PORTUGALIAE MATHEMATICA
Vol. 62 Fasc. 1 – 2005
Nova Série
PERIODIC SOLUTIONS OF A RESONANT
HIGHER ORDER EQUATION
P. Amster, P. De Nápoli and M.C. Mariani
Recommended by L. Sanchez
Abstract: We study the existence of periodic solutions for an ordinary differential
equation of resonant type. Under suitable conditions we prove the existence of at least
one periodic solution of the problem applying Mawhin Coincidence degree theory.
1 – Introduction
In the last years there has been an increasing interest in higher order problems
which have applications in many fields, such as beam theory [4], [5] and multi-ion
electrodiffusion problems [8].
In this work, we consider the problem:
Lx + g(x, x0 , ..., x(N −2) ) = p(t)
(1)
where
(2)
Lx = x(N ) + aN −1 x(N −1) + ... + a0 x
under periodic conditions
x(0) = x(2π)
x0 (0) = x0 (2π)
...
x(N −1) (0) = x(N −1) (2π)
for continuous and bounded g.
Received : August 29, 2003; Revised : November 6, 2003.
14
P. AMSTER, P. DE NÁPOLI and M.C. MARIANI
We assume that L is a resonant operator, i.e. that the homogeneous problem Lx = 0 admits nontrivial periodic solutions. Namely, we assume that the
polynomial
(3)
P (λ) = λN + aN −1 λN −1 + ... + a0
admits imaginary roots ±im (m ∈ Z).
For notational convenience, we introduce the n-dimensional symbolic vectors
V±± given by
V++ = (+∞, +∞, −∞, −∞, ...)
V+− = (+∞, −∞, −∞, +∞, ...)
V−+ = (−∞, +∞, +∞, −∞, ...)
V−− = (−∞, −∞, +∞, +∞, ...)
where the sequences of signs are 4-periodic.
Our main result is:
Theorem 1.1. Let us assume that
1. The polynomial (3) has exactly two roots ±im in iZ, which are simple.
2. g : RN −1 → R is a continuous bounded function such that the four limits
lim g(s) := g±±
s→V±±
exist.
Let p ∈ L2 (0, 2π) and consider the m-th Fourier coefficients of p, namely:
1
π
Z
1
bm (p) =
π
Z
am (p) =
2π
p(t) cos(mt) dt ,
0
2π
p(t) sin(mt) dt .
0
If furthermore, we assume that
(4)
a2m (p) + b2m (p) <
i
2 h
(g+− − g−+ )2 + (g++ − g−− )2
2
π
then equation (1) has at least one 2π-periodic solution in H N (0, 2π).
PERIODIC SOLUTIONS OF A RESONANT HIGHER ORDER EQUATION
15
Remark 1.1. The particular case N = 3 with g = g(x0 ) has been considered
in [1].
Remark 1.2. In the same way as in [1], Proposition 3.1, it is easy to see that
condition (4) is “almost” necessary: namely, if (1) admits a 2π-periodic solution
then
16
(5)
a2m (p) + b2m (p) ≤ 2 kgk2L∞ .
π
In particular, if g verifies
|g+− − g−+ | = |g++ − g−− | = 2 kgkL∞ ,
and inequality (5) holds stricly, then (4) holds.
Remark 1.3. We observe that (4) is a Landesman-Lazer type condition (see
e.g. [7],[10]). Thus, Theorem 1.1 can be regarded as a N th-order analogue of
Theorem 1 in [11].
2 – Auxiliary results
2.1. Mawhin coincidence degree theory
Let us briefly summarize some aspects of Mawhin theory. This technique has
been applied to many problems, see e.g. [2] and [6]. For further details see [9],
[3].
Let X and Y be real normed spaces, L : dom(L) → Y be a linear mapping,
and N : X → Y be a continuous mapping. The mapping L is called a Fredholm
mapping of index 0 if Im(L) is a closed subspace of Y and
dim(Ker(L)) = codim(Im(L)) < ∞ .
If L is a Fredholm mapping of index 0, then there exists continuous projectors
P : X → X and Q : Y → Y such that Im(P ) = Ker(L) and Ker(Q) = Im(L).
It follows that
LP = L|dom(L)∩Ker(P ) : dom(L) ∩ Ker(P ) → Im(L) = Ker(Q)
16
P. AMSTER, P. DE NÁPOLI and M.C. MARIANI
is one-to-one and onto Im(L). We denote its inverse by KP . If Ω is a bounded
open subset of X, N is called L-compact on Ω if QN (Ω) is bounded and
KP (I − Q)N : Ω → X is compact. Since Im(Q) is isomorphic to Ker(L), there
exists and isomorphism J : Im(Q) → Ker(L).
The following continuation theorem is due to Mawhin [9]:
Theorem 2.1. Let L be a Fredholm mapping of index zero and N be
L-compact on Ω. Suppose
1. For each λ ∈ (0, 1], x ∈ ∂Ω we have that Lx 6= λN x;
2. QN x 6= 0 for each x ∈ Ker(L) ∩ ∂Ω;
3. The Brouwer degree dB (JQN, Ω ∩ Ker(L), 0) 6= 0.
Then the equation Lx = N x has at least one solution in dom(L) ∩ Ω.
In our case, we shall work in the usual Sobolev space of periodic functions,
namely
K
Hper
(0, 2π) =
n
o
x ∈ H K (0, 2π) : x(j) (0) = x(j) (2π) ∀ j = 0, ..., K −1 .
N −1 (0, 2π), Y = L2 (0, 2π) and L the
More precisely, we shall consider X = Hper
N (0, 2π). Under
linear differential operator given by (2), with dom(L) = Hper
the assumption 1 of Theorem 1.1, it is immediate to see that Ker(L) = Em is
⊥ ; thus L is a
the subspace generated by sin(mt) and cos(mt), and Im(L) = Em
Fredholm mapping of index zero. Moreover, we may take Q as the orthogonal
N −1 (0, 2π).
projection Pm onto Em in L2 (0, 2π) and P as the restriction of Pm to Hper
N −1 (0, 2π). Then, if N x =
Ω will be an appropriate open bounded subset of Hper
(N
−2)
p(t) − g(x, ..., x
), it will follow from the estimates in section 2.2 that N is
L-compact on Ω.
2.2. Estimates for the linear operator L
We assume throughout this section that assumption 1 of Theorem 1.1 holds.
We recall the following lemma from [1]:
Lemma 2.2. There exists a constant c such that for any x ∈ H 2 (0, 2π) we
have
kx − Pm (x)kH 1 ≤ c kx00 + m2 xkL2
where Pm is the orthogonal projection on Em in L2 (0, 2π).
PERIODIC SOLUTIONS OF A RESONANT HIGHER ORDER EQUATION
17
Then we have:
Lemma 2.3. There exists a constant c such that
kx − Pm (x)kH N −1 ≤ c kLxkL2
as
N
∀ x ∈ Hper
(0, 2π) .
Proof: Writing P (λ) = (λ2 + m2 ) Pe (λ) we may decompose the operator L
e
L = L2 L
e is the differential operator associated to Pe .
where L2 x = x00 + m2 x and L
⊥
2
For x ∈ Em in L (0, 2π), let us write the equality Lx = f as a system:
(
L2 y = f
e = y .
Lx
e ∈ E ⊥ , from Lemma 2.2, we have that
Since y = Lx
m
kykH 1 ≤ c kf kL2 .
We claim that
(6)
kxkH N −1 ≤ c kykH 1 .
Indeed, as P (0) 6= 0, from Lemma 4.3 in [1] we know that (6) holds for N = 3.
Assume that (6) holds for N −1, and let a be any root of Pe . Thus, if Pe (λ) =
(λ − a)Pb (λ), by inductive hypothesis we have that
° 0°
°x °
H N −2
b 0 )k 1 ≤ c kLxk
b
≤ c kL(x
H
H2
and from Lemma 4.3 in [1] we know that
It follows that
b
kLxk
H 2 ≤ c kykH 1 .
° 0°
°x °
H N −2
≤ c kykH 1 .
On the other hand, it is easy to see that
for any z
For x
N (0, 2π), and
∈ Hper
⊥ , it suffices to
6∈ Em
¯Z
¯
¯
¯
¯
¯
e
z(t) dt¯¯ ≤ c kLzk
L2
2π
0
the claim follows from Wirtinger inequality.
write
kx − Pm (x)kH N −1 ≤ c kL(x − Pm (x))kL2 = c kLxkL2 .
18
P. AMSTER, P. DE NÁPOLI and M.C. MARIANI
Remark 2.1. It follows that the operator N x = p − g(x, ..., x(N −2) )
defined in the previous section is L-compact on Ω for any open bounded
N −1 (0, 2π).
Ω ⊂ Hper
3 – A priori estimates
Lemma 3.1. Assume that g : RN −1 → R is a continuous bounded function,
that the four limits g±± exist, and that
a2m (p) + b2m (p) 6=
i
2 h
(g+− − g−+ )2 + (g++ − g−− )2 .
2
π
Then the solutions of
³
L(x) = λ p(t) − g(x, ..., x(N −2) )
´
with λ ∈ (0, 1] are a priori bounded in H N −1 (0, 2π).
Proof: Let xn (t) be a solution of
³
−2)
Lxn = λn p(t) − g(xn , ..., x(N
)
n
´
with λn ∈ (0, 1] and assume that kxn kH N −1 → +∞. Let us write
xn = y n + z n
⊥ . We have that
where yn = Pm (xn ) and zn = xn − Pm (xn ) ∈ Em
°
°
°
°
−2)
kzn kH N −1 ≤ c1 °p(t) − g(xn , ..., x(N
)°
n
L2
≤ c2 .
From the compactness of the imbedding H N −1 ,→ C N −2 [0, 2π] there exists a
convergent subsequence, still denoted zn , with
zn → z
in C N −2 [0, 2π] .
We have that kyn kH N −1 → ∞. Since Pm (xn ) ∈ Em ,
yn (t) = αn cos(mt − βn )
PERIODIC SOLUTIONS OF A RESONANT HIGHER ORDER EQUATION
19
where αn ≥ 0, αn → +∞ and βn ∈ [0, 2π]. Taking a subsequence, we may
assume that βn → β. On the other hand, if ϕ ∈ Em and L∗ x = (−1)N x(N ) +
(−1)N −1 aN −1 x(N −1) + ... + a0 x, integrating by parts we have:
0 =
Z
2π
∗
xn L (ϕ) dt = λn
0
In particular, choosing ϕ = cos(mt)
am (p) =
Z
1
π
Z
0
2π h
2π
p(t) cos(mt) dt =
0
i
−2)
p(t) − g(xn , ..., x(N
) ϕ dt .
n
1
π
Z
2π
−2)
g(xn , ..., x(N
) cos(mt) dt
n
0
and choosing ϕ = sin(mt)
Z
1
1 2π
p(t) sin(mt) dt =
bm (p) =
π 0
π
From the identities
Z
2π
0
−2)
g(xn , ..., x(N
) sin(mt) dt .
n
xn (t) = zn0 (t) + αn cos(mt − βn )
x0n (t) = zn0 (t) − αn m sin(mt − βn )
x00n (t) = zn00 (t) − αn m2 cos(mt − βn )
...
we obtain:
³
´
−2)
g xn (t), ..., x(N
(t) →
n


g+−




 g−−

g−+




g
++
where
Cβ++ =
Cβ−+ =
Cβ−− =
Cβ+− =
n
n
n
n
if t ∈ Cβ++
if t ∈ Cβ−+
if t ∈ Cβ−−
if t ∈ Cβ+−
t ∈ [0, 2π] : cos(mt − β) > 0, sin(mt − β) > 0
t ∈ [0, 2π] : cos(mt − β) < 0, sin(mt − β) > 0
t ∈ [0, 2π] : cos(mt − β) < 0, sin(mt − β) < 0
o
o
o
o
t ∈ [0, 2π] : cos(mt − β) > 0, sin(mt − β) < 0 .
As n → ∞, by dominated convergence we obtain:
"
1
am (p) =
g+−
π
+ g−+
Z
Z
cos(mt) dt + g−−
++
Cβ
cos(mt) dt + g++
Cβ−−
Z
Z
cos(mt) dt
Cβ−+
cos(mt) dt
Cβ+−
#
20
P. AMSTER, P. DE NÁPOLI and M.C. MARIANI
and
"
1
bm (p) =
g+−
π
+ g−+
Z
Z
sin(mt) dt + g−−
++
Cβ
sin(mt) dt + g++
Cβ−−
Z
Z
sin(mt) dt
Cβ−+
#
sin(mt) dt .
Cβ+−
Moreover,
Z
eimt dt = eiβ
++
Cβ
Z
eims ds = eiβ m
++
C0
Z
π/2m
eims ds = (1 + i) eiβ
0
and in a similar way we compute:
Z
e
−+
imt
dt = e
iβ
Cβ
Z
e
−−
imt
dt = e
iβ
Cβ
Z
Z
eimt dt = eiβ
−+
Cβ
Z
e
−+
iβ
ds = e m
C0
e
−−
C0
Z
ims
ims
iβ
ds = e m
Z
eims ds = eiβ m
−+
C0
Z
π/m
eims ds = (−1 + i) eiβ
π/2m
3π/2m
eims ds = (−1 − i) eiβ
π/m
Z
2π/m
eims ds = (1 − i) eiβ .
3π/2m
Hence we have that
am (p) =
i
1h
(g+− − g−+ ) (cosβ − sin β) + (g++ − g−− ) (cosβ + sin β)
π
and
bm (p) =
i
1h
(g+− − g−+ ) (cosβ + sin β) + (g++ − g−− ) (cosβ − sin β) .
π
Then
ih
i
1h
2
2
2
2
(cosβ
−sin
β)
+(cosβ
+sin
β)
(g
−
g
)
+(g
−
g
)
+−
−+
++
−−
π2
i
2h
= 2 (g+− − g−+ )2 + (g++ − g−− )2 ,
π
a2m (p)+b2m (p) =
a contradiction.
PERIODIC SOLUTIONS OF A RESONANT HIGHER ORDER EQUATION
21
4 – A degree computation
In this section we compute the degree
³
d QN, Ω ∩ Ker(L), 0
where
Ω =
n
´
N −1
u ∈ Hper
(0, 2π) : kukH N −1 < R
o
for some R to be chosen. With this aim, let us consider the isomorphism
ψ : R2 → Em = Ker(L) given by
ψ(a, b) = a cos(mt) + b sin(mt)
and define
h = ψ −1 QN ψ .
Let us call Ω̃ = ψ −1 (Ω ∩ Em ) ⊂ R2 . As we shall prove, if R is large enough, then
h does not vanish on ∂ Ω̃, and hence
³
(7)
´
d QN, Ω ∩ Ker(L), 0 = d(h, Ω̃, 0) .
If we introduce polar coordinates a = r cos ω, b = r sin ω, a simple computation shows that the components of h = (h1 , h2 ) are
1
π
Z
1
= bm (p)−
π
Z
= am (p)−
h1 (r cos ω, r sin ω) =
2π ³
´
g r cos(mt−ω), −mr sin(mt−ω), −m2 r cos(mt−ω), ... cos(mt) dt
0
and
2π
³
h2 (r cos ω, r sin ω) =
´
g r cos(mt−ω), −mr sin(mt−ω), −m2 r cos(mt−ω), ... sin(mt) dt .
0
In the same way as before we obtain:
lim h(r cos ω, r sin ω) = (am (p), bm (p)) − C(ω)
r→∞
where C(ω) ∈ R2 is given by
C(ω) =
´
1³
A(cosω − sin ω) + B(cosω + sin ω), A(cosω + sin ω) + B(cosω − sin ω)
π
with A = g+− − g−+ and B = g++ − g−− .
22
P. AMSTER, P. DE NÁPOLI and M.C. MARIANI
Remark 4.1. In the same way as in [1] it can be proved that the previous
limits are uniform in ω ∈ [0, 2π).
Lemma 4.1. Assume that g : RN −1 → R is continuous and bounded, that
the four limits g±± exist, and that
a2m (p) + b2m (p) <
Then for R large enough
i
2 h
2
2
(g
−
g
)
+
(g
−
g
)
.
+−
−+
++
−−
π2
³
´
d QN, Ω ∩ Ker(L), 0 = −1 .
Proof: Let us introduce the function
h̃(x, y) =
³
µ
´


 am (p), bm (p) − T
x
y
,
k(x, y)k k(x, y)k
³
´


 am (p), bm (p) − T (x, y)
¶
if k(x, y)k ≥ 1
if k(x, y)k < 1
where T : R2 → R2 is the linear operator given by
T (x, y) =
´
1³
A(x − y) + B(x + y), A(x + y) − B(x − y)
π
with A and B as above. Next, define the homotopy
H(x, y, λ) = λ h(x, y) + (1 − λ) h̃(x, y) .
From the previous computations, for λ 6= 0 it is clear that H(x, y, λ) 6= 0 for
k(x, y)k large. On the other hand, for λ = 0, H = h̃. If h̃(x0 , y0 ) = 0 for some
(x0 , y0 ) with k(x0 , y0 )k ≥ 1, then
a2m (p) + b2m (p) =
2
(A2 + B 2 ) ,
π2
a contradiction. Note that dist(∂ Ω̃, 0) → +∞ for R → +∞. Hence, if R is large
enough, by the homotopy invariance and the excision property of the degree:
d(h, Ω̃, 0) = d(h̃, Ω̃, 0) = d(h̃, B1 , 0) .
But h̃|B1 has a unique zero at
(x0 , y0 ) =
³
´
π
(A+B)a
(p)+(A−B)b
(p),
−(A−B)a
(p)+(A+B)b
(p)
.
m
m
m
m
2(A2 +B 2 )
PERIODIC SOLUTIONS OF A RESONANT HIGHER ORDER EQUATION
23
Moreover,
1
Dh̃(x0 , y0 ) = −
π
Ã
A+B −(A−B)
A−B
A+B
!
.
Hence sgn(D h̃(x0 , y0 )) = −1, and we conclude that d(h, BR , 0) = −1.
Proof of Theorem 1.1: From Lemma 3.1 and Lemma 4.1, if we choose
R large enough, all the conditions of Theorem 2.1 are fulfilled, and the proof is
complete.
REFERENCES
[1] Amster, P.; Nápoli, P. and Mariani, M.C. – Periodic solutions of a resonant
third-order equation, Submitted for publication.
[2] Fang, H. and Wang, Z.C. – Periodic boundary value problems of impulsive
differential equations, Applied Mathematics E-Notes, 1 (2001), 77–85.
[3] Gaines, R. and Mawhin, J. – Lecture Notes in Mathematics, Vol. 586, SpringerVerlag, Berlin, 1997.
[4] Grossinho, M. and Ma, T.F. – Symmetric equilibria for a beam with a nonlinear
elastic foundation, Portugaliae Mathematica, 51 (1994), 375–393.
[5] Grossinho, M. and Tersian, S. – The dual variational principle and equilibria
for a beam resting on a discontinuous nonlinear elastic foundation. To appear in
Nonlinear Analysis, Theory, Methods, and Applications.
[6] Iyase, S. – Non-resonant oscillations for some fourth-order differential equations
with delay, Math. Proc. of the Royal Irish Academy, 99A(1) (1999), 113–121.
[7] Lazer, A. – A Second look at the first result of Landesman–Lazer type, Electron.
J. Diff. Eqns., Conf. 05 (2000), 113–119.
[8] Leuchtag, R. – Family of differential equations arising from multi-ion electrodiffusion, J. Math. Phys., 22(6) (1981), 1317–1320.
[9] Mawhin, J. – Topological degree methods in nonlinear boundary value problems,
NSF-CBMS Regional Conference in Mathematics no. 40, American Mathematical
Society, Providence, RI, (1979).
[10] Mawhin, J. – Landesman–Lazer conditions for boundary value problems: A nonlinear version of resonance, Bol. de la Sociedad Española de Mat. Aplicada, 16
(2000), 45–65.
[11] Ortega, R. and Sanchez, L. – Periodic Solutions of a forced oscillator with
several degrees of freedom, Bull. London Math. Soc., 34(3) (2002), 308–318.
24
P. AMSTER, P. DE NÁPOLI and M.C. MARIANI
P. Amster,
Departamento de Matemática, FCEyN – Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Buenos Aires – ARGENTINA
and
CONICET – Consejo Nacional de Investigaciones Cientı́ficas y Técnicas – ARGENTINA
and
P. De Nápoli,
Departamento de Matemática, FCEyN – Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Buenos Aires – ARGENTINA
and
M.C. Mariani,
Department of Mathematics, New Mexico State University,
P.O. Box 30001, Las Cruces, NM 88003-8001 – USA