Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 137084, 15 pages
doi:10.1155/2009/137084
Research Article
The Existence of Periodic Solutions for
Non-Autonomous Differential Delay Equations
via Minimax Methods
Rong Cheng
1, 2
1
College of Mathematics and Physics, Nanjing University of Information Science and Technology,
Nanjing 210044, China
2
Department 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.
Copyright q 2009 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 models 1, communication systems
2, and even in ecology 3 can be written as the following differential delay equation:
x
t
where f ∈ CR
n
, R
n
is odd and r>0 is a given constant. By using the pseudo index theory
in 24, they established the existence and multiplicity of periodic solutions of 1.2 with f
satisfying the following asymptotically linear conditions both at zero and at infinity:
f
x
B
0
x o
|
x
|
, as
|
x
|
−→ 0,
f
x
B
∞
−f
t, x
t − r
, 1.4
where f ∈ CR × R
n
, R
n
is odd with respect to x and satisfies the following superlinear
conditions both at zero and at infinity
lim
|
x
|
→0
f
t, x
|
x
|
0, uniformly in t,
1
,f
2
, ,f
n
. There exist constants μ>2andR
1
> 0 such that
0 <μ
x
i
0
f
i
t, x
1
, ,x
i−1
,y
i
,x
i1
, ,x
n
dy
i
≤ x
|
x
i
|
λ
1.7
for all x ∈ R
n
with |x
i
| >R
2
, for all t ∈ 0, 2r and i 1, 2, ,n.
Advances in Difference Equations 3
Then our main result can be read as follows.
Theorem 1.1. Suppose that ft, x ∈ CR × R
n
, R
n
satisfies 1.5 and the conditions H1–H3
hold. Then 1.4 possesses a nontrivial 4r-periodic solution.
Remark 1.2. We shall use a minimax theorem in critical point theory in 25 to prove our main
result. The ideas come from 25–27. Theorem 1.1 will be proved in Section 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. Let E be a Hilbert space with E E
1
⊕ E
2
.LetP
u − Φ
t, u
, 2.1
where Φ : 0, 2r × E → E
2
is compact.
Definition 2.1. Let S, Q ⊂ E, and Q be boundary. One calls S and ∂Q link if whenever ϕ ∈ Λ
and ϕt, ∂Q ∩ S ∅ for all t, then ϕt, Q ∩ S
/
∅.
Definition 2.2. A f unctional φ ∈ C
1
E, R satisfies PS condition, if every sequence that
{x
m
}⊂E, φ
x
m
→ 0andφx
m
being bounded, possesses a convergent subsequence.
Then 25, Theorem 5.29 can be stated as follows.
Theorem A. Let E be a real Hilbert space with E E
1
⊕E
2
is compact, and
I
3
there exists a subspace
E ⊂ E and sets S ⊂ E, Q ⊂
E and constants α>ωsuch that
i S ⊂ E
1
and φ|
S
≥ α,
ii Q is bounded and φ|
∂Q
≤ ω,
iii S and ∂Q link.
Then φ possesses a critical value c ≥ α.
Let
F
t, x
x
1
0
f
Then Ft, 00andF
t, xf
1
,f
2
, ,f
n
, where F
denotes the gradient of F with respect
to x. We have the following lemma.
4 Advances in Difference Equations
Lemma 2.3. Under the conditions of Theorem 1.1, the function F satisfies the following.
i Ft, x ∈ C
1
0, 2r ×R
n
, R is 2r-periodic with respect to t and Ft, x ≥ 0 for all t, x ∈
0, 2r × R
n
,
ii
lim
|
x
|
→0
F
n
with
|x| >Land |x
i
|≥R, i 1, 2, ,n, and t ∈ 0, 2r
0 <μF
t, x
≤
x, F
t, x
, 2.5
F
t, x
≤ c
2
|
,
···
x
n−1
r sin θ
1
sin θ
2
sin θ
3
···sin θ
n−2
cos θ
n−1
,
x
n
r sin θ
1
sin θ
2
sin θ
3
···sin θ
n−2
sin θ
n−1
.
2.7
Then |x|
√
nR
2
with |x
i
|≥R
2
,thatis,
f
2
1
t, x
1
··· f
2
n
t, x
n
≤ c
2
2
x
2
1
··· x
2
1
t, x
1
f
2
2
t, x
2
≤ c
2
1
|x
1
|
2λ
|x
2
|
2λ
.Let|x
1
|
λ
τ cos θ, |x
2
|
λ
τ sin θ.
By 1 <λ<2, 1 − sin
≥ τ
2
|
x
1
|
2λ
|
x
2
|
2λ
.
2.9
Advances in Difference Equations 5
By reducing method, we have
x
2λ
1
··· x
2λ
n
≤
x
2
,R
2
}. Then 2.5 and 2.6 hold with |x| >Land
|x
i
| >R.
Below we will construct a variational functional of 1.4 defined on a suitable Hilbert
space such that finding 4r-periodic solutions of 1.4 is equivalent to seeking critical points
of the functional.
Firstly, we make the change of variable
t −→
π
2r
t ν
−1
t.
2.11
Then 1.4 can be changed to
x
t
−νf
t, x
t −
π
2
a
m
cos mt b
m
sin mt
,
2.13
where a
m
,b
m
are n-vectors. H is the set of such functions that
x
2
|
a
0
|
2
∞
m1
m
|
a
m
,a
m
b
m
,b
m
,
2.15
where y a
0
∞
m1
a
m
cos mt b
m
t, x
t
dt.
2.16
6 Advances in Difference Equations
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
t
π
2
,y
t
Ax, x
ψ
x
, ∀x ∈ H.
2.19
According to a standard argument in 24, one has for any x, y ∈ H,
φ
x
,y
2π
0
1
2
x
t
π
dt.
2.20
Moreover according to 28, ψ
: H → H is a compact operator defined by
ψ
x
,y
ν
2π
0
f
t, x
t
,y
t
dt.
t
. 2.23
Lemma 2.4. Critical points of φ|
E
over E are critical points of φ on H,whereφ|
E
is the restriction of
φ over E.
Proof. Note that any x ∈ E is 2π-periodic and f is odd with respect to x. It is enough for us to
prove φ
x,y 0 for any y ∈ H and x being a critical point of φ in E.
Advances in Difference Equations 7
For any y ∈ H, we have
Γ
2
φ
x
,y
Γ
2
2π
0
x
t
π
2
,y
t − π
dt ν
2π
0
f
t, x
t
,y
t π
,y
t
dt
2π
0
−x
t
π
2
,y
t
dt ν
2π
0
dt − ν
2π
0
f
t, x
t
,y
t
dt
−φ
x
,y
.
2.24
This yields Γ
2
φ
x
,y
1
φ
x
,y
2
0. 2.25
The proof is complete.
Remark 2.5. By Lemma 2.4, we only need to find critical points of φ|
E
over E. Therefore in the
following φ will be assumed on E.
For x ∈ E, xt π−xt yields that a
0
0, where a
x
t
π
2
,y
t
dt
−
2π
0
x
t
,y
t −
π
2
dt
⊕E
−
. Letting E
1
E
, E
2
E
−
,weseethatI
1
of Theorem A holds.
Since ψ
is compact, I
2
of Theorem A holds. Now we establish I
3
of Theorem A by the
following three lemmas.
Lemma 2.6. Under the assumptions of Theorem 1.1, i of I
3
holds for φ.
Proof. From the assumptions of Theorem 1.1 and Lemma 2.3, one has
F
t, x
≤ c
, ∀t ∈
0,π
,
|
x
|
≤ δ.
2.28
Therefore, there is an M Mε > 0 such that
F
t, x
≤ ε
|
x
|
2
M
|
x
|
λ1
, ∀
t, x
∈
x
λ1
L
λ1
≤
εc
5
Mc
6
x
λ−1
x
2
.
2.30
Consequently, for x ∈ E
1
E
,
φ
x
1. Then for any x ∈ ∂B
ρ
∩ E
1
,
φ
x
≥
1
3
ρ
2
.
2.32
Thus φ satisfies i of I
3
with S ∂B
ρ
∩ E
1
and α 1/3ρ
2
.
Lemma 2.7. Under the assumptions of Theorem 1.1, φ satisfies ii of I
3
.
Proof. Set e ∈ S ∂B
ρ
1
,λ
−
inf
x∈E
−
,
x
1
Ax
−
,x
−
,λ
sup
x∈E
,
x
,sx
1
2
Asx
−
,sx
−
− ν
2π
0
F
t, sx
dt
≤−
1
2
λ
−
s
2
2
x
2
x
−
2
≤
1 γ
2
x
2
. 2.36
That is
x
|
≥ ε
1
}
≥ ε
1
. 2.38
Now for x x
x
−
∈
K,setΩ
x
{t ∈ 0,π : |xt|≥ε
1
}.By2.4, for a constant M
0
A/νε
3
1
> 0, there is an L
3
> 0 such that
F
t, z
≥ M
0
|
sxt
|
2
≥ M
0
s
2
ε
2
1
, ∀t ∈ Ω
x
.
2.40
For s ≥ s
1
, we have
φ
sx
1
2
s
2
s
2
− ν
Ω
x
F
t, sx
dt
≤
1
2
A
s
2
− M
0
s
2
ε
2
1
meas
Ω
∂Q
≤ 0. Then ii of I
3
holds.
Lemma 2.8. S and ∂Q link.
Proof. Suppose ϕ ∈ Λ and ϕ∂Q∩S ∅for all t ∈ 0,π. Then we claim that for each t ∈ 0,π,
there is a wt ∈ Q such that φt, wt ∈ S,thatis,
Pϕ
w
t
0,
w
t
ρ, 2.42
where P : E → E
−
is a projection. Define
G :
0,π
× Q −→ E
−
u se
− ρ
e. 2.44
It is easy to see that
G
t, u se
u
s − ρ
e
/
0, as u se ∈ ∂Q. 2.45
However,
G
1,u se
Pϕ
u se
; Q, 0
deg
G
0, ·
; Q, 0
deg
id
E
−
; E
−
∩ B
2s
1
, 0
deg
s − ρ,
0, 2s
1
} is bounded. If {x
m
} is not bounded, then by passing to a subsequence
if necessary, let x
m
→∞ as m → ∞.
By 2.4, there exists a constant M
> 0 such that Ft, x >c
7
|x|
2
as |x| >M
.By2.5,
one has
2φ
x
m
−
φ
x
m
,x
F
t, x
m
dt
≥ c
7
ν
μ − 2
2π
0
|
x
m
|
2
dt.
2.49
This yields
2π
0
|
x
m
, ∀
t, x
∈
0,π
× R
n
.
2.51
Therefore,
2π
0
F
t, x
m
k
dt ≤
2π
0
0
x
m
2κλ−1
dt
1/2
c
10
≤ c
11
2π
0
x
m
2
dt
1/2
x
m
κλ−1
≤
c
11
2π
0
x
m
2
dt
1/2
x
m
1/2
x
m
κλ−1
x
m
κ−1/2
,x
−
m
Ax
−
m
,x
−
m
−
2π
0
x
−
m
,F
t, x
m
dt
dt
≥
Ax
−
m
,x
−
m
−
2π
0
F
t, x
m
κ
dt
1/κ
C
κ
φ
x
m
x
−
m
x
m
x
−
m
2π
0
|
F
−
m
x
m
−→ 0asm −→ ∞.
2.56
12 Advances in Difference Equations
Similarly, we have
x
m
x
m
−→ 0asm −→ ∞.
2.57
Thus it follows from 2.56 and 2.57 that
1
x
m
x
: H → H is compact. Since {x
m
} is bounded, we may suppose that
ψ
x
m
−→ y as m −→ ∞. 2.59
Since A has continuous inverse A
−1
in E, it follows from
Ax
m
φ
x
m
ψ
x
m
2.60
that
x
Lemma A.1. There exists ε
1
> 0 such that, for all u ∈
K,
meas
{
t ∈
0,π
:
|
u
t
|
≥ ε
1
}
≥ ε
1
. A.1
Proof. If A.1 is not true, ∀k>0, there exists u
k
∈
K such that
meas
E. Notice that dimspan{e} < ∞and u
k
≤1. In the sense of subsequence,
we have
u
k
−→ u
0
∈ span
{
e
}
as k −→ ∞. A.3
Then 2.37 implies that
u
0
2
≥
1
1 γ
2
in L
2
,thatis,
π
0
|
u
k
− u
0
|
2
dt −→ 0ask −→ ∞.
A.6
By A.4 we know that u
0
> 0. Therefore,
π
0
|u
0
|
2
dt > 0. Then there exist δ
1
> 0,δ
2
> 0 such
u
0
t
|
≥
1
n
0, A.8
that is, meas{t ∈ 0,π : |u
0
t|≤1/n} 1, 0 <
π
0
|u
0
|
2
dt < 1/n
2
→ 0 as n → ∞.Wegeta
contradiction. Thus A.7 holds. Let Ω
0
{t ∈ 0,π : |u
0
t|≥δ
1
Ω
0
− meas
Ω
0
∩ Ω
⊥
k
≥ δ
2
−
1
k
.
A.9
Let k be large enough such that δ
2
− 1/k ≥ 1/2δ
2
and δ
1
− 1/k ≥ 1/2δ
1
. Then we have
|
u
k
π
0
|
u
k
− u
0
|
2
dt ≥
Ω
k
∩Ω
0
|
u
k
− u
0
|
2
dt ≥
1
2
δ
1
2
2
1
2
δ
2
> 0.
A.11
This is a contradiction to A.6. Therefore the lemma is true and A.1 holds.
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 Technology 20070049.
References
1 W. J. Cunningham, “A nonlinear differential-difference equation of growth,” Proceedings of the National
Academy of Sciences of the United States of America, vol. 40, no. 4, pp. 708–713, 1954.
2 T. Furumochi, “Existence of periodic solutions of one-dimensional differential-delay equations,” The
T
ˆ
ohoku Mathematical Journal, vol. 30, no. 1, pp. 13–35, 1978.
3 R. May, Stablity and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA,
1973.
4 G. S. Jones, “The existence of periodic solutions of f
x−αfxt − 1{1 fx},” Journal of
Mathematical Analysis and Applications, vol. 5, no. 3, pp. 435–450, 1962.
5 S N. Chow and H O. Walther, “Characteristic multipliers and stability of symmetric periodic
solutions of ˙x gxt − 1,” Transactions of the American Mathematical Society, vol. 307, no. 1, pp.
127–142, 1988.
point theory,” Journal of Differential Equations, vol. 218, no. 1, pp. 15–35, 2005.
18 J. Llibre and A A. Tart¸a, “Periodic solutions of delay equations with three delays via bi-Hamiltonian
systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2433–2441, 2006.
19 S. Jekel and C. Johnston, “A Hamiltonian with periodic orbits having several delays,” Journal of
Differential Equations, vol. 222, no. 2, pp. 425–438, 2006.
20 G. Fei, “Multiple periodic solutions of differential delay equations via Hamiltonian systems. I,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 1, pp. 25–39, 2006.
21 G. Fei, “Multiple periodic solutions of differential delay equations via Hamiltonian systems. II,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 1, pp. 40–58, 2006.
22 M. A. Han, “Bifurcations of periodic solutions of delay differential equations,” Journal of Differential
Equations, vol. 189, no. 2, pp. 396–411, 2003.
23 P. Dormayer, “The stability of special symmetric solutions of ˙xtαfxt − 1 with small
amplitudes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 8, pp. 701–715, 1990.
24 V. Benci, “On critical point theory for indefinite functionals in the presence of symmetries,”
Transactions of American Mathematical Society, vol. 247, no. 2, pp. 533–572, 1982.
25 P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,
vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence,
RI, USA, 1986.
26 G. Fei, “On periodic solutions of superquadratic Hamiltonian systems,” Electronic Journal of Differential
Equations
, vol. 8, pp. 1–12, 2002.
27 K. C. Chang, Critical Point Theory and Applications, Shanghai Science & Technology Press, Beijing,
China, 1986.
28 Y. Long and E. Zehnder, “Morse-theory for forced oscillations of asymptotically linear Hamiltonian
systems,” in Stochastic Processes in Physics and Geometry, pp. 528–563, World Scientific, Singapore, 1990.
29 J. Mawhin and M. Willem, “Critical point theory and Hamiltonian systems,” in Applied Mathematical
Sciences, vol. 74, Springer, Berlin, Germany, 1989.
Copyright: Tài liệu đại học ©