Nonlinear Analysis: Real World Applications 64 (2022) 103441 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa Periodic bouncing solutions of the Lazer–Solimini equation with weak repulsive singularity David Rojas a ,∗, Pedro J. Torres b a Departament d’ Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, 17003 Girona, Spain b Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain article info abstract Article history: We prove the existence and multiplicity of periodic solutions of bouncing type Received 17 May 2021 for a second-order differential equation with a weak repulsive singularity. Such Received in revised form 23 September solutions can be cataloged according to the minimal period and the number of 2021 elastic collisions with the singularity in each period. The proof relies on the Accepted 28 September 2021 Poincaré–Birkhoff Theorem. Available online 14 October 2021 © 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under Keywords: the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Periodic solution Bouncing Singularity Poincaré–Birkhoff theorem 1. Introduction Differential equations with singularities appear as mathematical models in many scientific areas and have been studied from many viewpoints [1]. In this paper, we consider the singular second order differential equation 1 ü − α = p(t), u > 0, (1) u with parameter α > 0 and p : R → R a continuous and 2π-periodic function. In a seminal paper, Lazer and Solimini [2] proved that when α ⩾ 1 Eq. (1) has a positive periodic solution if and only if p has negative mean value. The authors also showed that the statement is sharp with respect to the parameter α in the sense that if 0 < α < 1, a function p with negative mean value can be constructed in such a way that (1) has no periodic solutions. Later, [3, Example 3.9] provided an effective sufficient condition over p for the existence of a classical periodic solution in the weak repulsive case. The particular case α = 1/2 has been studied in [4] showing that the equation corresponds to a perturbed isochronous oscillator and resonance conditions on the forcing term p(t) are given. ∗ Corresponding author. E-mail addresses: david.rojas@udg.edu (D. Rojas), ptorres@ugr.es (P.J. Torres). https://doi.org/10.1016/j.nonrwa.2021.103441 1468-1218/© 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 In the mentioned references, existence of solutions is understood in the classical sense and collisions with the singularity are not allowed. The goal of the present paper is twofold. First, we aim to extend the notion of solutions of Eq. (1) for 0 < α < 1 admitting elastic collisions with the singularity at x = 0. Second, we prove the existence of harmonic and sub-harmonic bouncing solutions of Eq. (1) for any negative 2π-periodic forcing p(t). For the analogous equation with attractive nonlinearity (that is, changing the sign of the second term of the left-hand side of the equation), the notion of bouncing solution has been adequately defined and studied in a number of papers [5–12]. In contrast, it remains unexplored for the repulsive case. Our aim is to fill, at least partially, this gap. The structure of the paper is as follows. In Section 2, we analyze in detail the autonomous case (when the forcing term p(t) is constant), including the associated period function and the continuation of colliding orbits. In Section 3, we define rigorously the notion of bouncing solution, proving that the initial boundary value problem is well-defined and continuable to the whole real line. Section 4 begins with the definition of the so-called successor map, which is a section of the flux whose fixed points are equivalent to periodic solutions of the equation. It can be proved that this map is area-preserving and a suitable version of the Poincaré–Birkhoff Theorem can be applied by using the estimates from Section 2, leading to the main results. 2. The integrable weak-singular system Throughout this section we consider a general potential function V ∈ C 2 (I) defined in an open interval I = (α, +∞) satisfying lim V (u) = h∗ and lim V ′ (u) = −∞. u→α+ u→α+ Additionally we assume that u = 0 is the only local minimum of V . More precisely, V (0) = V ′ (0) = 0, V ′′ (0) > 0 and uV ′ (u) > 0 if u ̸= 0. Under these hypotheses it is clear that there exists β > 0 with lim V (u) = lim V (u) = h∗ > 0, u→β u→α+ and the equation ü + V ′ (u) = 0, u ∈ I, (2) has a center at the origin with a bounded period annulus, namely P, which projection over the u-axis is the open interval (α, β) (see Fig. 1.) The associated first order differential system is a Hamiltonian system with Hamiltonian function H(u, u̇) = 21 u̇2 + V (u). In particular, the energy at the outer boundary of the period annulus is h∗ . From the first integral H we have that periodic orbits inside the period annulus correspond to energy levels h ∈ (0, h∗ ). On the other hand, energies greater than h∗ correspond to solutions that collide with the singularity. More precisely, if u(t) is a solution of (2) with H(u, u̇) = h ⩾ h∗ and initial condition u̇(0) = 0, there exists t0 > 0 such that u is defined on the interval (−t0 , t0 ) and limt→t− u(t) = limt→−t+ u(t) = α. Moreover, by conservation of energy the limits limt→t− u̇(t) = u̇(t− 0) 0 0 0 − and limt→−t+ u̇(t) = u̇(−t+ + 0 ) exist and the equality u̇(t0 ) = −u̇(−t0 ) holds. That is, the collision 0 with the singularity can be interpreted as an elastic collision. In particular, one can understand solutions with collisions as generalized periodic solutions, also known as bouncing periodic solutions. Indeed, the − continuation is done by taking u̇(t+ 0 ) = −u̇(t0 ). 2 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 Fig. 1. On the left, potential function with a weak singularity at x = α. On the right, the phase portrait of the potential system. The gray region corresponds to the period annulus. Its outer boundary is emphasized in bold. 2.1. The period function and its extension to bouncing solutions The previous discussion shows that all solutions of Eq. (2) (classical and bouncing type) are periodic. The period function parametrized by the energy, Tp : (0, h∗ ) −→ (0, +∞), is the function that, for each 0 < h < h∗ , assigns the period of the periodic solution u(t) inside the energy level h = H(u, u̇). Due to the symmetry of the system with respect to the u-axis the function Tp (h) is given by √ ∫ u+ (h) du Tp (h) = 2 √ , h ∈ (0, h∗ ), u− (h) h − V (u) where α < u− (h) < 0 < u+ (h) are the negative and positive solutions of the equality h − V (u) = 0. That is, the endpoints of the projection of the orbit over the u-axis. When h ⩾ h∗ solutions of Eq. (2) are of bouncing type. In this case, we can define the time between two consecutive collisions of the solution u(t) as √ ∫ u+ (h) du Tb (h) = 2 √ , h ⩾ h∗ . α h − V (u) The function Tb can be understood as an extension of the period function Tp outside the period annulus. Therefore we define the function { Tp (h) if 0 < h < h∗ , T (h) = Tb (h) if h ⩾ h∗ , as the extended period function. Lemma 2.1. The function T (h) is C 1 (0, +∞). Proof . The function Tp is C 1 (0, h∗ ) by the classical theory of the period function. In addition, the function Tb is C 1 (h∗ , +∞) since V ′ (u) > 0 for all u > 0 and V (α) < h∗ . Let us show that the result is true for h = h∗ . To do so let us define ⎧ ( ⎨ V (u) 1/2 ) u u2 if u ̸= 0, g(u) = ⎩0 if u = 0. √ The change of variable u = g −1 ( h sin θ) transforms the integral expression of Tb into π √ ∫ 2 −1 ′ √ Tb (h) = 2 √ ∗ (g ) ( h sin θ)dθ. − arcsin h h 3 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 The same change of variables transforms Tp into π √ ∫ 2 √ Tp (h) = 2 (g −1 )′ ( h sin θ)dθ. −π 2 √ ∗ At this point the continuity becomes clear since limh→(h∗ )+ arcsin hh = π2 . We differentiate with respect to h to obtain ∫ π √ ′ 1 2 −1 ′′ √ h∗ −1 ′ Tb (h) = √ √ ∗ (g ) ( h sin θ) sin θdθ − √ lim √ (g ) (z). 2h − arcsin hh 2h h − h∗ z→− h∗ Using the identity g 2 = V , we notice that 1 2g (g −1 )′ (z) = = ′ (g −1 (z)). g ′ (g −1 (z)) V √ The previous equality together with limu→α+ g(u) = − h∗ implies that −1 ′ 2g(u) lim √ (g ) (z) = lim =0 z→− h∗ u→α+ V ′ (u) since V ′ (u) → −∞ as u → α+ . Consequently, on account of the previous limit together with the notation √ u = g −1 ( h sin θ), the derivative of Tb writes π −g ′′ (u) ∫ 1 2 Tb′ (h) =√ sin θdθ. (3) h∗ g ′ (u)3 √ 2h − arcsin h This shows the continuity of the derivative at h = h∗ . □ Lemma 2.2. If u/V ′ (u) → +∞ as u → +∞ and V ′′ (u) > 0 for all u > β then the function T (h) tends to infinity as h tends to infinity. Proof . For h > h∗ the extended period function writes √ ∫ β √ ∫ u+ (h) ∫ u+ (h) du du du T (h) = 2 √ + 2 √ ⩾ √ , α h − V (u) β h − V (u) β h − V (u) where (α, β) is the projection over the u-axis of the period annulus and u+ (h) is the positive solution of h − V (u) = 0 with V ′ (u+ (h)) > 0. By the mean value theorem, there exists β < c(h) < u+ (h) such that 1 1 √ =√ √ , h − V (u) V ′ (c(h)) u+ (h) − u and so u+ (h) √ u+ (h) − β ∫ du 2 √ = √ . β h − V (u) V ′ (c(h)) From the hypothesis V (u) > 0 for all u > β, we have V ′ (c(h)) < V ′ (u+ (h)) and the previous equality ′′ yields to ∫ u+ (h) √ du 2 u+ (h) − β √ ⩾ √ β h − V (u) V ′ (u+ (h)) Since limh→+∞ u+ (h) = +∞ and limu→+∞ u/V ′ (u) = +∞ the previous inequality proves the result. □ 4 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 The following result is an extension of Schaaf’s monotonicity criterion [13] for the period function of planar potential systems with a weak singularity. The proof follows similarly to the original and here we only include some comments for the sake of brevity. Theorem 2.3. If V ∈ C 4 (I) satisfies that (a) (5(V ′′′ )2 − 3V ′′ V (4) )(u) > 0 for all u ∈ I where V ′′ (u) > 0, (b) V ′ (u)V ′′′ (u) < 0 if V ′′ (u) = 0, then T ′ (h) > 0 for all h > 0. Proof . From Schaaf’s monotonicity criterion [13] we already know that the hypothesis in the statement (0, h∗ ). To show that Tb (h) is monotone increasing on (h∗ , +∞) imply that Tp (h) is monotone increasing on√ −1 we perform the change of variable u = g ( h sin θ) on the expression in (3) and the derivative of Tb writes √ β u+ (h) ϕ(u)V ′ (u)du ϕ(u)V ′ (u)du ∫ ∫ 2hTb′ (h) = √ + √ , (4) α h − V (u) β h − V (u) ′ 2 ′′ where ϕ(u) = (V )(V−2V ′ )3 V (u) and (α, β) is the projection of the period annulus over the u-axis. The original proof of Schaaf shows that the first integral is positive but the same arguments employed in [13, Lemma 1] prove that ϕ(u) > 0 for u > 0 and so the second integral is also positive. □ Lemma 2.4. If (1 − 2V V ′′ /(V ′ )2 )(u) ⩾ ℓ > 0 for all u large enough, u/V ′ (u) → +∞ as u → +∞ and V ′′ (u) > 0 for all u > β then there exists h > h∗ such that T (h) is monotone increasing on (h, +∞). Proof . Denoting by u the point such that (1 − 2V V ′′ /(V ′ )2 )(u) ⩾ ℓ if u ⩾ u, we split the expression (4) in two parts: √ ∫ u ∫ u+ (h) ′ ψ(u)du ψ(u)du 2hTb (h) = √ + √ , α h − V (u) u h − V (u) where ψ(u) = (1 − 2V V ′′ /(V ′ )2 )(u). For the singular value h = V (u) the first integral corresponds to the period of the bouncing solution with initial conditions u(0) = u, u̇(0) = 0. That is, the integral is bounded for h = V (u). Clearly this bound is uniform if h is increased since the numerator is bounded on the fixed interval of integration and the denominator increases as the energy does. We claim that the second integral tends to infinity as h tends to infinity. Indeed, we have ∫ u+ (h) ∫ u+ (h) ψ(u)du du √ >ℓ √ . u h − V (u) u h − V (u) The integral at the right-hand side of the inequality tends to infinity as h tends to infinity, as we have already √ shown in Lemma 2.2. Consequently, limh→+∞ 2hTb′ (h) = +∞ and the result follows. □ 2.2. The power-like integrable system In this section we recover system (1) and analyze it when p(t) ≡ p0 is a negative constant. That is, 1 ü − = p0 , u > 0. (5) uα Eq. (5) has an associated potential energy function given by u1−α V (u) := −p0 u − , u ⩾ 0. 1−α 5 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 A direct study of the potential shows that the associated first order differential system of Eq. (5) exhibits a center at ((−p0 )−1/α , 0) with bounded period annulus, which projection over the u-axis is (0, ((α−1)p0 )−1/α ). The total energy function is denoted by H(u, u̇) := 12 u̇2 + V (u). In particular, h = 0 is the energy at the collision point with zero velocity (i.e. the energy at the outer boundary of the period annulus). We denote by u(t; u0 ) the solution of (5) with initial conditions u(0; u0 ) = u0 > 0 and u̇(0; u0 ) = 0. Solutions with initial conditions u0 ∈ (0, ((α − 1)p0 )−1/α ) (that is, with negative energy) are globally defined and periodic. On the other hand, solutions with u0 ⩾ ((α−1)p0 )−1/α (positive energy) are no longer globally defined since they reach the singularity u = 0 in finite time. From the expression of the energy function these orbits reach the singularity with finite velocity and the energy is conserved so the singularity can be interpreted as an elastic collision. More precisely, there exists t0 > 0 such that u(t0 ; u0 ) = u(−t0 ; u0 ) = 0 and u̇(t0 ; u0 ) = −u̇(−t0 ; u0 ). In the following Lemma we describe the period function of Eq. (5) using the results presented in the previous section. Notice that the potential V in Section 2 has a minimum at u = 0 and the energy h ⩾ 0. However, the potential treated in this section has a minimum at u = (−p0 )−1/α > 0 and the energy h ⩾ V ((−p0 )−1/α ). Through the proof of the following Lemma, a translation is performed to enter the setting of Section 2. Lemma 2.5. Consider the extended period function T (h) associated to Eq. (5) and any p0 < 0. (a) If α > 1/2 then T (h) is monotone increasing and tends to infinity. (b) If α = 1/2 then T (h) is constant for h < 0, and monotone increasing and tends to infinity for h > 0. (c) If 0 < α < 1/2 then T (h) is monotone decreasing for h < 0 and there exists h > 0 such that T (h) is monotone increasing in (h, +∞) and tends to infinity. Proof . The result in (a) follows applying Theorem 2.3. Indeed, elementary computations lead to V ′′ (u) = αu−(α+1) > 0 for all u > 0 and (5(V ′′′ )2 − 3V ′′ V (4) )(u) = α2 (α + 1)(2α − 1)u−2(α+2) . (6) Therefore assumptions in Theorem 2.3 are fulfilled when α > 1/2 so the extended period function is monotone increasing. Moreover, u u lim = lim = +∞ u→+∞ V ′ (u) u→+∞ −p0 − u−α since α > 0 and p0 < 0. Then Lemma 2.2 implies that T (h) tends to infinity. To show (c) we first employ the classical Schaaf’s criterion in [13] for monotone decreasing period function. In this case the condition to be satisfied is expression in (6) to be negative. Clearly this is so for 0 < α < 1/2 and then T (h) is monotone decreasing for h < 0. For h > 0 we notice that ( 1−α ) ( 2V V ′′ ) 2α p0 u + u1−α lim 1− (u) = lim 1 + 1+α = 1, u→+∞ (V ′ )2 u→+∞ u (p0 + u−α )2 so assumptions in Lemma 2.4 hold and the result in (c) follows. Finally, we prove (b) by direct computation. Indeed, for α = 1/2 the integral of the expression of the period function writes √ ∫ du √ ∫ du 2 √ = 2 √ √ . h − V (u) h + p0 u + 2 u The change of variables u = z 2 yields to √ ∫ du √ ∫ zdz 2 √ =2 2 √ , h − V (u) h− 1 + p0 (z + 1/p0 )2 p0 6 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 which can be explicitly integrated, giving √ ⎛ √ ⎞ √ √ 1 1 2 2 −p0 (z + p10 ) 2 2 h− p0 + p0 (z + p0 ) 3 arctan ⎝ √ ⎠+ . (−p0 ) 2 h − p10 + p0 (z + 1 p0 p0 ) It is then a computation to show that, if h < 0, the evaluation of the previous function on both endpoints of the interval of integration give √ 2 2π Tp (h) ≡ 3 . (−p0 ) 2 On the other hand, if h > 0 the left-hand endpoint is z = 0 and so √ 2 2 (π 1 ) 2√2h −2 Tb (h) = 3 + arctan((−p 0 h) ) − . (−p0 ) 2 2 p0 √ 3 In particular, limh→0+ Tb (h) = 2 2π(−p0 )− 2 . The properties on the statement are easily checked using the expression of Tb . □ 3. Regularization of collisions and bouncing solutions We now return to system (1). The periodicity of p(t) is not needed at this moment so we assume that p : R → R is a continuous and bounded negative function satisfying p2 ⩽ p(t) ⩽ p1 < 0 (7) for all t ∈ R. Let us consider the first order differential system associated to (1). That is, 1 X : u̇ = v, v̇ = + p(t). uα We also denote by X1 and X2 the first order differential systems associated to (1) taking p(t) ≡ p1 and p(t) ≡ p2 , respectively. Notice that X1 and X2 are both integrable first order differential systems associated to an equation of the form (5). We denote by H1 and H2 the energy functions associated to X1 and X2 . We also define η = ((α − 1)p1 )−1/α for convenience. Lemma 3.1. Let us assume that p(t) is a continuous and bounded negative function satisfying (7). Then every classical solution u of Eq. (1) with initial conditions u(0) > η and u̇(0) = 0 has a finite maximal interval of definition (t0 , t1 ) such that u(t0 ) = u(t1 ) = 0. Moreover, u̇(t0 ) > 0 and u̇(t1 ) < 0 are finite. Proof . Let u(t) be a maximal solution of Eq. (1) with initial conditions u(0) = u0 > 0 and u̇(0) = 0 and let u1 (t) and u2 (t) be the maximal solutions of Eq. (1) with same initial conditions as u(t) taking p(t) ≡ p1 and p(t) ≡ p2 , respectively. Since p1 and p2 are negative and the initial conditions satisfy ui (0) > η and u̇i (0) = 0, i = 1, 2, by the discussion of the previous section, both u1 and u2 are solutions defined in a bounded interval, (t01 , t11 ) and (t02 , t12 ) respectively, and they reach the singularity. The function u1 is 1−α contained in the level curve H1 (u, v) = 12 v 2 − p1 u − u1−α = H1 (u0 , 0) whereas u2 is contained in the level 1−α curve H2 (u, v) = 12 v 2 − p2 u − u1−α = H2 (u0 , 0). Notice that H2 (u0 , 0) − H1 (u0 , 0) = (p1 − p2 )u0 > 0. First we show that t12 < t11 . To do so let us argue by contradiction assuming that t11 ⩽ t12 . Since u1 (0) = u2 (0) > η and u̇1 (0) = u̇2 (0) = 0 the difference function ω(t) = u1 (t) − u2 (t) satisfies ω(0) = 0, ω̇(0) = 0 and 1 1 ω̈(t) = u1 (t)α − u (t)α + p1 − p2 . In particular, ω̈(0) = p1 − p2 > 0 and so u1 (t) > u2 (t) for small t positive. 2 7 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 Therefore, if t11 ⩽ t12 , there exists some 0 < t∗ ⩽ t11 such that u1 (t∗ ) = u2 (t∗ ) with |u̇1 (t∗ )| > |u̇2 (t∗ )|. Using the energy level of each solution, we have that H2 (u2 (t∗ ), u̇2 (t∗ )) − H1 (u1 (t∗ ), u̇1 (t∗ )) = (p1 − p2 )u0 . That is, 1 (u̇2 (t∗ )2 − u̇1 (t∗ )2 ) = (p1 − p2 )(u0 − u1 (t∗ )) > 0. 2 Then u̇2 (t∗ )2 > u̇1 (t∗ )2 reaching contradiction. Similarly one can show that t01 < t02 . Therefore (t02 , t12 ) ⊂ (t01 , t11 ). Let us show now that the maximal interval of definition of u(t) is (t0 , t1 ) finite satisfying t01 ⩽ t0 ⩽ t02 < 0 < t12 ⩽ t1 ⩽ t11 (8) and that u2 (t) ⩽ u(t) ⩽ u1 (t) for all t in the common interval of definition. Indeed, ⟨X, ∇H1 ⟩ = v(p(t) − p1 ) which, since p(t) ⩽ p1 , has opposite sign than v. Similarly, ⟨X, ∇H2 ⟩ = v(p(t) − p2 ) has the same sign than v. In particular, the trajectory (u(t), u̇(t)) is confined in the region delimited by H1 (u, v) = H1 (u0 , 0) and H2 (u, v) = H2 (u0 , 0) for all time t in the interval of definition of u(t). We point out that the outer boundary of the region is given by H2 (u0 , 0), whereas the inner boundary is given by H1 (u0 , 0). Moreover, at (u0 , 0) the vector field X is vertical and points down. This implies that the function u(t) for t ⩾ 0 is decreasing and u̇(t) cannot tend to zero. Thus u(t) reaches the singularity u = 0 in finite time and −∞ < u̇1 (t11 ) ⩽ u̇(t1 ) ⩽ u̇2 (t12 ) < 0. Finally assume that t1 > t11 . Then there exists t∗ in such a way u(t∗ ) = u1 (t∗ ) and u̇(t∗ ) > u̇1 (t∗ ). This contradicts the fact that u(t) is inside the region mentioned before. Thus t1 ⩽ t11 . Respective arguments with u2 show t1 ⩾ t12 . The result holds backwards in time similarly. □ √ Lemma 3.2. Let t0 and v0 be two numbers with v0 > 2(p1 − p2 )η. Assume that p(t) is Lipschitz- continuous. Then there exists a unique maximal solution of (1) defined in (t0 , t1 ) with t0 < t1 < +∞ satisfying lim u(t) = lim u(t) = 0 t→t+ 0 t→t− 1 and lim u̇(t) = v0 and lim u̇(t) = v1 , t→t+ 0 t→t− 1 for some real number v1 < 0. Proof . Let us consider the system √ √ u̇ = + 2(w − V (u)), ẇ = p(t) 2(w − V (u)). For the classical theory of differential equations the Cauchy problem u(0) = 0, w(0) = h0 has a local unique solution if h0 > 0 = V (0). That function u = u(t) is defined in some open interval (0, t∗ ) in which it is also √ solution of Eq. (1) with initial conditions u(0) = 0 and u̇(0) = + 2h0 . Indeed, from the first equation of the previous system, 1 2 u̇ = w − V (u). 2 Multiplying by u̇ and integrating the equation we get ü(t)u̇(t) = (p(t) − V ′ (u))u̇(t), so we recover Eq. (1) due to u̇(t) ̸= 0 for t ∈ (0, t∗ ). The previous system then acts as a regularization of the collision of Eq. (1). Indeed, by uniqueness of the initial value problem Eq. (1) has a unique solution defined in (0, t∗ ) and coinciding with u(t) in that interval. This is enough to ensure that they coincide everywhere in the interval of definition (t0 , t1 ). Here t1 may be infinite. Arguing similarly as in the proof of Lemma 3.1, the condition on v0 in the statement implies that the solution reaches a local maximum u∗ > η. Therefore, the solution of Eq. (1) satisfies the assumptions in Lemma 3.1 and so t1 is finite and u(t) has a collision with finite velocity at t = t1 . □ 8 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 The previous results allow to define a bouncing solution of (1) as a continuous function u : R → [0, +∞) satisfying (a) Z = {t ∈ R : u(t) = 0} is discrete, (b) for any open interval I ⊂ R \ Z the function u is in C 2 (I) and satisfies Eq. (1) on I, − (c) for each t0 ∈ Z the limits limt→t+ u̇(t) = u̇(t+ + 0 ) and limt→t− u̇(t) = u̇(t0 ) exist and satisfy u̇(t0 ) = 0 0 −u̇(t− 0 ). Remark 3.3. We point out that the last item implies that the limit lim H(u(t), u̇(t)) = h0 t→t0 exists. This limit is taken from both sides of t0 and hence the energy function has a well defined value at t = t0 . That is, the energy is preserved at √ the collision. In all arguments velocity and energy play an analogous role. For instance, condition v0 > 2(p1 − p2 )η can be replaced by h0 > (p1 − p2 )η. √ Condition v0 > 2(p1 − p2 )η is enough to ensure the occurrence of at least one collision, but subsequent impacts are not ensured. The following lemma implies that solutions with sufficiently high energy exhibit an arbitrary number of collisions. Lemma 3.4. Assume that p(t) is Lipschitz-continuous. For any n ∈ N, there exists γn > 0 such that if t0 and v0 are two numbers with v0 > γn , the unique solution of (1) with initial conditions u(t0 ) = 0 and u̇(t0 ) = v0 has at least n impacts. √ Proof . Let us take γ1 > 2(p1 − p2 )η and consider the solution of X1 that reaches the point (0, −γ1 ). This solution crosses the v-axis backwards in time at some u = u1 > 0. Now consider the solution of X2 that crosses the point (u1 , 0). This solution collides with the singularity u = 0 backwards in time with some velocity v = γ2 . First, we point out that γ2 > γ1 . Indeed, H2 (u1 , 0) − H1 (u1 , 0) = √ (p1 − p2 )u1 > 0 and H1 (u1 , 0) = H1 (0, γ1 ) = 21 γ12 and H2 (u1 , 0) = H2 (0, γ2 ) = 21 γ22 . In particular, γ2 > 2(p1 − p2 )η and Lemma 3.2 implies that the solution of (1) with initial conditions u(t0 ) = 0 and u̇(t0 ) = v0 > γ2 has at least one collision at some time t = t1 . (See Fig. 2) Fig. 2. Scheme of the proof of Lemma 3.4 for n = 4. Lines in black (bold and dashed) correspond to level curves of the differential system X1 . Lines in gray (bold and dashed) correspond to level curves of the differential system X2 . Arrows represent the direction of the vector field X on the level curves of X1 and X2 . 9 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 Second, since ⟨X, ∇H1 ⟩ = v(p(t) − p1 ) and ⟨X, ∇H2 ⟩ = v(p(t) − p2 ), property (7) implies that the solution of (1) impacts the singularity with velocity −v1 < −γ1 at time t = t1 . Thus the bouncing solution is continued by the solution of (1) with initial conditions u(t1 ) = 0, u̇(t1 ) = v1 > γ1 , which ensures at least one collision more. Thus the solution of (1) with u(t0 ) = 0 and u̇(t0 ) = v0 > γ2 has at least two collisions. This procedure generates a succession γ1 < γ2 < · · · < γn and the solution of (1) with initial conditions u(t0 ) = 0 and u̇(t0 ) = v0 > γn has at least n collisions with the singularity. □ 4. The successor map and the twist condition Now we recover the periodicity property of the forcing term p(t). For convenience, let us call γ := √ 2(p1 − p2 )η. For a given t0 ∈ R and v0 > γ, Lemma 3.2 assures that there exists a unique solution u(t) of (1) such that u(t0 ) = 0, u̇(t0 ) = v0 . Moreover, such solution has a finite interval of definition and vanishes at some time t1 . We define the successor map S : R × (γ, +∞) → R × R+ (t0 , v0 ) → S(t0 , v0 ) = (t1 , −u̇(t− 1 )). In the following, we denote S1 (t0 , v0 ) = t1 , S2 (t0 , v0 ) = −u̇(t− 1 ). The map S is one-to-one and continuous in its domain. Moreover, by the 2π-periodic dependence of the equation, one has S(t0 + 2π, v0 ) = S(t0 , v0 ) + (2π, 0), and then (t0 , v0 ) can be seen as polar coordinates. 4.1. The generalized Poincaré–Birkhoff Theorem For completeness, in this subsection we enunciate the version of the Poincaré–Birkhoff Theorem that is used in the proofs. This version was presented in [6] as a variant of the main result of [14]. Theorem 4.1 (Twist Theorem). Let A = S1 × [a1 , a2 ], B = S1 × [b1 , b2 ] be two annuli in the plane with A ⊂ B. Assume that f : A → B is an area-preserving homeomorphism such that the area of the two connected components of the complement of f (A) in B is the same as the area of the corresponding components of the complement of A in B. Assume also that f has a lift f˜ : R × [a1 , a2 ] → R × [b1 , b2 ] of the form θ′ = θ + h(θ, ρ), ρ′ = g(θ, ρ), where h, g are continuous and 2π-periodic in θ. Then, if the boundary twist condition h(θ, a1 ) · h(θ, a2 ) < 0 for any θ ∈ [0, 2π] holds, f has at least two geometrically distinct fixed points. 4.2. The successor map is area-preserving Consider a sequence {εn }n⩾0 with εn > 0 and εn ↘ 0, and for each n ⩾ 0 the second order differential equation 1 ü − = p(t), (9) (u + εn )α 10 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 which is a translation on the u-axis of the original Eq. (1) with the singularity placed at u = −εn . Every solution un of Eq. (9) is a translation of a solution of Eq. (1), un (t) = u(t) − εn . By Lemma 3.1 every classical solution un with initial conditions un (0) = u(0) − εn with u(0) > η and u̇n (0) = 0 has a finite maximal interval of definition and reaches the singularity u = −εn forwards and backwards in time with finite velocity. In particular, there exist times t0n and t1n in which un (t0n ) = un (t1n ) = 0 satisfying t0 < t0n < 0 < t1n < t1 , where (t0 , t1 ) is the maximal interval of definition of the classical solution of Eq. (1) with initial conditions u(0) > η, u̇(0) = 0. Continuous dependence on εn of un (t) is deduced from the explicit formula un (t) = u(t) − εn . Let us define wn (t) as un (t) for t ∈ (t0n , t1n ) and identically zero for t ∈ (t0 , t0n ] ∪ [t1n , t1 ). With all the previous comments in mind the following lemma is straightforward. Lemma 4.2. The sequences {wn }n⩾0 and {ẇn }n⩾0 tend respectively to u and u̇ uniformly on (t0 , t1 ). Moreover, t0n → t0 and t1n → t1 as n → +∞. Let us now consider the sequence of successor mappings corresponding to functions wn . That is, for each (t0 , v0 ) ∈ R × (γ, +∞) we define Sn (t0 , v0 ) = (t1n , −ẇn (t1n )). It is clear from Lemma 4.2 that {Sn }n⩾0 converges point-wise to S. A time reversion argument also shows that {Sn−1 }n⩾0 converges point-wise to S −1 . The extension of the area-preserving property of {Sn }n⩾0 to S is now verbatim the case in [10], what leads to the following statement. Proposition 4.3. The successor mapping S is area-preserving. 4.3. Existence of bouncing periodic orbits Now, given natural numbers m, n, our objective is to find fixed points of the map S n (t0 , v0 ) − (2mπ, 0), that are identified as the initial conditions of 2mπ-periodic solution with exactly n impacts in each period. Our first main result is the following one. Theorem 4.4. Assume that 0 < α < 1 and p(t) is a Lipschitz-continuous and 2π-periodic function with negative values. Then, there exists m1 ∈ N such that for any m ⩾ m1 , Eq. (1) has at least two 2mπ-periodic solutions with exactly 1 impact in the period interval [0, 2mπ). The proof relies on the version of Poincaré–Birkhoff Theorem presented in Section 4.1 and the following key technical lemma. Lemma 4.5. lim S1 (t0 , v0 ) − t0 = +∞ uniformly in t0 ∈ [0, 2π]. v0 →+∞ Proof . From the equation and (7), ü(t) > p2 for all t ∈ (t0 , t1 ). Then, an integration from t0 to t1 gives u̇(t− 1 ) − v0 > p2 (t1 − t0 ). 11 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 Now, considering that u̇(t− 1 ) < 0, we have v0 < −p2 (t1 − t0 ), and the conclusion is clear if we remember that p2 < 0 blue and S1 (t0 , v0 ) = t1 . □ Proof of Theorem 4.4. By continuity, we can fix m1 such that S1 (t0 , γ + 1) − t0 < 2m1 π (10) for any t0 ∈ [0, 2π]. Now, for a given m ⩾ m1 , by Lemma 4.5 there exists v+ > γ + 1 such that S1 (t0 , v+ ) − t0 > 2mπ for any t0 ∈ [0, 2π]. Now, the result is a direct consequence of the Poincaré–Birkhoff Theorem. □ □ One of the main differences with the attractive case studied in [5,6] is that the successive iterations of S are not necessarily well-defined in the repulsive case. However, Lemma 3.4 proves the existence of γn such that the nth iterate S n (t0 , v0 ) is well-defined for all v0 > γn . With this observation in mind, we can prove the following result. Theorem 4.6. Assume that 0 < α < 1 and p(t) is a continuous and 2π-periodic function with negative values. Then, for any natural number n ⩾ 2, there exists mn ∈ N such that, for any m ⩾ mn , Eq. (1) has at least one 2mπ-periodic solution with exactly n impacts in the period internal [0, 2mπ). Proof . By the observation above, S n (t0 , v0 ) is well-defined for v0 > γn . Then we can follow exactly the same argument as in the proof of Theorem 4.4, working now with the first component of S n (t0 , v0 ). Again, Poincaré–Birkhoff Theorem provides two fixed points of the successor map, the difference is that they may correspond to the same bouncing solution. □ The previous results ensure the existence of sub-harmonic bouncing solutions. In order to guarantee the existence of harmonic bouncing solution we need an accurate statement. Theorem 4.7. Assume that 0 < α ⩽ 1/2 and p(t) is a continuous and 2π-periodic function satisfying ( ) α p(t) ⩽ p1 < 0. If α1 1+α < −p1 then Eq. (1) has at least two 2π-periodic solutions with exactly 1 impact in the period interval [0, 2π). Proof . The proof is verbatim the one of Theorem 4.4 with the difference that in this case we need to ensure that the inequality (10) is verified for m1 = 1. To do so we first notice that, due to Lemma 3.1, for any (t0 , v0 ) ∈ R × (γ, +∞) the quantity S1 (t0 , v0 ) − t0 = t1 − t0 is bounded by the length of the interval of definition of the solution of (1) with p(t) ≡ p1 (see (8)). Therefore it is enough to prove that the integrable equation with p(t) ≡ p1 has a bouncing solution with period less than 2π. 1 Eq. (1) with p(t) ≡ p1 has a center located at ((−p1 )− α , 0) (see Section 2.2) and the value of the period function at the center itself is given by 2π 2π √ )=√ . ( 1 −α 1+α V ′′ (−p ) 1 α(−p1 ) α For 0 < α ⩽ 1/2, Lemma 2.5 guarantees that the period function is either constant or monotone decreasing (1) with p(t) ≡ p1 near the boundary of the inside the period annulus. Therefore, bouncing solutions of Eq.√ 1+α period annulus have bouncing period less than or equal to 2π/ α(−p1 ) α by a continuity argument (see Lemma 2.1). The condition in the statement implies that the previous value is less than 2π. □ 12 D. Rojas and P.J. Torres Nonlinear Analysis: Real World Applications 64 (2022) 103441 The fact that the period function for 1/2 < α < 1 is monotone increasing does not allow to ensure the existence of harmonic bouncing solutions using the value of the period at the center. In order to derive an analogous result for this range of α we need the expression of the period at the outer boundary of the period annulus. However the authors are not able to obtain its expression with their methods available. Acknowledgments We are grateful to Rafael Ortega for bringing to our attention the position-energy system that has been crucial for the regularization of collisions. We also thank the anonymous referees for their valuable comments and suggestions that contributed to improve the initial version of the paper. This work has been realized thanks to the Agencia Estatal de Investigación, Spain and Ministerio de Ciencia, Innovación y Universidades, Spain grants MTM2017-82348-C2-1-P and MTM2017-86795-C3-1-P. References [1] P.J. Torres, Mathematical Models with Singularities - a Zoo of Singular Creatures, Atlantis Press, ISBN: 978-94-6239-105-5, 2015. [2] A.C. Lazer, S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. 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