Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 115, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
MULTIPLICITY AND MINIMALITY OF PERIODIC SOLUTIONS
TO DELAY DIFFERENTIAL SYSTEM
HUAFENG XIAO, ZHIMING GUO
Abstract. In this article, we study periodic solutions of a class of delay differential equations. By restricting our discussion on generalized Nehari Manifold,
some sufficient conditions are obtained to guarantee the existence of infinitely
many pairs of periodic solutions. Also, there exists at least one periodic solution with prescribed minimal period.
1. Introduction
The existence of periodic solutions to delay differential equations have been investigated since 1962. Various methods, such as fixed point theory, Kaplan-Yorke
method, coincidence degree theory, the Hopf bifurcation theorem, the PoicaréBendixson theorem and critical point theory, have been used to study such a problem. We refer to[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 26, 27].
As far as the authors know, there are only a few results concerning with periodic
solutions with prescribed minimal period to delay differential equations. In 1974,
Kaplan and Yorke [10] translated the problem of existence of periodic solutions to
delay differential equations to that of ordinary differential equations. Then, they
showed the existence of solutions with minimal period to delay differential equations with one or two delays. In 1978, by using a completely different approach,
Nussbaum [22] extended Kaplan and Yorke’s result to differential equation with
arbitrary delays. In 2012, by making use of rigorous analysis techniques, Yu [26]
proved the existence, nonexistence, multiplicity and minimality of periodic solutions to differential equation with single delay. In 2013, Yu and the author [27]
made use of Maslov-type index and Fourier series showing the existence of multiplicity periodic solutions with the same minimal period to a class of nonautonomous
differential equation with single delay.
Because of lacking effective tools, no more results on solutions with prescribed
minimal period to delay differential system have been obtained so far. As we know,
critical point theory is a useful tool to prove the minimality of periodic solutions to
ordinary differential systems and difference systems. A natural question is weather
or not critical point theory can be used to study the minimality of period for
periodic solutions to delay differential system.
2000 Mathematics Subject Classification. 34K13, 58E05.
Key words and phrases. Nehari Manifold; periodic solution; delay differential equation;
minimal period.
c
2014
Texas State University - San Marcos.
Submitted February 21, 2014. Published April 21, 2014.
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Nehari manifold method, introduced by Nehari [20, 21] and generalized by Szukin
and Weth [24], has been widely used to study the existence of ground state solutions to partial differential equations. We refer to [2, 23]. Recall that a ground
state solution is a solution which minimizes the variational functional on the set of
nontrivial solutions. When a variational functional is restricted on a Nehari Manifold, it possesses some minimal characteristic. Such a characteristic can be used to
prove the minimality of periodic solutions. Thus Nehari manifold had been used
to study solutions with minimal period to Hamiltonian systems, see for example
[1, 16]. In this article, we devote to making use of critical point theory combining
with generalized Nehari manifold to study the existence of solutions with prescribed
minimal period to delay differential system.
Before going too far, let us introduce some notation. Denote by N, Z, R∗ , R the
set of all natural numbers, integers, nonnegative real numbers and real numbers,
respectively. For N ∈ N, denote by RN the N −deminsional Hilbert space with
the usual inner product (·, ·) and the usual norm | · |. Let S 1 = R/(2πZ) and ϕ ∈
C 1 (X, R). A sequence {xn } ⊂ X is called (P S)c sequence (resp. (P S) sequence) of
ϕ if it satisfies ϕ(xn ) → c and ϕ0 (xn ) → 0 (resp. ϕ(xn ) is bounded and ϕ0 (xn ) → 0)
as n → ∞. We say that ϕ satisfies (P S)c condition (resp. (P S) condition) if every
(P S)c sequence (resp. (P S) sequence) has a convergent subsequence. Obviously,
ϕ satisfying (P S) condition implies ϕ satisfying (P S)c condition for each c ∈ R.
Consider the delay differential equation
π
π
(1.1)
x0 (t) = −Ax(t − ) − f (x(t − )), x(t) ∈ RN .
2
2
We use the following assumptions:
(A1) A is a symmetric, nonnegative definite matrix, {(−1)j+1 (2j − 1) : j ∈
N} ∩ σ(A) = ∅, where σ(A) denotes the spectral of matrix A;
(F1) f is odd, i.e., f (−x) = −f (x), for any x ∈ RN ;
(F2) there exists a function F ∈ C 1 (RN , R) such that F (0) = 0 and the gradient
of F is f ; i.e., for any x ∈ RN , ∇F (x) = f (x);
(F3) there exist constants s > 1 and a > 0 such that
F (x) ≤ a(1 + |x|s ),
∀x ∈ RN ;
(F4) there exists µ > 2 such that
0 < µF (x) ≤ (x, f (x)),
∀ x ∈ RN \ {0};
(F5) (f (x), y)(x, y) ≥ 0 for all x, y ∈ RN ;
(F6) F (x) = F (y) and (f (x), y) ≤ (f (x), x) if |x| = |y|;
(F7) (f (x), y) 6= (f (y), x) if |x| =
6 |y| and (x, y) 6= 0.
Remark 1.1. Denote M = max|x|=1 F (x) and m = min|x|=1 F (x). Then (F4)
implies that F (x) ≤ M |x|µ , when |x| ≤ 1; F (x) ≥ m|x|µ , when |x| ≥ 1.
Our main result reads as follows.
Theorem 1.2. Assume that (A1), (F1)–(F7) hold. Then (1.1) has infinitely many
pairs of periodic solutions. Also, (1.1) possesses a solution having 2π as its minimal
period.
The rest of this paper is organized as follows: in section 2, variation functional
will be established and some useful lemmas will be given; in section 3, generalized
Nehari manifold will be defined and our main results will be proved.
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MULTIPLICITY AND MINIMALITY OF PERIODIC SOLUTIONS
3
2. Preliminaries
The space H = H 1/2 (S 1 , RN ) consists of 2π-periodic vector-valued functions,
which possess square integrable derivative of order 1/2. For any x ∈ H, it has
Fourier expansion
∞
1 X
a0
(aj cos(jt) + bj sin(jt))
x(t) = √ + √
π j=1
2π
where a0 ∈ RN , aj , bj ∈ RN , j ∈ N. The space H is a Hilbert space with norm and
inner product as follows
kxk21
2
= |a0 | +
∞
X
j(|aj |2 + |bj |2 ),
j=1
hx, yi1 = (a0 , c0 ) +
∞
X
j[(aj , cj ) + (bj , dj )],
j=1
√
√ P∞
where y = c0 / 2π + 1/ π j=1 (cj cos(jt) + dj sin(jt)), c0 ∈ RN , cj , dj ∈ RN ,
j ∈ N.
Let x, y ∈ L2 (S 1 , RN ). If for every z ∈ C ∞ (S 1 , RN ),
Z 2π
Z 2π
(x(t), z 0 (t))dt = −
(y(t), z(t))dt,
0
0
then y is called a weak derivative of x, denoted by ẋ.
We define the variational functional defined J on H as follows,
Z 2π
1
π
1
(ẋ(t − ), x(t)) − (Ax(t), x(t)) − F (x(t)) dt.
J(x) =
2
2
2
0
(2.1)
Using the argument in [7], we can prove the following lemma.
Lemma 2.1. Assume that (A1), (F2), (F3) hold. Then J is continuous differentiable on H and
Z 2π
1
π
π
hJ 0 (x), hi =
[ (ẋ(t − ) − ẋ(t + ), h(t)) − (Ax(t), x(t)) − (f (x(t)), h(t))]dt,
2
2
2
0
for all h ∈ H. Moreover, ϕ0 : H → H ∗ is a compact mapping defined as follows:
Z 2π
hϕ0 (x), hi =
(f (x(t), h(t))dt, ∀h ∈ H.
0
Set
E = {x ∈ H : x(t − π) = −x(t)}.
Then E is a closed subspace of H. If x ∈ E, it has Fourier expansion
∞
1 X
x(t) = √
[aj cos(2j − 1)t + bj sin(2j − 1)t].
π j=1
It is easily to verify the following lemma.
Lemma 2.2. Assume that (A1), (F1)–(F4) hold. Then the critical points of J
restricted to E are critical points of J on the whole space H.
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For the rest of this article, J is considered as a functional restricted to E. For
simplicity, we denote J the restriction of J to E.
Define an operator on E by extending the bilinear forms
Z 2π
π
hLx, yi1 =
[(ẋ(t − ), y(t)) − (Ax(t), y(t))]dt.
2
0
It is easy to verify that L is a linear, bounded and self-adjoint operator. Suppose
that there exists a function x ∈ E \ {0} such that Lx = νx, where ν ∈ R. Then for
any y ∈ E, we have hLx, yi1 = νhx, yi1 . By a direct computation,
hLx, yi1 =
∞
∞
X
X
(−1)j+1 (2j − 1)[(aj , cj ) + (bj , dj )] −
[(Aaj , cj ) + (Abj , dj )],
j=1
j=1
νhx, yi1 = ν
∞
X
(2j − 1)[(aj , cj ) + (bj , dj )].
j=1
√
√
For any j ∈ N, take y(t) = 1/ π cos[(2j − 1)t]ei and y(t) = 1/ π sin[(2j − 1)t]ei ,
where {ei : i = 1, 2, . . . , N } denotes the canonical basis of RN . Then the theory of
Fourier series implies that
(−1)j+1 (2j − 1)aj − Aaj = ν(2j − 1)aj
and
(−1)j+1 (2j − 1)bj − Abj = ν(2j − 1)bj .
Thus, for some j ∈ N, ν(2j − 1) is an eigenvalue of (−1)j+1 (2j − 1)I − A. By the
definition of the space E, one can check that ν is an eigenvalue of L if and only if
ν(2j − 1) is an eigenvalue of (−1)j+1 (2j − 1)I − A for some j ∈ N.
Since A is a symmetric matrix, all eigenvalues are real numbers. It follows that
eigenvalues of matrix (−1)j+1 (2j − 1)I − A and then operator L are real numbers.
Since {(−1)j+1 (2j − 1) : j ∈ N} ∩ σ(A) = ∅, then 0 6∈ σ(L). Denote the eigenvalues
of L on E by
· · · ≤ λ−2 ≤ λ−1 < 0 < λ1 ≤ λ2 ≤ . . . .
Let {e±j }j∈N be the eigenvectors of L corresponding to {λ±j }j∈N , respectively.
Define
E + = span{ej : j ∈ N}, E − = span{e−j : j ∈ N}.
Hence there exists an orthogonal decomposition E = E + ⊕ E − . Clearly, x ∈ E can
be written as x = x+ + x− , where x+ ∈ E + and x− ∈ E − . Define an equivalent
inner product in E, denoted by h·, ·i and defined by
hx, yi = hLx+ , y + i1 − hLx− , y − i1 .
Hence, we have
Z 2π
π
[(ẋ(t − ), x(t)) − (Ax(t), x(t))]dt = hLx, xi1 = kx+ k2 − kx− k2 ,
2
0
where k · k denotes the norm induced by h·, ·i.
Now, J can be rewritten as
1
1
J(x) = kx+ k2 − kx− k2 − ϕ(x),
2
2
At the end of this section, we state a useful lemma.
∀x ∈ E.
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MULTIPLICITY AND MINIMALITY OF PERIODIC SOLUTIONS
5
Lemma 2.3 ([24]). If X is infinite-dimensional and S is the unit sphere of X,
ψ ∈ C 1 (S, R) is even, bounded below and satisfies (P S) condition, then ψ has
infinitely many pairs of critical points.
3. Main results and their proofs
Define the generalized Nehari Manifold
M = {x ∈ E \ E − : hJ 0 (x), xi = 0 and hJ 0 (x), yi = 0 for all y ∈ E − }.
Proposition 3.1. Nontrivial critical points of J belong to E \ E − .
Proof. Assume that x0 ∈ E − is a nontrivial critical point of J. Then J 0 (x0 ) = 0.
It follows that
Z 2π
hJ 0 (x0 ), x0 i = −kx0 k2 −
(f (x0 (t)), x0 (t))dt = 0,
0
which implies that
Z
2π
(f (x0 (t)), x0 (t))dt = −kx0 k2 ≤ 0.
(3.1)
0
On the other hand,
with (3.1).
R 2π
0
(f (x0 (t)), x0 (t))dt > 0 since (f (x), x) > 0, which contradicts
We denote
S = {x ∈ E : kxk = 1}, S + = S ∩ E + ,
and for any x ∈ E, denote
b
E(x) = Rx ⊕ E − ≡ Rx+ ⊕ E − , E(x)
= R∗ x ⊕ E − ≡ R∗ x + ⊕ E − .
(3.2)
Proposition 3.2. For any x ∈ E, there exists R1 large enough such that J(y) ≤ 0
b
for all y ∈ E(x)
\ BR1 (0), where Br (0) denotes the circle in E with center 0 and
radius r.
b
Proof. Suppose, to the opposite, that there exists a sequence {xn } ⊂ E(x)
such
that J(xn ) > 0 and kxn k → ∞ as n → ∞. Let
yn =
xn
,
kxn k
z=
x+
.
kx+ k
Then kyn k = 1. Obviously, there exists a sequence {sn } such that yn+ = sn z.
Passing to a subsequence if necessary, {yn } converges weakly to some point, denoted
by y0 . Suppose that y0 6= 0. Since {yn } converges weakly to y0 , then {yn } converges
strongly to y0 in Lµ (S 1 , RN ). It follows that {yn } is convergence in measure to y0
(cf. [25, Theorem 4.2.2]). Thus there exists a subsequence {ynk } of {yn } such that
{ynk } converges almost everywhere to {y0 } (cf. [25, Theorem 3.2.3]). For any δ > 0,
there exists a measurable subset Mδ of [0, 1] such that meas{[0, 1] − Mδ } < δ, where
meas{M } denotes the length of set M , and {ynk } converges uniformly to y0 (cf.
[25, Theorem 3.2.8]). Choosing 0 < δ0 < 1/2, then meas{Mδ0 } ≥ 1/2. It follows
from (F4) and Remark 1.1 that
Z
Z 2π
F (kxnk k · ynk (t))
F (kxnk k · ynk (t))
dt ≥
dt
2
kx
k
kxnk k2
nk
0
Mδ0
Z
mkxnk kµ · |ynk (t)|µ + M
dt → +∞,
≥
kxnk k2
Mδ0
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EJDE-2014/115
as k → ∞. Consequently,
0≤
1
J(xnk )
1
= kyn+k k2 − kyn−k k2 −
2
kxnk k
2
2
Z
0
2π
F (kxnk k · ynk (t))
dt → −∞,
kxnk k2
(3.3)
which is a contradiction. Thus y0 = 0 and yn * 0. Since F (x) ≥ 0, we obtain from
(3.3) that kyn+ k > kyn− k. If yn+ → 0, then also yn− → 0. Hence, yn = yn+ + yn− → 0,
which contradicts with the fact that kyn k = 1. Thus yn+ → 0 and kyn+ k ≥ λ for
some λ > 0, possibly after passing to a subsequence. Thus kyn+ k = ksn zk = sn is
bounded and bounded away from 0. Passing to a subsequence if necessary, yn+ → sz
for some s > 0, which contradicts with the fact that yn * 0. Thus, there exists R1
b
large enough such that J(x) ≤ 0 for all E(x)
\ BR1 (0).
b
Lemma 3.3. For every c > 0, J satisfies (P S)c condition on E(x).
b
Proof. Let {xn } ⊂ E(x)
be a (P S)c sequence. Because of Proposition 3.2, {xn } is
0
bounded. Since ϕ : H → H ∗ is compact and
−
0
J 0 (xn ) = x+
n − xn − ϕ (xn ) → 0,
−
we conclude that {x+
n − xn } has a convergent subsequence. Thus {xn } has a
convergent subsequence.
b
Lemma 3.4. E(x)
∩M=
6 ∅ for any x ∈ E \ E − .
b
b + ) = E(x
b + /kx+ k), we assume that
Proof. Let x ∈ E \ E − . Since E(x)
= E(x
+
+
b
x ∈ S . Let y ∈ E(x) ∩ E . Since
Z 2π
1
F (y(t))dt,
J(y) = kyk2 −
2
0
it follows from Remark 1.1 that there exists r0 > 0 small enough such that J(y) ≥
kyk2 /4 for kyk < r0 . On the other hand, Proposition 3.2 implies that
0<
r02
J(y) < ∞.
≤ sup J(y) =
sup
16 y∈E(x)
b
b
y∈E(x),kyk≤R
1
Since J satisfies the (P S)c condition for every c > 0, the supremum supy∈E(x)
J(y)
b
is attained at some point x. Since J(0) = 0, then x 6= 0 and x ∈ M.
Now, we give two lemmas. The idea comes from [3, 24].
b
Lemma 3.5. If x ∈ M, then J(x + y) < J(x) whenever x + y ∈ E(x),
y 6= 0.
Hence, x is the unique global maximum of J|E(x)
.
b
Proof. Let x + y = (1 + s)x + z, where s ≥ −1, z ∈ E − . Then
J(x) − J(x + y)
Z 2π
Z 2π
1
1
F (x(t))dt − h(x + y), (x + y)i +
F (x(t) + y(t))dt
= hx, xi −
2
2
0
0
Z 2π
s2 + 2s
1
=−
hx, xi − (1 + s)hx, zi − hz, zi −
F (x(t))dt
2
2
0
Z 2π
+
F (x(t) + y(t))dt
0
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MULTIPLICITY AND MINIMALITY OF PERIODIC SOLUTIONS
s
1
= −hx, s( + 1)x + (1 + s)zi + kzk2 −
2
2
Z
2π
Z
F (x(t))dt +
0
7
2π
F (x(t) + y(t))dt
0
Since x ∈ M, hJ 0 (x), xi = hJ 0 (x), zi = 0. It follows that hJ 0 (x), s(s/2 + 1)x + (1 +
s)z)i = 0; i.e.,
Z 2π
s
s
hx, s( + 1)x + (1 + s)zi −
(f (x(t)), s( + 1)x(t) + (1 + s)z(t))dt = 0.
2
2
0
Thus
J(x) − J(x + y) =
Z 2π
1
s
kzk2 −
(f (x(t)), s( + 1)x(t) + (1 + s)z(t))dt
2
2
0
Z 2π
Z 2π
−
F (x(t))dt +
F (x(t) + y(t))dt
0
0
The result follows if
F (x(t) + y(t)) − F (x(t)) − (f (x(t)), s(s/2 + 1)x(t) + (1 + s)z(t)) ≥ 0.
This will be proved in the following lemma.
Lemma 3.6. Assume that (F4)–(F7) hold. Let x, z ∈ RN and s ∈ R such that
s ≥ −1 and let y = z + sx. Then
s
F (x + y) − F (x) − (f (x), s( + 1)x + (1 + s)z) ≥ 0.
2
Proof. Set u = (1 + s)x + z. Then u = x + y. Denote
s
g(s) := F ((1 + s)x + z)) − F (x) − (f (x), s( + 1)x + (1 + s)z).
2
We only need check that g(s) ≥ 0.
Case I: (x, u) ≤ 0. It follows from (F4) and (F5) that
s
g(s) = F (u) − F (x) − (f (x), s( + 1)x + (1 + s)(u − (1 + s)x))
2
1
s2
> F (u) − (f (x), x) + ( + s + 1)(f (x), x) − (1 + s)(f (x), u)
2
2
1
2
= F (u) + (s + 1) (f (x), x) − (1 + s)(f (x), u) ≥ 0.
2
Case II: (x, u) > 0. Obviously, g(−1) > F (z) > 0. It follows from Remark 1.1
that F (sx) ≥ sµ F (x) when s ≥ 1. Thus g(s) → +∞ as s → +∞. By a directly
computation,
g 0 (s) = (f (u), x) − (f (x), u).
If there exists s1 such that g(s1 ) < 0, then there exists s2 such that g 0 (s2 ) =
(f (u), x) − (f (x), u) = 0. It follows from (F7) that |x| = |u|. Hence (F6) implies
that
s
g(s) := F ((1 + s)x + z)) − F (x) − (f (x), s( + 1)x + (1 + s)z)
2
s2
= F (u) − F (x) − (f (x), (1 + s)u − ( + s + 1)x)
2
s2
≥ −(1 + s)(f (x), u) + ( + s + 1)(f (x), x)
2
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H. XIAO, Z. GUO
=
EJDE-2014/115
s2
(f (x), x) ≥ 0.
2
This completes the proof.
From Lemmas 3.4 and 3.5, we know that for each x ∈ E \ E − there exists
a unique nontrivial critical point m(x)
b
of J|E(x)
. Moreover, m(x)
b
is the unique
b
global maximum of J|E(x)
. Let
b
m
b : E \ E − → M,
b : E + \ {0} → R,
Ψ
m := m|
b S + : S + → M.
b
Ψ(x)
:= J(m(x)),
b
b S+ .
Ψ := Ψ|
Lemma 3.7. Assume that (A1), (F1)–(F7) hold. Then Conditions (B1)–(B3) in
[24] hold.
Proof. It follows from (F4) and Lemmas 2.1, 3.4, 3.5 that (B1) and (B2) hold.
Now, we verify that (B3) holds. For any x ∈ E \ E − , Proposition 3.2 implies
that
sup J(y) ≥ r02 /16 > 0.
b
y∈E(x)
+
+
Since J(m(x)
b
) ≥ J(m(x)),
b
then there exists δ > 0 such that m(x)
b
≥ δ.
Let W ∈ E \{0} be a weakly compact set and let {xn } ⊂ W . Then, after passing
to a subsequence if necessary, xn * x 6= 0. Thus, xn (t) → x(t) a.e. for t ∈ [0, 2π].
If sn → ∞ as n → ∞, then |sn xn (t)| → ∞ as n → ∞. It follows that
Z 2π
F (sn xn (t))
ϕ(sn xn )
=
|xn (t)|2 dt → ∞, as n → ∞.
s2n
s2n |xn (t)|2
0
Hence ϕ(sx)/s2 → ∞ uniformly for x on weakly compact subsets of E \ {0} as
s → ∞. Let W 0 ⊂ S + be a compact set and let {yn } ⊂ W 0 . Since J(sy)/s2 =
1/2 − ϕ(syn )/s2 , then {J(syn )} must be bounded from above. Thus {J(m(y
b n ))} is
bounded from above. Hence {m(y
b n )} is bounded from above and (B3) holds. Lemma 3.8 ([24]). The mapping m
b is continuous and m is a homeomorphism
between S + and M.
b Ψ ∈ C 1 (S + , R) and
Lemma 3.9 ([24]). (1) Ψ,
+
km(x)
b
k 0
hΨ (m(x)),
b
yi for all x, y ∈ E + , x 6= 0,
kxk
hΨ0 (x), yi = km(x)+ khΨ0 (m(x)), yi for all y ∈ Tx (S + ),
b 0 (x), yi =
hΨ
where Tx (S + ) is the tangent space of S + at x.
(2) If {xn } is a (PS) sequence for Ψ, then {m(xn )} is a (PS) sequence for J. If
{xn } ⊂ M is a bounded (PS) sequence for J, then {m−1 (xn )} is a (PS) sequence
for Ψ.
(3) x is a critical point of Ψ if and only if m(x) is a nontrivial critical point of J.
Moreover, the corresponding values of Ψ and J coincide and inf S + Ψ = inf M J.
(4) If J is even, then so is Ψ.
Lemma 3.10. J satisfies the (P S) condition on M.
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9
Proof. If x ∈ M, by Lemmas 3.4 and 3.5,
J(x) = sup J(y) ≥
b
y∈E(x)
r02
> 0.
16
Suppose that {xn } ⊂ M is a (P S) sequence. Suppose that {xn } is unbounded.
Then, passing to a subsequence, kxn k → ∞ as n → ∞. Set
yn =
xn
,
kxn k
zn =
x+
n
.
kx+
nk
Similarly as in the proof of Proposition 3.4, we can prove that yn * 0 and kyn+ k ≥ λ
for some λ > 0, possibly after passing to a subsequence. By the assumption of {xn },
there exists d > 0 such that
1
1
(3.4)
d ≥ J(xn ) ≥ J(syn+ ) ≥ s2 λ2 − ϕ(syn+ ) → s2 λ2 ,
2
2
for all s > 0, which is a contradiction. So {xn } is bounded. Similarly as in the
proof of Lemma 3.3, {xn } has a convergent subsequence.
Proof of Theorem 1.2. Set
c = inf J(x) = inf
x∈M
max J(y) = inf+ max J(y).
b
x∈E\F y∈E(x)
x∈S
(3.5)
b
y∈E(x)
Obviously, c ≥ r02 /16 > 0.
Let {yn } ⊂ S + be a (P S) sequence for Ψ. Set xn = m(yn ) for n ∈ N. Then {xn }
is a (P S) sequence for J. According to Lemma 3.10, passing to a subsequence if
necessary, xn → x0 and yn → m−1 (x0 ). Thus Ψ satisfies (P S) condition.
Let {yn } be a minimizing sequence for Ψ. By Ekeland’s variational principle, we
may assume that Ψ0 (yn ) → 0. By the (P S) condition, yn → y0 . Hence x0 = m(y0 )
is a critical point for J. Since J(0) = 0, x0 is a nonconstant solution of (1.1).
Since F (−x) = F (x), then both J and Ψ are even. Since inf S Ψ = inf M J =
c > 0, Ψ is bounded from below. Since Ψ satisfies (P S) condition, it follows from
Lemma 2.3 that Ψ and then J has infinitely many pairs of solutions.
Lemma 3.11. The minimal period of x0 is 2π.
Proof. Suppose, to the opposite, that x0 has minimal period 2π/m, where m > 1
is an integer.
Claim: m is odd. If there exists k ∈ N such that m = 2k, then
2π
· k) = x0 (t),
m
which implies x0 (t) ≡ 0. This contradicts with the fact that x0 is a nonconstant
solution. Thus m must be odd.
Denote y0 (t) = x0 (t/m). Obviously, y0 (t) has minimal period 2π.
Case I: m = 4k + 1 for some k ∈ N. Then
π
π
2π
4k + 1
π
x0 (t −
) = x0 (t −
−
k) = x0 (t −
π) = x0 (t − ).
2m
2m
m
2m
2
It follows that
π
π
π
x0 (t − ) = x0 (t −
−
) = x0 (t − π) = −x0 (t),
(3.6)
m
2m 2m
−x0 (t) = x0 (t − π) = x0 (t −
10
H. XIAO, Z. GUO
EJDE-2014/115
Thus
t−π
t
π
t
) = x0 ( − ) = −x0 ( ) = −y0 (t),
(3.7)
m
m m
m
which implies that y0 ∈ E. Since x0 ∈ M, y0 ∈ E \ E − . Denote y 0 := m(y
b 0) =
sy0+ + z, where z ∈ E − and s > 0. Then J(y 0 ) ≥ J(x0 ) = inf x∈M J(x). Setting
z(t/m) = z(t), we have y 0 (t) = sx+
0 (t/m) + z(t/m). Also
y0 (t − π) = x0 (
z(
t
π
mπ
π
π
t
π
− ) = z(t −
) = z(t − 2kπ − ) = z(t − ) = z( −
).
m
2
2
2
2
m 2m
Thus z(t/m − π/(2m)) = z(t/m − π/2).
b
Let y(t) = sx+
0 (t) + z(t). Then y ∈ E(x0 ). Computing directly,
J(y 0 )
Z 2π
Z
Z
π
1 2π
1 2π
F (y 0 (t))dt
=
(ẏ 0 (t − ), y 0 (t))dt −
(Ay 0 (t), y 0 (t))dt −
2 0
2
2 0
0
Z 2π
1
π
t
π
t
t
t
=
−
) + ż( −
), sx+
) + z( ))dt
(sẋ+
0(
0(
2m 0
m 2m
m 2m
m
m
Z
Z 2π
t
1 2π
t
t
t
t
t
) + Az( ), sx+
) + z( ))dt −
) + z( ))dt
−
(sAx+
F (sx+
0(
0(
0(
2 0
m
m
m
m
m
m
0
Z 2π
1
π
t
π
t
t
t
− ) + ż( − ), sx+
) + z( ))dt
(sẋ+
=
0(
0(
2m 0
m
2
m
2
m
m
Z
Z 2π
1 2π
t
t
t
t
t
+ t
−
)
+
Az(
),
sx
(
)
+
z(
))dt
−
) + z( ))dt
(sAx+
(
F (sx+
0
0
0(
2 0
m
m
m
m
m
m
0
Z 2π
1
π
π
=
(sẋ+
) + ż(τ − ), sx+
0 (τ −
0 (τ ) + z(τ ))dt
2m 0
2
2
Z 2π
Z 2π
1
+
+
−
(sAx0 (τ ) + Az(τ ), sx0 (τ ) + z(τ ))dt −
F (sx+
0 (τ ) + z(τ ))dt
2 0
0
Z 2π
Z
Z 2π
1
π
1 2π
=
(ẏ(τ − ), y(τ ))dτ −
(Ay(τ ), y(τ ))dt −
F (y(τ ))dτ.
2m 0
2
2 0
0
Since J(y 0 ) > 0 and A is a nonnegative definite matrix, then
Z 2π
Z
Z 2π
1
π
1 2π
(ẏ(τ − ), y(τ ))dτ >
(Ay(τ ), y(τ ))dt +
F (y(τ ))dτ > 0.
2m 0
2
2 0
0
It follows that
Z 2π
Z
Z 2π
1
π
1 2π
(ẏ(τ − ), y(τ ))dτ −
(Ay(τ ), y(τ ))dt −
F (y(τ ))dτ
2m 0
2
2 0
0
Z
Z
Z 2π
π
1 2π
1 2π
(ẏ(τ − ), y(τ )dτ −
(Ay(τ ), y(τ ))dt −
F (y(τ ))dτ
<
2 0
2
2 0
0
= J(y) ≤ J(x0 ) = inf J(x),
J(y 0 ) =
x∈M
which contradicts with the fact that J(y 0 ) ≥ inf x∈M J(x).
Case II: m = 4k − 1 for some k ∈ N. Then
x0 (t −
π
π
2π
4k − 1
π
π
) = x0 (t −
+
k) = x0 (t +
π) = x0 (t + ) = −x0 (t − ).
2m
2m
m
2m
2
2
EJDE-2014/115
MULTIPLICITY AND MINIMALITY OF PERIODIC SOLUTIONS
11
It follows that
x0 (t −
π
π
π
) = x0 (t −
−
) = x0 (t − π) = −x0 (t),
m
2m 2m
Thus
t−π
t
π
t
) = x0 ( − ) = −x0 ( ) = −y0 (t),
m
m m
m
which implies that y0 ∈ E. Since
y0 (t − π) = x0 (
0 ≤ ky0+ k2
Z 2π
Z 2π
π
+
+
=
(ẏ0 (t − ), y0 (t))dt −
(Ay0+ (t), y0+ (t))dt
2
0
0
Z
Z 2π
1 2π + t
π
t
t
t
+
=
(ẋ0 ( −
), x0 ( ))dt −
(Ax+
), x+
))dt
0(
0(
m 0
m 2m
m
m
m
0
Z 2π
Z 2π
t
π
t
t
t
1
(ẋ+
− ), x+
))dt −
(Ax+
), x+
))dt
=−
0(
0(
0(
0(
m 0
m
2
m
m
m
0
Z
Z 2π
1 2π +
π
+
=−
(ẋ0 (τ − ), x+
(τ
))dt
−
(Ax+
0
0 (τ ), x0 (τ ))dt
m 0
2
0
Z 2π
1
1
+
2
k
−
(1
−
)
(Ax+
= − kx+
0 (τ ), x0 (τ ))dt ≤ 0,
m 0
m 0
+
−
−
−
it follows that x+
0 = 0. Then x0 = x0 + x0 = x0 ∈ E , which contradicts the fact
−
that all nontrivial critical points belong to E \ E . Consequently, x0 has minimal
period 2π.
Acknowledgements. This project is supported by National Natural Science Foundation of China (No. 11031002 and No. 11301102) and by Program for Changjiang
Scholars and Innovative Research Team in University (No. IRT1226).
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Huafeng Xiao (corresponding author)
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou,
510006, China.
Key Laboratory of Mathematics and Interdisciplinary Science of Guangdong, Higher
Education Institutes, Guangzhou University
E-mail address: huafeng@gzhu.edu.cn
Zhiming Guo
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou,
510006, China.
Key Laboratory of Mathematics and Interdisciplinary Science of Guangdong, Higher
Education Institutes, Guangzhou University
E-mail address: guozm@gzhu.edu.cn