Abstract
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.
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1 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 = n2 for some integer n. On the contrary, if \(\lambda \notin \{n^{2}: n \in {\mathbb {N}}\}\), 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 \(2\pi\)-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, \(J = \left( {\begin{array}{*{20}c} 0 & { - 1} \\ 1 & 0 \\ \end{array} } \right)\) is the standard symplectic matrix, the Hamiltonian function \(H:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is continuously differentiable, and \(r:{\mathbb {R}}\times {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2\) 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 \(\nabla H(z)\) 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 \(2\pi\)-periodic solution, unless the period \(2\pi\) of the forcing term r(t) interferes with the periods of the large amplitude free oscillations, meaning that T(E) approaches \(2\pi /n\), 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
It is the main objective of this paper to present conditions under which the inequality (1.5) guarantees the existence of a \(2\pi\)-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 \(2\pi\)-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 \(2\pi\)-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 \(+\infty\) as \(E\rightarrow +\infty\). 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.
In the following, we always denote by \(\langle \cdot \,,\cdot \rangle\) the Euclidean scalar product in \({\mathbb {R}}^2\), with associated norm \(|\cdot |\).
2 General setting and preliminaries
In the first part of this section, we list some conditions on the Hamiltonian function \(H:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) that will be assumed to hold throughout the paper. In the second part, we define some notation, and we place emphasis on a result that will be useful in the sequel.
2.1 The structural assumptions
We recall that the function \(H:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is supposed to be continuously differentiable. We start with two basic assumptions and their consequences for the autonomous equation
- A1.:
-
The Hamiltonian function H is coercive:
$$\begin{aligned} \lim _{|z| \rightarrow \infty } H(z) = + \infty \,. \end{aligned}$$(2.2) - A2.:
-
There exists a number \(\rho >0\) such that
$$\begin{aligned} \nabla H(z)\ne 0\,, \;\textit{ for } |z| \ge \rho . \end{aligned}$$
With those hypotheses, there is uniqueness for the solutions of the associated initial value problems with a starting point of sufficiently large norm (cf. [28]), and it results from the Poincaré–Bendixson theory that all these solutions are periodic. Because of (2.2), we also see that the solutions of large amplitude circle the origin, and that the corresponding trajectories are oriented clockwise; there exists thus an annulus of closed orbits, extending to infinity. Notice that these orbits are not necessarily star-shaped. Among them, we select a particular one, denoted by \(\Gamma _1\) and, by convention, we take
We assume that \(\Gamma _1\) has been chosen in such a way that \(|z| \ge \rho ,\) for all \(z \in \Gamma _1.\)
We want to parametrize the solutions of large amplitude of equation (2.1) by the energy. More precisely, we make the following structural assumption.
A3. There exists a differentiable function \(\varphi :{\mathbb {R}}\times \,]1,+\infty [\,\rightarrow {\mathbb {R}}\) such that
and
As a consequence, the system (2.1) being conservative, we have that
and, differentiating this relation,
2.2 About condition A3
Let us describe a standard way to construct the function \(\varphi\). Fix a point \(z^*_0\) in \(\Gamma _1\), and consider a solution \(w(\tau )\) 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 \(\varphi (\cdot \,;E)\) be the solution of the Hamiltonian system (2.1) such that \(\varphi (0;\,E) = w(E)\). It is clear that both equalities in A3 hold true. The regularity of \(\varphi\) 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 \(\varphi (t;\,E)\) having some convenient properties.
From the definition of \(\varphi (t;\,E),\) we also have that
this property of \(\varphi (0;\,E)\) 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
and
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 \({\widehat{T}}\). (For the reader’s convenience, this fact will also be proved below, as an easy consequence of Lemma 2.1.) Assumptions A1, A2 are readily verified. Concerning A3, let us show that in this case it is possible to choose \(z_0^*\) in (2.4) so that the resulting function \(\varphi (t;\,E)\) satisfies
Indeed, if z is a solution of the autonomous equation (2.1) of energy equal to 1, because the function \(t \mapsto |z(t)|^2\) reaches its extremal values in \([0,{\widehat{T}}],\) there exists a number \(t^* \in [0,{\widehat{T}}]\) such that
Consequently, since \(\langle z, \nabla H(z) \rangle =2H(z) >0,\) for \(z \ne 0,\) there exists \(\nu >0\) such that \(z(t^*) = \nu \, \nabla H(z(t^*)) .\) Using the fact that \(\nabla H(\lambda z) = \lambda \nabla H(z)\) for any \(\lambda \ge 0,\) if we then take \(z_0^*= z(t^*)\) in (2.4), we note that a solution of this system is given by \(w(\tau ) =\sqrt{\tau } \, z(t^*),\) implying that \(\varphi (0;\,E)= \sqrt{E}\,\varphi (0;1)\), from which (2.10) follows. As a consequence, in this case we have
Notice that, as a particular case, we could have \(H(z)={\textstyle \frac{1}{2}}\langle {{\mathbb {A}}}z,z\rangle\), with a positive definite symmetric matrix \({{\mathbb {A}}}.\)
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 \(\varphi (t;1)\) is a solution of (2.1) of energy 1, we observe that
where
is also a solution of (2.1), implying that, for \(\mu >0,\) the minimal period tends to 0 when E tends to \(+\infty ,\) whereas, for \(\mu <0,\) it tends to \(+\infty .\) On the other hand, if \(\mu =0,\) i.e., if
the autonomous system is isochronous, and
If we choose \(\varphi (0;1)= ((1/a)^{1/p},0),\) it can be checked that (2.5) is satisfied.
2.3 A basic property
Let us denote by \(\mathrm{int}(\Gamma _1)\) and \(\mathrm{ext}(\Gamma _1)\) the bounded and the unbounded connected components of \({\mathbb {R}}^2\setminus \Gamma _1\), respectively. For every \(z_0\in {\mathrm{ext}(\Gamma _1)},\) let \({{\mathcal {T}}}(z_0)\) be the minimal period of the solution issuing from it. We define the continuous function \(T :\,]1, +\infty [\, \rightarrow {\mathbb {R}}\) as
it expresses the period as a function of the energy. Moreover, for \(E>1,\) we introduce the open bounded set
Notice that, for E sufficiently large, \(\Omega (E)\) is the bounded set delimited by the level curve \(H^{-1}(E)\). The following lemma expresses a fundamental relation between the area a(E) of \(\Omega (E)\) and the minimal period T(E).
Lemma 2.1
Let the assumptions A1 to A3 hold. Then,
Proof
Given \(E>1,\) let us consider the open sets
Notice that \({{\mathcal {B}}}\) differs from \({\mathbb {R}}^2\setminus \Omega (1)\) by a set of zero Lebesgue measure. Define the function \(\Phi :{{\mathcal {A}}}\rightarrow {{\mathcal {B}}}\) as \(\Phi (\tau ,e)=\varphi (\tau ;e)\). It is one-to-one and onto. Using (2.3), we have
for every \((\tau ,e)\in {{\mathcal {A}}}\), so that \(\Phi\) is a diffeomorphism. For \(E>1\), the area of \(\Omega (E)\) is then given by
and the conclusion directly follows. \(\square\)
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 \(\Omega (E)=\sqrt{E}\;\Omega (1)\), for any \(E \ge 0,\) so that \(a(E) = E \, a(1).\) Consequently, the period is given by \(T(E)=a'(E) = a(1).\)
Remark 2.2
Given a continuous function \(T:[1,+\infty [\,\rightarrow \,]0,+\infty [\) , it is always possible to construct a Hamiltonian function \(H:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) for which T(E) is the time map for the orbits of energy \(E\ge 1.\) Indeed, define \(a :[1,+\infty [\,\rightarrow \,[\pi ,+\infty [\) by
This function is strictly increasing and onto, hence invertible; consider a \(C^1\)-extension \(a:[0,+\infty [\,\rightarrow [0,+\infty [\) , for which we keep the same notation, with \(a(0)=0\) and \(a'(E)>0\) for every \(E\ge 0\). Define \(f :[0,+\infty [\,\rightarrow [0,+\infty [\) by \(f(r) = a^{-1}(\pi \, r^2),\) 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 \(E\ge 1,\) the minimal period of the corresponding solution is T(E).
3 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 \(H:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is differentiable with a locally Lipschitz continuous gradient.
- L2.:
-
The function \(r:{\mathbb {R}}\times {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2\) is continuous, \(2\pi\)-periodic in its first variable, and locally Lipschitz continuous in its second variable.
3.1 Statement of the existence result
Here is the main result of this section.
Theorem 3.1
Let the assumptions A1 to A3 hold, as well as L1, L2, and the following nonresonance conditions:
- A4. :
-
The function T(E) is controlled as follows:
$$\begin{aligned} \limsup _{E \rightarrow +\infty } T(E) >0\,,\quad \, \liminf _{E \rightarrow +\infty } T(E) < +\infty \,. \end{aligned}$$ - A5. :
-
For any integer n,
$$\begin{aligned} \lim _{E \rightarrow +\infty } T(E) \ne \frac{2\pi }{n}\,. \end{aligned}$$Assume also that:
- A6. :
-
There is a constant \(C>0\) such that
$$\begin{aligned} \limsup _{|z| \rightarrow \infty } \frac{\big |\langle J \nabla H(z) , r(t,z) \rangle \big |}{H(z)} \le C\,, \hbox { uniformly in } t\in [0,2\pi ] \,, \end{aligned}$$and
- A7. :
-
For any given compact interval \(I\subseteq {\mathbb {R}}\),
$$\begin{aligned} \lim _{E \rightarrow +\infty } \left\langle \frac{\partial \varphi }{\partial E}(s;\,E), r(t, \varphi (s;\,E))\right\rangle = 0\,,\hbox { uniformly for } (t,s)\in [0,2\pi ]\times I\,. \end{aligned}$$Then, Eq. (1.3) admits at least one \(2\pi\)-periodic solution.
Before going to the proof of Theorem 3.1, we make some comments and draw some consequences.
3.2 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.
- A4’.:
-
The function a(E)/E is controlled as follows:
$$\begin{aligned} \limsup _{E \rightarrow +\infty } \frac{a(E)}{E}>0\,,\,\quad \liminf _{E \rightarrow +\infty } \frac{a(E)}{E} < +\infty \,. \end{aligned}$$ - A5’.:
-
For any integer n,
$$\begin{aligned} \lim _{E \rightarrow +\infty } \frac{a(E)}{E} \ne \frac{2\pi }{n} \,. \end{aligned}$$
We can thus state the following immediate consequence of Theorem 3.1.
Corollary 3.2
Let the assumptions A1 to A3 hold, as well as L1, L2, A6, and A7. Suppose that the nonresonance conditions A4’, A5’ are satisfied. Then, Eq. (1.3) admits at least one \(2\pi\)-periodic solution.
The interest of the above corollary lies in the observation that, under hypotheses A1 to A3, A6 and A7, a \(2\pi\)-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 \(\liminf _{E \rightarrow +\infty } a(E)/E\) and \(\limsup _{E \rightarrow +\infty } a(E)/E\) 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.
3.3 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.
Corollary 3.3
Assume that L1, L2, and (2.9) hold. Let \({\widehat{T}}\) be the minimal period of the solutions of the isochronous system (2.1). If \(\,{\widehat{T}} \ne 2\pi /n\), for all integers n, and the forcing term satisfies (3.3), then equation (1.3) admits at least one \(2\pi\)-periodic solution.
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.
A8. There exist a continuous function \({{\mathcal {D}}}: [1,+\infty ) \rightarrow \mathcal{GL}({\mathbb {R}}^2)\) (the group of invertible \(2\times 2\) real matrices) and a continuous function \(\kappa : {\mathbb {R}}^2 \rightarrow \,]0,+\infty [\) such that, for \(E>1,\)
and
Moreover, for any given compact interval \(I\subseteq {\mathbb {R}}\),
Let us show that A8 implies A7. Indeed, from the variational equation
we see that
and we deduce by (3.4) and Gronwall Lemma that
for every \(t \in {\mathbb {R}}.\) Using (2.5) and (3.5), we now see that A7 results from (3.6).
Based on the above considerations, the following corollary is an immediate consequence of Theorem 3.1.
Corollary 3.4
Let the assumptions A1 to A3, and L2 hold, as well as A6 and A8, H being twice continuously differentiable. Then, if the nonresonance conditions A4 and A5 are satisfied, Eq. (1.3) admits at least one \(2\pi\)-periodic solution.
A noteworthy situation is the case where \(H''\) is globally bounded. Taking as \({{\mathcal {D}}}(E)\) the identity matrix, we see that (3.4) is plainly satisfied with \(\kappa\) being a constant function, whereas (3.6) holds if, for any given compact interval \(I\subseteq {\mathbb {R}}\),
Under this condition, we will show that hypothesis A6 also holds, obtaining the following corollary.
Corollary 3.5
Let the assumptions A1 to A3 hold, as well as L2. Assuming H to be twice continuously differentiable, let \(H''\) be globally bounded and suppose that (3.8) holds. Then, if the nonresonance conditions A4 and A5 are satisfied, Eq. (1.3) admits at least one \(2\pi\)-periodic solution.
Proof
We first observe that, with \(H''\) globally bounded, (3.4) implies
hence, by Gronwall Lemma,
for every \(s \in {\mathbb {R}}.\) Taking (3.8) into account, it then follows that condition A6 will hold if
But, this inequality is a consequence of the boundedness property of \(H''.\) Indeed, differentiating \(|\nabla H(\varphi (0;\,E))|^2\) with respect to E, we get, using (2.5),
Since \(H''\) is assumed to be globally bounded, it follows that there exist constants \(C_1,\) \(C_2\) such that
implying (3.10). The proof is thus completed. \(\square\)
Remark 3.6
Because of (3.9), it is clear that, when \(H''\) is globally bounded, condition (3.8) will hold if
Remark 3.7
Consider the scalar second-order equation
with g continuously differentiable, \(g'\) being globally bounded, and r continuous and \(2\pi\)-periodic. Denoting by G a primitive of g, we can associate with this equation the Hamiltonian function defined by (2.6). If we assume that, for some \(d>0,\)
it is clear that A1, A2 hold. Condition A3 also holds since g is continuously differentiable. Moreover, if we take \(\varphi (0;\,E) = (0, \sqrt{E})\), it is immediate that (3.8) is satisfied. Consequently, we are in a situation where Corollary 3.5 applies, and we may conclude that (3.12) admits at least one \(2\pi\)-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 \(g(x) /x \ge c,\) for \(|x|>d.\)
The above remark can be adapted to the equation
provided that
Assuming the existence of \(\eta >0\) such that
we see that (3.8) is satisfied if we take \(\varphi (0;\,E) = (0, \sqrt{E})\). 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, \(2\pi\)-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 \(2\pi\)-periodic solution.
4 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.
4.1 Comparison with isochronous Hamiltonians
Suppose that, for some number \(\rho >0\),
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 \(\Omega _1(E)\), \(\Omega (E)\), \(\Omega _2(E)\) the bounded sets delimited by the curves \(H_1^{-1}(E),\) \(H^{-1}(E),\) \(H_2^{-1}(E)\), and by \(a_1(E)\), a(E), \(a_2(E)\) their areas, respectively. Then,
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 \(T_2\le T_1\). This leads to the following consequence of Theorem 3.1.
Corollary 4.1
Let \(H,H_1,H_2\) satisfy (4.1) and the hypotheses A1 to A3, the Hamiltonians \(H_1,H_2\) being isochronous with respective minimal periods \(T_1,\) \(T_2\). Assume moreover that H and r satisfy L1, L2, A6, A7. Then, Eq. (1.3) admits at least one \(2\pi\)-periodic solution, provided that, either \(T_2> 2\pi\), or, for some integer \(n_0,\)
If we now recall the situation considered in Corollary 3.5, we immediately get the following.
Corollary 4.2
Let \(H,H_1,H_2\) satisfy (4.1) and the hypotheses A1 to A3, the Hamiltonians \(H_1,H_2\) being isochronous with respective minimal periods \(T_1,\) \(T_2\). With H twice continuously differentiable and r satisfying L2, let \(H''\) be globally bounded, and assume that (3.8) holds. Then, Eq. (1.3) admits at least one \(2\pi\)-periodic solution, provided that, either \(T_2> 2\pi\), or, for some integer \(n_0,\) condition (4.2) holds.
Notice that, for condition (3.8), advantage can be taken of the freedom of choice in the construction of \(\varphi (0;\,E).\)
4.2 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, \(2\pi\)-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 \(\varepsilon >0\) 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 \(\rho\) sufficiently large. On the other hand, the minimal periods \(T_{1,\varepsilon },\) \(T_{2,\varepsilon }\) 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 \(\eta >0\) such that \(G(x)\ge \eta \,x^2\) for |x| sufficiently large. Using the fact that \(\varepsilon\) 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 \(G_{\pm },G^{\pm }\) 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, \(2\pi\)-periodic in t, locally Lipschitz continuous in x, and satisfies (3.15). Then, Eq. (3.14) admits at least one \(2\pi\)-periodic solution.
The conditions (4.4) can be interpreted in terms of the Fučík spectrum for the \(2\pi\)-periodic boundary value problem. They amount to requiring that the rectangle \([G_+,G^+] \times [G_-,G^-]\) 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 \(x \rightarrow \pm \infty ,\) 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
4.3 Hamiltonians with separated variables
We now consider systems with Hamiltonian functions of the form
we will suppose that \(G, K :{\mathbb {R}}\rightarrow {\mathbb {R}}\) 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 \(C_1\), \(C_2\) such that
From there, it is easy to deduce constants \(C_1'\), \(C_2'\) such that
We will also assume that for some positive constants \(c_1\), \(c_2,\)
The above hypotheses imply that there exist constants \(L \ge \ell >0\) 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, \(2\pi\)-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 \(\alpha _1,\) \(\alpha _2,\) \(\beta _1,\) \(\beta _2\) 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 \(\mu =0\)); 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,
\({\widetilde{T}}_1,\) \({\widetilde{T}}_2\) 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.
Corollary 4.4
Let \(H(x,y)= G(x) + K(y),\) with G, K twice continuously differentiable 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 \(2\pi\)-periodic solution, provided that, either \(T_2>2\pi\), or there exists an integer \(n_0\) such that (4.2) holds.
Proof
We have seen that assumptions A1 to A3 are satisfied. Moreover, it results from (4.7), 4.9), (4.13), and (4.14), using Young’s inequality again, that condition A6 is satisfied.
Let us show that, if we take \({{\mathcal {D}}}(E) = \hbox {diag}(E^{1/q},E^{1/p}) ,\) condition A8 will be satisfied, implying that the same holds true for A7.
Let \(\varphi (t;\,E)\) 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 \(\varphi _x(t;\,E),\) \(\varphi _y(t;\,E),\) the components of the solution \(\varphi (t;\,E)\) we compute, with \(v =(v_1,v_2),\)
The inequality (3.4) then holds if we define
We now have to to show that, with this definition, \(\kappa\) satisfies (3.5). Let us consider, for instance, the case where \(p\ge 2\) (the case \(q \ge 2\) being analogous). Using (4.6), (4.9) and the relation (2.13) between p and q, we observe that \(E^{1-2/q}\, g'(\varphi _x(t;\,E))\) is bounded, independently of t and E. The same property holds for \(E^{1-2/p} \,k'(\varphi _y(t;\,E)),\) as long as \(|\varphi _y(t;\,E)| \ge 1.\) To prove (3.5), it remains to show that, given any compact interval I,
where \(\Sigma (E) =\{t \in I: |\varphi _y(t;\,E)| \le 1\}\). The curve \(H^{-1}(E)\) being star-shaped for E large, and the motion being clockwise, the set \(\Sigma (E)\) is contained in the union of intervals, corresponding to transitions between the values \(-1\) and \(+1\) for the function \(\varphi _y(t;\,E).\) Because of (4.12), the number of those intervals can be assumed to be finite. Considering for instance one of those intervals, let \(t_1, t_2 \in [0,T(E)] ,\) with \(t_1 <t_2,\) be such
Assuming E sufficiently large, we have \(\varphi _x(t;\,E) >0,\) for \(t \in [t_1,t_2].\) Moreover, by (4.8) and (4.9), we see that if \(H(x,y) = E\) and if \(|y| \le 1,\) then \(| g(x) | \ge c_0\, E^{1-2/p}\) for some constant \(c_0>0,\) so that the equation \(-y'=g(x)\) leads to
It then follows that \(t_2-t_1=O\big (E^{\frac{2}{p}-1}\big )\) for \(E \rightarrow +\infty .\) Consequently, since \(|\varphi _y(t;\,E)| \le 1\) for \(t \in [t_1,t_2],\) we have that
This, combined with the observations made above, finally proves that the function \(\kappa\) 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 \(\varphi (0;\,E)\) 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 \(x \ge 1,\) so that, for \(E \ge G(1) + K(y_0),\) we may define \(\varphi (0;\,E)\) by
with \(x_E \ge 1\) such that \(G(x_E)=E-K(y_0).\) Since \(k(y_0)=0,\) we have \(\nabla H(\varphi (0;\,E)) = (g(x_E),0),\) and it is easily checked that \(\varphi (0;\,E)\) 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. \(\square\)
The case where \(g(x)= \alpha _0|x|^p,\) \(k(y)=|y|^q\) 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.
5 Proof of Theorem 3.1
The proof is based upon degree arguments. We will use the homotopy
with \(\lambda \in [0,1],\) and denote by \(P^{(\lambda )}_{2\pi }\) the Poincaré map for the period \(2\pi ,\) associated with the above equation. We look for fixed points of \(P^{(1)}_{2\pi },\) which correspond to \(2\pi\)-periodic solutions of (1.3).
We first need to prove that \(P^{(\lambda )}_{2\pi }\) is well-defined for \(\lambda \in [0,1],\) and continuous. Since we will assume \(\nabla H(z)\) 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 \(z^{(\lambda )}(t)\) denotes a solution and if we define \(e^{(\lambda )}(t) = H(z^{(\lambda )}(t)),\) we have
We then deduce from A6 that, for some positive constants \(C', C'',\)
showing that \(e^{(\lambda )}(t)\) remains bounded on any compact interval. It then follows, by the coercivity condition (2.2), that \(z^{(\lambda )}(t)\) 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 \(E_0>1\),
Lemma 5.1
Let the assumptions A1 to A3 hold, as well as L1, L2, and let \(E_0>1\) be such that \(T(E_0) \ne 2\pi /n,\) for any integer n. Then, we have \(\deg (I-P^{(0)}_{2\pi } , \Omega (E_0), 0)=1\). Moreover, if
then \(\deg (I-P^{(1)}_{2\pi } , \Omega (E_0), 0)=1\) and, consequently, equation (1.3) admits at least one \(2\pi\)-periodic solution.
Proof
The degree \(\deg (I-P^{(0)}_{2\pi } , \Omega (E_0), 0)\) is clearly well-defined if \(T(E_0) \ne 2\pi /n,\) for all integer n. Indeed, we then have \(P^{(0)}_{2\pi }(z_0) \ne z_0,\) for all \(z_0\) belonging to the boundary of \(\Omega (E_0).\) Moreover, the closed set \(\overline{\Omega (E_0)},\) which is homeomorphic to a closed ball, is mapped into itself by \(P^{(0)}_{2\pi },\) so that the result concerning \(I-P^{(0)}_{2\pi }\) 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 \(P^{(1)}_{2\pi }\) has a fixed point in \(\Omega (E_0).\) \(\square\)
To apply the above lemma for proving Theorem 3.1, we need to estimate \(P^{(\lambda )}_{2\pi } (z_0).\) 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
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 \(\tau ^{(\lambda )}(0)=\tau _0,\) \(e^{(\lambda )}(0)=E_0\) by \((\tau ^{(\lambda )}(t;\tau _0,E_0),e^{(\lambda )}(t;\tau _0,E_0))\), or briefly by \((\tau ^{(\lambda )}(t),e^{(\lambda )}(t)),\) 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 \(\tau _0 \in [0,T(E_0)],\) and some \(\lambda \in [0,1],\) we have
By assumptions A4 and A5, it is always possible to find a sequence \((E_k)_k,\) with \(\lim _k E_k = +\infty ,\) such that \(T(E_k)\) converges to some strictly positive finite value \(T^*,\) with \(T^* \ne 2\pi /n\) for any integer n. Hence, a number \(\eta >0\) exists such that, for k sufficiently large, \(|2\pi - n\,T(E_k) | \ge \eta ,\) 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 \(E_k.\) \(\square\)
6 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 \(+\infty\) as \(E\rightarrow +\infty\). Then, the critical case when the limit of T(E) is equal to some \(2\pi /n_0\) 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.
6.1 The case when \(\lim _{E \rightarrow +\infty }T(E) = +\infty\)
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 \(\varphi _0(0;\,E)\) defined by an equation of the type (2.4).
Theorem 6.1
Let the assumptions A1 to A3 hold, as well as L1, L2, A6, A7, and
Assume also the following condition:
A9. There exist a differentiable function \(H_0:{\mathbb {R}}^2\rightarrow {\mathbb {R}},\) satisfying the assumptions A1 to A3, and a constant \(E^*>0\) such that
Then, Eq. (1.3) admits at least one \(2\pi\)-periodic solution.
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 \(P^{(\lambda )}_{2\pi }\) denotes, as before, the Poincaré map for the period \(2\pi ,\) associated with Eq. (5.1). Assume by contradiction that this is not true. Let \(z^{(\lambda )}(t;0,z_0)\) denote the solution of (5.1) corresponding to the initial condition \(z(0)=z_0\,;\) that solution is assumed to be \(2\pi\)-periodic. Arguing as in the proof of Theorem 3.1, and using hypothesis A6, we can show that \(H(z^{(\lambda )}(t;0,z_0))\) can be made arbitrarily large by choosing \(E_0\) large enough, uniformly for \(t \in [0,2\pi ].\) Provided that \(E_0\) is taken sufficiently large, it then results from A9 that, when the trajectory of the solution \(z^{(\lambda )}(t;0,z_0)\) crosses the gradient curve \(E \mapsto \varphi _0(0;\,E),\) associated with the Hamiltonian \(H_0,\) the crossing occurs in the clockwise direction. That solution being, by assumption, \(2\pi\)-periodic, we then conclude that, on the interval \([0,2\pi ],\) 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 \(+\infty ,\) for \(E \rightarrow +\infty .\) We will therefore manage to consider only values of the argument s of \(\partial \varphi (s;\,E)/\partial E\) in an interval slightly larger than \([0,2\pi ].\)
The curve \([0,2\pi ] \rightarrow {\mathbb {R}}^2:t \mapsto z^{(\lambda )}(t;0,z_0)\), making at least one turn in the phase plane, must also cross all the gradient curves associated with the Hamiltonian H, provided that \(E_0=H(z_0)\) is large enough. It will cross in particular the curve \(E\mapsto \varphi (0;\,E)\). Therefore, we can find a value \(t^*\in [0,2\pi ]\) such that
Define then
a solution of
with the initial condition \({\widetilde{z}}(0) = z^{(\lambda )}(t^*;0,z_0)= \varphi (0;H({\widetilde{z}}(0))).\) We have already observed that \(H(z^{(\lambda )}(t^*;0,z_0))\) can be made arbitrarily large by choosing \(E_0\) large enough. We will now compare \({\widetilde{z}} (t)\) with \(\varphi (t; H({\widetilde{z}}(0))\) for \(t \in [0,2\pi ],\) and show that, because of hypothesis A7, the difference in “phase” remains “small”. More precisely, letting
we see that the function \(\tau ^{(\lambda )}\) satisfies an equation similar to (5.5), i.e.,
with the initial conditions \(\tau ^{(\lambda )}(0)=0,\) \(e^{(\lambda )}(0) = H({\widetilde{z}}(0)).\) The hypothesis of the contradiction argument would then imply that
for some integer \(n \ne 0.\) Using A7 (with s in an interval slightly larger than \([0,2\pi ])\) and working as in the proof of Theorem 3.1, it can be proved that
showing that the equality (6.3) is impossible if \(E_0\) is taken sufficiently large, since \(\lim _{E \rightarrow +\infty } T(E) = +\infty .\) \(\square\)
It is, of course, admissible to choose \(H_0=H\) in assumption A9. Therefore, that condition is fulfilled if there exists a \(E^*>1\) such that
This observation, together with those made at the end of Sect. 3.3, lead to the following corollary.
Corollary 6.2
Let the assumptions A1 to A3 hold, as well as (6.2). With H twice continuously differentiable and r satisfying L2, let \(H''\) be globally bounded, and assume that (3.8) holds. Then, equation (1.3) admits at least one \(2\pi\)-periodic solution.
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].
6.2 Approaching resonance
When
Theorem 3.1 cannot be invoked to prove the existence of \(2\pi\)-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.
Theorem 6.3
Let the Hamiltonian H satisfy assumptions A1 to A3, and be such that (6.4) holds. Assume that L1, L2 hold, that
A6’.
and that
A7’. there exists a number \(\gamma >0,\) and a constant \(C \ge 0\) such that, for any given compact interval \(I\subseteq {\mathbb {R}}\),
Then, Eq. (1.3) admits at least one \(2\pi\)-periodic solution, provided that, either
or
Proof
By hypotheses (6.5), (6.6), it is possible to find a sequence \((E_k)_k,\) with \(\lim _k E_k=+\infty ,\) such that
so that, by (6.4) and Lemma 5.1, the degree \(\deg (I-P^{(0)}_{2\pi } , \Omega (E_k), 0)\) is equal to 1. Working as in the proofs of Theorem 3.1 and Theorem 6.1, we want to show that, for k sufficiently large,
where \(P^{(\lambda )}_{2\pi }\) denotes, as usual, the Poincaré map for the period \(2\pi\) associated with Eq. (5.1). Writing, as in the proof of Theorem 3.1, the solutions of equation (5.1) under the form
we have to find values \(E_k\) such that, for any integer n,
As already explained, the choice of the interval \([0,4\pi /n_0]\) is somehow arbitrary, as long as it contains \(2\pi /n_0\) in its interior. Notice that, A6’, A7’ being stronger than A6, A7, it is clear that
so that, by (6.4), the above inequality is certainly verified if \(n\ne n_0\). Referring to the system (5.5)–(5.6), and using the same notation \(\tau ^{(\lambda )}(t;\tau _0,E_0),\) \(e^{(\lambda )}(t;\tau _0,E_0)\) 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 \(n=n_0\). \(\square\)
We notice that similar nonresonance results have been proposed by Hao and Ma in [21] for the second-order equation (1.1), with \(\gamma =1/2.\) A somewhat related approach to resonance was also proposed by Omari and Zanolin in [26].
We now provide an example of application of Theorem 6.3 to the second-order equation
where m is an integer, \(p>2,\) \(a>0,\) and where q(t, x) is assumed to be continuous and \(2\pi\)-periodic in t, with
This condition is surely satisfied if q(t, x) is globally bounded. The Hamiltonian function associated to the unperturbed equation is
and the minimal periods of the free oscillations are given by
where \(T^-(E)\) is the transit time in the negative phase plane. It can be computed that
where \(\Gamma (\cdot )\) is the Euler gamma function. Consequently,
The condition (6.5) is thus fulfilled with \(\gamma =(1/2)-(1/p)\,,\) for some positive constant C. Consider now the solution \(\varphi (t;\,E)\) associated with the autonomous equation for the Hamiltonian (6.11), built from the initial value \(\varphi (0;1) = (0,\sqrt{2}).\) It is fairly immediate that
On the other hand, denoting by \(\varphi _x, \varphi _y\) the components of \(\varphi ,\) and adapting (2.12) with \(\mu = 1-(1/2)-(1/p)\) , we see that
From there, it can be checked that \(\sqrt{E}\, \partial \varphi _x(t;\,E)/\partial E\) remains bounded for \(E \rightarrow +\infty ,\) uniformly for t in compact sets. Hence, using (6.10), it follows that A7’ holds with \(\gamma =(1/2)-(1/p)\), 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 \(2\pi\)-periodic solution.
Remark 6.4
Actually, the conclusion still holds for the equation
no matter what value the coefficient \(k>0\) takes. Indeed, if \(k \ne m^2/4\) for any integer m, Theorem 3.1 applies.
Remark 6.5
Similar situations have been considered in [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.
6.3 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 \(2\pi\)-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 \(2\pi /n_0,\) for some positive integer \(n_0.\) In that case, it is no longer possible to resort directly to Lemma 5.1, since \(\deg (I-P^{(0)}_{2\pi } , \Omega (E_0), 0)\) is no longer defined. But the arguments still work with an adapted homotopy, i.e.,
the choice of \(\sigma \ne 0,\) its sign in particular, being explained below. It is immediate that, for \(\lambda =0,\) the minimal period of the large solutions now becomes \(2\pi /(n_0(1+\sigma )).\) Hence, if \(P^{(\lambda )}_{2\pi }\) now denotes the Poincaré map for the period \(2\pi\) associated with Eq. (6.12), the arguments used in Lemma 5.1 tell us that, provided that \(\sigma \ne 0\) is taken small enough, the degree \(\deg (I-P^{(0)}_{2\pi } , \Omega (E_0), 0)\) 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.
Theorem 6.6
Let the assumptions A1 to A3 hold, as well as L1, L2, A6’ and A7’. Let H be isochronous with period \(2\pi /n_0.\) With \(\gamma\) the constant appearing in assumption A7’, assume that there exists \(\eta >0\) such that either, for every \(\tau _0\in [0,4\pi /n_0]\),
or, for every \(\tau _0\in [0,4\pi /n_0]\),
Then, equation (1.3) admits at least one \(2\pi\)-periodic solution.
Proof
As in the proof of Theorem 3.1, we write the solutions of equation (6.12) under the form
Adapting (5.5), (5.6) leads to
Assume that (6.14) holds. In this case, since \(T(E_0)=2\pi /n_0\), we will take \(\sigma >0\) small enough, in order to show that, for any integer n,
Using A6’ in (6.17), we have that
while using A7’ in (6.16), we get, for any compact interval \(I\subseteq {\mathbb {R}}\),
and hence, by Lebesgue’s Theorem,
Then, taking \(\sigma >0\) small enough, we see that (6.18) could hold only if \(n=n_0\); moreover, by (6.14),
for every \(\tau _0\in [0,4\pi /n_0]\). It then follows from Fatou’s Lemma and A7’ that there exists a number \(\eta >0\) such that
Hence, if \(E_0\) is taken large enough,
showing that (6.18) cannot hold even if \(n=n_0\).
The case when (6.15) holds can be treated similarly. \(\square\)
Remark 6.7
We notice that it is not really necessary that H be isochronous of period \(2\pi /n_0;\) 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]).
As an illustration, consider the case of the Hamiltonian function
with
\(n_0\) 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 \(2\pi /n_0\). Denoting by \(\varphi _x, \varphi _y\) the components of \(\varphi ,\) using (2.11), condition (6.14) with \(\gamma =1/2\) reduces to
for every \(\tau _0\in [0,2\pi /n_0]\). Since \(\eta\) can be taken arbitrarily small, if we assume \(r_x(t,x)\) to be globally bounded, this condition is fulfilled when, for every \(\tau _0\in [0,2\pi /n_0]\),
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.
7 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 \(z(0)= z_0.\)
Lemma 7.1
Let the function \(H: {\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) be twice continuously differentiable. Assume that the hypotheses of Theorem 3.1hold, except for A5, not needed here. Then, for any given compact interval \(I\subseteq {\mathbb {R}}\),
Proof
Let us write
Since the function H is twice continuously differentiable, it is well-known that the function \(E \mapsto T'(E)\) is continuously differentiable (see, for instance, [27, Theorem 3.4.1]). Since \(\Delta (0;z_0) = 0,\) 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 \(\varphi (t;\,E)\) being, as before, a solution of the autonomous system (2.1), as introduced in Sect. 2.1. Since \(\varphi (t;\,E)\) 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 \(t \in {\mathbb {R}},\) and all E sufficiently large, we can also write
Taking the scalar product with r(t, z(t)) and remembering that
and
the conclusion follows from (7.2) and assumption A7. \(\square\)
We are now in a position to state a multiplicity result based on a generalized version of the Poincaré–Birkhoff Theorem.
Theorem 7.2
Let the function \(H: {\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) be twice continuously differentiable and satisfy
Assume that the hypotheses of Theorem 3.1hold, with \(r(t,z)=\nabla _z R(t,z)\), except for the nonresonance condition A5, replaced by A5 \(''\). Moreover, let
Then, system (7.1) admits an infinite number of periodic solutions.
Proof
By assumption A5\(''\), it is possible to find positive integers m, n, and a positive number \(\varepsilon ,\) such that
We then build increasing sequences \((E_k)_k\), \((E^*_k)_k\), with \(E_k \rightarrow +\infty\), such that \(E_{2k}< E^*_{2k}<E^*_{2k+1} < E_{2k+1}\) and
We may assume that \(\varepsilon >0\) has been chosen small enough so that
which implies that
To the above defined sequences we associate, for large values of k, the annuli
Notice that \({{\mathcal {A}}}^*_k\) is contained in \({{\mathcal {A}}}_k.\)
For some sufficiently large k, to be fixed below, we now modify our system. The idea is to leave it unchanged in \({{\mathcal {A}}}^*_k,\) and to cancel the forcing term on \({\mathbb {R}}^2 \setminus {{\mathcal {A}}}_k,\) the transition in the zone \({{\mathcal {A}}}_k \setminus {{\mathcal {A}}}^*_k\) being built as described below. We thus replace R(t, z) by
with \(\chi : {\mathbb {R}}\rightarrow {\mathbb {R}}\) a \(C^\infty\)-function, built in such a way that
We may assume that \(|\chi '(s) |\le 4/\varepsilon\), for every \(s\in {\mathbb {R}}\).
We now consider the modified system
Since
it is readily seen that \({{\widetilde{r}}}(t,z):=\nabla _z{{\widetilde{R}}}(t,z)\) inherits from r(t, z) the regularity properties required for the application of Theorem 3.1. Moreover, using hypothesis (7.5), we observe that \(\nabla _z{{\widetilde{R}}}(t,z)\) 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 \({\widetilde{z}}(t;z_0)\) 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 \({\widetilde{z}}(t;z_0)\) of system (7.7) issued from a point \(z_0\) in \({{\mathcal {A}}}^*_k\) does not enter the zone where the equation has been modified, when \(t\in [0, 2\pi m]\). Indeed, with \(z_0 \in {{\mathcal {A}}}^*_k,\) assume that for some \(t_0\in [0, 2\pi m]\) either \(H({\widetilde{z}}(t_0;z_0))= E_{2k}^*\), or \(H({\widetilde{z}}(t_0;z_0)) = E_{2k+1}^*\), and
Applying Lemma 7.1 with \({\widetilde{z}}(t_0;z_0)\) instead of \(z_0\), if k is taken large enough, then
from which follows that
This means that, for any \(t \in [0,2\pi m],\) either \({\widetilde{z}}(t;z_0)\) belongs to \({{\mathcal {A}}}^*_k,\) or \(\chi (T(H({\widetilde{z}}(t;z_0)) ))=1.\) In any case, we have that \({\widetilde{r}}(t, {\widetilde{z}}(t;z_0)) = r(t, {\widetilde{z}}(t;z_0)),\) for \(t \in [0, 2\pi m],\) meaning that \({\widetilde{z}}(t;z_0)\) is actually a solution of the original system (7.1).
On the other hand, we claim that there are no \(2\pi m\)-periodic solutions \({\widetilde{z}}(t;z_0)\) of system (7.7) starting from a point \(z_0\) in \({{\mathcal {A}}}_k \setminus {{\mathcal {A}}}^*_k\), with k large enough. Indeed, if \({\widetilde{z}}(t;z_0)\) 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 \({{\mathcal {A}}}_k\) are strictly star-shaped with respect to the origin; moreover, if \(z_0\in {{\mathcal {A}}}_k\), then \(z(t;z_0)\ne (0,0)\), for every \(t\in [0,2\pi m]\). 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 \({{\mathcal {A}}}_k\). For k sufficiently large, by (7.4) the function \(\arg z(\cdot \,;z_0)\) is strictly decreasing when \(z_0\in H^{-1}(E_{2k})\), 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 \(2\pi m\)-periodic solutions \(z^{(1)}(t)\), \(z^{(2)}(t)\), starting from the interior of \({{\mathcal {A}}}_k\), such that
By the above arguments, these are \(2\pi m\)-periodic solutions of the original system (7.1), and the proof is thus completed. \(\square\)
We end this section with two remarks.
Remark 7.3
When the system satisfies assumption A8, it is possible to deduce condition (7.5) in Theorem 7.2 from conditions based more directly on H(z) and R(t, z). In particular, if H is twice differentiable and \(H''\) globally bounded, using (3.7) (with \({{\mathcal {D}}}(E)\) the identity matrix) and (3.9), it results from (7.3) that there exists a constant such that
so that condition (7.5) 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 \(\gamma >0\) the same constant as in (6.5) and (6.6). The resulting periodic solutions would then be of period \(2\pi ,\) making \(n_0\) turns around the origin on a time interval of length \(2\pi .\)
8 More general differential equations
In this section, we consider the planar system
We will assume throughout that F(t, z) is continuous, \(2\pi\)-periodic in t, and locally Lipschitz continuous in z. We will provide existence conditions for \(2\pi\)-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).
Lemma 8.1
Let \(H^*:{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) satisfy conditions A1 to A3, and let \(\varphi ^*(t;\,E)\) be the function associated with \(H^*\) by hypothesis A3. Assume that \(\varphi ^*(t;\,E)\) is of minimal period \(T^*\) in t, the period being independent of E, for \(E>1,\) and that \(\varphi ^*(0;\,E)\) satisfies (2.5). Consider a parametric curve \(t \mapsto z(t)=\varphi ^*(\sigma (t);e(t)),\) with \(z(T)=z(0)\), the functions \(\sigma (t)\), e(t) being differentiable on \({\mathbb {R}},\) and such that \(e(t) >1,\) for all \(t\in [0,T].\) Assume that
If the curve \(t \mapsto z(t)\) makes n turns around the origin on the interval [0, T], then
Conversely, if the above equality holds for some nonnegative integer n, the curve \(t \mapsto z(t)\) makes n turns around the origin on the interval [0, T], in the clockwise sense.
Proof
The curve \(z:t \mapsto z(t)\) crosses the gradient curve \(E \mapsto \varphi ^*(0;\,E),\) when and only when \(\sigma (t) = 0\) mod \(T^*.\) The condition (8.2) means that the curve z is transversal to the gradient curve \(\varphi ^*(0;\cdot ),\) 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 \(\{z \in {\mathbb {R}}^2 \mid H(z) >1\}\) for \(t \in [0,T],\) the number of turns of z around the origin on the interval [0, T) is equal to the number of crossings with the curve \(\varphi ^*(0;\cdot ).\) Because of (8.2) and (2.5), we have \(\sigma '(t) >0\) when \(\sigma (t) = 0\) 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]. \(\square\)
Notice that the above result also makes sense with \(n=0.\)
8.1 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 \(\nabla H_1,\) \(\nabla H_2,\) whereas, in Sect. 3, the hypotheses concern more directly the functions \(H_1,\) \(H_2\) themselves, through the relations (4.1).
Let the three Hamiltonian functions H, \(H_1,\) \(H_2\) satisfy conditions A1 to A3. Assume that the functions \(\varphi _1(t;\,E),\) \(\varphi _2(t;\,E),\) associated respectively with \(H_1,\) \(H_2,\) satisfy Eq. (2.5). The respective minimal periods of H, \(H_1,\) \(H_2,\) will be denoted by T(E), \(T_1,\) \(T_2,\) the Hamiltonian functions \(H_1\) and \(H_2\) being assumed to be isochronous. Notice that \(\nabla H_1(z),\) \(\nabla H_2(z)\) are not required to be locally Lipschitz continuous.
Theorem 8.2
Assume that the functions \(H, H_1,\) \(H_2\) satisfy the assumptions A1 to A3, that \(\nabla H\) is locally Lipschitz continuous, and that the Hamiltonian functions \(H_1\) and \(H_2\) are isochronous. Assume that
With
assume that
for \(\lambda \in \{0,1\},\) \(t \in [0,2\pi ],\) and E sufficiently large. Assume moreover that, for any given compact interval \(I\subseteq {\mathbb {R}}\),
for \(\lambda \in \{0,1\},\) uniformly for \((t, s)\in [0,2\pi ]\times I\). If for some integer \(n_0,\)
then Eq. (8.1) admits at least one \(2\pi\)-periodic solution.
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.
Proof
We denote by \(P^{(\lambda )}_{2\pi }\) the Poincaré map for the period \(2\pi ,\) associated with Eq. (8.4); reasoning as in the proof of Theorem 3.1, this map can be shown to be well defined. Indeed, letting \(z^{(\lambda )}(t;z_0)\) denote the solution of (8.4) for the initial condition \(z(0)=z_0,\) and defining \(e^{(\lambda )}(t;z_0) = H(z^{(\lambda )}(t;z_0)),\) we have
We then deduce from (8.5) that \(e^{(\lambda )}(t;z_0)\) remains bounded on any compact interval, from which follows, by the coercivity condition A1, that \(z^{(\lambda )}(t;z_0)\) can be extended to the whole real line and \(|z^{(\lambda )}(t;z_0) |\) tends to \(+\infty ,\) for \(E_0 \rightarrow +\infty ,\) uniformly for \(z_0 \in H^{-1}(E_0).\) Notice that, by hypothesis A1 for \(H_1\) and \(H_2,\) both \(H_1(z^{(\lambda )}(t;z_0))\) and \(H_2(z^{(\lambda )}(t;z_0))\) also tend to \(+\infty\) for \(E_0 \rightarrow +\infty ,\) uniformly for \(z_0 \in H^{-1}(E_0)\), \(t \in [0,2\pi ]\), \(\lambda \in [0,1].\)
The theorem will be proved if we can show that, for \(E_0\) sufficiently large,
In order to prove that (8.13) holds for \(E_0\) sufficiently large, we will look at the number of turns of possible \(2\pi\)-periodic solutions of (8.4) around the origin. For that purpose, we will, in a first stage, consider those solutions as perturbations of the solutions of equation
We therefore write them under the form
Taking (2.3) into account, simple calculations lead to the system
The solution of the above system for the initial conditions \(\tau _1^{(\lambda )}(0)=\tau _0,\) \(e_1^{(\lambda )}(0)=H_1(z_0),\) where \(\tau _0\) is such that \(z_0= \varphi _1(\tau _0, H_1(z_0)),\) will be denoted by \(\tau _1^{(\lambda )}(t;\tau _0,H_1(z_0)),\) \(e_1^{(\lambda )}(t;\tau _0,H_1(z_0)).\)
Condition (8.13) will be satisfied unless, for some integer n (which may be positive, negative or zero), some \(\tau _0 \in [0,T_1],\) some \(\lambda \in [0,1],\) we have
Using Lemma 8.1 with \(H^*=H_1,\) we see that n is the number of turns of the solution around the origin, counted positively in the clockwise sense. Indeed, condition (8.2) follows from hypothesis (8.7), for any \(\lambda \in [0,1]\). Using (8.9), we deduce from (8.15) that, given any \(\eta >0,\) for sufficiently large values of \(E_0,\)
for any \(z_0 \in H^{-1}(E_0)\) (remember that \(\tau _0\) depends on \(z_0).\) Hence, by (8.11), the periodicity condition (8.17) can hold only if \(n >n_0.\)
A similar argument, with the solutions of (8.4) considered as perturbations of the solutions of
shows that \(n<n_0+1,\) so that (8.17) is impossible, n being an integer. \(\square\)
We illustrate the above theorem by considering the simple case where \(H_1\) and \(H_2\) are positively homogeneous of degree 2, i.e., they satisfy (2.9). Due to the homogeneity property of \(H_1,H_2,\) as observed in Sect. 2.2, we can build \(\varphi _1(;\,E),\) \(\varphi _2(t;\,E)\) in such a way that
(see (2.10) and (2.11)). Moreover, for \(i=1,2,\) the gradients \(\nabla H_i(\varphi _i(0;\,E))\), \(i=1,2\), are then positive multiples of \(\varphi _i(0;\,E)\) so that conditions (8.7), (8.8) are satisfied if
On the other hand, conditions (8.9), (8.10) hold if
Notice that, by the homogeneity property of \(H_1,\) \(H_2,\) the conditions (8.19) imply that, for any \(\varepsilon >0,\) we have \(H_1(z) \le (1+\varepsilon ) H_2(z),\) for |z| sufficiently large. Using, for instance, the arguments of Sect. 4.1, this in turn entails that \(T_2 \le T_1.\) We can deduce the following corollary from Theorem 8.2.
Corollary 8.3
Assume that the functions H, \(H_1,\) \(H_2\) satisfy the assumptions A1 to A3, that \(\nabla H\) is locally Lipschitz continuous, and that the functions \(H_1,\) \(H_2\) satisfy (2.9), and hence are isochronous, their respective minimal periods being denoted by \(T_1,\) \(T_2.\) Assume moreover that (8.5) is satisfied. If conditions (8.19) hold and if, for some integer \(n_0,\)
then equation (8.1) admits at least one \(2\pi\)-periodic solution.
Remark 8.4
When \(H_1\) is positively homogeneous of degree 2, the computation of the “rotation number” associated with the auxiliary function \(H_1,\) as introduced, for instance, in [1, 33], is equivalent to the computation of \(2\pi + \tau _1^{(\lambda )}(t;\tau _0,H_1(z_0)) - \tau _0,\) with \(\tau _1^{(\lambda )}\) being defined by (8.15).
It is interesting to compare the above corollary to results of Sect. 4. We will do this for the equation
with r continuous and \(2\pi\)-periodic, assuming that H is twice continuously differentiable, that \(H''\) is globally bounded, and that
the Hamiltonians \(H_1\) and \(H_2\) satisfying (2.9). Notice that condition (8.5) is then automatically satisfied. With \(T_1,\) \(T_2\) denoting the minimal periods of the nontrivial solutions of the systems associated with \(H_1,\) \(H_2,\) respectively, assume that (8.20) holds. The application of Corollary 8.3 requires that
whereas such conditions on \(\nabla H\) are not needed in Corollary 4.2, the condition (8.22) combined with the hypotheses on \(T_1\), \(T_2\) being sufficient. This difference is explained by the fact that Corollary 8.3 does not exploit the Hamiltonian structure of the autonomous equation associated with (8.21). The application of the above corollary to equation (8.21) allows to recover results obtained long ago by Sȩdziwy [30].
8.2 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.
Theorem 8.5
Assume that the functions H, \(H_2\) satisfy the hypotheses A1 to A3, and that \(\nabla H\) is locally Lipschitz continuous. Let H(z), F(t, z) be such that (8.5) holds. Assume that
for \(t \in [0,2\pi ],\) |z| sufficiently large. Assume moreover that, for any given compact interval \(I\subseteq {\mathbb {R}}\),
uniformly for \((t, s)\in [0,2\pi ]\times I\). If
then Eq. (8.1) admits at least one \(2\pi\)-periodic solution.
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.
8.3 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, \(2\pi\)-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 \(\alpha _2\ge \alpha _1>0\) and \(\beta _2\ge \beta _1>0\) such that
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 \(+\infty\) and for x going to \(-\infty ,\) 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 \(|\nabla R(z)| = o(|z|)\) for \(|z| \rightarrow +\infty .\) Denoting, as before, by \(\varphi _1(t;\,E),\) \(\varphi _2(t;\,E)\) 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 \(\alpha _1, \beta _1\) 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 \(T_2>2\pi\), or for some integer \(n_0,\) condition (8.20) holds, then the system
admits a \(2\pi\)-periodic solution.
A result close to the above corollary can be found in [6], for the case of a second-order 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.
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Acknowledgements
We would like to thank Andrea Sfecci and Fabio Zanolin for their helpful comments and suggestions. The second author has been partially supported by the Italian research association GNAMPA–INdAM.
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Fabry, C., Fonda, A. A systematic approach to nonresonance conditions for periodically forced planar Hamiltonian systems. Annali di Matematica 201, 1033–1074 (2022). https://doi.org/10.1007/s10231-021-01148-9
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DOI: https://doi.org/10.1007/s10231-021-01148-9