xm
xm ≤
x
mx−m
xm −→
0 asm−→ ∞, 2.58
which is a contradiction. Hence{xm}is bounded.
Below we show that{xm}has a convergent subsequence. Notice that kerA|E{0}and
ψ:H → His compact. Since{xm}is bounded, we may suppose that
ψxm−→y asm−→ ∞. 2.59
SinceAhas continuous inverseA−1inE, it follows from
Axmφxm ψxm 2.60
that
xmA−1
φxm ψxm
−→A−1y asm−→ ∞. 2.61
Henceforth{xm}has a convergent subsequence.
Now we are ready to proveTheorem 1.1.
Proof ofTheorem 1.1. It is obviously thatTheorem 1.1holds from Lemmas2.3,2.4,2.6,2.7,2.8, and2.9and Theorem A.
Appendix
The purpose of this appendix is to prove the following lemma. The main idea of the proof comes from26.
Lemma A.1. There existsε1>0such that, for all u∈K,
meas{t∈0, π:|ut| ≥ε1} ≥ε1. A.1
Proof. IfA.1is not true,∀k >0, there existsuk∈K such that
meas t∈0, π:|ut| ≥ 1
k !
≤ 1
Writeuku−kuk ∈E. Notice that dimspan{e}<∞anduk ≤1. In the sense of subsequence, we have
uk−→u0 ∈span{e} ask−→∞. A.3
Then2.37implies that
u
0 2 ≥ 1
1γ2 >0. A.4
Note thatu−k ≤1, in the sense of subsequenceu−k u−0 ∈E−ask → ∞. Thus in the sense of subsequence,
uk u0u−0u0 ask−→∞. A.5
This means thatuk → u0inL2, that is, π
0
|uk−u0|2dt−→0 ask−→∞. A.6
ByA.4we know thatu0>0. Therefore, π
0|u0|2dt >0. Then there existδ1 >0, δ2>0 such
that
meas{t∈0, π:|u0t| ≥δ1} ≥δ2. A.7
Otherwise, for alln >0, we must have
meas t∈0, π:|u0t| ≥ 1 n
!
0, A.8
that is, meas{t∈0, π:|u0t| ≤1/n}1, 0< π
0|u0|2dt <1/n2 → 0 as n → ∞. We get a
contradiction. ThusA.7holds. LetΩ0{t∈0, π:|u0t| ≥δ1},Ωk {t∈0, π:|u0t| ≤
1/k}, andΩ⊥k 0, π\Ωk. ByA.2, we have
measΩk∩Ω0 meas Ω0−Ω0∩Ω⊥k
≥measΩ0−meas Ω0∩Ω⊥k
≥δ2−
1
k. A.9
Letkbe large enough such thatδ2−1/k≥1/2δ2andδ1−1/k≥1/2δ1. Then we have
|uk−u0|2≥ "
δ1−
1
k #2
≥ "
1 2δ1
#2
This implies that
This is a contradiction toA.6. Therefore the lemma is true andA.1holds.
Acknowledgments
This work is supported by the Specialized Research Fund for the Doctoral Program of Higher Education for New Teachers and the Science Research Foundation of Nanjing University of Information Science and Technology20070049.
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