Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 39, pp. 1–13.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.39

CARATHÉODORY PERIODIC PERTURBATIONS OF

DEGENERATE SYSTEMS

ALESSANDRO CALAMAI, MARCO SPADINI

Abstract. We study the structure of the set of harmonic solutions to T -
periodically perturbed coupled differential equations on differentiable mani-

folds, where the perturbation is allowed to be of Carathéodory-type regularity.

Employing degree-theoretic methods, we prove the existence of a noncompact
connected set of nontrivial T -periodic solutions that, in a sense, emanates from

the set of zeros of the unperturbed vector field. The latter is assumed to be

“degenerate”: Meaning that, contrary to the usual assumptions on the leading
vector field, it is not required to be either trivial nor to have a compact set

of zeros. In fact, known results in the “nondegenerate” case can be recovered

from our ones. We also provide some illustrating examples of Liénard- and
ϕ-Laplacian-type perturbed equations.

1. Introduction and preliminaries

In this article we study the set of harmonic solutions of Carathéodory-type pe-
riodic perturbations of autonomous systems on a smooth constraining manifold
M ⊆ Rd. Namely, equations of the form

ẋ = G(x) + λF (t, x), λ ≥ 0,

where G : M → Rd and F : R×M → Rd are tangent vector fields to M, meaning
that G(p) ∈ TpM and F (t, p) ∈ TpM for all (t, p) ∈ R × M, and the perturbing
term F is T -periodic in t for some given T > 0. Here TpM denotes the tangent
space to M at p. Assuming that G is continuous and F is Carathéodory, we aim
to study the structure of the set of the pairs (λ, x), where x : R → M is an
absolutely continuous T -periodic function such that ẋ(t) = G(x(t)) + λF (t, x(t))
for a.e. t ∈ R. For this purpose we follow a topological approach, based on the
concept of topological degree of a tangent vector field. This approach was initially
pursued by Furi and Pera (see, e.g., [8, 9, 10, 11]) and afterwards applied to other
situations. We mention in particular the papers [19, 20] that we wish here, in a
sense, to generalize in a unified manner.

In fact, an interesting – and difficult to study – situation presents itself when the
autonomous field G is “degenerate” in the sense that its set of zeros is a noncompact
submanifold of the constraint. Indeed, the “boundary” cases, that is when G−1(0)

2020 Mathematics Subject Classification. 34C25, 34C40, 34C23, 47H11.
Key words and phrases. Coupled differential equations on manifolds; topological degree;

branches of periodic solutions; Carathéodory vector field.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted January 12, 2024. Published July 9, 2024.

1



2 A. CALAMAI, M. SPADINI EJDE-2024/39

is compact and when G = 0, are well understood see, e.g., [11, 20]. The present
research is an attempt at filling the “gap” between the results of these two papers.
We concentrate on the family of systems where the constraining manifold is of the
form M = M × N , the cartesian product of two smooth boundaryless manifolds
M ⊆ Rk and N ⊆ Rs, and G : M ×N → Rk ×Rs is of the form (0, g), i.e., the first
component is identically zero and g : M × N → Rs is such that g−1(0) compact.
To do so we follow the technique introduced in [20], compare also [2] and [15].

More precisely, in this article, we study the following system of coupled equations,
depending on the parameter λ ≥ 0, on the product manifoldM×N , whereM ⊆ Rk

and N ⊆ Rs are (smooth, boundaryless) differentiable manifolds:

ẋ = λf
(
t, x, y, λ

)
,

ẏ = g(x, y) + λh
(
t, x, y, λ

)
.

(1.1)

We set our investigation under Carathéodory-type assumptions. Note that a
similar problem in the continuous setting has been considered in [2]: however, unlike
(1.1), in that paper the first equation is coupled with a periodic perturbation of a
particular nonautomous differential equation.

It is worth mentioning that our results cannot be deduced from analogous ones
in [20], in which the continuity of the involved functions is required. The milder
conditions needed here in the Carathéodory setting may allow to study general
ϕ-Laplacian-like equations: cp. Example 3.6. The study of periodic solutions for
equations involving the ϕ-Laplacian via topological methods has been pursued re-
cently by many authors. We cite, for instance, [1, 3, 4, 6, 7, 18] where the problem
is set under Carathéodory assumptions. We will follow this line of investigation in
a forthcoming paper.

We will make the following assumptions on the maps f, g, h that appear in (1.1).
Hereafter by TpM ⊆ Rk we mean the tangent space of M at a point p of M ,
respectively by TqN ⊆ Rs we denote the tangent space of N at q ∈ N . By L1

T (R)
we denote the space of L1

loc(R), T -periodic maps γ : R → R.
• The map f : R ×M ×N × [0,∞) → Rk is a Carathéodory, T -periodic vector

field tangent to M , meaning that:

(F1) f(t+ T, p, q, λ) = f(t, p, q, λ) ∈ TpM , for all (p, q, λ) ∈M ×N × [0,∞) and
for a.e. t ∈ R;

(F2) t 7→ f(t, p, q, λ) is measurable, for all (p, q, λ) ∈M ×N × [0,∞);
(F3) (p, q, λ) 7→ f(t, p, q, λ) is continuous, for a.e. t ∈ R;
(F4) for any compact setK ⊆M×N×[0,∞), there exists a function ϕK ∈ L1

T (R)
such that |f(t, p, q, λ)| ≤ ϕK(t), for all (p, q, λ) ∈ K and for a.e. t ∈ R.

• Analogously the map h : R×M×N×[0,∞) → Rs is a Carathéodory, T -periodic
vector field tangent to N , namely:

(H1) h(t+ T, p, q, λ) = h(t, p, q, λ) ∈ TqN , for all (p, q, λ) ∈M ×N × [0,∞) and
for a.e. t ∈ R;

(H2) t 7→ h(t, p, q, λ) is measurable, for all (p, q, λ) ∈M ×N × [0,∞);
(H3) (p, q, λ) 7→ h(t, p, q, λ) is continuous, for a.e. t ∈ R;
(H4) for any compact set C ⊆M×N×[0,∞), there exists a function ψC ∈ L1

T (R)
such that |h(t, p, q, λ)| ≤ ψC(t), for all (p, q, λ) ∈ C and for a.e. t ∈ R.

• The map g : M ×N → Rs is a continuous, autonomous vector field tangent to
N ; that is, g(p, q) ∈ TqN , for all (p, q) ∈M ×N .



EJDE-2024/39 CARATHÉODORY PERIODIC PERTURBATIONS 3

By a solution of system (1.1) we mean a function pair (x, y) ∈ W 1,1
loc (M × N)

such that the equalities

ẋ(t) = λf
(
t, x(t), y(t), λ

)
,

ẏ(t) = g(x(t), y(t)) + λh
(
t, x(t), y(t), λ

)
hold for a.e. t ∈ R.

Since any solution of system (1.1) is (absolutely) continuous, as crucially pointed
out in [11, 19], it is convenient to investigate the properties of the T -periodic solu-
tions of (1.1) in the metric space of the continuous functions.

Some further notation is in order. We denote by CT (M × N) the set of the
M ×N -valued, T -periodic, continuous functions with the topology induced by the
Banach space CT (Rk+s). We will say that (λ, x, y) ∈ [0,∞) × CT (M × N) is a
T -triple for (1.1) if equalities (1.1) hold identically. A T -triple (λ, x, y) is called
trivial if (x, y) is constant and λ = 0. Given (p, q) ∈M ×N , by p and q we denote
the functions constantly equal to p and q, respectively. Thus, a T -triple is trivial if
and only if it is of the form (0, p, q) with (p, q) ∈ g−1(0).

Let w :M ×N → Rk be the mean value vector field defined by

w(p, q) =
1

T

∫ T

0

f(t, p, q, 0) dt

and observe that this is a continuous, autonomous vector field tangent toM ; mean-
ing that w(p, q) ∈ TpM , ∀(p, q) ∈M ×N

Let now ν :M ×N → Rk+s be defined as

ν(p, q) = (w(p, q), g(p, q)) , (1.2)

note that, being w and g vector fields tangent, respectively, to M and N , by
definition ν is a vector field tangent to the product manifoldM×N ⊆ Rk+s. Let Ω
be an open subset of [0,∞)×CT (M ×N). The main result of this paper, Theorem
3.1, establishes a topological condition in terms of the degree of ν in Ω for the
existence of a connected set of nontrivial T -triples that in a sense “emanates” from
the set of zeros of ν in Ω and is not contained in any compact subset of Ω.

Before providing a precise statement and proof we need to recall some basic no-
tions, see Section 2. Section 3 contains our main result, and we close the paper with
some illustrating examples, showing the possible shape of the connected “branches”
of T -triples in some concrete situation.

2. Topological degree of a tangent vector field

The topological degree of a tangent vector field (sometimes called rotation or
characteristic) plays a crucial role in this paper. Although this is a very well
known concept, we summarize here the definitions and properties that are most
relevant for our argument.

Assume that M ⊆ Rd be a smooth boundaryless manifold. Take a tangent
vector field v : M → Rd and let p ∈ v−1(0) so, by definition, v(p) ∈ TpM ⊆ Rd.

When v is C1 it is known, see e.g. [17], that also the image of the Fréchet
derivative v′(p) : TpM → Rd of v at p is contained into TpM. That is, when p
is a zero of v, v′(p) is an endomorphism of TpM. In particular, the determinant
det v′(p) is well defined. When p ∈ v−1(0) and det v′(p) ̸= 0 we say that p is a
nondegenerate zero of v. In this case we define its index i(v, p) as sign

(
det v′(p)

)
.



4 A. CALAMAI, M. SPADINI EJDE-2024/39

Let us now briefly look at the construction of the degree. Let v : M → Rd be
a continuous tangent vector field on M, and V ⊆ M be an open subset such that
v−1(0) ∩ V is compact. In this case we say that v is admissible (for the degree) in
V or that the pair (v, V ) is admissible. We associate to the admissible pair (v, V )
the integer deg(v, V ) which, so to speak, counts algebraically the number of zeros
of v in V (see e.g. [14, 17] and references therein). To give a precise meaning to the
last sentence consider the case when v is smooth and v−1(0)∩V consists of a finite
number of nondegenerate zeros, in this situation we define deg(v, V ) as the sum of
the indices of these zeros. That is

deg(v, V ) =
∑

p∈v−1(0)∩V

i(v, p) =
∑

p∈v−1(0)∩V

sign
(
det v′(p)

)
. (2.1)

In the general case when v is admissible in V the degree is defined by taking
a sufficiently close smooth approximation of v with finitely many nondegenerate
zeros in V (see e.g. [12]).

The degree of a tangent vector field enjoys many of the properties of the clas-
sical Brouwer degree such as solution, excision, additivity, homotopy invariance,
normalization etc. Indeed, When M = Rd, W is a bounded open neighborhood
of w−1(0) ∩ V whose closure is contained in V , deg(v,W ) is equal to the Brouwer
degree (see, e.g. the classical [14, 16, 17] or the more recent [5]) deg(v,W, 0), of v
at 0. We do not list these properties as they are easily found in any of the above
references. We only mention that the properties of normalization, additivity and
homotopy invariance can be used as axioms to uniquely determine to the notion of
degree of a tangent vector field (see [13]).

Remark 2.1. By the Poincaré-Hopf Theorem, when M is a compact manifold,
deg(v,M) coincides with the Euler-Poincaré characteristic of M, so it is indepen-
dent of v. Observe, in particular, that when M = {p} is a singleton, one has

deg(0,M) = 1 (2.2)

where 0 denotes the zero vector field.

Remark 2.2. Let U1 ⊆M and U2 ⊆ N be open and let U = U1×U2. Consider the
tangent vector field on M ×N given by v(p, q) =

(
v1(p), v2(q)

)
for (p, q) ∈M ×N ,

where v1 : M → Rk and v2 : N → Rs are tangent vector fields. As a consequence
of (2.1) and the construction of degree outlined above, one can prove that when
(v1, U1) and (v2, U2) are admissible for the degree of tangent vector fields, then so
is (v, U) and we have

deg(v, U) = deg(v1, U1) deg(v
2, U2). (2.3)

3. Main result

In the sequel, given an open subset Ω of [0,∞)× CT (M ×N), we let

ΩM×N =
{
(p, q) ∈M ×N : (0, p, q) ∈ Ω

}
.

We are now in a position to state and prove our main result.

Theorem 3.1. Let f , g and h be as in system (1.1), let ν be as in (1.2), and let
Ω be an open subset of [0,∞)× CT (M ×N). Assume that deg

(
ν,ΩM×N

)
is well-

defined and nonzero. Then there exists a connected set Γ of nontrivial T -triples in



EJDE-2024/39 CARATHÉODORY PERIODIC PERTURBATIONS 5

Ω of (1.1) whose closure in [0,∞)× CT (M ×N) intersects{
(0, p, q) ∈ [0,∞)× CT (M ×N) : (p, q) ∈ ν−1(0) ∩ ΩM×N

}
and is not contained in any compact subset of Ω. In particular, if M ×N is closed
in Rk+s and Ω = [0,∞)× CT (M ×N), then Γ is unbounded.

Remark 3.2. By formulas (2.2) and (2.3), by taking M = {p} and N = {q} and
comparing the notions of T -triple with that of T -pairs in [19] and [11], respectively,
one can use Theorem 3.1 to retrieve the main results of these papers. Namely,
theorem [19, Th. 3.1] and [11, Th. 2.2] that separately ensure the existence of a
connected set of nontrivial T -triples as in Theorem 3.1 for equations of the form

ẋ = λf(t, x), or ẏ = g(y) + λh(t, y),

respectively: That is, for equations equivalent to system (1.1) when N , resp.M , is a
singleton. Such results, although similar, are not directly comparable. In particular,
the former is not a consequence of the latter since the vector field (p, q) 7→

(
0, g(q)

)
is not admissible for the degree, unless M is compact.

The proof of Theorem 3.1 is rather elaborate and requires some preliminary
steps.

We introduce some further notation. A triple (λ, p, q) ∈ [0,∞)×M ×N is called
a starting point (of T -periodic solutions) of (1.1) if the Cauchy problem

ẋ = λf
(
t, x, y, λ

)
,

ẏ = g(x, y) + λh
(
t, x, y, λ

)
,

x(0) = p,

y(0) = q,

(3.1)

has a T -periodic solution. A starting point is trivial when λ = 0 and g(p, q) = 0.
Roughly speaking, the concept of starting point is the finite-dimensional counter-
part of that of T -triple. The set of all the starting points of (1.1) will be denoted
by S. The investigation of this set is a crucial step towards the proof of our main
result. To this end, it is convenient to place ourselves under uniqueness conditions.
Thus, we assume provisionally that g is C1 and that the following Lipschitz-like
conditions on the maps f and h hold in addition to the (F1)–(F4) and (H1)–(H4):

(F5) For any compact set K ⊆ M ×N × [0,∞), there exists γK ∈ L1
T (R) such

that

|f(t, p1, q1, λ1)− f(t, p2, q2, λ2)| ≤ γK(t) (|p1 − p2|+ |q1 − q2|+ |λ1 − λ2|) ,

for all (p1, q1, λ1), (p2, q2, λ2) ∈ K and for a.e. t ∈ R.
(H5) For any compact set C ⊆ M × N × [0,∞), there exists ηC ∈ L1

T (R) such
that

|h(t, p1, q1, λ1)− h(t, p2, q2, λ2)| ≤ ηC(t) (|p1 − p2|+ |q1 − q2|+ |λ1 − λ2|) ,

for all (p1, q1, λ1), (p2, q2, λ2) ∈ C and for a.e. t ∈ R.
We point out that properties (F5) and (H5) are, in a sense, generic. This is

crucial to develop our argument in a Carathéodory setting. We explicitly state this
observation about f ; an analogous remark holds for h as well.



6 A. CALAMAI, M. SPADINI EJDE-2024/39

Remark 3.3. Assume that f satisfies (F1)–(F4). Then, there exists a sequence
{fn} of equi-Carathéodory, T -periodic vector fields tangent to M that can be as-
sumed to satisfy (F5) such that if pn → p0 then, for all (q, λ) ∈ N × [0,∞) and for
a.e. t ∈ R,

fn(t, pn, q, λ) → f(t, p0, q, λ).

Namely (cf. [11, 19]):

fn(t, p, q, λ) = πp

(∫
M

φn(p, u)f(t, u, q, λ)du
)
,

where πp : Rk → TpM is the orthogonal projection and φn : M ×M → R is a
smooth function such that:

(1) φn(p, u) ≥ 0 for all (p, u) ∈M ×M ;
(2) φn(p, u) = 0 whenever |p− u| > 1/n;
(3)

∫
M
φn(p, u)du = 1 for any p ∈M .

Note that, when g is of class C1 and under the assumptions (F1)–(F5) and
(H1)–(H5), by continuous dependence, the set

D = {(λ, p, q) ∈ [0,∞)×M ×N : the solution of (3.1) is defined on [0, T ]}

is open. Note that the set S of all the starting points of (1.1) is a closed subset of
D, even if it could be not so in [0,∞)×M ×N ; therefore, S is locally compact.

The next intermediate result, roughly speaking, requires more regularity of the
involved vector fields compared to our main Theorem 3.1. In particular, we assume
that g is C1, and that f and h satisfy (F5) and (H5), respectively. Such extra
assumptions will be removed in the main theorem via an approximation procedure.
The proof can be carried out following closely [20, Theorem 4.5], and therefore we
omit it.

Theorem 3.4. Let f , g, h be as in (1.1) and let ν be as in (1.2). Assume that
g is of class C1 and that conditions (F1)–(F5) and (H1)–(H5) hold. Let U be an
open subset of D such that ν−1(0)∩U0 is compact. If deg

(
ν, U0

)
̸= 0, then the set

of the nontrivial starting points in U admits a connected subset whose closure in U
meets {0} × (ν−1(0) ∩ U0) and is not compact.

The next lemma is a Whyburn-type topological result which is crucial in the
proof of Theorem 3.1.

Lemma 3.5 ([10]). Let Y0 be a compact subset of a locally compact metric space Y .
Assume that every compact subset of Y containing Y0 has nonempty boundary.
Then Y \Y0 contains a connected set whose closure in Y is non-compact and inter-
sects Y0.

We are now in the position to prove our main result.

Proof of Theorem 3.1. Let X be the set of T -triples of (1.1). One can prove that
X is closed in [0,∞)× CT (M ×N).

Let Ω be as in the statement. Let us prove that Ω contains a connected set Γ of
nontrivial T -triples, whose closure in X ∩ Ω intersects{

(0, p, q) ∈ [0,∞)× CT (M ×N) : (p, q) ∈ ν−1(0) ∩ ΩM×N

}
and is not compact.



EJDE-2024/39 CARATHÉODORY PERIODIC PERTURBATIONS 7

Assume first that g is of class C1 and that conditions (F1)–(F5) and (H1)–(H5)

hold. Let us denote by S the set of all the starting points of (1.1), and by Ŝ the set
of the starting points (λ, p, q) such that the corresponding T -triple (λ, x, y), where

(x, y) is a solution of (3.1), is contained in Ω. Note that Ŝ is an open subset of S;
thus, we can find an open subset U of D such that S ∩U = Ŝ. By construction, we
have

ν−1(0) ∩ ΩM×N = ν−1(0) ∩ Ŝ0 = ν−1(0) ∩ U0;

and thus, by definition of degree, deg
(
ν, U0

)
= deg

(
ν,ΩM×N

)
̸= 0. Therefore,

Theorem 3.4 applies, yielding the existence of a connected set Σ ⊆ U , made up of
nontrivial starting points of (1.1), whose closure in U is not compact and meets
{0} × (ν−1(0) ∩ U0).

Let now
h : X → S

be the map that assigns to any T -triple (λ, x, y) the starting point (λ, x(0), y(0)).
Observe that h is continuous, onto and, by the assumptions on f , g, h, it is also
one to one. Moreover, by the continuous dependence on initial data, we get the
continuity of the inverse h−1 : S → X. Therefore h maps X ∩Ω homeomorphically
onto S ∩U , and the trivial T -triples correspond to the trivial starting points under
this homeomorphism. This implies that Υ := h−1(Σ) satisfies the requirements.

Let us now remove the additional assumptions on f , g and h. Let Y = X ∩ Ω
and

Y0 =
{
(0, p, q) ∈ [0,∞)× CT (M ×N) : (p, q) ∈ ν−1(0) ∩ ΩM×N

}
.

Then, to prove our result, it is sufficient to apply Lemma 3.5.
Assume, by contradiction, that the pair (Y, Y0) does not satisfy the hypothesis

of Lemma 3.5. That is, there exists a compact subset K of Y containing Y0 whose
boundary, in Y , is empty. Then, there exists an open subset W of Ω, with closure
W contained in Ω and such that W ∩ Y = K, ∂W ∩ Y = ∅. Observe that being
K compact and [0,∞)×M ×N locally compact, we can choose W in such a way
that the set

{(λ, x(t), y(t)) ∈ [0,∞)×M ×N : (λ, x, y) ∈W, t ∈ [0, T ]}

is contained in a compact subset K̃ of [0,∞) ×M × N . This implies that W is
bounded with complete closure in Ω, and WM×N is a relatively compact subset of
ΩM×N . Hence, in particular, ν is nonzero on the boundary of WM×N (relative to
M ×N), and the same is true for its components, w and g.

Let us now approximate, respectively, g by a a sequence {gn} of smooth (au-
tonomous) vector fields tangent to N , uniformly converging to g on compact subsets
of M ×N , and f by a sequence {fn} of equi-Carathéodory, T -periodic vector fields
tangent to M , as in Remark 3.3. For each n ∈ N, let

wn(p, q) =
1

T

∫ T

0

fn(t, p, q, 0) dt

be the mean value vector field, tangent toM , associated to fn; by construction, the
sequence {wn(p, q)} converges uniformly to w(p, q) on compact subsets of M ×N .

Now, let
νn(p, q) = (wn(p, q), gn(p, q)).

Note that, since the zeros of ν in ΩM×N lie in a compact subset of WM×N , for
n large enough, the homotopy H(s, p, q) = sνn(p, q) + (1 − s)ν(p, q), s ∈ [0, 1], is



8 A. CALAMAI, M. SPADINI EJDE-2024/39

admissible for the degree in WM×N . Thus, deg
(
νn,WM×N

)
is well-defined and, by

the homotopy invariance property of the degree, it coincides with deg
(
ν,WM×N

)
.

Hence, by excision,

deg
(
νn,ΩM×N

)
= deg

(
νn,WM×N

)
= deg

(
ν,WM×N

)
= deg

(
ν,ΩM×N

)
̸= 0.

Therefore, for n sufficiently large, the first part of the proof can be applied to system

ẋ = λfn
(
t, x, y, λ

)
,

ẏ = gn(x, y) + λhn
(
t, x, y, λ

) (3.2)

where, again, {hn} a sequence of equi-Carathéodory, T -periodic vector fields tan-
gent to N , as in Remark 3.3.

Let Xn denote the set of T -triples of (3.2). By the above argument, there exists
a connected set Γn of nontrivial T -triples whose closure in Ω is noncompact and
meets {

(0, p, q) ∈ [0,∞)× CT (M ×N) : (p, q) ∈ ν−1
n (0) ∩ ΩM×N

}
.

Since W is bounded with complete closure, any Γn must intersect the complement
ofW in Ω, which implies the existence of a triple (λn, xn, yn) ∈ ∂W∩Γn. Therefore,

since for any n ∈ N and t ∈ R we have (λn, xn(t), yn(t)) ∈ K̃, the compactness of
K implies the existence of a pair of functions γ, η in L1

T (R) such that

|ẋn(t)| = |λnfn
(
t, xn(t), yn(t), λn

)
| ≤ γ(t),

|ẏn(t)| = |gn(xn(t), yn(t)) + λnhn
(
t, xn(t), yn(t), λn

)
| ≤ η(t)

for all n ∈ N and a.a. t ∈ R. Consequently, the sequences {xn} and {yn} are
equicontinuous, so that, by Ascoli’s theorem we may assume that (xn, yn) → (x0, y0)
in CT (M ×N) and, without loss of generality, λn → λ0, so that (λ0, x0, y0) ∈ ∂W .
Moreover, by the assumptions on the sequences {fn}, {gn}, {hn} we have, for a.a.
t ∈ [0, T ],

gn(xn(t), yn(t)) → g(x0(t), y0(t)),

fn
(
t, xn(t), yn(t), λn

)
→ f

(
t, x0(t), y0(t), λ0

)
,

hn
(
t, xn(t), yn(t), λn

)
→ h

(
t, x0(t), y0(t), λ0

)
.

Therefore, by the dominated convergence theorem, for a.a. t ∈ [0, T ],

ẋ0(t) = x0(0) + λ0

∫ t

0

f
(
s, x0(s), y0(s), λ0

)
ds,

ẏ0(t) = y0(0) +

∫ t

0

[
g(x0(s), y0(s)) + λ0h

(
s, x0(s), y0(s), λ0

)]
ds

In other words, (x0, y0) is a T -periodic solution of the system

ẋ = λ0f
(
t, x, y, λ0

)
,

ẏ = g(x, y) + λ0h
(
t, x, y, λ0

)
.

Thus, (λ0, x0, y0) is a T -triple of (1.1).
Now, if λ0 > 0, then (λ0, x0, y0) ∈ Y . Otherwise, if λ0 = 0, an argument

similar to the one used in proving the necessary condition for bifurcation (see [11,
Theorem 2.1]) shows that (0, p, q) ∈ Y0. Therefore, in any case, (λ0, x0, y0) ∈
∂W ∩ Y , which is a contradiction. Consequently, a straightforward application of
Lemma 3.5 to the pair (Y, Y0) implies the first part of our assertion.



EJDE-2024/39 CARATHÉODORY PERIODIC PERTURBATIONS 9

To complete the proof, assume M × N closed in Rk+s and take Ω = [0,∞) ×
CT (M × N). Then, there exists a connected set Γ of nontrivial T -triples of (1.1)
whose closure is not compact and meets {0} × ν−1(0). We need to prove that Γ is
unbounded. Assume the contrary.

Note that, as a consequence of the Ascoli–Arzelà Theorem, when M ×N closed
in Rk+s any bounded closed set of T -triples is compact. Thus, in this case, the
closure of Γ in [0,∞) × CT (M × N) is compact. This is a contradiction, and the
assertion follows. □

We close this article with two illustrating examples. In the first one we deal with
a Liénard-type equation while in the second we study a ϕ-laplacian like equation.

-0.1
-0.05

 0
 0.05

 0.1
 0.15

 0.2
 0.25

 0.3 -0.3

-0.2

-0.1

 0

 0.1

 0.2

 0.3

 0

 0.2

 0.4

 0.6

 0.8

 1

λ

x(0)

y(0)
-0.1

-0.05
 0

 0.05
 0.1

 0.15
 0.2

 0.25
 0.3 -0.3

-0.2

-0.1

 0

 0.1

 0.2

 0.3

 0

 0.2

 0.4

 0.6

 0.8

 1

λ

x(0)

y(0)

(a) θ(t, λ) = 2π cos(2πt) (b) θ(t, λ) = 2π sin(2πt)

Figure 1. Starting points of (3.4) with ϕ = 1, ψ(t, y, λ) =(
1
2 + sin(2πt)

)
y, a = b = 0, for two different choices of θ. One

easily checks that those contained in the xy-plane are trivial and
so correspond to trivial T -triples

Example 3.6. Consider the Liénard-type perturbed differential equation

ÿ + ϕ(y)ẏ + λ
(
ψ(t, y, λ) + θ(t, λ)

)
= 0, λ ≥ 0, (3.3)

where ϕ : R → R is continuous and ψ : R× R× [0,∞) → R and θ : R× [0,∞) → R
satisfy Carathéodory conditions and are T -periodic in t, for T > 0 given. We also
assume that θ has zero average on [0, T ] for all λ. We can rewrite (3.3) in the
Liénard plane as follows:

ẋ = −λψ(t, y, λ),
ẏ = x− Φ(y)− λΘ(t, λ),

(3.4)

where Φ and Θ can be taken, respectively, as Φ(y) =
∫ y

a
ϕ(s) ds and Θ(t, λ) =∫ t

b
θ(τ, λ) dτ with a, b ∈ R arbitrary constants. Notice that, since θ has zero average,

Θ is T -periodic. Let Ω = [0,∞)× CT (R× R) and

ν(p, q) =
( 1

T

∫ T

0

ψ(t, q, 0) dt , p− Φ(q)
)
.

According to our notation, we have ΩR×R = R × R. When deg(ν,R2) ̸= 0, Theo-
rem 3.1 yields an unbounded connected set Γ of nontrivial T -triples whose closure
intersects the set

T := {(0, p, q) ∈ [0,∞)× CT (R× R) : ν(p, q) = 0}.



10 A. CALAMAI, M. SPADINI EJDE-2024/39

It is not difficult to prove that the choice of the constants a and b do not affect the
degree of ν. In fact, b has no influence on ν at all, whereas changing the value of a
only induces a translation of the set of zeros along the p-axis because the zeros of

the map q 7→
∫ T

0
ψ(t, q, 0) dt remain unchanged.

Let ϖ : [0,∞)×CT (R×R) → [0,∞)×CT (R) the projection given by ϖ(λ, x, y) =
(λ, y) and let Υ = ϖ(Γ). Clearly Υ is a connected set, consisting of pairs (λ, y) ∈
[0,∞) × CT (R) with y a solution of (3.3). Notice also that if (0, y) ∈ Υ then y is
not constant. To check the latter assertion we proceed by contradiction: Assume
y constant and let (0, x, y) ∈ Γ be any T -triple such that ϖ(0, x, y) = (0, y) and
observe that when λ = 0, (3.4) implies that x is constant as well. Thus (0, x, y) is
a trivial T -triple, a contradiction.

Let {(λn, xn, yn)}n∈N ⊆ Γ be a sequence converging to (0, p, q) with (p, q) ∈
ν−1(0). (such a sequence exists because the closure Γ of Γ intersects T). Thus q

satisfies
∫ T

0
ψ(t, q, 0) dt = 0. The sequence {(λn, yn)}n∈N ⊆ Υ converges to (0, q)

which, hence, is contained in the closure Υ of Υ in [0,∞)×CT (R). In other words,

Υ intersects the set of pairs (0, q) such that
∫ T

0
ψ(t, q, 0) dt = 0.

In conclusion, when deg(ν,R2) ̸= 0, there exists an unbounded connected set Υ
of pairs (λ, y), with y a T -periodic solution of (3.3) that is not constant for λ = 0,
whose closure in [0,∞)× CT (R) intersects the set{

(0, q) ∈ [0,∞)× CT (R) :
∫ T

0
ψ(t, q, 0) dt = 0

}
.

It is worth mentioning that the perturbing term θ, independent of y, does not
enter in the definition of ν. As a consequence, the mere existence of the set Υ does
not depend on the choice of θ or, to put it differently, it is impossible to destroy Υ
by selecting a suitable θ. Taking different functions θ, though, may actually change
Υ. To illustrate this fact, consider figure 1 where it is represented a portion of the
set of starting points of (3.4) with ϕ = 1, ψ(t, y, λ) =

(
1
2 + sin(2πt)

)
y, a = b = 0,

and two different choices of θ.

-0.2
-0.15

-0.1
-0.05

 0
 0.05

 0.1
 0.15

 0.2 -0.2

-0.15

-0.1

-0.05

 0

 0.05

 0.1

 0.15

 0.2

 0

 0.2

 0.4

 0.6

 0.8

 1

λ

u(0)

v(0)
-0.2

-0.15
-0.1

-0.05
 0

 0.05
 0.1

 0.15
 0.2 -0.2

-0.15

-0.1

-0.05

 0

 0.05

 0.1

 0.15

 0.2

 0

 0.2

 0.4

 0.6

 0.8

 1

λ

u(0)

v(0)

(a) h(t, λ) = cos(2πt) + λ sin(2πt) (b) h(t, λ) = cos(2πt)− λ sin(2πt)

Figure 2. Starting points of (3.6) with ϕ(s) = s|s| + 2s,
f(t, s, r, λ) = sin(2πt) + s, for two different choices of h. One
easily checks that those contained in the xy-plane are trivial and
so correspond to trivial T -triples.

Example 3.7. Let ϕ : R → R be C1 with continuous inverse, and let f : R × R ×
R× [0,∞) → R be a map that satisfy Carathéodory assumptions and is T -periodic



EJDE-2024/39 CARATHÉODORY PERIODIC PERTURBATIONS 11

-0.2
-0.15

-0.1
-0.05

 0
 0.05

 0.1
 0.15

 0.2 -0.2

-0.15

-0.1

-0.05

 0

 0.05

 0.1

 0.15

 0.2

 0

 0.2

 0.4

 0.6

 0.8

 1

λ

u(0)

v(0)
-0.2

-0.15
-0.1

-0.05
 0

 0.05
 0.1

 0.15
 0.2 -0.2

-0.15

-0.1

-0.05

 0

 0.05

 0.1

 0.15

 0.2

 0

 0.2

 0.4

 0.6

 0.8

 1

λ

u(0)

v(0)

(a) h(t, λ) = cos(2πt) + λ sin(2πt) (b) h(t, λ) = cos(2πt)− λ sin(2πt)

Figure 3. Starting points of (3.6) with ϕ(s) = −20s,
f(t, s, r, λ) = sin(2πt) + s, for two different choices of h. Again,
one easily checks that those contained in the xy-plane are trivial
and so correspond to trivial T -triples.

in t. Consider the ϕ-Laplacian-like equation

d

dt
[ϕ
(
v̇(t) + λh(t, λ)

)
] = λf

(
t, v(t), v̇(t), λ

)
, λ ≥ 0, (3.5)

where h is a T -periodic C1 function. Setting u(t) = ϕ
(
v̇(t) + λh(t, λ)

)
or, equiva-

lently, v̇(t) = ϕ−1
(
u(t)

)
− λh(t, λ), we can rewrite (3.5) as the system

u̇(t) = λf
(
t, v(t), ϕ−1

(
u(t)

)
− λh(t, λ), λ

)
,

v̇(t) = ϕ−1
(
u(t)

)
− λh(t, λ).

(3.6)

Let Ω = [0,∞)× CT (R× R) so that ΩR×R = R× R, and

ν(p, q) =
( 1

T

∫ T

0

f
(
t, q, ϕ−1(p), 0) dt , ϕ−1(p)

)
.

By Theorem 3.1, when deg(ν,R2) ̸= 0, there exists an unbounded connected set Γ
of nontrivial T -triples whose closure intersects the set

T := {(0, p, q) ∈ [0,∞)× CT (R× R) : ν(p, q) = 0}.
Assuming that deg(ν,R × R) ̸= 0, similar considerations to example 3.6 show the
existence of an unbounded connected set Υ ⊆ [0,∞)×CT (R) of pairs (λ, v), where
v is a T -periodic solution of (3.5) that is not constant when λ = 0, such that the
closure Υ of Υ in [0,∞)× CT (R) intersects the set{

(0, q) ∈ [0,∞)× CT (R) :
∫ T

0
f(t, q, 0, 0) dt = 0

}
.

As in Example 3.6, observe that the set Υ depends on the perturbing term h
although no choice of h can make Υ disappear. Figure 2 represents a portion of
the set of starting points of (3.5) with ϕ(s) = s|s|+ 2s, f(t, s, r, λ) = sin(2πt) + s,
and two different choices of h.

Notice in passing that, ϕ being an isomorphism, by (2.1) and using the indicated
construction of the degree, it is not difficult to prove that∣∣deg(ν,R× R)

∣∣ = |deg(w,R)|, (3.7)

where w denotes the map q 7→ 1
T

∫ T

0
f
(
t, q, 0, 0) dt. A complete proof of this formula

however, would take us too far from the scope of the paper, so we omit it.



12 A. CALAMAI, M. SPADINI EJDE-2024/39

Combining formula (3.7) with the above arguments, one one sees that what is
really necessary for the existence of Υ is the condition deg(w,R) ̸= 0 instead of
deg(ν,R × R) ̸= 0. This shows that, although Υ may depend on the diffeomor-
phism ϕ, its mere existence is not affected by the actual choice of this map. An
investigation of this phenomenon will be pursued elsewhere; here we only show an
example with ϕ(s) = −20s, see figure 3, illustrating how figure 2 is altered by a
different choice of ϕ.

Acknowledgements. The authors would like to thank the referee for the careful
reading of the manuscript and the constructive comments. The authors are mem-
bers of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro
Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
This study was supported by: Research project of MUR (Italian Ministry of Univer-
sity and Research) PRIN 2022 “Nonlinear differential problems with applications
to real phenomena” (Grant Number: 2022ZXZTN2).

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Alessandro Calamai

Dipartimento di Ingegneria Civile, Edile e Architettura, Università Politecnica delle
Marche, Via Brecce Bianche, I-60131 Ancona, Italy

Email address: a.calamai@univpm.it

Marco Spadini

Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze,

Via S. Marta 3, I-50139 Florence, Italy
Email address: marco.spadini@unifi.it


	1. Introduction and preliminaries
	2. Topological degree of a tangent vector field
	3. Main result
	Acknowledgements

	References