A systematic approach to nonresonance conditions for periodically forced planar Hamiltonian systems

In the first part of the paper we consider periodic perturbations of some planar Hamiltonian systems. In a general setting, we detect conditions ensuring the existence and multiplicity of periodic solutions. In the second part, the same ideas are used to deal with some more general planar differential systems.


Introduction
The meaning of the word resonance is well understood for a linear equation of the type where λ is a positive constant and q(t) is a 2π-periodic forcing: resonance occurs when all the solutions are unbounded, both in the past and in the future. This may happen only when λ = n 2 for some integer n. On the contrary, if ∉ {n 2 ∶ n ∈ ℕ} , then all solutions of the differential equation are bounded, and among them there is a 2π-periodic solution, for any 2π-periodic forcing term q(t).
For a more general nonlinear equation the meaning of resonance does not appear so clearly. However, it seems to be commonly accepted to consider as nonresonance conditions on the function g(x) those ensuring that the differential equation admits at least one 2π-periodic solution, for any 2π-periodic forcing term q(t).
Life becomes still more complicated if we consider equations of the type where q(t, x), which is 2π-periodic in its first variable, is considered as some kind of perturbation of the autonomous equation. There is a huge literature on the existence of periodic solutions for this type of equations (see e.g. [13] and the references therein). In this case, "nonresonance conditions" necessarily involve both the functions g(x) and q(t, x), and they are supposed to guarantee the existence of at least one 2 -periodic solution of the differential equation.
In this paper, we are looking for "nonresonance conditions" for more general planar systems of the type Here, and throughout the paper, J = 0 −1 1 0 is the standard symplectic matrix, the Hamiltonian function H ∶ ℝ 2 → ℝ is continuously differentiable, and r ∶ ℝ × ℝ 2 → ℝ 2 is assumed to be continuous, and 2π-periodic in its first variable. We search for conditions on H(z) and r(t, z) guaranteeing that system (1.3) has at least one 2π-periodic solution.
Since we need uniqueness of solutions, we will typically assume ∇H(z) to be locally Lipschitz continuous, and r(t, z) to have the same regularity property with respect to its second variable. However, we will also have to consider autonomous Hamiltonian systems without necessarily assuming the gradient of their Hamiltonian function to be locally Lipschitz continuous. For the autonomous system associated with (1.3), it will be assumed that all large amplitude solutions are periodic. More precisely, we will assume that, for large energy levels E, the sets H −1 (E) are closed curves corresponding to periodic solutions of (1.4) with some minimal period T(E). In our approach for the study of system (1.3), we will consider it as some kind of perturbation of the autonomous system (1.4). Through a change of variables, we transform (1.3) into a system of differential equations having, as variables, the energy and a phase. By the systematic use of the energy as a parameter, our aim is to obtain sharp nonresonance results and to provide new insights into the nonresonance problem. When the forcing term does not depend on z, i.e., when r(t, z) = r(t) , one expects that the system admits a 2 -periodic solution, unless the period 2 of the forcing term r(t) interferes with the periods of the large amplitude free oscillations, meaning that T(E) approaches 2 ∕n , for some integer n, when E goes to infinity. Consequently, we expect that a nonresonance condition should be of the type including the case where (1.2) x �� + g(x) = q(t, x),  It is the main objective of this paper to present conditions under which the inequality (1.5) guarantees the existence of a 2 -periodic solution for the system (1.3). However, we believe that further restrictions on the Hamiltonian function H are required in order for (1.5) to become a valid nonresonance condition. We remark that, even for the particular case of the scalar equation (1.1), it has been shown in [7] that a nonresonance condition of type (1.5) is sufficient for the existence of a 2 -periodic solution provided that g is differentiable, with a globally bounded derivative. Some other set of restrictions on g can be found in [3].
The paper is organized as follows.
In Sect. 2, we present the general setting, showing how to cover some classical situations, like asymmetric oscillators, positively homogeneous Hamiltonians of degree 2, or Hamiltonians with separated variables, arising, e.g., from differential equations involving the scalar p-Laplacian operator.
Section 3 is devoted to the statement of our first existence theorem; it provides fairly general conditions which, combined with hypothesis (1.5), ensure that system (1.3) admits at least one 2 -periodic solution. Due to their generality, the assumptions of that theorem need to be analyzed in further detail. The proof of that existence theorem is carried out in Sect. 5. In Sect. 4, we develop an approach to the nonresonance condition (1.5) by a comparison between the Hamiltonian function of equation (1.4), and two other Hamiltonians, which would typically be isochronous. Examples are provided by scalar second-order equations and systems where the Hamiltonian function H has separated variables.
In Sect. 6, we extend our existence theorem in several directions. First, we consider the case when the limit of period function T(E) is +∞ as E → +∞ . Then, we obtain an existence result in the critical case when, for some positive integer n 0 , assuming that the approach to resonance is "not too fast". As an example, we can deal with an equation of the type with p > 2 and q(t, x) satisfying a growth condition at infinity. In the last part of the section, we also show how to formulate some Landesman-Lazer type conditions in our setting.
In Sect. 7, assuming that our system has a Hamiltonian structure, we show how the Poincaré-Birkhoff Theorem can be applied to provide multiplicity of periodic solutions when the time map has an oscillatory behaviour, i.e., when (1.6) holds. In order to be brief, we limit our attention to this situation, where only the asymptotic behaviour of the time map is considered. Other situations could also be dealt with, following a similar approach; further developments along these lines are expected in the future.
Finally, in Sect. 8, we consider the more general system and still obtain existence conditions for periodic solutions through a comparison with Hamiltonian systems.
and, differentiating this relation,

About condition A3
Let us describe a standard way to construct the function . Fix a point z * 0 in Γ 1 , and consider a solution w( ) of the Cauchy problem Since integrating on [1, E] we see that In other words, E corresponds to the "energy" at the point w(E), which motivates the notation E. Now, for any E > 1 , let (⋅ ;E) be the solution of the Hamiltonian system (2.1) such that (0; E) = w(E) . It is clear that both equalities in A3 hold true. The regularity of is surely guaranteed if H is twice continuously differentiable, but we will see that it is satisfied also in some more general situations. As will appear below, advantage can be taken of the possibility of carefully choosing z * 0 , in order to get a function (t; E) having some convenient properties.
From the definition of (t; E), we also have that this property of (0; E) will play an important role in the sequel, and we will assume it to hold throughout. As a first example, consider a Hamiltonian function of the type with G(0) = 0. The autonomous system (2.1) is then equivalent to the scalar second-order equation where g(x) = G � (x) . Notice that conditions A1 and A2 will be satisfied assuming

3
In the above example, the regularity assumption in A3 is surely satisfied if g is continuously differentiable. However, it is also satisfied if, e.g., for some positive constants a + , a − (here, as usual, x + = max{x, 0} and x − = max{−x, 0}) . This last example leads us to study more carefully the case of positively homogeneous systems of degree 2.
Let the Hamiltonian function H be such that In this case, it is well known that the autonomous system (2.1) is isochronous, all the nonconstant orbits having the same minimal period T . (For the reader's convenience, this fact will also be proved below, as an easy consequence of Lemma 2. Consequently, since ⟨z, ∇H(z)⟩ = 2H(z) > 0, for z ≠ 0, there exists > 0 such that z(t * ) = ∇H(z(t * )). Using the fact that ∇H( z) = ∇H(z) for any ≥ 0, if we then take z * 0 = z(t * ) in (2.4), we note that a solution of this system is given by w( ) = √ z(t * ), implying that (0; E) = √ E (0;1) , from which (2.10) follows. As a consequence, in this case we have Notice that, as a particular case, we could have H(z) = 1 2 ⟨ z, z⟩ , with a positive definite symmetric matrix .
In the sequel, we will also frequently refer to Hamiltonian functions of the form with a and b positive constants, p > 1 and q > 1 . It must be kept in mind that the gradient of such a function is not necessarily locally Lipschitz continuous. Nevertheless, its properties will prove useful when comparing the minimal period of solutions of various Hamiltonian systems. If (t;1) is a solution of (2.1) of energy 1, we observe that ⟨z(t * ), z � (t * )⟩ = −⟨z(t * ), J∇H(z(t * ))⟩ = 0 .

A basic property
Let us denote by int(Γ 1 ) and ext(Γ 1 ) the bounded and the unbounded connected components of ℝ 2 ⧵ Γ 1 , respectively. For every z 0 ∈ ext(Γ 1 ), let T(z 0 ) be the minimal period of the solution issuing from it. We define the continuous function Notice that B differs from ℝ 2 ⧵ Ω(1) by a set of zero Lebesgue measure. Define the function Φ ∶ A → B as Φ( , e) = ( ;e) . It is one-to-one and onto. Using (2.3), we have for every ( , e) ∈ A , so that Φ is a diffeomorphism. For E > 1 , the area of Ω(E) is then given by . ;

3
and the conclusion directly follows. ◻ As a first example of application we can show that, when H satisfies (2.9), the system (2.1) is isochronous. Indeed, the homogeneity property implies that (1). Consequently, the period is given by T(E) = a � (E) = a(1).

Remark 2.2 Given a continuous function
, and consider the associated Hamiltonian system (2.1), with H(z) = f (|z|) . It is clear that the orbit of energy E > 0 of this system is a circle with area a(E). Consequently, by Lemma 2.1, for every E ≥ 1, the minimal period of the corresponding solution is T(E).

Existence of periodic solutions
In this section, we first state our existence theorem, its proof being postponed to Sect. 5. We then make some remarks on the assumptions of the theorem, and derive some useful corollaries.
We will make use of the following regularity conditions on H and r.

L1.
The function H ∶ ℝ 2 → ℝ is differentiable with a locally Lipschitz continuous gradient. L2. The function r ∶ ℝ × ℝ 2 → ℝ 2 is continuous, 2 -periodic in its first variable, and locally Lipschitz continuous in its second variable.

Statement of the existence result
Here is the main result of this section.

A4. The function T(E) is controlled as follows:
A5. For any integer n, Assume also that: Before going to the proof of Theorem 3.1, we make some comments and draw some consequences.

About conditions A4 and A5
It is to be understood that assumptions A4 and A5 hold if and only if Therefore, assuming A4 and A5 is equivalent to assuming the existence of a sequence (E k ) k such that Notice that, by Lemma 2.1 and the general l'Hôpital rule, so that conditions A4 and A5 will surely be verified if the following two hold.

A5'. For any integer n,
We can thus state the following immediate consequence of Theorem 3.1. The interest of the above corollary lies in the observation that, under hypotheses A1 to A3, A6 and A7, a 2 -periodic solution will exist if, for some integer n 0 , This condition is less stringent than the condition it is also likely to be easier to check, since estimates on lim inf E→+∞ a(E)∕E and lim sup E→+∞ a(E)∕E are deduced from estimates on a(E), which, in turn, can be obtained by comparing the Hamiltonian H to other Hamiltonians, as will be shown below.

About conditions A6 and A7
In order to better understand conditions A6 and A7, let us first consider the particular case when the Hamiltonian function is positively homogeneous of degree 2, i.e., when (2.9) holds. In this case, taking into account (2.10), condition A6 holds if there exists a constant c > 0 such that On the other hand, taking into account (2.11), condition A7 holds if We thus have the following. In order to deal with more general situations, we introduce an assumption which will ensure that A7 is satisfied for a function H which is twice continuously differentiable.
Based on the above considerations, the following corollary is an immediate consequence of Theorem 3.1. A noteworthy situation is the case where H ′′ is globally bounded. Taking as D(E) the identity matrix, we see that (3.4) is plainly satisfied with being a constant function, whereas (3.6) holds if, for any given compact interval I ⊆ ℝ,

Corollary 3.4 Let
for all t ∈ ℝ and v ∈ ℝ 2 , it is clear that A1, A2 hold. Condition A3 also holds since g is continuously differentiable. Moreover, if we take (0; E) = (0, Consequently, we are in a situation where Corollary 3.5 applies, and we may conclude that (3.12) admits at least one 2 -periodic solution, provided that the nonresonance conditions A4 and A5 are satisfied. This result had been obtained in [7], under a more restrictive condition than (3.13), namely, that there exists a positive constant c > 0 such that g(x)∕x ≥ c, for |x| > d.
The above remark can be adapted to the equation provided that Assuming the existence of > 0 such that The following corollary can thus be deduced from Corollary 3.5.

Corollary 3.8 Let the function g(x) be continuously differentiable, with a globally
bounded derivative, and such that (3.13) and (3.16) hold. Assume moreover that q(t, x) is continuous, 2 -periodic in t, locally Lipschitz continuous in x, and satisfies (3.15). Then, if the nonresonance conditions A4 and A5 are satisfied, equation (3.14) admits at least one 2 -periodic solution.

Comparison between Hamiltonians
In this section, we provide some corollaries of the results of the previous sections. The main idea is to compare the Hamiltonian function H with other Hamiltonians for which the assumptions are easier to check.

Comparison with isochronous Hamiltonians
Suppose that, for some number > 0, the three functions H, H 1 , H 2 being continuously differentiable and satisfying the hypotheses A1 to A3 of Sect. 2. We do not require however the gradients of H 1 , H 2 to be locally Lipschitz continuous. For E large enough, we denote by Ω 1 (E) , Ω(E) , Ω 2 (E) the bounded sets delimited by the curves H −1 , and by a 1 (E) , a(E), a 2 (E) their areas, respectively. Then, (3.14)

and hence
So, if for some integer n 0 , one has that it is clear that (3.1) is satisfied. Moreover, if H 1 , H 2 are isochronous Hamiltonians with respective minimal periods T 1 , T 2 , by Lemma 2.1 we will have that a 1 (E) = T 1 E + C 1 and a 2 (E) = T 2 E + C 2 , for some constants C 1 , C 2 , and necessarily T 2 ≤ T 1 . This leads to the following consequence of Theorem 3.1. Notice that, for condition (3.8), advantage can be taken of the freedom of choice in the construction of (0; E).

Scalar second-order equations
We illustrate the above results with an application to the second-order equation (3.14). As before, we assume the function g(x) to be continuously differentiable, and q(t, x) to be continuous, 2 -periodic in t, and locally Lipschitz continuous in x. We denote by G a primitive of g, and associate with this equation the Hamiltonian function defined by (2.6). Assume that G has a quadratic growth; more precisely, suppose that there exist numbers G − , G + , G − , G + such that We fix a small > 0 and define The Hamiltonians H 1 , H 2 being positively homogeneous, conditions A1 to A3 are satisfied, and we see that (4.1) holds for sufficiently large. On the other hand, the minimal periods T 1, , T 2, associated with H 1 , H 2 are such that In order to apply Corollary 3.8, we assume that g � (x) is globally bounded. Hypothesis (3.16) of that corollary is satisfied because there exists a constant > 0 such that G(x) ≥ x 2 for |x| sufficiently large. Using the fact that can be chosen arbitrarily small, we then deduce the following result.

Corollary 4.3 Let the function g(x) be continuously differentiable, with a globally
bounded derivative, and such that (3.13) holds. Denoting by G(x) a primitive of g(x), assume that positive numbers G ± , G ± exist for which (4.3) holds, these numbers being such that, either or, for some integer n 0 ,

Assume moreover that q(t, x) is continuous, 2 -periodic in t, locally Lipschitz continuous in
x, and satisfies (3.15). Then, Eq. (3.14) admits at least one 2 -periodic solution.
The conditions (4.4) can be interpreted in terms of the Fučík spectrum for the 2 -periodic boundary value problem. They amount to requiring that the rectangle [G + , G + ] × [G − , G − ] lies between two successive Fučík curves (or below the first one). This is an improvement with respect to the "classical" conditions of Drábek and Invernizzi [8] for the equation considered here, which is a perturbation of a Hamiltonian equation. Indeed, our hypotheses are based on the limits, for x → ±∞, of the ratio 2 G(x)∕x 2 , rather than on the limits of g(x)/x.
Notice that the assumption (4.4) does not necessarily imply that condition (3.2) holds for the periods T(E). We illustrate this with the equation to which we can associate a Hamiltonian function like the one in (2.6), with We deduce from the above expression that so that (4.4) is satisfied for n 0 = 1. However, numerical computations show that

Hamiltonians with separated variables
We now consider systems with Hamiltonian functions of the form we will suppose that G, K ∶ ℝ → ℝ are twice continuously differentiable, their first derivatives being denoted respectively by g(x), k(y). We assume that, for some numbers p > 1, q > 1 related by (2.13), More precisely, we introduce numbers We will also assume that for some positive constants c 1 , c 2 , The above hypotheses imply that there exist constants L ≥ > 0 and C > 0 such that and it can be checked that the function H satisfies the assumptions A1 to A3. We can also observe that the curves H −1 (E) are star-shaped for E large.
Writing r(t, x, y) = (r x (t, x, y), r y (t, x, y)) , we want to apply Corollary 4.1 to the system assuming r x , r y to be continuous, 2 -periodic in t, and locally Lipschitz continuous in (x, y). For simplicity, we only deal here with symmetric conditions on G, K, i.e. with bounds for the limits independent of the signs of x and y. We therefore take positive numbers 1 , 2 , 1 , 2 such that and consider the Hamiltonian functions H 1 , H 2 defined by When the exponents p, q satisfy condition (2.13), both H 1 and H 2 are isochronous (cf. (2.12), with = 0 ); their respective minimal periods are given by where (see, e.g., [6,24]). It is clear that In order to apply Corollary 4.1, we need to show that T(E) denoting, as before, the minimal period of solutions of (2.1), for E > 1. Actually, using (4.7) and (4.8), it can be shown that T(E) is bounded away from 0, as well as bounded above. More precisely, T 1 , T 2 being the minimal period of the nontrivial solutions of the equations associated with the Hamiltonians where c 1 , c 2 , C ′ 1 , C ′ 2 are the constants appearing in (4.7) and (4.8). We omit the proof, for briefness, the arguments being similar to those of Theorem 8.2 below. Notice that strict inequalities being possible.
Considering that (4.12) holds, we deduce the following result from Corollary 4.1.
functions, their first derivatives being denoted by g(x), k(y), respectively. Assume that the conditions (4.6), (4.8) hold, and that the exponents p, q satisfy condition (2.13). Assume moreover that Then, system (4.10) admits at least one 2 -periodic solution, provided that, either T 2 > 2 , or there exists an integer n 0 such that (4.2) holds.
Let us show that, if we take D(E) = diag(E 1∕q , E 1∕p ), condition A8 will be satisfied, implying that the same holds true for A7.
Let (t; E) be a solution of energy E for the Hamiltonian system As explained below, we will need later a solution built in a particular way; but, at this stage, this particular construction plays no role. Denoting by

The inequality (3.4) then holds if we define
We now have to to show that, with this definition, satisfies (3.5). Let us consider, for instance, the case where p ≥ 2 (the case q ≥ 2 being analogous). Using (4.6), (4.9) and the relation (2.13) between p and q, we observe that To prove (3.5), it remains to show that, given any compact interval I, The curve H −1 (E) being star-shaped for E large, and the motion being clockwise, the set Σ(E) is contained in the union of intervals, corresponding to transitions between the values −1 and +1 for the function y (t; E). Because of (4.12), the number of those intervals can be assumed to be finite. Considering for instance one of those intervals, let t 1 , t 2 ∈ [0, T(E)], with t 1 < t 2 , be such . Moreover, by (4.8) and (4.9), we see that if H(x, y) = E and if |y| ≤ 1, then |g(x)| ≥ c 0 E 1−2∕p for some constant c 0 > 0, so that the equation −y � = g(x) leads to It then follows that This, combined with the observations made above, finally proves that the function indeed satisfies (3.5).
To establish A8, we still have to prove that (3.6) holds. From (4.13) and (4.14) we deduce that so that (3.6) will be satisfied if We will show that this is the case if the function (0; E) has been built in an appropriate way. Notice first that, by (4.8), there exists y 0 such that k(y 0 ) = 0. Moreover, the function G is strictly increasing for x ≥ 1, so that, for E ≥ G(1) + K(y 0 ), we may define (0; E) by and it is easily checked that (0; E) satisfies (2.5), at least for E "large". Condition (4.15) then amounts to finding a constant C > 0 such that Taking into account the relation (2.13) between p and q, the fact that such a constant can be found is then deduced from (4.7), (4.8). Condition A8 is thus satisfied, and the conclusion then follows from Corollary 4.1. ◻ The case where g(x) = 0 |x| p , k(y) = |y| q is covered by results of Jiang [22], who also deals with asymmetric functions g(x). (See also [2] for a more general system.) It must be emphasized again that our hypotheses are based on the limits of the ratios G(x)∕|x| p , K(y)∕|y| q , yielding less restrictive conditions with respect to more classical assumptions based on the limits of the ratios x g(x)∕|x| p , y k(y)∕|y| q . By Theorem 6.1 below, it will also be possible to deal with the case where (1∕p) + (1∕q) > 1 ; in that situation, besides conditions (4.6), (4.8) and (4.13)-(4.14), no further hypotheses will be needed.

Proof of Theorem 3.1
The proof is based upon degree arguments. We will use the homotopy with ∈ [0, 1], and denote by P ( ) 2 the Poincaré map for the period 2 , associated with the above equation. We look for fixed points of P (1) 2 , which correspond to 2 -periodic solutions of (1.3).
We first need to prove that P ( ) 2 is well-defined for ∈ [0, 1], and continuous. Since we will assume ∇H(z) and r(t, z) to be locally Lipschitz continuous in z, uniqueness of the solutions of (5.1) and continuity with respect to initial conditions are guaranteed. It remains to show that the solutions of (5.1) do not escape to infinity. This is a consequence of condition A6. Indeed, if z ( ) (t) denotes a solution and if we define e ( ) (t) = H(z ( ) (t)), we have We then deduce from A6 that, for some positive constants C ′ , C ′′ , showing that e ( ) (t) remains bounded on any compact interval. It then follows, by the coercivity condition (2.2), that z ( ) (t) can be extended to the whole real line.
We are now in a position to formulate a lemma which describes the guiding idea of the proof of Theorem 3.1. We recall that, for E 0 > 1, for all integer n. Indeed, we then have P (0) 2 (z 0 ) ≠ z 0 , for all z 0 belonging to the boundary of Ω(E 0 ). Moreover, the closed set Ω(E 0 ), which is homeomorphic to a closed ball, is mapped into itself by P (0) 2 , so that the result concerning I − P (0) 2 follows from Brouwer's theorem. Finally, using hypothesis (5.4), the property of invariance of the degree with respect to a homotopy implies that so that P (1) 2 has a fixed point in Ω(E 0 ). ◻ To apply the above lemma for proving Theorem 3.1, we need to estimate P ( ) 2 (z 0 ). This will be done by considering the large amplitude solutions of equation (5.1) as perturbations of the solutions of the autonomous equation (1.4). We therefore write the solutions of (5.1) under the form so that Simple calculations making use of (2.3) then lead to the system (5.2) (e ( ) ) � (t) = ⟨J∇H(z ( ) (t)), r(t, z ( ) (t))⟩ .
= r t, t + ( ) (t);e ( ) (t) . Notice by the way that this last equation is just a rewrite of (5.2). Let us denote the solution of the above system for the initial conditions ( ) (0) = 0 , e ( ) (0) = E 0 by ( ( ) (t; 0 , E 0 ), e ( ) (t; 0 , E 0 )) , or briefly by ( ( ) (t), e ( ) (t)), when there is no risk of ambiguity in omitting the initial conditions. The basic point for the proof of our existence results is the observation that (5.4) will be satisfied unless, for some integer n, some 0 ∈ [0, T(E 0 )], and some ∈ [0, 1], we have By assumptions A4 and A5, it is always possible to find a sequence (E k ) k , with lim k E k = +∞, such that T(E k ) converges to some strictly positive finite value T * , with T * ≠ 2 ∕n for any integer n. Hence, a number > 0 exists such that, for k sufficiently large, |2 − n T(E k )| ≥ , for any integer n. On the other hand, it follows from (5.3) that and, considering (5.5), we deduce from A7 that (the above choice of the interval [0, 2T * ] is somehow arbitrary; what is needed, is just an interval going beyond T * ). Consequently, the second equality in (5.7) is impossible for any integer n, when E 0 is replaced by a sufficiently large element E k . ◻

Some extensions of Theorem 3.1
In this section, we extend Theorem 3.1 in several directions. First, in Sect. 6.1, we consider the case when the limit of the period function T(E) is +∞ as E → +∞ . Then, the critical case when the limit of T(E) is equal to some 2 ∕n 0 is considered, with two different approaches: in Sect. 6.2 we approach resonance, but "not too fast", while in Sect. 6.3 we add some conditions of Landesman-Lazer type.

The case when lim E→+∞ T(E) = +∞
The nonresonance conditions A4 -A5 are satisfied when so that Theorem 3.1 can be invoked to deal with such situations. On the other hand, Theorem 3.1 does not apply when However, it is still possible to obtain existence conditions for this last case, as shown by the next theorem, where an auxiliary Hamiltonian function H 0 is introduced. We associate with it a function 0 (0; E) defined by an equation of the type (2.4).
(5.6) (e ( ) ) � = ⟨ J∇H t + ( ) ;e ( ) , r t, t + ( ) ;e ( ) ⟩ .  Proof As already observed, we only need to consider the case when (6.1) holds. Referring to Lemma 5.1, we aim to show that, for sufficiently large values of E 0 , where P ( ) 2 denotes, as before, the Poincaré map for the period 2 , associated with Eq. (5.1). Assume by contradiction that this is not true. Let z ( ) (t;0, z 0 ) denote the solution of (5.1) corresponding to the initial condition z(0) = z 0 ; that solution is assumed to be 2 -periodic. Arguing as in the proof of Theorem 3.1, and using hypothesis A6, we can show that H(z ( ) (t;0, z 0 )) can be made arbitrarily large by choosing E 0 large enough, uniformly for t ∈ [0, 2 ]. Provided that E 0 is taken sufficiently large, it then results from A9 that, when the trajectory of the solution z ( ) (t;0, z 0 ) crosses the gradient curve E ↦ 0 (0; E), associated with the Hamiltonian H 0 , the crossing occurs in the clockwise direction. That solution being, by assumption, 2 -periodic, we then conclude that, on the interval [0, 2 ], the trajectory must make at least one turn around the origin, in the clockwise direction. We will show that the other hypotheses prevent this possibility.
For this aim, we want to use the same arguments as in the proof of Theorem 2. Some modification is needed however, because condition A7 holds only for s in a compact set, whereas the natural domain of this variable is [0, T(E)], with the period T(E) of the free oscillations going to +∞, for E → +∞. We will therefore manage to consider only values of the argument s of (s; E)∕ E in an interval slightly larger than [0, 2 ].
now compare z(t) with (t;H(z(0)) for t ∈ [0, 2 ], and show that, because of hypothesis A7, the difference in "phase" remains "small". More precisely, letting we see that the function ( ) satisfies an equation similar to (5.5), i.e., with the initial conditions ( ) (0) = 0, e ( ) (0) = H(z(0)). The hypothesis of the contradiction argument would then imply that for some integer n ≠ 0. Using A7 (with s in an interval slightly larger than [0, 2 ]) and working as in the proof of We remark that a similar situation has been considered by Fernandes and Zanolin in [12] for a second-order scalar equation of the type (1.1). See also [19,20].

Approaching resonance
When Theorem 3.1 cannot be invoked to prove the existence of 2 -periodic solutions for Eq. (1.3). But, adapting the arguments of the proof of Theorem 3.1, it is still possible to provide some existence conditions. This is the object of the next theorem.
, and any ∈ [0, 1] .  so that, by (6.4), the above inequality is certainly verified if n ≠ n 0 . Referring to the system (5.5)-(5.6), and using the same notation ( ) (t; 0 , E 0 ), e ( ) (t; 0 , E 0 ) as in the proof of Theorem 3.1, we deduce from A6' that Using that result, it follows from A7' that The combination of (6.8) with (6.5) or (6.6) then implies that large values E k can be found for which (6.7) holds also with n = n 0 . ◻ We notice that similar nonresonance results have been proposed by Hao and Ma in [21] for the second-order equation (1.1), with = 1∕2. A somewhat related approach to resonance was also proposed by Omari and Zanolin in [26].
We now provide an example of application of Theorem 6.
Consider now the solution (t; E) associated with the autonomous equation for the Hamiltonian (6.11), built from the initial value (0;1) = (0, . It is fairly immediate that On the other hand, denoting by x , y the components of , and adapting (2.12) with = 1 − (1∕2) − (1∕p) , we see that From there, it can be checked that √ E x (t; E)∕ E remains bounded for E → +∞, uniformly for t in compact sets. Hence, using (6.10), it follows that A7' holds with = (1∕2) − (1∕p) , the limit being equal to 0. Moreover, under (6.10), hypothesis A6' is also satisfied. We conclude by Theorem 6.3 that equation (6.9) admits at least one 2 -periodic solution.  [4,5,15,31] for more general nonlinearities. Since problems "near resonance" are concerned, some restrictions must be imposed on the nonlinearity (in our approach, condition (6.10)). They may take the form of conditions of Landesman-Lazer type.

Landesman-Lazer conditions
A huge literature exists concerning existence conditions for periodic solutions based on the so-called Landesman-Lazer conditions (see, for instance, [13] and [25] for references). We want to discuss briefly the relation between those conditions and the results presented above.
Since the forcing term r(t, z) in Eq. (1.3) is 2 -periodic in t, Landesman-Lazer conditions would typically concern situations where the Hamiltonian function H is isochronous (at least for solutions of large amplitude), with solutions having a minimal period of the form 2 ∕n 0 , for some positive integer n 0 . In that case, it is no longer possible to resort directly to Lemma 5.1, since deg(I − P (0) 2 , Ω(E 0 ), 0) is no longer defined. But the arguments still work with an adapted homotopy, i.e., the choice of ≠ 0, its sign in particular, being explained below. It is immediate that, for = 0, the minimal period of the large solutions now becomes 2 ∕(n 0 (1 + )). Hence, if P ( ) 2 now denotes the Poincaré map for the period 2 associated with Eq. (6.12), the arguments used in Lemma 5.1 tell us that, provided that ≠ 0 is taken small enough, the degree deg(I − P (0) 2 , Ω(E 0 ), 0) is well-defined for E 0 large, and equal to 1. In order to obtain an existence result, it remains once more to find conditions ensuring that, for well-chosen values E 0 , Here is our result. Assume that (6.14) holds. In this case, since T(E 0 ) = 2 ∕n 0 , we will take > 0 small enough, in order to show that, for any integer n, Using A6' in (6.17), we have that (6.12) Jz � = [1 + (1 − ) ] ∇H(z) + r(t, z) , (6.13) P ( ) 2 (z 0 ) ≠ z 0 , for any z 0 ∈ H −1 (E 0 ) and any ∈ [0, 1] .

Remark 6.7
We notice that it is not really necessary that H be isochronous of period 2 ∕n 0 ; under the above conditions, it suffices that Remark 6.8 Conditions like (6.14) and (6.15) appear in [1,14,17] for Hamiltonians which are positively homogeneous of degree 2. Also the "double resonance" situation has been considered there (see also [9][10][11]).
H(x, y) = 1 2 a + [x + ] 2 + a − [x − ] 2 + y 2 , n 0 being a positive integer, and let r(t, x, y) = (r x (t, x), 0). We are thus dealing with the scalar second-order differential equation The solutions of the autonomous equation all have the same minimal period 2 ∕n 0 . Denoting by x , y the components of , using (2.11), condition (6.14) with = 1∕2 reduces to for every 0 ∈ [0, 2 ∕n 0 ] . Since can be taken arbitrarily small, if we assume r x (t, x) to be globally bounded, this condition is fulfilled when, for every 0 ∈ [0, 2 ∕n 0 ], This is the classical Landesman-Lazer condition (first introduced in [23] for the Dirichlet problem). For the example considered here, the problem has been treated by Dancer [4,5] (in the case where r does not depend on x). See also [13] and the references therein.

Multiplicity of periodic solutions
When the Eq. (1.3) has a Hamiltonian structure, i.e., when the perturbation r(t, z) is a gradient, with respect to z, of a function R(t, z), it is possible to obtain multiplicity results for the periodic solutions by the use of a generalized version of the Poincaré-Birkhoff Theorem, cf. [13,18,29]. Let us then consider the system We will prove the existence of an infinite number of periodic solutions under the assumptions of Theorem 3.1, only replacing A5 by the following.

A5 ′′ . The function T(E) is such that
The idea of a using a condition on the period of the free oscillations in order to apply the Poincaré-Birkhoff Theorem can also be found, for instance, in [16].
We need a preliminary result concerning the forced system (1.3); it does not require r(t, z) to be a gradient. In the following, we will denote by z(t;z 0 ) the solution of (7.1) with initial condition z(0) = z 0 .

Proof Let us write
Since the function H is twice continuously differentiable, it is well-known that the function E ↦ T � (E) is continuously differentiable (see, for instance, [27,Theorem 3.4.1]). Since Δ(0;z 0 ) = 0, the lemma will be proven if we can show that Writing z(t) for z(t;z 0 ), since z(t) is a solution of (1.3), we have Assuming E 0 to be sufficiently large, working as in Theorem 3.1, we can write z(t) under the form where e(t) = H(z(t)), and (t; E) being, as before, a solution of the autonomous system (2.1), as introduced in Sect. 2.1. Since (t; E) is, by definition, of period T(E) in t, we have obviously from which follows, differentiating with respect to E, or Since the above relation holds for all t ∈ ℝ, and all E sufficiently large, we can also write Taking the scalar product with r(t, z(t)) and remembering that z(t) = (t + (t);e(t)) , −J∇H(z(t)) T � (e(t)) = E (t + (t);e(t)) − E (t + (t) + T(e(t));e(t)) .

lim
We may assume that > 0 has been chosen small enough so that which implies that To the above defined sequences we associate, for large values of k, the annuli (t + (t) + T(e(t));e(t)) = (t + (t);e(t)) , (7.4) ⟨∇H(z), z⟩ > 0 , for H(z) sufficiently large .
for any integer n * and any E ∈ [E 2k , Notice that A * k is contained in A k . For some sufficiently large k, to be fixed below, we now modify our system. The idea is to leave it unchanged in A * k , and to cancel the forcing term on ℝ 2 ⧵ A k , the transition in the zone A k ⧵ A * k being built as described below. We thus replace R(t, z) by with ∶ ℝ → ℝ a C ∞ -function, built in such a way that We may assume that | � (s)| ≤ 4∕ , for every s ∈ ℝ.
We now consider the modified system Since it is readily seen that r(t, z) ∶= ∇ zR (t, z) inherits from r(t, z) the regularity properties required for the application of Theorem 3.1. Moreover, using hypothesis (7.5), we observe that ∇ zR (t, z) satisfies the same conditions A6 and A7 as r(t, z). Consequently, arguing as in the proof of Theorem 3.1, if we write a solution z(t;z 0 ) of the modified system (7.7) under the form we can show that, if k is large enough, we will have Let us first show that, if k is sufficiently large, any solution z(t;z 0 ) of system (7.7) issued from a point z 0 in A * k does not enter the zone where the equation has been modified, when t ∈ [0, 2 m] . Indeed, with z 0 ∈ A * k , assume that for some t 0 ∈ [0, 2 m] either H(z(t 0 ;z 0 )) = E * 2k , or H(z(t 0 ;z 0 )) = E * 2k+1 , and Applying Lemma 7.1 with z(t 0 ;z 0 ) instead of z 0 , if k is taken large enough, theñ � .
On the other hand, we claim that there are no 2 m-periodic solutions z(t;z 0 ) of system (7.7) starting from a point z 0 in A k ⧵ A * k , with k large enough. Indeed, if z(t;z 0 ) is such a solution, writing it under the form (7.8), we have for some integer n * . But, using (7.6) and (7.9), we see that, if k is large enough, this equality is impossible, for any integer n * .
We will now apply the version of Poincaré-Birkhoff Theorem presented in [18,Theorem 1.2]. To this aim, first notice that, by assumption (7.4), for k large enough the two curves delimiting the set A k are strictly star-shaped with respect to the origin; moreover, if z 0 ∈ A k , then z(t;z 0 ) ≠ (0, 0) , for every t ∈ [0, 2 m] . This allows us to consider continuous determinations arg z(t;z 0 ) of the argument function along these trajectories, and to define their rotation numbers Let us now estimate these rotation numbers when z 0 belongs to the inner and to the outer boundary of the annulus A k . For k sufficiently large, by (7.4) the function arg z(⋅ ;z 0 ) is strictly decreasing when z 0 ∈ H −1 (E 2k ) , and by the definition of E 2k we get By a similar argument, using the definition of E 2k+1 we see that, for k sufficiently large, Then, by [18,Theorem 1.2], system (7.7) has at least two 2 m-periodic solutions z (1) (t) , z (2) (t) , starting from the interior of A k , such that By the above arguments, these are 2 m-periodic solutions of the original system (7.1), and the proof is thus completed. ◻ We end this section with two remarks.

reduces to
For the second-order equation with g differentiable and g ′ globally bounded, this allows to recover multiplicity results obtained in [7] (see also [32] and the references therein).

Remark 7.4
In a situation of resonance, i.e. when for some positive integer n 0 , conditions of existence of infinitely many periodic solutions can still be obtained, the idea being then to use the hypotheses of Theorem 6.3 rather than those of Theorem 3.1. We will not give the details here, but simply indicate that a "twist condition" would be met by asking the conditions (6.5) and (6.6) to hold simultaneously, whereas (7.5) should be replaced by the stronger condition with > 0 the same constant as in (6.5) and (6.6). The resulting periodic solutions would then be of period 2 , making n 0 turns around the origin on a time interval of length 2 .

More general differential equations
In this section, we consider the planar system We will assume throughout that F(t, z) is continuous, 2 -periodic in t, and locally Lipschitz continuous in z. We will provide existence conditions for 2 -periodic solutions, through a comparison with Hamiltonian systems. We first state a preliminary result concerning the number of turns, around the origin, of a closed curve; il will be applied below to trajectories of (possible) periodic solutions of equations like (8.1).
Conversely, if the above equality holds for some nonnegative integer n, the curve t ↦ z(t) makes n turns around the origin on the interval [0, T], in the clockwise sense.
Proof The curve z ∶ t ↦ z(t) crosses the gradient curve E ↦ * (0; E), when and only when (t) = 0 mod T * . The condition (8.2) means that the curve z is transversal to the gradient curve * (0;⋅), the crossing occurring in the clockwise direction. This gradient curve extends from a point on the closed curve H(z) = 1 to infinity. Since the curve z remains in the unbounded set {z ∈ ℝ 2 | H(z) > 1} for t ∈ [0, T], the number of turns of z around the origin on the interval [0, T) is equal to the number of crossings with the curve * (0;⋅). Because of (8.2) and (2.5), we have � (t) > 0 when (t) = 0 mod T * . Hence, each revolution of the curve z corresponds to an increase T * of the parameter t. Therefore, the number n in (8.3) is necessarily the number of turns of the curve z, around the origin, on the interval [0, T]. ◻ Notice that the above result also makes sense with n = 0.

An existence result
We now state an existence result for Eq. (8.1) based on a comparison between F(t, z) and the gradients of two Hamiltonian functions H 1 , H 2 . The proof is based again on degree arguments, a homotopy being used. The hypotheses on the third Hamiltonian function H, which appears in that equation, aim essentially at controlling the energy of the possible periodic solutions, whereas the hypotheses on H 1 , H 2 concern their number of revolutions in the plane. Many existence results have been obtained in the past by considering this number of turns, namely by means of a so-called "rotation number" (see, for instance, [1,33]). Our aim, here, is to present a result in the line of our approach of Sect. 3, transforming again the system (8.4) into a system whose variables are the energy and the phase. An objective will then be to compare, for an equation like (1.3), the results presented below to the results of Sect. 3. As will be seen, the main difference is that, in the present section, the hypotheses used for the comparison are based essentially on ∇H 1 , ∇H 2 , whereas, in Sect. 3, the hypotheses concern more directly the functions H 1 , H 2 themselves, through the relations (4.1).
Let   The hypotheses in the above theorem may seem awkward, but have been stated at that level of generality to allow applications in a large variety of situations. We will provide below various sets of conditions ensuring that these hypotheses are satisfied.

The case when n 0 = 0
In the case when n 0 = 0 , the function H 1 is superfluous, and the following result can be stated. The proof is analogous to the one of Theorem 8.2, hence we omit it, for brevity. Notice however the difference between conditions (8.23) and (8.8). The conditions (8.23) ensure that a possible periodic solution of (8.4) crosses all the gradient curves associated with H 2 , whereas in Theorem 8.2, condition (8.8), which concerns only one particular gradient curve, suffices. Notice also that it is not necessary here to assume the Hamiltonian function H 2 to be isochronous.

Application to equations with separated variables
Consider the case where the variables are "separated" in the planar system (8.1), i.e., the right-hand side has the form We will assume that g, k are continuous, 2 -periodic in t, locally Lipschitz continuous in x, y, and that, for some p > 1, q > 1 related by condition (2.13), we have uniformly in t. More precisely, we will assume that there exist numbers 2 ≥ 1 > 0 and  the limits being assumed to be uniform in t. Actually, we could write more general results by considering separately the limits for x going to +∞ and for x going to −∞, and analogously for y. This involves no particular difficulty, but makes the formulation of the hypotheses more involved. In order to apply Theorem 8.2 or 8.5, we will use the following functions as references for the comparison: We also need a Hamiltonian function H having the properties required for Theorem 8.2 and 8.5; it is possible to choose a function, whose gradient is locally Lipschitz continuous, and which is of the form with |∇R(z)| = o(|z|) for |z| → +∞. Denoting, as before, by 1 (t; E), 2 (t; E) the functions associated with the Hamiltonians H 1 , H 2 , and remembering (2.14), i.e., it can be checked that all the conditions for the application of Theorem 8.2 or 8.5 are satisfied. Altogether, the following corollary is obtained. Corollary 8.6 Assume that, for some p > 1, q > 1 related by (2.13), the conditions (8.24), (8.25) hold (with 1 , 1 positive). Let the Hamiltonians H 1 , H 2 then be defined by (8.26), the minimal period of their nontrivial solutions being denoted by T 1 , T 2 , respectively. If either T 2 > 2 , or for some integer n 0 , condition (8.20) holds, then the system admits a 2 -periodic solution.
A result close to the above corollary can be found in [6], for the case of a secondorder equation with a p-Laplacian operator, corresponding to the choice k(t, y) = c |y| q , for some constant c > 0.
As already observed above in the case of systems with positively homogeneous Hamiltonians of degree 2, sharper results can be obtained for systems of the form by exploiting the Hamiltonian structure and resorting to the results of Section 4, the Hamiltonian being defined by H(x, y) = G(x) + K(y), where G and K denote some primitives of g and k, respectively.