PERIODIC SOLUTIONS OF THE BRILLOUIN ELECTRON BEAM FOCUSING EQUATION

. Quite unexpectedly with respect to the numerical and analyti-cal results found in literature, we establish a new range for the existence of 2 π − periodic solutions of the Brillouin focusing beam equation This is possible thanks to suitable nonresonance conditions acting on the ro- tation number of the solutions in the phase plane.

1. Introduction and main results. This note deals with the 2π-periodic boundary value problem for the equation x where b is a positive constant. We will be interested in the existence of positive 2πperiodic solutions of (1) (thus avoiding taking into account solutions with collisions).
The physical meaning of equation (1) arises in the context of Electronics, since it governs the motion of a magnetically focused axially symmetric electron beam under the influence of the Brillouin flow, as shown in [1]. From a mathematical point of view, (1) is a singular perturbation of a Mathieu equation, as we will explain below.
Motivated by some numerical experiments realized in [1], where it was conjectured that, if b ∈ (0, 1/4), equation (1) should have a 2π-periodic solution, in the last fifty years the work of many mathematicians has given birth to an extensive literature about this topic. Although at the moment the conjecture has not been correctly proven yet, many advances in this line have been obtained, allowing to understand that proving the existence of 2π-periodic solutions of (1) when b ∈ (0, 1/4) can be a really delicate issue, and arising doubts on the validity of the result conjectured in [1].
The first analytic work on the periodic solvability of (1) was realized by T. Ding in [7]. There, it was proven that if b ∈ (0, 1/16), then equation (1) has at least a 2π-periodic solutions. Later, other works showed that uniqueness holds under the previous hypothesis (see for instance [14,15,21]).
To the best of our knowledge, the following step towards the resolution of the conjecture was done by M. Zhang, extending existence to the interval (0, 0.1532) using a contraction argument applied to a positive linear operator (see [19]).
A couple of years later, the same author determined in [20] the best range of b actually known for the 2π-periodic solvability of (1), using a non-resonance hypothesis for the associated Mathieu equation (a particular Hill type equation) x + b(1 + cos t)x = 0. ( In order to prove such a result, the author considered there the function K : [1, +∞] → R defined by with this definition, a sufficient condition in order for the Dirichlet problem (x(0) = 0 = x(2π)) associated with (2) to have a unique solution is that Under this non-resonance condition, i.e., if b ∈ (0, 0.16448), equation (1) has at least one 2π-periodic solution. This last result has been extended to equations where the singularity may be of weak type (see [16]). Actually, the function K is a powerful tool in order to study the existence of periodic solutions for more general versions of equation (1) (see [5,17]).
An important result to understand the difficulty of showing the validity of the conjecture proposed in [1] was proven in [21]. In that paper, it was established an unanimous relation between the stability intervals for the Mathieu equation (2) and the existence of periodic solutions for the Yermakov-Pinney equation In particular, [ Figure 1]). This suggests that, in order to obtain a correct proof of the conjecture by V. Bevc, J. L. Palmer and C. Süsskind, one has to take into account some feature of equation (1) which equation (3) does not have. Indeed, if we assumed that (3) has at least one 2π−periodic solution whenever b ∈ (0, 1/4), then one could take b sufficiently close to 1/4 in such a way that it does not belong to any stability interval of (2), and this would contradict [21,Theorem 2.1]. It has to be said that, in [13,Theorem 3.2], it was obtained that (1) has a periodic solution when b ∈ (0, 1/4). Unfortunately, the same theorem seems applicable to equation (3) obtaining the same conclusion, but this it is not possible according to the previous discussion. This contradiction seems to leave the conjecture in [1] still open.
However, in this work we are not able to prove or disprove the result which was conjectured in [1], but we will show that (1) may have periodic solutions also when the parameter b belongs to intervals other than (0, 1/4). In fact, we will prove the following. This result seems in some sense unexpected, according to the numerical ones obtained in [1], where it was observed that when the parameter b begins to cross the umbral 1/4, the 2π-periodic solvability of (1) is not clear. It seems indeed that Theorem 1.1 is the first result of existence for (1) when b does not belong to the first stability interval of equation (2) (notice that we are dealing with values of the parameter b belonging to the second stability interval of equation (2), agreeing with [21, Theorem 2.1]). Moreover, as it can be seen in Remark 2.3, our result is in some sense optimal, when some additional control is required on the nonlinearity. Theorem 1.1 follows from a general existence result, Theorem 2.1 below, obtained thanks to suitable nonresonance assumptions which can be traced back to the work [8] by Fabry, as explained in Remark 2.2. The main abstract tool involved in its proof is the Poincaré-Bohl fixed point theorem.
The structure of this short note is as follows: in Section 2, we will prove Theorem 2.1 concerning strong singular perturbations of a Mathieu equation. As a consequence, in Section 3 we will prove a general proposition (Proposition 3.1) allowing to prove Theorem 1.1.

2.
A non-resonance theorem for singular perturbations of a Mathieu equation. The proof of Theorem 1.1 is based on a non-resonance result which involves nonlinearities with "atypical" linear growth, and could have interest by itself.
for some natural number n. Then (1) has at least one 2π-periodic solution.
Before introducing the main tools to prove the theorem, a couple of remarks are in order.
Remark 2.1. With the aim of keeping the exposition at a rather simple level, and taking into account that we mainly want to study the existence of 2π-periodic solutions of (1), we will always consider equation (1) as a starting point. However, the result can be extended, with the same approach and similar computations, to more general equations likeẍ where a(t) is continuous and 2π-periodic, and g : [0, 2π] × (0, +∞) → R has a similar behavior as −1/x γ , with γ ≥ 1, near x = 0, being allowed to grow at most sublinearly at infinity. For instance, as in [11], one can assume that there exist σ > 0 and a continuous function f : Of course, in this case a(t) will replace b(1 + cos t). (4) and (5) were introduced by Fabry in [8] for the equationẍ + g(t, x) = 0, with

Remark 2.2. Conditions
asking that where λ j is the j-th eigenvalue of the considered 2π-periodic problem. Such conditions are usually coupled with the sign assumption lim inf |x|→+∞ sgn x g(t, x) > 0 (see for instance [9]), which, however, in the model case , with lim |x|→+∞ f (t, x) = 0, is not satisfied. This is one of the main difficulties of the problem considered in the present paper.
As it is easy to see, (4) and (5) are the counterpart of Fabry's conditions for the Dirichlet spectrum (which is the natural one to consider when dealing with problems with a singularity, see [20]). Remark 2.3. As a consequence of Theorem 2.1, we can obtain the main results in [6,11]. Indeed, assume that there exist positive constants A + , B + such that Then, according to [6,11], there exists at least one 2π-periodic solution of (6) under the nonresonance assumption where n ∈ N. It is easy to obtain this result from Theorem 2.1, since from (7) we deduce that Under (7), from the point of view of resonance, the results in [6,11] are optimal, in view of the counterexample produced in [2], according to which there exist forcing terms e(t) such that the equation has no 2π-periodic solutions. Thus, Theorem 2.1 seems to be optimal whenever we are able to control a(t) with estimates like (7) and (8), essentially requiring, in this case, a nonresonance assumption. On the other hand, the mean conditions (4) and (5) do not ask that a(t) is controlled like in (7) and (8), allowing it to possibly cross some eigenvalues (cf. [3]) as in our case, being 0 ≤ a(t) ≤ 2b.
We are now going to prove Theorem 2.1. As it was mentioned previously, we will have to overcome the difficulty of working with nonlinearities with atypical linear growth, since, in our concrete case, the nonlinearity grows linearly towards the function b(1 + cos t)x, which vanishes at some times. For this reason, classical arguments in literature (like the ones in [9,10,11]) do not extend as they are to (1), because it is not possible to construct an admissible spiral which allows to control the dynamics of the solutions.
We will prove Theorem 2.1 by means of some preliminary lemmas. To this aim, denoting by Λ the half-plane with positive abscissa, i.e. Λ = {(x, y) ∈ R 2 : x > 0}, it will be convenient to define the application Moreover, for a fixed number c, we will denote the corresponding level curve of N (x, y) by γ c , i.e., γ c = {(x, y) ∈ Λ : N (x, y) = c}. It is worth observing that the function N (x, y) reaches its minimum in the point , the level curves of N turn around P 0 , being the union of two symmetric arcs joining on the x-axis.
For our problem, the function N (x, y) is the analogue of a norm, as usual in problems with a singularity (see, for instance, [11]). In particular, a solution of equation (1) is considered "large" if N (x,ẋ) is "large", and this can happen if the C 1 -norm of x(t) is large, or if x(t) approaches the singularity. Usually, functions similar as N (x, y) are involved in proving the validity of some elastic property, thanks to the construction of a suitable spiral-like curve controlling the solutions; this is strictly connected to the possibility of giving precise estimates on the rotation number (in the phase plane) of the solutions having large norm. In our problem the situation is slightly more complicated, but through the use of N (x, y) we still reach useful estimates on the solutions of (1), as we are going to see.
We will look at the solutions of (1) in the phase plane, considering thus the couple (x,ẋ). As already mentioned, we are interested in positive solutions, so that we will take into account the dynamics of the solutions in the right half-plane. The first lemma ensures the global continuability, i.e., shows that the maximal domain of every solution of (1) is [0, +∞).
Proof. Since For every solution x(t) of (1), we define the function where I is the maximal domain of x(t). We are going to prove that I = [0, +∞). Since from which it can be deduced that Indeed, if t ≥ 0 is such that x(t) ≥ K 0 , then (9) implies that U (t) ≤ CU (t). On the contrary, if x(t) ≤ K 0 , we deduce that either x(t) ≤ 1, and then U (t) ≤ CU (t), or 1 ≤ x(t) ≤ K 0 , and thus (10) holds. Now, according to the Gronwall-Bellman Lemma, the result is proven.
As it was mentioned in the previous discussion, equations like (1) do not admit the existence of an admissible spiral controlling the solutions. However, the following result ensures that (1) has the "property of elasticity", at least locally. Roughly speaking, this means that if there is a time when the norm of the solution is large enough, then, for every preceding time instant, the solution had to be large (in norm). Precisely, we have the following. Lemma 2.3. Let ρ 0 > 0 be sufficiently large. Then, there exists R 1 > ρ 0 such that, for every solution x(t) of (1) satisfying Proof. We first observe that there exists a constant M > 0 such that for every (x, y) ∈ Λ. Now, choosing ρ 0 > 0 sufficiently large, there exist where F 0 : (0, +∞) → R is a function such that F 0 (u − 0 ) = F 0 (u + 0 ) = 0, having constant sign on (u − 0 , u + 0 ). Let us fix L 1 satisfying bx 2 − 2 ln x + 2ρ 0 , and consider γ 2L1 : explicitly, Thus, there exist u − 1 < u − 0 < u + 0 < u + 1 such that, similarly as before, where F 1 : (0, +∞) → R is defined by F 1 (x) = 2L 1 − (bx 2 − 2 ln x) (and consequently vanishes in u − 1 , u + 1 ). On the other hand, we take L 2 > e 2πM L 1 , and consider the level curve γ 2L2 , which is explicitly given by Finally, we fix R 1 > 2L 2 , so that Assume that there exists x(t) solving (1) such that N (x(t 1 ),ẋ(t 1 )) ≥ R 1 , but there is t * ∈ [0, t 1 ) such that N (x(t * ),ẋ(t * )) ≤ ρ 0 . By continuity, we can assume that there exist t * < t * such that (x(t * ),ẋ(t * )) ∈ γ 2L1 and (x(t * ),ẋ(t * )) ∈ γ 2L2 ; setting, as in Lemma 2.2, U (t) = N (x(t),ẋ(t))/2, this explicitly means that According to (11) and (12), from the definition of U (t) we deduce that which implies, thanks to the Gronwall-Bellman Lemma, that This, however, contradicts (12), in view of the definition of L 2 . Now, intuitively speaking, we will prove that either the solutions of (1) have the global elasticity property, or their norm in the instant t = 2π is lower than in the initial one. This useful property is similar to the one introduced in [10]. Lemma 2.4. Let ρ 0 > 0 be sufficiently large. Then, there exists R 2 > ρ 0 such that, for every solution x(t) of (1) fulfilling it is either N (x(t),ẋ(t)) ≥ ρ 0 for every t ∈ [0, 2π], (14) Proof. Let us take R 1 as in the statement of Lemma 2.3, for the fixed ρ 0 . In the same way, we apply again Lemma 2.3, this time with R 1 playing the role of ρ 0 , finding the corresponding R 2 for which the statement holds. Assume now that there exists a solution x(t) of (1) satisfying (13), for which it is Since there exists t 2 ∈ [0, 2π] such that N (x(t 2 ),ẋ(t 2 )) ≥ R 2 , Lemma 2.3 implies that N (x(0),ẋ(0)) ≥ R 1 , so that, in view of (16), N (x(2π),ẋ(2π)) ≥ R 1 . Consequently, using again Lemma 2.3, we obtain that N (x(t),ẋ(t)) ≥ ρ 0 for t ∈ [0, 2π].
We are now able to show that an adaptation of the arguments in [9,11] to our equation allows to prove that the global elasticity property cannot be fulfilled for solutions of (1) with large norm which perform an integer number of revolutions when t goes from 0 to 2π. Proof. In view of (4) and (5), there exists a positive number δ such that In correspondence of δ, we can find K δ > 0 such that, for every x ∈ [1, +∞) and every t ≥ 0, Moreover, we choose ρ 1 and B + large, in such a way that the following relations hold: In order to perform the estimates leading to the result, we first fix ρ 0 > 0 sufficiently large and apply Lemma 2.4 in order to find R 2 > ρ 0 such that the statement therein holds. Then, we fix a solution x(t) of (1) satisfying (13) and such that, in the phase plane, the couple (x(t),ẋ(t)) performs an integer number of revolutions around (1, 0) -say k ∈ N -during the time interval [0, 2π]. Thus, assume by contradiction that (15) is not satisfied; then, in view of Lemma 2.4, x(t) fulfills (14). We are now going to estimate the time needed by (x(t),ẋ(t)) to rotate k times around the point (1, 0), by dividing the half-plane Λ in vertical strips and analyzing the behavior of (x(t),ẋ(t)) in each strip, following the procedure used in [11]. As a first step, we perform our estimates in the strip {x > 1}. Passing to modified polar coordinates around (1, 0) by writing where µ > 0, we obtain Setting in view of the properties of the modified rotation numbers (see, for instance, [8]) we have that Consequently, in view of (17), Taking into account that the function is non-decreasing for α ≤ β, Without loss of generality, we can assume (up to enlarging ρ 0 ) that R 2 is sufficiently large, so that B + (x − 1) 2 +ẋ 2 ≥ ρ 1 , for every t ∈ J + . Therefore, (22) implies We now pass to compute the time spent by (x(t),ẋ(t)) to perform k/2 revolutions on the "left" half phase plane, i.e. when x ∈ (0, 1]. Preliminarily, we fix and observe that, since In this way it is possible to define both the sets 1} and, correspondingly, the time instants t 1 , t 2 , t 3 and t 4 (as in Figure 1) such that, in the time t 4 − t 1 , the couple (x(t),ẋ(t)) performs half a turn in the "left" half phase plane (x ∈ (0, 1]), and Passing to usual polar coordinates around (1, 0), i.e., x = 1 + ρ cos ϑ,ẋ = ρ sin ϑ, In view of (25), we deduce that −θ(t) > K cos 2 ϑ(t) + sin 2 ϑ(t), t ∈ [t 2 , t 3 ], so that ds K cos 2 s + sin 2 s ds According to (24), it follows that t 3 − t 2 < η/2; repeating the argument for every revolution made by (x,ẋ) around (1, 0) yields In order to compute t 2 − t 1 , we observe that, thanks to (26), it holds where C = max x∈ [d,1] 2bx + 1/x. Again, we assume that ρ 0 is large enough, so that −θ(t) > 1/2 on [t 1 , t 2 ], and t 2 − t 1 < η/4. Analogously, one can prove that t 4 − t 3 < η/4, having thus that Thus, in view of (24) and (27), we deduce that This relation, together with (23), gives On the other hand, reasoning on (20) with a similar argument and taking (17) into account, we have Since the function Ψ defined in (21) is non-increasing whenever α ≥ β, choosing µ = A + and taking α = max b(1+cos t)+δ A+ , 1 (x − 1) 2 and β = (x − 1) 2 , we obtain Again, we can assume ρ 0 (and thus R 2 ) so large that We are now able to conclude the proof. Assume first that x(t) − 1 has at most 2n zeros. Then k ≤ n, but this contradicts (18) and (28). On the contrary, if x(t) − 1 has at least 2n + 2 zeros (notice that, in any case, the number of zeros is even), since k ∈ N it has to be k ≥ n + 1. However, this contradicts (19) and (29). The proof is completed.
Remark 2.4. In [3], the relationships between conditions (4) and (5) and the rotation number of "large" solutions of a first order planar system were highlighted. This perfectly agrees with what we have seen in the proof which has just been performed; indeed, conditions (4) and (5) force "large" solutions of the Cauchy problems associated with (1) not to perform an integer number of turns around (1, 0) in the time interval [0, 2π]. Thus, they turn to be hypotheses on the number of rotations made by the solutions of equation (1) in the phase plane.
Using the previous results, a basic application of the Poincaré-Bohl Theorem allows to prove Theorem 2.1.
3. Proof of Theorem 1.1. In order to prove Theorem 1.1 it will be convenient, for any n ∈ N, to define the absolutely continuous functions F n , G n : (0, +∞) × (0, +∞) → R by Such functions are non-decreasing with respect to the variable b. Moreover, if there exists n ∈ N such that inf x>0 G n (b, x) < 0 and sup x>0 F n (b, x) > 0, then Theorem 2.1 implies that (1) has at least one 2π-periodic solution. Therefore, we have the following proposition.
Proposition 3.1. Assume that there exists n ∈ N such that b ∈ inf b > 0 : sup Then, (1) has at least one 2π-periodic solution.
Let us first observe that, in view of the continuity and the monotonicity of the functions F n , G n in the variable b, there exist b n 0 and b n 1 such that b > 0 : sup x>0 F n (b, x) > 0 = (b n 0 , +∞), and b > 0 : inf x>0 G n (b, x) < 0 = (0, b n 1 ). The point is to prove that these two intervals contain common points, i.e., b n 0 < b n 1 . We will show this in the case when n = 0 and n = 1, and the estimates performed in this last case will allow to achieve the new result consisting in Theorem 1.1. In particular, a gross estimation of the interval in (30) would lead to prove existence for b ∈ n 2 2 , (n + 1) 2 4 π 1 + π 2 .

Summary
In the present paper, we have studied the 2π-periodic solvability of the Brillouin focusing beam equation