Results Math (2021) 76:91

c 2021 The Author(s), under exclusive licence to

Springer Nature Switzerland AG
1422-6383/21/020001-23
published online April 12, 2021
https://doi.org/10.1007/s00025-021-01401-w Results in Mathematics

Non-resonance and Double Resonance for a

Planar System via Rotation Numbers
Chunlian Liu, Dingbian Qian, and Pedro J. Torres

Abstract. We consider a planar system z  = f (t, z) under non-resonance

or double resonance conditions and obtain the existence of 2π-periodic so-
lutions by combining a rotation number approach together with Poincaré-
Bohl theorem. Firstly, we allow that the angular velocity of solutions of
z  = f (t, z) is controlled by the angular velocity of solutions of two posi-
tively homogeneous system z  = Li (t, z), i = 1, 2, whose rotation numbers
satisfy ρ(L1 ) > n and ρ(L2 ) < n + 1, namely, nonresonance occurs in the
sense of the rotation number. Secondly, we prove the existence of 2π-
periodic solutions when the nonlinearity is allowed to interact with two
positively homogeneous system z  = Li (t, z), i = 1, 2, with ρ(L1 ) ≥ n and
ρ(L2 ) ≤ n + 1, which gives rise to double resonance, and some kind of
Landesman–Lazer conditions are assumed at both sides.

Mathematics Subject Classification. 34C25, 34B15, 34D15.

Keywords. Periodic solution, resonance and non-resonance, Landesman–

Lazer conditions, rotation number, Poincaré–Bohl theorem.

1. Introduction and Main Results

In this paper, we are interested in the study of existence of periodic solutions
of the planar system
z  = f (t, z), z = (x, y) ∈ R2 , (1.1)
from a rotation number viewpoint. We assume that f : R × R → R2 is 2

continuous, 2π-periodic with respect to the first variable, and locally Lipschitz-
continuous with respect to the second variable in order that uniqueness for
the associated Cauchy problems is guaranteed. However, this framework can
be extended to the Carathéodory setting without difficulty.
91 Page 2 of 23 C. Liu et al. Results Math

There are many interesting results on the existence of periodic solutions of

resonant and nonresonant equations, see, for instance, [1–12] and the references
therein. In particular, Fonda and Habets [7] studied the existence of solutions
of Eq. (1.1) when f is asymptotically positively homogeneous. More recently,
Fonda and Sfecci [10] introduced a condition consisting of a control of the
angular velocity of solutions of Eq. (1.1), which is as follows,
   
Jf (t, λz) Jf (t, λz)
ψ1 (w) ≤ lim inf , z ≤ lim sup , z ≤ ψ2 (w),
λ→+∞ λ λ→+∞ λ
z→w z→w

where ψ1 , ψ2 : S \{w1 , w2 , . . . , wm } →]0, +∞] are two positive functions not

1

identically equal to +∞, and w1 , w2 , . . . , wm ∈ S1 .

Our result for the nonresonant problem aims at the indefinite equations,
that is the component of f (t, z) may take positive and negative signs multi-
ple times for t ∈ [0, 2π]. We will show in Example 1.1 that the nonresonant
condition described in the sense of rotation number is different from that in
[7,10].
For the double resonant problem with Landesman–Lazer conditions, we
observe that in the recent paper [9], Fonda and Garrione studied the existence
of periodic solutions for equation (1.1) under Landesman–Lazer conditions,
where f is controlled by two positively homogeneous functions for which res-
onance occurs, and they assumed that f has the following decomposition
f (t, z) = −(1 − γ(t, z))J∇H1 (z) − γ(t, z)J∇H2 (z) + r(t, z),
 
0 −1
being J = the standard 2 × 2 symplectic matrix, 0 ≤ γ(t, z) ≤ 1
1 0
and r(t, z) is bounded by an L2 -function. Nonlinearity is indeed allowed to
interact with two positively homogeneous Hamiltonians, both at resonance,
and a condition of Landesman–Lazer type is assumed at both sides. H1 , H2
are two C 1 -functions that are positively homogeneous of degree 2, hence the
origin is an isochronous center for the systems Jz  = ∇Hi (z), i = 1, 2, and then
the nonzero solutions ϕ and ψ of Ju = ∇Hi (z), i = 1, 2 have minimal periods
τϕ and τψ , respectively. It is assumed that there exists a positive integer N
such that NT+1 ≤ τψ < τϕ ≤ N T
, hence double resonance occurs when equalities
hold. We will use the rotation number approach to describe double resonance
and improve the results in [9]. Let us mention [11,12] for related results on
problems under double resonance.
The rotation number is an important index of dynamics of second order
linear periodic equations with variable coefficients. In [13], Gan and Zhang in-
vestigated the relationship between the rotation numbers and characteristics
of the second order linear periodic equation and showed many useful esti-
mates on twist dynamics by the rotation numbers. Zhang [14] also studied
the periodic Fučik spectrum by rotation number approach. Zanini [15] anal-
ysed a relation obtained in [13] between rotation numbers and eigenvalues of
Vol. 76 (2021) Non-resonance and Double Resonance for a Planar System Page 3 of 23 91

Hill’s equation and obtained multiplicity results of periodic solutions by using

Poincaré–Birkhoff theorem. Recentely, Margheri, Rebelo and Torres [16] used
Morse index and rotation numbers to obtain the multiplicity of periodic so-
lutions of linear asymptotic equations. Their results were improved by Qian,
Torres and Wang [17] via the phase-plane analysis of the spiral properties of
the solutions. We also refer to [18,19] for related works using the rotation
number approach.
To describe the non-resonance from a rotation number viewpoint we sup-
pose in this paper that the angular velocity of solutions of (1.1) is controlled by
the angular velocity of solutions of two positively homogeneous system. More
precisely, we suppose that f (t, z) satisfies the following conditions
(H0 ) f ∈ C(R × R2 , R2 ) is continuous, 2π-periodic with respect to the first
variable and the solution of the Cauchy problem associated to (1.1) is
unique and exists globally;
(H1 ) JL1 (t, z), z ≤ lim inf  Jf (t,λz)
λ , z ≤ lim sup Jf (t,λz)
λ , z ≤ JL2 (t, z), z
λ→+∞ λ→+∞

holds uniformly for a.e. t ∈ [t0 , t0 + 2π], z ∈ S1 ;

(H2 ) ρ(L1 ) > n, ρ(L2 ) < n + 1,
where L1 , L2 : R × R2 → R2 are L2 -Carathéodory functions, 2π-periodic with
respect to the first variable, locally Lipschitz-continuous and positively homo-
geneous of degree 1 with respect to the second variable. Let ρ(Li ), i = 1, 2
denote the rotation numbers of equations z  = Li (t, z), i = 1, 2. A precise def-
inition of the rotation number is given in Sect. 2. Our first main result is as
follows.

Theorem 1.1. Suppose (H0 ), (H1 ) and (H2 ), then Eq. (1.1) has a 2π-periodic
solution.

Example 1.1. Consider the following equations

x + ai (t)x = 0, x + bi (t)x = 0, i = 1, 2,

where ai (t), bi (t) ∈ L1 ([0, 2π]), ai (t) < bi (t), i = 1, 2, are 2π-periodic functions
defined by
 
(2n + 1)2 , t ∈ [0, π], (2n + δ)2 , t ∈ [0, π],
a1 (t) = a2 (t) =
c(t), t ∈ [π, 2π], −λ2 , t ∈ [π, 2π],
 
(2n + 3/2) , t ∈ [0, π],
2
(2n + 1) , t ∈ [0, π],
2
b1 (t) = b2 (t) =
−μ2 , t ∈ [π, 2π], d(t), t ∈ [π, 2π],

where n ∈ N, c(t), d(t) ∈ L1 ([π, 2π]), d(t) ≥ 0, δ ≥ 2(2n + δ) arctan |λ|/π > 0
and arctan μ ≥ π/2 − π/(4(2n + 3/2)). Then it can be proved (see Sect. 4)

n < ρ(a1 ), ρ(a2 ), ρ(b1 ), ρ(b2 ) < n + 1.

91 Page 4 of 23 C. Liu et al. Results Math

Assume that g(t, x) is continuous, 2π-periodic with respect to t and locally

Lipschitz-continuous with respect to x such that
g(t, x) g(t, x)
ai (t) ≤ lim inf ≤ lim sup ≤ bi (t) for a.e. t ∈ [0.2π],
|x|→+∞ x |x|→+∞ x

For i = 1 or i = 2. Then, the equation x + g(t, x) = 0 has a 2π-periodic

solution by using Theorem 1.1.
Comparing with the nonresonant condition described in Fonda and Ha-
bets [7], note that it would need a constant u0 ∈ [ai (t), bi (t)] for all t ∈ [0, 2π]
such that the related topological degree d(L − N0 , Ω0 )
= 0 using a result
in [20] (see condition (iv) in Theorem 2-Theorem 4 in [7]). Such condition
is not satified by the examples of ai (t), bi (t), i = 1, 2 given above. Also,
in the nonresonant condition described in Boscaggin and Garrione [18], it is
 2π
needed 0 a1 (t)dt > 0. But if we let a1 (t) = (2n + 1)2 for t ∈ [0, π] and
 2π
a1 (t) = −(2n + 2)2 for t ∈ (π, 2π], we have 0 a1 (t)dt < 0. Therefore, the
nonresonant condition described in the sense of rotation number is different
from the previous nonresonant conditions.

Secondly, we consider a double resonance scenario in the case of two non-

Hamiltonian functions Li (t, z) instead of Hamiltonian functions Hi (z), i = 1, 2
like in [9]. Now, the nonzero solutions of z  = Li (t, z), i = 1, 2 may not be
periodic. We suppose
(H3 ) There are two L2 -Carathéodory functions Li : R × R2 → R2 , i = 1, 2
being 2π-periodic with respect to the first variable, locally Lipschitz-
continuous and positively homogeneous of degree 1 with respect to the
second variable such that
JL1 (t, z), z ≤ JL2 (t, z), z, for (t, z) ∈ R × R2 . (1.2)
Moreover, for any t and z,
JL1 (t, z), z = JL2 (t, z), z ⇒ L1 (t, z) = L2 (t, z), (1.3)
and the argument function of each solution of
z  = L1 (t, z) (1.4)
and
z  = L2 (t, z) (1.5)
is 2π-periodic and there is a positive integer n such that the rotation
numbers of (1.4) and (1.5) satisfy
ρ(L1 ) = n and ρ(L2 ) = n + 1, (1.6)
respectively.
Vol. 76 (2021) Non-resonance and Double Resonance for a Planar System Page 5 of 23 91

(H4 ) For every ν ∈ S1 , let uν (t) and vν (t) be the solutions of (1.4) and (1.5)
respectively satisfying uν (t0 ) = ν = vν (t0 ). Then for every ω ∈ S1 , the
following relations are satisfied
 t0 +2π  
Jf (t, λuν (t)), uν (t) JL1 (t, uω (t)), uω (t)
lim inf −λ dt > 0,
t0 λ→+∞ |uν (t)|2 |uω (t)|2
ν→ω
(1.7)
 t0 +2π  
JL2 (t, vω (t)), vω (t) Jf (t, λvν (t)), vν (t)
lim inf λ − dt > 0.
t0 λ→+∞ |vω (t)|2 |vν (t)|2
ν→ω
(1.8)

Remark 1.1. Let Hi (z), i = 1, 2 be two C 2 -continuous and positively homo-

geneous functions of degree 2, then
Hi (λz) = λ2 Hi (z), for λ > 0, i = 1, 2.
Taking derivation of λ on both sides of the above formula, we have
∇Hi (λz), z = 2λHi (z), for λ > 0, i = 1, 2.
Let λ = 1 we obtain 2Hi (z) = ∇Hi (z), z, i = 1, 2. In (H3 ), if Li (t, z) =
−J∇Hi (z), i = 1, 2. Then Hi (z) = ∇Hi (z), z = JLi (t, z), z, i = 1, 2.
Therefore, (1.3) holds.

Our second main result is as follows.

Theorem 1.2. Suppose that (H3 ) and (H4 ) hold. Moreover, suppose that there
exist two L2 -carathéodory functions, 2π-periodic with respect to the first vari-
able and locally Lipschitz-continuous with respect to the second variable, γ :
R × R2 → [0, 1] and r : R × R2 → R2 such that
f (t, z) = (1 − γ(t, z))L1 (t, z) + γ(t, z)L2 (t, z) + r(t, z), (1.9)
with r satisfying
|r(t, z)| ≤ η(t), (1.10)
for a suitable η ∈ L (R), for every t ∈ R, z ∈ R . Then, system (1.1) has a
2 2

2π-periodic solution.

Example 1.2. From Remark 5.1 in Section 5, Theorem 1.2 is an improvement of

Theorem 2.1 in [9] in the regular case. Moreover, we can consider the following
positively homogeneous systems
 
x = y − αx+ ,
(Li )
y  = −a2i x+ + b2i x− ,
where α > 0, a2i = ( α2 )2 + (n − 1 + i)2 , b2i = (n − 1 + i)2 , n ∈ N, i = 1, 2.
Such systems are not Hamiltonian, however (1.2) and(1.3) are easily verified
91 Page 6 of 23 C. Liu et al. Results Math

by observing that
JL2 (t, z), z = a22 (x+ )2 + b22 (x− )2 + y 2 − αyx+
≥ a21 (x+ )2 + b21 (x− )2 + y 2 − αyx+ = JL1 (t, z), z,
and
JL1 (t, z), z = JL2 (t, z), z ⇒ (a22 − a21 )(x+ )2 + (b22 − b21 )(x− )2 = 0 ⇒ x = 0
⇒ L1 (t, z), z = L2 (t, z), z.
The argument function θi (t) of the solutions of (Li ) satisfy
−θi = a2i ((cos θ)+ )2 + b2i ((cos θ)− )2 + sin2 θ − α sin θ(cos θ)+ ,
respectively. It is easy to check that θi (t) < 0 for t ∈ R, i = 1, 2. Moreover,
θi (t + 2π) = θi (t) − 2(n − 1 + i)π for t ∈ R, i = 1, 2, which implies
ρ(L1 ) = n, ρ(L2 ) = n + 1,
and (H3 ) holds.
Choose Li (t, z), i = 1, 2 as above and assume that f (t, z) satisfies (1.9),
(1.10) and (H4 ). Then the system (1.1) has a 2π-periodic solution from Theo-
rem 1.2.
Let us briefly describe how the paper is organized. In Sect. 2, we give a
general setting for the application of the Poincaré–Bohl fixed point theorem
in this context. Then the notions of rotation number and t-rotation number of
solutions of positive homogeneous systems are introduced in Sect. 3. Sections 4
and 5 are devoted to the proofs and applications of Theorem 1.1 and Theo-
rem 1.2, respectively. Finally, we give the detailed proofs of some technical
lemmas in Sect. 6.

2. A General Setting
The proofs of our main results are applications of the Poincaré–Bohl fixed
point theorem, which we recall here for the reader’s convenience.
Theorem 2.1. (Poincaré–Bohl) Let Ω ⊂ Rm be an open bounded set containing
the origin, and ϕ : Ω → Rm be a continuous function such that
ϕ(u)
= λu, for every u ∈ ∂Ω and λ > 1.
Then, ϕ has a fixed point in Ω.
A direct consequence of the Poincaré–Bohl fixed point theorem on the
Eq. (1.1) is the following result.
Theorem 2.2. Suppose that f : R × R2 → R2 is a L2 -Carathéodory function,
2π-periodic with respect to the first variable and locally Lipschitz-continuous
with respect to the second variable. Moreover, assume
Vol. 76 (2021) Non-resonance and Double Resonance for a Planar System Page 7 of 23 91

(H5 ) There exists R > 0 such that, any solution z : [0, 2π] → R2 of (1.1)
satisfying |z(0)| ≤ R exists globally. Moreover, for |z(0)| = R, one has that,
either |z(2π)| < |z(0)|, or
 2π
Jf (t, z(t)), z(t)
|z(t)| > 0 for t ∈ [0, 2π] and dt
∈ 2πZ.
0 |z(t)|2
Then, the Eq. (1.1) has a 2π-periodic solution.
Actually, the locally Lipschitz continuity of f with respect to the second
variable implies that the solution of (1.1) is continuous with respect to the
initial values (see Theorem 5.3 in Ch.I in [21]). Together with the global exis-
tence of the solutions for (1.1), it means that the Poincaré map P of (1.1) is
well defined and homeomorphic. Moreover, hypothesis (H5 ) implies that given
Ω = {|z| ≤ R}, then P(z)
= λz for every z ∈ ∂Ω and λ > 1. Thus, P has
at least one fixed point, which is equivalent to the existence of 2π-periodic
solution of (1.1).
Remark 2.1. There are many sufficient conditions to guarantee the global ex-
istence of the solutions to (1.1). For example, if f (t, z) satisfies the quasi-linear
growth condition, that is, there is a, b ∈ L1 (t0 , t0 + 2π) such that
|f (t, z)| ≤ a(t)|z| + b(t), for a. e. t ∈ [0, 2π] and z ∈ R.
In [10], Fonda and Sfecci introduce a rotating regular spiral assumption and
an angular speed controlling assumption to guarantee the global existence of
the solutions to (1.1), see assumptions (H1 ) and (H3 ) in Theorem 2.3 in [10].

3. Rotation Number of a Planar Positive Homogeneous System

In this section, we introduce the definition of rotation number of a planar
positive homogeneous system. The main discussions come from [22] and are
similar to the arguments in [17,23].
Consider a planar positive homogeneous system of the form

z  = L(t, z), z = (x, y) ∈ R2 , (3.1)

where L : R × R → R is an L -Carathéodory function, 2π-periodic with
2 2 1

respect to the first variable, ocally Lipschitz-continuous and positively homo-

geneous of degree 1 with respect to the second variable, namely
L(t, λz) = λL(t, z), ∀ λ ≥ 0, ∀z ∈ R2 .
In the following, the set of all the functions with the above properties is denoted
by L.
We suppose the initial value problem of Eq. (3.1) has a unique solution,
then a solution z(t; z0 ) with initial value z0
= (0, 0) can be written in polar
coordinates
x(t) = r(t) cos θ(t), y(t) = r(t) sin θ(t).
91 Page 8 of 23 C. Liu et al. Results Math

Then we obtain
Jz  , z JL(t, z), z
−θ (t) = =
|z| 2 |z|2
= JL(t, (cos θ, sin θ)), (cos θ, sin θ)
=: Θ(t, θ; L). (3.2)
We denote by θ(t; t0 , θ0 ) the unique solution of Eq. (3.2) satisfying the initial
value condition θ(t0 ; t0 , θ0 ) = θ0 . Since Θ(t, θ; L) is 2π-periodic both for t and
θ when L(t, z) is 2π-periodic for t, (3.2) is a differential equation on a torus.
We have
θ(t + 2mπ; t0 , θ0 ) = θ(t; t0 , θ(2mπ; t0 , θ0 )),
θ(t; t0 , θ0 + 2jπ) = θ(t; t0 , θ0 ) + 2jπ
for arbitrary θ0 , t0 ∈ R and m, n, j ∈ Z.
Therefore, the rotation number of Eq. (3.2) defined by
θ0 − θ(t0 + t; t0 , θ0 ) θ0 − θ(t0 + 2kπ; t0 , θ0 )
ρ(L) := lim = lim
t→∞ t k→∞ 2kπ
exists, and it is independent of t0 and θ0 , see Theorem 2.1 in Chapter 2 of
Hale [21]. By extension, we refer to ρ(L) as the rotation number of the planar
positively homogeneous system (3.1).
On the other hand, if the solution z(t) of Eq. (1.1) satisfies z(t)
= 0 for
t ∈ [t0 , t], we can define the t-rotation number of z(t) as
 t
θ(t0 ) − θ(t) 1 Jf (t, z(t)), z(t)
Rotf (z(t); [t0 , t]) = = dt.
2π 2π t0 |z(t)|2
For a given vector v ∈ S1 , denote by z(·; lv) the solution of Eq. (3.1)
satisfying the initial condition z(t0 ; lv) = lv, l > 0. Since z(t; lv) = lz(t; v), the
rotation number of solutions of Eq. (3.1) is independent of l, for any l > 0.
Therefore, we can let l = 1 when we estimate the rotation number of solutions
of Eq. (3.1). In the following, we write by Rotf (t; z(t)) = Rotf (z(t); [t0 , t]) and
RotL (t; v) = RotL (z(t); [t0 , t]) for short, respectively, where z(t0 ) = v ∈ S1 .
Next, we discuss the relationship between the rotation number ρ(L) of
Eq. (3.1) and the rotation number RotL (t0 + 2π; v) of the solution z(t) with
z(t0 ) = v. In a similar way to that in [13,17,22], we can prove the following
lemmas. The proof of Lemma 3.1 is almost same as in [17]. The details of
proofs for Lemmas 3.2 and 3.3 will be given in Sect. 6.

Lemma 3.1. For n ∈ Z, we have

(i) ρ(L) < n ⇒ RotL (t0 + 2π; v) < n, ∀(t0 , v) ∈ R × S1 ;
(ii) ρ(L) > n ⇒ RotL (t0 + 2π; v) > n, ∀(t0 , v) ∈ R × S1 .
(iii) ρ(L) = n if and only if there is at least one nontrivial 2π-periodic solution
θ(t; θ0 ) of equation (3.2) with θ0 − θ(2π; θ0 ) = 2nπ.
 Vol. 76 (2021) Non-resonance and Double Resonance for a Planar System Page 9 of 23 91

Lemma 3.2. Suppose that Li (t, z) ∈ L, i = 1, 2 and

JL2 (t, z) − JL1 (t, z), z ≥ 0 (3.3)
holds for all z ∈ R2 and a.e. t ∈ [t0 , t0 + 2π]. Then,
RotL2 (t; v) ≥ RotL1 (t; v), ∀ t ∈ [t0 , t0 + 2π], ∀ v ∈ S1 .

Lemma 3.3. Suppose that Li (t, z) ∈ L, i = 1, 2. Then for each ε > 0, there
exists δ > 0 such that, if
JL2 (t, z), z ≥ JL1 (t, z), z − δ|z|2 (3.4)
holds for all z ∈ R and a.e. t ∈ [t0 , t0 + 2π], then
2

RotL2 (t; v) ≥ RotL1 (t; v) − ε, ∀ t ∈ [t0 , t0 + 2π], ∀ v ∈ S1 . (3.5)

4. Nonresonance from the Viewpoint of Rotation Number

From (H1 ) we have that ∀δ > 0, ∃ λδ > 0, such that
 
Jf (t, λz)
, z ≥ JL1 (t, z), z − δ
λ


for t ∈ [t0 , t0 + 2π], λ ≥ λδ and z ∈ S1 . Since f is continuous then  Jf (t,λz)
λ , z
is bounded by a positive constant Mδ for t ∈ [t0 , t0 +2π], λ ∈ [1, λδ ] and z ∈ S1 .
Let l(t) ∈ L1 and l(t) ≥ 0 such that
l(t) ≥ λ2 (JL1 (t, z), z − δ + Mδ )
for t ∈ [t0 , t0 + 2π], λ ∈ [1, λδ ] and z ∈ S1 . Therefore, l(t) ∈ L1 satisfies
 
Jf (t, λz) l(t)
, z ≥ JL1 (t, z), z − δ − 2 ,
λ λ
for t ∈ [t0 , t0 + 2π], λ ≥ 1 and z ∈ S1 . An analogous reasoning can be done
with L2 (t, z). In conclusion, from (H1 ), we deduce the following two properties
(H1l ) There exists L1 (t, z) ∈ L satisfying: ∀ δ > 0, ∃ l(t) ∈ L1 ([t0 , t0 + 2π], R+ )
such that
 
Jf (t, λz) l(t)
, z ≥ JL1 (t, z), z − δ − 2 (4.1)
λ λ
holds for a.e. t ∈ [t0 , t0 + 2π], ∀ z ∈ S1 and λ ≥ 1;
(H1r ) There exists L2 (t, z) ∈ L satisfying: ∀ δ > 0, ∃ l(t) ∈ L1 ([t0 , t0 + 2π], R+ )
such that
 
Jf (t, λz) l(t)
, z ≤ JL2 (t, z), z + δ + 2 (4.2)
λ λ
holds for a.e. t ∈ [t0 , t0 + 2π], ∀ z ∈ S1 and λ ≥ 1.
91 Page 10 of 23 C. Liu et al. Results Math

Similarly to that in [17,22,23], we can use Lemma 3.2 and Lemma 3.3 to
obtain the following relationships of the rotation numbers of solutions between
the positive homogeneous system (3.1) and nonlinear system (1.1). The proof
of Lemma 4.1 is in Sect. 6.
Lemma 4.1. Suppose (H0 ) and Li (t, z) ∈ L, i = 1, 2, then
(i) If (H1l ) holds, for each ε > 0, there exists Rε > 0 such that
Rotf (t; z) > RotL1 (t; v) − ε, ∀ t ∈ [t0 , t0 + 2π], v = z(t0 )/|z(t0 )| (4.3)
hods for every solution of Eq. (1.1) satisfying |z(t)| ≥ Rε , ∀ t ∈ [t0 , t0 +
2π].
(ii) If (H1r ) holds, for each ε > 0, there exists Rε > 0 such that
Rotf (t; z) < RotL2 (t; v) + ε, ∀ t ∈ [t0 , t0 + 2π], v = z(t0 )/|z(t0 )| (4.4)
holds for every solution of Eq. (1.1) satisfying |z(t)| ≥ Rε , ∀ t ∈ [t0 , t0 +
2π].
Combining Lemmas 3.1 and 4.1, we have the following result.
Lemma 4.2. Suppose (H0 ) and Li (t, z) ∈ L, i = 1, 2, then
(i) If ρ(L1 ) > n and (H1l ) holds, then there exists R1 > 0 such that Rotf (t0 +
2π; z) > n holds for every solution z(t) of Eq. (1.1) satisfying |z(t)| ≥
R1 , ∀ t ∈ [t0 , t0 + 2π].
(ii) If ρ(L2 ) < n + 1 and (H1r ) holds, then there exists R2 > 0 such that
Rotf (t0 +2π; z) < n+1 holds for every solution z(t) of Eq. (1.1) satisfying
|z(t)| ≥ R2 , ∀ t ∈ [t0 , t0 + 2π].
Now we have all the ingredients to prove Theorem 1.1.
Proof of Theorem 1.1. Suppose that z(t; z0 ) = (x(t; z0 ), y(t; z0 ) is a solution
of Eq. (1.1) satisfying the initial value condition z(t0 ; z0 ) = z0 = (x0 , y0 ). Take
ΩM = {z ∈ R2 : |z| < M }, and define the Poincaré map
P : ΩM → R2 ,
(x0 , y0 ) → (x(t0 + 2π; z0 ), y(t0 + 2π; z0 )).
From (H0 ) we have the global existence of the solution, thus P is well defined.
Moreover, the uniqueness of the solution implies that the solution depends
continuously on the initial value. Therefore P is continuous. In addition, P
has a fixed point z0 ∈ R2 if and only if Eq. (1.1) has a 2π-periodic solution.
Our aim is to prove that P has a fixed point for M > 0 sufficiently large by
the Poincaré–Bohl theorem.
Applying again the global existence of the solution of Eq. (1.1) and con-
tinuous dependence of the solution with respect to the initial values, we can
prove the elastic properties of the solutions of Eq. (1.1). That is, for Ri , i = 1, 2
as in Lemma 4.2, there exists MR > 0 such that |z(t; z0 )| > Ri , i = 1, 2 for
t ∈ [t0 , t0 + 2π] when |z0 | ≥ MR .
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Let M = MR . For any z0 ∈ ∂ΩM , we have |z(t; z0 )| > Ri , i = 1, 2 for

t ∈ [t0 , t0 + 2π]. Then by condition (H1 ), (H2 ) and using Lemma 4.2 we have
n < Rotf (t0 + 2π; z) < n + 1,
where n ∈ N, which implies that z(t; z0 ) is not able to rotate any integer
number of turns for t ∈ [t0 , t0 + 2π]. Hence P(z0 )
= λz0 , for z0 ∈ ∂ΩM and
λ > 1. Using Poincaré–Bohl theorem, we have that P has a fixed point. The
proof is thus completed. 
To prepare the application of Theorem 1.1 we need the following orien-
tation preserving lemma for the torus differential Eq. (3.2), which indicates
that the solutions are ordered. The proof is trivial by using the uniqueness of
solutions of (3.2) with respect to the initial value.
Lemma 4.3. Let θ(t; t0 , θ0 ) and θ(t; t0 , θ1 ) be two solutions of (3.2) with the
initial values θ(t0 ; t0 , θ0 ) = θ0 > θ1 = θ(t0 ; t0 , θ1 ). Then
θ(t; t0 , θ0 ) > θ(t; t0 , θ1 ) for t > t0 .
Then, we can give the following lemmas related the estimation of the
rotation number.
Lemma 4.4. Suppose that a(t), b(t) ∈ L1 ([0, 2π], R) in equations
x + a(t)x = 0 (4.5)
and
x + b(t)x = 0 (4.6)
are of the form
 
a+ (t), t ∈ [0, π], b+ (t), t ∈ [0, π],
a(t) = b(t) =
a− (t), t ∈ [π, 2π], b− (t), t ∈ [π, 2π],
where a+ (t) ≥ m2 , m2 ≤ b+ (t) ≤ (m + 1)2 , b− (t) ≤ 0, m ∈ N. Then
ρ(a) ≥ m/2, ρ(b) ≤ (m + 1)/2,
where ρ(a) and ρ(b) denote the rotation numbers of equations (4.5) and (4.6),
respectively.
Proof. From Proposition 1 (i) in [17], we have that ρ(a) ≥ m/2. Next we prove
ρ(b) ≤ m+1
2 . Equation (4.6) is equivalent to
x = y, y  = −b(t)x. (4.7)
Using the polar coordinates transformation we can obtain
−θ = b(t) cos2 θ + sin2 θ.
Let θ(t; 0, 0) be the solution such that θ(0; 0, 0) = 0. When t ∈ [0, π], from
m2 ≤ b(t) ≤ (m + 1)2 and
 θ(π;0,0)  −(m+1)π
dθ dθ
2 = −π = ,
0
2
b(t) cos θ + sin θ 0 (m + 1) cos2 θ + sin2 θ
2
91 Page 12 of 23 C. Liu et al. Results Math

we have θ(π; 0, 0) ≥ −(m + 1)π. Next we will prove θ(2π; 0, 0) ≥ −(m + 1)π.
Without loss of generality, let m = 2l+1, l ∈ Z, θ(π; 0, 0) ∈ [−(m+1)π, −mπ−
π/2). Assume that there exist tε , tε ∈ [π, 2π], such that
θ(tε ; 0, 0) = −(m + 1)π, θ(tε ; 0, 0) = −(m + 1)π − ε
and
θ(t; 0, 0) ∈ [−(m + 1)π − ε, −(m + 1)π), for t ∈ (tε , tε ),
where ε > 0 is small enough. The solution (x(t; 0, 0), y(t; 0, 0)) of (4.7) satisfies
y(tε ; 0, 0) = 0, y(tε ; 0, 0) < 0,
and
x(t; 0, 0) > 0 for t ∈ (tε , tε ).
But, using y  = −b(t)x ≥ 0, we know that y(tε ; 0, 0) − y(tε ; 0, 0) ≥ 0. This is a
contradiction. Therefore, we coclude that θ(2π; 0, 0) ≥ −(m + 1)π.
Using Lemma 4.3 and a discussion like in Proposition 1 (i) of [17], we
have
−θ(2kπ; 0, 0) ≤ k(m + 1)π, for k ∈ N.
Now, by the definition of rotation number,
−θ(2kπ; 0, 0) k(m + 1)π m+1
ρ(b) = lim ≤ lim = . 
k→∞ 2kπ k→∞ 2kπ 2
Lemma 4.5. If in (4.5), a(t) = −λ2 for t ∈ [π, 2π], then
− 2 arctan |λ| < θ(π) − θ(2π) < 2(π/2 − arctan |λ|). (4.8)

Proof. For t ∈ [π, 2π], the Eq. (4.5) is of the from x − λ2 x = 0. The solutions
of (4.5) are
x(t) = c1 exp(λt) + c2 exp(−λt),
where c1 and c2 are constants. When c1 = 0 or c2 = 0, the corresponding
solution lies in the half lines y = ±λx of the phase plane. The other solutions
are located in the conical region surrounded by these half-lines. Thus we have
−2 arctan |λ| < θ(t0 ) − θ(t) < 2(π/2 − arctan |λ|), for t, t0 ∈ [π, 2π], t > t0 .
In particular, we obtain (4.8). 

Now we can analyze in detail Example 1.1. Let 2π-periodic functions

ai (t), bi (t) ∈ L1 ([0, 2π]), i = 1, 2 be defined as in Example 1.1. From Lemma
4.4 we know that
ρ(a1 ) ≥ n + 1/2 > n, ρ(a2 ) ≥ n, ρ(b1 ) ≤ n + 1, ρ(b2 ) ≤ n + 1/2 < n + 1.
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If ρ(a2 ) = n, from Lemma 3.1, there is at least one nontrivial 2π-periodic

solution θ2 (t) of the torus differential equation −θ = sin2 θ + a2 (t) cos2 θ with
θ2 (2π) − θ2 (0) = −2nπ. Using a simple computation,
  
δ
θ2 1− π − θ2 (0) = −2nπ.
2n + δ

 
δ
Then θ2 (2π) − θ2 1 − 2n+δ π = 0. Using Lemma 4.5, we have that θ2 (π) −
θ2 (2π) > −2 arctan |λ|, which implies
  
δ
θ2 (π) − θ2 1− π > −2 arctan |λ|.
2n + δ
Since
  
δ
−θ2 2 2
= sin θ2 + (2n + δ) cos θ2 > 1, 2
for t ∈ 1− π, π ,
2n + δ
then,
  
δ δπ
θ2 (π) − θ2 1− π <− .
2n + δ 2n + δ
Thus, δπ < 2(2n + δ) arctan |λ|. This contradicts to the definitions of δ and λ
in Example 1.1. Therefore, ρ(a2 ) > n.
Moreover, suppose that ρ(b1 ) = n + 1 and θ1 (t) is a 2π-periodic solution
of −θ = sin2 θ + b1 (t) cos2 θ with θ1 (2π) − θ1 (0) = −2(n + 1)π. Taking t =
π/(2(2n + 3/2)), a simple computation gives
θ1 (π − t ) − θ1 (0) = −(2n + 1)π, θ1 (π) − θ1 (π − 2t ) = −π,
and
θ1 (π − t ) − θ1 (π − 2t ) < −t .
Thus
π
θ1 (π) − θ1 (0) > −2(n + 1)π + .
2(2n + 3/2)
Using Lemma 4.5, θ1 (π) − θ1 (2π) < 2(π/2 − arctan μ), which implies that
π π
< 2( − arctan μ),
2(2n + 3/2) 2
again in contradiction with the definitions of δ and λ in Example 1.1. Therefore,
ρ(b1 ) < n + 1.

5. The Existence of Periodic Solutions Under Landesman–Lazer

Conditions
In this section, we consider the existence of periodic solutions of system (1.1)
under Landesman-Lazer conditions. We begin with some preparatory lemmas.
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Lemma 5.1. Let g : R × R2 → R2 be an L1 -Carathéodory function such that

|g(t, z)| ≤ c(t)(1 + |z|),
for almost every t ∈ [t0 , t0 +2π] and every z ∈ R2 , being c(t) a suitable function
in L1 (t0 , t0 + 2π). Then, for every R0 > 0 there exists R1 ≥ R0 such that, if z
satisfies
z  = g(t, z), (5.1)
and |u(t̄)| ≤ R0 for some t̄ ∈ [t0 , t0 + 2π], then |u(t)| ≤ R1 for every t ∈
[t0 , t0 + 2π].
The proof of lemma 5.1 is similar as that in [9], so we omit it here.
Lemma 5.2. Suppose that L : R × R2 → R2 is an L1 -Carathéodory function
being 2π-periodic in the first variable, locally Lipschitz-continuous and positive
homogeneous of degree 1 in the second variable. Let uω (t) be the solution of
the equation u = L(t, u) with the initial value uω (t0 ) = ω ∈ S1 . Then any
continuous function z(t) can be expressed as z(t) = r(t)uω(t) (t), where r(t)
and ω(t) are continuous functions and r(t) ≥ 0.
Proof. Let Pt : R2 → R2 , ζ → u(t; ζ) be the t-flow of the equation u = L(t, u),
where u(t; ζ) is the solution of u = L(t, u) with the initial condition u(t0 ; ζ) =
ζ. Since L(t, u) is an L1 -Carathéodory function, and positively homogeneous
of degree 1 with respect to the second variable, we know that solutions of
equation u = L(t, u) exists globally. Indeed, from the existence and uniqueness
of solutions we can write the solution z(t; z0 ) with initial value z0
= (0, 0) in
polar coordinates x(t) = r(t) cos θ(t), y(t) = r(t) sin θ(t), then
r = rL(t, (cos θ, sin θ)), (cos θ, sin θ).
By Cauchy-Schwartz inequality, |r | ≤ r|L(t, (cos θ, sin θ))|. Since L(t, z) is an
L1 -Carathédory function, there exists η ∈ L1 (t0 , t0 + 2π), such that |L(t, v)| ≤
η(t) holds for every t ∈ [t0 , t0 + 2π] and every v = (cos θ, sin θ) ∈ S1 . Thus,
|r | ≤ η(t)r.
Hence, there is M > 0 such that
r(t0 )e−2πM ≤ r(t) ≤ r(t0 )e2πM , for t ∈ [t0 , t0 + 2π].
So r(t) is large enough when r(t0 ) is large enough, which implies the global
existence of the solutions of equation u = L(t, u). Then, Pt is well defined.
The uniqueness of the solution with respect to the initial value implies that
the solutions depend continuously on the initial values. Therefore Pt is a
global homeomorphism on R2 . Moreover, from the positive homogeneity of
degree 1 of L(t, u) for u, we know that Pt (rω) = rPt (ω) for r ≥ 0 and
ω ∈ S1 . Thus, for any t ∈ R, we can define the generalized polar coordi-
nates (r(t), Pt (ω(t))) ∈ R+ × Pt (S1 ), that is, for z(t) we can find r ≥ 0 and
ω ∈ S1 , such that Pt (ω(t))/|Pt (ω(t))| = z(t)/|z(t)| and z(t) = r(t)Pt (ω(t)).
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Therefore, ω(t) = P−t (z(t)/r(t)). It is easy to see that r(t) and ω(t) are con-
tinuous functions provided that z(t) is a continuous function. 

Lemma 5.3. Suppose that Li (t, w), i = 1, 2 satisfy (H3 ) and w ∈ H 1 (t0 , t0 +2π)
satisfies
w = ξ(t)L1 (t, w) + (1 − ξ(t))L2 (t, w) := L(t, w), (5.2)
where ξ ∈ L2 (t0 , t0 + 2π) with 0 ≤ ξ(t) ≤ 1 for almost every t ∈ [t0 , t0 + 2π],
the argument function of w(t) is 2π-periodic, then w(t) solves either (1.4) or
(1.5).

Proof. First of all, we observe that L(t, w) is positively homogeneous of de-

gree 1 for z. Then the rotation number ρ(L) of (3.2) exists. Writing w(t) =
r(cos θ, sin θ), we have
ρ(L) = lim −θ(t)/t.
t→+∞

Moreover,
JL1 (t, (cos θ(t), sin θ(t))), (cos θ(t), sin θ(t)) ≤ −θ (t)
≤ JL2 (t, (cos θ(t), sin θ(t))), (cos θ(t), sin θ(t)) (5.3)
holds for almost every t ∈ R. Let θ1 (t) and θ2 (t) be the solutions of
−θ = JL1 (t, (cos θ(t), sin θ(t))), (cos θ(t), sin θ(t))
and
−θ = JL2 (t, (cos θ(t), sin θ(t))), (cos θ(t), sin θ(t)),
with the initial conditions θ1 (t0 ) = θ(t0 ) and θ2 (t0 ) = θ(t0 ), respectively.
Integrating on the inequalities (5.3) from t0 to t, we have
θ1 (t) ≥ θ(t) ≥ θ2 (t) for t ≥ t0 . (5.4)
It follows that
n = ρ(L1 ) ≤ ρ(L) ≤ ρ(L2 ) = n + 1.
Since θ(t) is 2π-periodic, ρ(L) is an integer number, which implies that either
ρ(L) = n or either ρ(L) = n+1. Without loss of generality, let fix ρ(L) = n+1.
As a consequence of (5.4), that
θ(t) − θ(t0 ) = θ2 (t) − θ2 (t0 ) for t ∈ [t0 , t0 + 2π].
Now we can write w(t) = l(t)vω (t), where l(t) is a H 1 -function and vω (t) is
the solution of (1.5) with vω (t0 ) = ω ∈ S1 . We want to prove that l (t) = 0
for t ∈ [t0 , t0 + 2π], which implies that w(t) = R0 vω (t) and w(t) solves (1.5).
Indeed, w(t) and vω (t) perform at the same speed around the origin, so
Jw (t), w(t) Jvω (t), vω (t)
= .
|w(t)|2 |vω (t)|2
91 Page 16 of 23 C. Liu et al. Results Math

Using that Li (t, w) are positively homogeneous of degree 1 and w(t) = l(t)vω (t),
this last equation means
ξ(t)(JL1 (t, vω (t)), vω (t) − JL2 (t, vω (t)), vω (t)) = 0. (5.5)
Since {z; Jz} is an orthogonal basis of R2 , the validity of (1.3) is equivalent
to
JL1 (t, z), z = JL2 (t, z), z ⇒ L1 (t, z), z = L2 (t, z), z.
Then using (1.3) in hypothesis (H3 ) and (5.5), we obtain
ξ(t)(L1 (t, vω (t)), vω (t) − L2 (t, vω (t)), vω (t)) = 0. (5.6)
On the other hand,
ξ(t)L1 (t, w) + (1 − ξ(t))L2 (t, w), vω (t) = w (t), vω (t)
= l (t)vω (t) + l(t)vω (t), vω (t). (5.7)
Hence (5.6) and (5.7) imply
l (t)|vω (t)|2 = l(t)ξ(t)(L1 (t, vω (t)) − L2 (t, vω (t)), vω (t)) = 0,
which means l (t) = 0 for t ∈ [t0 , t0 + 2π].
Therefore l(t) = R0 and w(t) = R0 vω (t) solves (1.5). The lemma is thus
proved. 

Proof of Theorem 1.2. Since L1 (t, z) and L2 (t, z) are positively homogeneous
of degree 1 and r(t, z) is bounded, f (t, z) has an at most linear growth of z.
Then the global existence and the uniqueness of the solutions to the Cauchy
problem associated to (1.1) is guaranteed and the Poincaré map associated
to (1.1) is well-defined. Let us see that the Poincaré–Bohl theorem (Theorem
2.2) can be applied. By contradiction and using the elstic property of the
solutions, we assume that there is a sequence (zn )n of solutions such that
min{|zn (t)| : t ∈ [t0 , t0 + 2π]} → +∞, and zn (t0 + 2π) = σn zn (t0 ), for some
σn > 1. By setting wn = zn /zn ∞ , we have
r(t, zn )
wn = (1 − γ(t, zn ))L1 (t, wn ) + γ(t, zn )L2 (t, wn ) + . (5.8)
zn ∞
Since (wn )n is bounded in L2 (t0 , t0 +2π), (5.8) implies that (wn )n is bounded in
H 1 (t0 , t0 +2π) and so there exists a function w ∈ H 1 (t0 , t0 +2π) such that (up to
subsequence) wn → w uniformly and wn  w weakly in H 1 (t0 , t0 + 2π). Then
w∞ = 1. Moreover, the sequence (γ(·, zn (·)))n is bounded in L2 (t0 , t0 + 2π),
so (extracting a new subsequence) they weakly converge to a given function
β ∈ L2 (t0 , t0 + 2π). As
{ζ ∈ L2 (t0 , t0 + 2π) | 0 ≤ ζ(t) ≤ 1 for almost every t ∈ [t0 , t0 + 2π]}
is a convex and closed subset of L2 (t0 , t0 + 2π), it is weakly closed and then
0 ≤ β(t) ≤ 1 for almost every t ∈ [t0 , t0 + 2π]. Passing to the weak limit in
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(5.8) and noticing that the last term vanishes thanks to the L2 -boundedness
of r(t, z), we get

w = (1 − β(t))L1 (t, w) + β(t)L2 (t, w). (5.9)

Moreover, the argument function of w(t) is 2π-periodic, so by Lemma 5.3,

w(t) performs n or n + 1 clockwise turns around the origin in the time interval
[t0 , t0 + 2π]. Let us assume this last case (the other being similar). Then,
w(t) = R0 vω (t) for suitable R0 > 0 and vω (t) is the solution of (1.5) with
vω (t0 ) = ω ∈ S1 . In the generalized polar coordinates introduced in Lemma
5.2, zn (t) = rn (t)vωn (t) (t), with ωn (t) ∈ S1 , for every n. Then,

wn (t) = zn (t)/zn ∞ → R0 vω (t), rn (t)/zn ∞ → R0 and ωn (t) → ω

uniformly as n → ∞. Since the argument functions of wn are 2π-periodic and

wn tends to w uniformly, wn performs the same clockwise turns around the
origin as w performs in the time interval [t0 , t0 + 2π] for n sufficiently large,
that is, wn and zn perform n + 1 turns around the origin in the time interval
[t0 , t0 + 2π] for n sufficiently large. Then we have
 t0 +2π  t0 +2π
Jf (t, zn ), zn (t) JL2 (t, vω (t)), vω (t)
dt = dt, (5.10)
t0 |z n (t)|2
t0 |vω (t)|2
which implies
  zn (t) 
t0 +2π
JL2 (t, R0 zn ∞ vω (t)), vω (t) Jf (t, zn (t)), (R0 zn ∞ ) 
− dt ≤ 0,
t0 |vω (t)|2 |zn (t)|2 /(R0 zn ∞ )2
that is
 t0 +2π  zn (t) 
R0 zn ∞ JL2 (t, rn (t)vω (t)), vω (t) Jf (t, zn (t)), rn (t) 
− dt ≤ 0.
t0 rn (t) |vω (t)|2 |zn (t)|2 /(rn (t))2
Hypotheses (1.9) and (1.10) now allow us to apply Fatou’s lemma, which gives
 t0 +2π 
R0 zn ∞ JL2 (t, rn (t)vω (t)), vω (t)
lim inf
t0 n→+∞ rn (t) |vω (t)|2
zn (t) 
Jf (t, zn (t)), rn (t) 
− dt ≤ 0.
|zn (t)|2 /(rn (t))2
Using standard properties of the inferior limit and taking into account that

wn (t) = zn (t)/(R0 zn ∞ ) → vω (t), rn (t)/(R0 zn ∞ ) → 1,

uniformly, we can assume without loss of generality that ωn (t) → ω uni-

formly, if necessary, passing to a further subsequence. Thus, for every fixed
t ∈ [t0 , t0 + 2π] we are computing the inferior limit which appears in (1.8)
91 Page 18 of 23 C. Liu et al. Results Math

along the particular subsequence (rn (t), ωn (t)), for which ωn (t) → ω and
rn (t) → +∞. We deduce that
 t0 +2π  
JL2 (t, vω (t)), vω (t) Jf (t, λvν (t)), vν (t)
lim inf λ − dt ≤ 0,
t0 λ→+∞ |vω (t)|2 |vν (t)|2
ν→ω
(5.11)
in contradiction with hypothesis (1.8). 
Remark 5.1. From lemma 4 in [24], we know that if a 2π-periodic solution z(t)
of z  = f (t, z) satisfies that z(t)
= 0 for t ∈ [t0 , t0 + 2π] and z(t) performs k(z)
turns around the origin in the time 2π then
 2π
1 Jz  (t), z(t)
k(z) = dt
A(H, 1) 0 H(z(t))
is independent upon the particular positive definite function H, where H =
H(z) is a continuous and positively homogeneous function of degree 2 and
A(H, 1) = Area{z : H(z) ≤ 1}.
In Theorem 1.2, if Li (t, z) = −J∇Hi (z), i = 1, 2, where Hi (z), i = 1, 2
are two continuous and positively homogeneous functions of degree 2. Then
we can compute k(z) in the proof of Theorem 1.2 using H1 or H2 . That is,
replace (5.10) with
 t0 +2π  t0 +2π
Jf (t, zn ), zn (t) ∇H2 (vω (t)), vω (t)
dt = dt.
t0 H2 (zn (t)) t0 H2 (vω (t))
Then, we deduce
 t0 +2π  
lim inf λ − Jf (t, λvν (t)), vν (t) dt ≤ 0,
t0 λ→+∞
ν→ω

in contradiction with hypothesis (18) in [9]. Hence, in the case of Li (t, z) =

−J∇Hi (z), i = 1, 2, hypotheses (1.7) and (1.8) in Theorem 1.2 are consistent
with (17) and (18) in [9].

6. Proof of Technical Lemmas

Proof of Lemma 3.2. For each θ0 ∈ R, let θ(t; θ0 , L) be the unique solution of
(3.2) satisfying θ(t0 ; θ0 , L) = θ0 . Denote ϑ(t) =: θ2 (t) − θ1 (t), where θi (t) =
θ(t; θ0 , Li ), for i = 1, 2. Note that ϑ(t0 ) = 0. We have
dϑ(t)
− = Θ(t, θ2 ; L2 ) − Θ(t, θ1 ; L1 )
dt
= Θ(t, θ2 ; L2 ) − Θ(t, θ1 ; L2 ) + Θ(t, θ1 ; L2 )) − Θ(t, θ1 ; L1 )
=: a(t)ϑ(t) + b(t),
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where a(t) = ∂Θ ∂θ (t, ξ; L2 ), ξ = ξ(t) ∈ [θ1 (t), θ2 (t)], b(t) = J(L2 (t, (cos θ1 , sin θ1 ))−
L1 (t, (cos θ1 , sin θ1 ))), (cos θ1 , sin θ1 ). Therefore,
 t  t 
ϑ(t) = − b(s) exp −a(τ )dτ ds.
t0 s
It follows from (3.3) that b(t) ≥ 0 for a.e. t ∈ [t0 , t0 + 2π]. Thus ϑ(t) ≤ 0 for
all t ∈ [t0 , t0 + 2π], that is,
θ2 (t) − θ0 ≤ θ1 (t) − θ0 , ∀ t ∈ [t0 , t0 + 2π].
According to the definition of RotL (t; v), we obtain
RotL2 (t; v) ≥ RotL1 (t; v), ∀ t ∈ [t0 , t0 + 2π], ∀ v ∈ S1 ,
which completes the proof of Lemma 3.2. 
Proof of Lemma 3.3. Let ε > 0 be fixed. For δ > 0 be small enough, suppose
that (3.4) holds for all z ∈ R2 and a.e. t ∈ [t0 , t0 +2π]. According to Lemma 3.2,
we know that
RotL2 (t; v) ≥ RotLδ (t; v), ∀ t ∈ [t0 , t0 + 2π], ∀ v ∈ S1 ,
where RotLδ (t; v) is the t-rotation number of the solution z(t) for the planar
system z  = L1 (t, z) + δJz. Hence, to prove the result, it will be sufficient to
prove that there exists a δ = δ(ε) > 0, it follows that
RotLδ (t; v) ≥ RotL1 (t; v) − ε, ∀ t ∈ [t0 , t0 + 2π], ∀ v ∈ S1 . (6.1)
If not, for each n ∈ N, there would be tn ∈ [t0 , t0 + 2π] and vn ∈ S such that 1

RotL1/n (tn ; vn ) < RotL1 (tn ; vn ) − ε. (6.2)

Without loss of generality, we can assume that tn → τ ∈ [t0 , t0 + 2π] and
vn → w ∈ S1 . Since RotL1/n (·; ·) is continuous on [t0 , t0 + 2π] × S1 , by passing
to the upper limit on both sides of (6.2),
lim sup RotL1/n (tn ; vn ) ≤ RotL1 (τ ; w) − ε. (6.3)
n→∞
Let zn (t), z(t) be, respectively, the solutions of
1
z  = L1 (t, z) + Jz and z  = L1 (t, z),
n
with zn (t0 ) = vn , z(t0 ) = w. According to the theorem of continuous depen-
dence of the solutions, it follows that zn (t) → z(t), uniformly on t ∈ [t0 , t0 +2π].
Hence,
 tn
1 Jzn , zn 
RotL1/n (tn ; vn ) = dt
2π t0 |zn |2
 tn
1 JL1 (t, zn ), zn  + n1 |zn |2
= dt
2π t0 |zn |2
 τ
1 JL1 (t, z), z
→ dt = RotL1 (τ ; w),
2π t0 |z|2
91 Page 20 of 23 C. Liu et al. Results Math

which is in contradiction with (6.3). Then, (6.1) is proved and this completes
the proof of Lemma 3.3. 

Proof of Lemma 4.1. Let ε > 0 be fixed. By Lemma 3.3, there is δ > 0 small
enough such that
ε
RotLδ (t; v) ≥ RotL1 (t; v) − , ∀ t ∈ [t0 , t0 + 2π], ∀ v ∈ S1 , (6.4)
2
where RotLδ (t; v) is the t-rotation number of the solution z(t) of the planar
system z  = L1 (t, z) + δJz with z(t0 ) = v. By assumption (H1l ), for such a δ,
there exists l(t) ∈ L1 ([t0 , t0 + 2π], R+ ) such that (4.1) holds.
Next, assume, by contradiction, that assertion (i) is not true. This implies
that, for each n ∈ N, there is a solution zn (t) of (1.1) defined on [t0 , t0 +
2π] with |zn (t)| ≥ n, ∀ t ∈ [t0 , t0 +2π] and such that, for some tn ∈ [t0 , t0 +2π],
Rotf (tn ; zn ) ≤ RotL1 (tn ; vn ) − ε, (6.5)
where
zn (t0 )
vn = = (cos αn , sin αn ).
|zn (t0 )|
Without loss of generality, we can assume that if n → ∞, then tn → τ ∈
[t0 , t0 + 2π] and vn → ω = (cos α, sin α) ∈ S1 with αn → α. Since RotL1 (·; ·) is
continuous on [t0 , t0 + 2π] × S1 , by passing to the upper limit on both sides
of (6.5), we have
lim sup Rotf (tn ; zn ) ≤ RotL1 (τ ; ω) − ε. (6.6)
n→∞

On the other hand, it is easy to see that zn (t)

= (0, 0), ∀ t ∈ [t0 , t0 + 2π].
Using polar coordinates
xn (t) = rn (t) cos θn (t), yn (t) = rn (t) sin θn (t),
we know that
Jzn , zn  Jf (t, zn ), zn 
−θn (t) = =
|zn | 2 |zn |2
l(t)
≥ JL1 (t, eiθn (t) ), eiθn (t)  − δ −
n2
holds for a.e. t ∈ [t0 , t0 + 2π], where eiθn (t) = (cos θn (t), sin θn (t)) and θn (t0 ) =
αn . By a basic differential inequality (see for instance [25]), we have
θn (t0 ) − θn (t) αn − ϑn (t)
Rotf (t; zn ) = ≥ , (6.7)
2π 2π
where ϑn (t) is the solution of

l(t)
−θ (t) = JL1 (t, eiθ(t) ), eiθ(t)  − δ − 2 ,
n
θ(t0 ) = αn .
Vol. 76 (2021) Non-resonance and Double Resonance for a Planar System Page 21 of 23 91

By the continuous dependence of the solutions, we obtain that, as n → ∞,

ϑn (t) → θ(t), uniformly on t ∈ [t0 , t0 + 2π], where θ(t) is the solution of

−θ (t) = JL1 (t, eiθ(t) ), eiθ(t)  − δ,
θ(t0 ) = α.
In particular, we know from the uniform convergence that
ϑn (tn ) → θ(τ ) for n → ∞.
Thus, it follows from (6.7) that
α − θ(τ ) θ(t0 ) − θ(τ )
lim inf Rotf (tn ; zn ) ≥ = = RotLδ (τ ; ω).
n→∞ 2π 2π
By (6.6),
RotLδ (τ ; ω) ≤ RotL1 (τ ; ω) − ε,
which is in contradiction with (6.4).
Hence, the first claim (i) in Lemma 4.1 is proved. The proof of assertion
(ii) is analogous to that of assertion (i). 

Acknowledgements
The authors are grateful to an anonimous referee for a careful reading of a first
version of this paper. This work is supported by the National Natural Science
Foundation of China (No. 12071327), Spanish ERDF project MTM2017-82348-
C2-1-P and the Natural Science Foundation of the Jiangsu Higher Education
Institutions of China (No. 19KJD100004).

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Vol. 76 (2021) Non-resonance and Double Resonance for a Planar System Page 23 of 23 91

Chunlian Liu and Dingbian Qian

School of Mathematical Sciences
Soochow University
Suzhou 215006
People’s Republic of China
e-mail: clliu06@163.com;
dbqian@suda.edu.cn

Pedro J. Torres
Departamento de Matematica Aplicada
Universidad de Granada
18071 Granada
Spain
e-mail: ptorres@ugr.es

Chunlian Liu
College of Xinglin
Nantong University
Nantong 226008
People’s Republic of China

Received: June 3, 2020.

Accepted: March 25, 2021.

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