Special Issue in honor of Alan C. Lazer Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 203–212. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu REMARKS ON PERIODIC RESONANT PROBLEMS WITH NONLINEAR DISSIPATION LUÍS SANCHEZ, JOÃO G. SILVA In memory of Professor Alan C. Lazer Abstract. We consider the periodic problem for a 2nd order ODE with non- invertible linear part, and mild nonlinear dissipation term. The motivation for this study is a paper by Lazer [9]. We add a bounded restoring force g(u) and show that the sufficient condition (of Landesman-Lazer type) given in [9] still implies the existence of a periodic solution in our case. We also comment on some variants of the problem and on the existence of bounded solutions. 1. Introduction Lazer [9] gave a simple proof of necessary and sufficient conditions for the exis- tence of a 2π-periodic solution to the second order ordinary equation with a non- linear dissipative term u′′(t) + u(t) + d dt F (u(t)) = e(t) (1.1) where F is a bounded C1 function, and e is continuous and 2π-periodic function. The conditions include an inequality involving the size of the projection of e onto the kernel of the linear operator u′′ + u in the space of 2π-periodic functions, and the gap between the limits of F at ±∞, namely 2(F (∞)− F (−∞)) > √ e2s + e2c (1.2) where ec = ∫ 2π 0 cosx e(x) dx, es = ∫ 2π 0 sin(x) e(x) dx. (1.3) Among other features, the proof in [9] invokes the Brouwer fixed point theorem for a disk in the plane. Inequality (1.2) is a condition of Landesman-Lazer type; see [7] for the original paper of Landesman and Lazer. As stated in [9], the applicability of that condition to the resonant periodic problem had appeared in articles by Lazer and Leach [10], and Frederickson and Lazer [4]. 2010 Mathematics Subject Classification. 34B15, 34C11, 34C25. Key words and phrases. Periodic solution; resonance; bounded solution. c©2021 This work is licensed under a CC BY 4.0 license. Published October 6, 2021. 203 204 L. SANCHEZ, J. G. SILVA EJDE/SI/01 In this note we consider the slightly more general problem u′′(t) + u(t) + g(u) + d dt F (u(t)) = e(t) (1.4) where g is continuous and bounded. That is, we are interested in a bounded pertur- bation of equation (1.1). Basically, we intend to show that (1.2) remains applicable to (1.4) and we propose an alternative, although basically equivalent, method of proof. We stress that, beyond the papers already mentioned, there exists an extensive and rich literature concerning equations similar to (1.1), covering existence and stability of periodic, almost periodic or bounded solutions, under a variety of as- sumptions. A handful of material, which is significant and representative of the research about these problems can be found in works by Ahmad [1], Ezeilo [3], Mawhin [11], Ortega [13], Ortega and Tineo [14], Fonda and Zanolin [5], Mawhin and Ward [12], and, of course, in their references. In this note, as in [9], our arguments use a shooting method and continuity with respect to parameters. They are as simple as possible if we assume that g is a locally Lipschitz function, although weaker conditions may be considered via approximate problems. On the other hand, in our proof the Brouwer fixed point theorem is naturally re- placed with the particular version of the following Brouwer-Bohl existence principle (see e.g. [11]). Theorem 1.1. Let Bn = {x ∈ Rn : ‖x‖ ≤ 1}. If V : Bn → Rn is a continuous vector field such that for all x ∈ ∂Bn, V (x) · x > 0, then the equation V (x) = 0 has a solution in Bn. To deal with the perturbation g we need in addition the following property of sequences of oscillatory integrands. Lemma 1.2. Let T = 2nπ, n ∈ N, {WR,φ(t)} (R > 0, φ ∈ R) be a family of C1 functions in [0, T ], bounded independently of R; φ, with their derivatives and g be a continuous bounded function. Then∫ T 0 g(R sin(t+ φ) +WR,φ(t)) cos(t+ φ)dt −→ 0 as R→∞ uniformly with respect to φ ∈ R. Proof. Let G be an antiderivative of g. We have G(R sin(φ) +WR,φ(2nπ))−G(R sin(φ) +WR,φ(0)) = ∫ 2nπ 0 d dt (G(R sin(t+ φ) +WR,φ(t)))dt = R ∫ 2nπ 0 g(R sin(t+ φ) +WR,φ(t)) cos(t+ φ)dt + ∫ 2nπ 0 g(R sin(t+ φ) +WR,φ(t))W ′R,φ(t)dt From our assumptions and the mean-value theorem we can conclude that R ∫ 2nπ 0 g(R sin(t+ φ) +WR,φ(t)) cos(t+ φ)dt EJDE-2021/SI/01 REMARKS ON PERIODIC RESONANT PROBLEMS 205 is bounded, and therefore lim R→∞ ∫ T 0 g(R sin(t+ φ) +WR,φ(t)) cos(t+ φ)dt = 0. � In section 2 we present the main result of this note with some remarks on the analogous (simpler) problem where the term u is dropped in (1.4) (that is, the case of resonance at the eigenvalue zero). In the final section, following mainly [4], we refer to the existence of bounded solutions. 2. Main result Theorem 2.1. Let e(t) be continuous and 2π-periodic, F : R → R a bounded C1 function such that F (∞), F (−∞) exist, and g : R → R a locally Lipschitz continuous, bounded function. Defining ec, es as in (1.3), the condition (1.2) is sufficient for the existence of a 2π-periodic solution of (1.4). Proof. We want to prove the existence of a solution to the problem u′′(t) + u(t) + d dt F (u(t)) + g(u(t)) = e(t) u(0) = u(2π), u′(0) = u′(2π) (2.1) Let u be a function of class C2. Consider the decomposition u(x) = A cos(x) +B sin(x) +W (x,A,B), where A = u(0) and B = u′(0), so that W (0) = W ′(0) = 0. Hence u is a solution of (2.1) if, and only if W is a solution of W ′′(t, A,B) +W (t, A,B) + F (u(t))′ + g(u(t)) = e(t), (2.2) W (2π) = 0, W ′(2π) = 0. (2.3) By an elementary formula, the solution of (2.2) such that W (0) = W ′(0) = 0 is given by W (t, A,B) = ∫ t 0 sin(t− x)(e(x)− F (u(x))′ − g(u(x)) dx (2.4) Integrating by parts and using well known results on integral equations with parameters (see e.g. [16]) we see that there exists in fact a unique solutionW (t, A,B) of (2.4), continuous with respect to A, B, and (2.4) shows that it is bounded in [0, 2π] independently of A and B. We now prove the existence of a point (A,B) in the plane which is a solution of (2.2)-(2.3). We have W ′(t, A,B) = ∫ t 0 cos(t− x)(e(x)− F (u(x))′ − g(u(x)) dx and we look for a solution of the system W (2π,A,B) = 0, W ′(2π,A,B) = 0 which is equivalent to∫ 2π 0 sin(2π − x)(e(x)− F (u(x))′ − g(u(x)) dx = 0, 206 L. SANCHEZ, J. G. SILVA EJDE/SI/01∫ 2π 0 cos(2π − x)(e(x)− F (u(x))′ − g(u(x)) dx = 0 which is equivalent to∫ 2π 0 sin(x)(F (u(x))′ + g(u(x)) dx− es = 0, − ∫ 2π 0 cos(x)(F (u(x))′ + g(u(x)) dx+ ec = 0 . Using integration by parts we obtain the system es + ∫ 2π 0 cos(x)F (u(x)) dx− ∫ 2π 0 sin(x)g(u(x)) = 0, ec − ∫ 2π 0 sin(x)F (u(x)) dx− ∫ 2π 0 cos(x)g(u(x)) dx = 0 . (2.5) We define the vector field (X(A,B), Y (A,B)): X(A,B) = ec − ∫ 2π 0 sin(x)F (u(x)) dx− ∫ 2π 0 cos(x)g(u(x)) dx, Y (A,B) = es + ∫ 2π 0 cos(x)F (u(x)) dx− ∫ 2π 0 sin(x)g(u(x)) . Let (A,B) ∈ R2, R = √ A2 +B2 and φ ∈ R, such that cos(φ) = B√ A2 +B2 , sin(φ) = A√ A2 +B2 . Then we consider the dot product (−Y (A,B), X(A,B)) · (A,B) = Bec −Aes − ∫ 2π 0 (A cos(x) +B sin(x))F (A cos(x) +B sin(x) +W (x,A,B)) dx + ∫ 2π 0 (A sin(x)−B cos(x))g(A cos(x) +B sin(x) +W (x,A,B)) dx = Bec −Aes − ∫ 2π 0 R sin(x+ φ)F (R sin(x+ φ) +W (x,A,B)) dx + ∫ 2π 0 R cos(x+ φ)g(R sin(x+ φ) +W (x,A,B)) dx ≤ ‖(A,B)‖ · ‖(es, ec)‖ − ∫ 2π 0 R sin(x+ φ)F (R sin(x+ φ) +W (x,A,B)) dx + ∫ 2π 0 R cos(x+ φ)g(R sin(x+ φ) +W (x,A,B)) dx = R √ e2c + e2s − ∫ 2π 0 R sin(x+ φ)F (R sin(x+ φ) +W (x,A,B)) dx + ∫ 2π 0 R cos(x+ φ)g(R sin(x+ φ) +W (x,A,B)) dx = R (√ e2c + e2s − ∫ 2π 0 sin(x+ φ)F (R sin(x+ φ) +W (x,A,B)) dx ) EJDE-2021/SI/01 REMARKS ON PERIODIC RESONANT PROBLEMS 207 +R ∫ 2π 0 cos(x+ φ)g(R sin(x+ φ) +W (x,A,B)) dx Using the assumptions on F and g and using Lemma 1.2, we obtain lim R→∞ (√ e2c + e2s − ∫ 2π 0 sin(x+ φ)F (R sin(x+ φ) +W (x,A,B)) dx + ∫ 2π 0 cos(x+ φ)g(R sin(x+ φ) +W (x,A,B)) dx ) = √ e2c + e2s − 2(F (∞)− F (−∞)) < 0, uniformly with respect to φ ∈ R. Note that by (2.4) and since F and g are bounded, W is a bounded function in [0, 2π], independently of A and B. Therefore there is R0 ∈ R+ such that for all (A,B) ∈ R2 with ‖(A,B)‖ ≥ R0, we have (−Y (A,B), X(A,B)) · (A,B) < 0. So that by Theorem 2.1 there is a (A∗, B∗) ∈ R2 with ‖(A∗, B∗)‖ < R0 such that X(A∗, B∗) = Y (A∗, B∗) = 0. Hence we have a solution of (2.5). � It is interesting to note that in the above theorem we did not use asymptotic assumptions on g; the condition involving the gap of F is sufficient, precisely as in the case where g is absent as in [9]. However, in the absence of F , the argument of the above proof may be repeated (with slightly simpler calculations) to obtain the Lipschitz case of [10, Theorem 1.1] Theorem 2.2. Let e be a 2π-periodic continuous function, and g a bounded con- tinuous function in R such that g(∞) and g(−∞) exist. By setting ec = ∫ 2π 0 cos(x)e(x) dx, es = ∫ 2π 0 sin(x)e(x) dx, we have that 2(g(∞)−g(−∞)) > √ e2c + e2s is a sufficient condition for the existence of a 2π-periodic solution of the differential equation u′′(t) + u(t) + g(u(t)) = e(t). Remark 2.3. Korman and Li [6] studied 2π-periodic solutions to the equation that is analogous to (1.1) when resonance occurs at the nth eigenvalue, that is u′′(t) + n2u(t) + d dt F (u(t)) = e(t) (2.6) where n ∈ N. Setting ec,n = ∫ 2π 0 cos(nx) e(x) dx, es,n = ∫ 2π 0 sin(nx) e(x) dx. (2.7) they found the condition 2n(F (∞)− F (−∞)) > √ e2c,n + e2s,n (2.8) for the existence of a solution. Following our procedure in the proof of Theorem 2.1 it may be seen that the condition (2.8) still implies existence when a bounded nonlinear term g(u) is included in the equation (2.6). Without going to the details, let us mention that it suffices to use the decomposition u(x) = A cos(nx) + B n sin(nx) +W (x,A,B), 208 L. SANCHEZ, J. G. SILVA EJDE/SI/01 where A = u(0), B = u′(0), so that W satisfies W (t, A,B) = 1 n ∫ t 0 sin n(t− x)(e(x)− F (u(x))′ − g(u(x)) dx (2.9) and in the final estimate we choose R and φ so that A = R sin φ, B = nR cos φ. Remark 2.4. In [9] it is mentioned that in a previous paper [8] the case where the linear operator reduces to u′′ (that is, when the zero eigenvalue is considered) and the dissipative term is of the form cu′, c > 0, was discussed. Let us then consider u′′(t) + g(u) + d dt F (u(t)) = e(t) (2.10) where e is continuous and T -periodic. First, it is obvious that if (2.10) has a T -periodic solution and m ≤ g ≤M for some constants m ≤M and setting ē = 1 T ∫ T 0 e(t) dt for the mean value of e, it turns out that m ≤ ē ≤ M . Moreover, letting g± = lims→±∞ g(s) the condition g− < ē < g+ (2.11) is sufficient to the existence of a T -periodic solution of the differential equation (2.10). This has been observed in [11] (even with less demanding inequalities). We note that in the special case where F and g are locally Lipschitz functions, the outline of our proof in section 2 may be followed to yield the partial converse; the situation here is simpler and the argument relies on the Poincaré-Miranda theorem. In fact, we write any solution u of (2.10) as u(t) = A+Bt+ w(t) where A = u(0), B = u′(0) and w satisfies the integral equation w(t) = ∫ t 0 (t− s)[e(s)− g(A+Bs+ w(s))]− d ds F (A+Bs+ w(s))] ds (2.12) Now there exists a solution w(t, A,B) of (2.12), continuous in the set of its variables. Therefore we must show that there exist A, B so that u(T ) = A, u′(T ) = B, that is, w(T ) = −BT and w′(T ) = 0. Since w′(t) = ∫ t 0 [e(s)− g(A+Bs+ w(s))− d ds F (A+Bs+ w(s))] ds we obtain the following system of equations for A and B,∫ T 0 (T − s)[e(s)− g(A+Bs+ w(s))] ds − ∫ T 0 F (A+Bs+ w(s)) ds+ TF (A) +BT = 0, (2.13) ∫ T 0 e(s) ds− ∫ T 0 g(A+Bs+ w(s)) ds = 0. (2.14) We define a vector field (X,Y ) in the plane such thatX(A,B) (respectively Y (A,B)) is the left-hand side of (2.13) (respectively (2.14)). Invoking the boundedness of g EJDE-2021/SI/01 REMARKS ON PERIODIC RESONANT PROBLEMS 209 and F it is clear that there exists b > 0 such that X(A,B) > 0 (respectively < 0) if B ≥ b (respectively B ≤ −b), ∀A ∈ R. On the other hand, using the boundedness of g again, together with (2.11), ± lim A→±∞ Y (A,B) < 0 uniformly in B ∈ [−b, b]. We infer that we may choose a > 0 so that the field (X(A,B), Y (A,B)) satisfies the conditions of the Poincaré-Miranda theorem in the rectangle [−a, a]× [−b, b]. It follows that the vector field (X,Y ) has a zero in this rectangle and the proof is complete. Remark 2.5. If g ≡ 0 in equation (2.10), we can be more specific. Namely, it is not difficult to see that ē = 0 is necessary and sufficient to the existence of a periodic solution of u′′(t) + d dt F (u(t)) = e(t) (2.15) Moreover, the initial value u(0) of the periodic solution can be arbitrarily prescribed. In fact, let E(t) = ∫ t 0 e(s) ds. The T -periodic solution u of (2.15) with A = u(0), B = u′(0) solves the parametric first order initial value problem u′ + F (u) = E(t) +B + F (A), u(0) = A (2.16) subject to the condition∫ T 0 E(s) ds+BT + F (A)T − ∫ T 0 F (u(s)) ds = 0. (2.17) Clearly, (2.17) has a solution B for every A ∈ R, but more can be said: if F is C1 and increasing, (2.17) defines a C1 function B = B(A). To see this, we check that the partial derivative of the left-hand side of (2.17) with respect to B does not vanish. Setting v = ∂u ∂B and f = F ′, we have v′ + f(u)v = 1, v(0) = 0 (2.18) so that v(T ) + ∫ T 0 f(u(s))v(s) ds = T. (2.19) Since v(T ) > 0 our claim is proved. It follows that, under the stated conditions, in addition to the periodic solutions defined by (2.16), the problem (2.15) has a family of solutions bounded to the right, given by u′ + F (u) = E(t) +B(A) + F (A), u(0) = C. (2.20) Moreover, if we denote by zA the periodic solution of (2.15) given by (2.16)-(2.17), it is easily seen that for each solution u as in (2.20) the function |u(t) − zA(t)| is decreasing for t ≥ 0; therefore there exists a constant K such that lim t→+∞ (u(t)− zA(t)) = K. Remark 2.6. We can discard the Lipschitz continuity of g in theorem 2.1 or in the setting of Remark 2.4. Let us briefly describe how to proceed with respect to g in the simpler case of Remark 2.4. Let L± = lims→±∞ g(s). For each n ∈ N let ±αn± > 0 be chosen so that s ≥ αn+ (resp. s ≤ αn−) implies g(s) ≥ L+ − 1 n (resp g(s) ≤ L− + 1 n ). Then construct a Lipschitz function gn such that gn(s) = g(αn+) (resp. gn(s) = g(αn−)) ∀s ≥ αn+ (resp. ∀s ≤ αn−) and max[αn−,αn+] |g− gn| < 1 n . Consider a sequence of 210 L. SANCHEZ, J. G. SILVA EJDE/SI/01 approximation problems where g is replaced with gn in equation (2.10). Each one of these problems has a T -periodic solution un whose sequence of initial conditions (An, Bn) is bounded, as the corresponding systems (2.13)-(2.14) show. Using the sequence of differential equations it is easy to obtain uniform C1-estimates for (un). By the Ascoli-Arzelà’s theorem, a subsequence of un converges uniformly to a function u which, clearly, is a periodic solution of (2.10). 3. Boundedness of solutions Following [4, 9] a naturally related problem is the existence of bounded or al- most periodic solutions when e is a continuous almost periodic function. For such functions the mean value exists. Let us consider the case where the following limit exists and is finite, uniformly in a ∈ R+: P (e) = lim T→+∞ 1 T ((∫ a+T a e(t) cos t dt )2 + (∫ a+T a e(t) sin t dt )2)1/2 . (3.1) Lemma 3.1. The condition F (+∞)− F (−∞) > π P (e) is sufficient for the existence of a solution of (1.4) which is bounded to the right. Proof. Following the argument in [4], let us set, for solutions such that u and u′ + F (u) do not vanish simultaneously, u = r cosϕ, u′ + F (u) = r sinϕ (r > 0), and replace (2.1) with the system r′ = − cosϕF (r, cosϕ)− g(r cosϕ) sinϕ+ e sinϕ (3.2) ϕ′ = −1− sinϕF (r, cosϕ) r − cosϕg(r cosϕ) r + e cosϕ r (3.3) A solution with r(t0) = R and ϕ(t0) = θ is well defined in an arbitrary interval [t0, t0 + T ] provided that R is sufficiently large, since r′ is bounded. In fact r(t) = R+α(t) and ϕ(t) = θ− t+β(t)/R where α and β are bounded functions. Fix ε > 0 so that F (+∞)− F (−∞) > π(P (e) + 2ε). (3.4) By definition of P (e) there exists T̄ > 0 so that if T > T̄ we have for every t0 ≥ 0 and every α, ∫ t0+T t0 sin(α− s)e(s) ds < T (P (e) + ε). (3.5) Next we choose T̄ > 1 so that if T is a multiple of 2π such that T̄ ≤ T ≤ T̄ + 2π, and then taking R sufficiently large, we have∫ t0+T t0 cosϕ(s)F (r(s) cosϕ(s)) ds > T π (F (+∞)− F (−∞)) as well as (by Lemma 1.2)∣∣ ∫ t0+T t0 g(r cosϕ(t)) sinϕ(t) ∣∣ < ε. EJDE-2021/SI/01 REMARKS ON PERIODIC RESONANT PROBLEMS 211 Using the differential equation (3.2), the two preceding inequalities and the assump- tion of the Lemma we conclude that r(t0 + T ) < R. Since r′ is bounded and t0 is arbitrary, the proof is complete. � On the basis of this property, almost periodicity and compactness arguments (see [2, 4]) allow to obtain a bounded solution of (2.1) in the whole real line, provided the limit (3.1) is uniform in a ∈ R and e(t) is almost periodic. Figure 1. Solution of (3.6) with F (u) = arctanu and E(t) = − cos t− sin(πt). Remark 3.2. Perhaps the simplest context where the above argument may be displayed consists in dealing with a first order equation of the type (2.16), say u′ + F (u) = E(t) (3.6) where E has a mean value, at least in the sense that Ē := lim T→+∞ 1 T ∫ a+T a E(s) ds exists uniformly in a ≥ 0. Then it is easy to see that the condition F (−∞) < Ē < F (+∞) implies that all the solutions of (3.6) are bounded to the right. In fact, fix ε > 0 so that F (−∞) + 2ε < Ē < F (+∞)− 2ε, and take T > 0 so that for all a ≥ 0,∣∣ 1 T ∫ a+T a E(s) ds− Ē ∣∣ < ε, and R > 0 such that x > R (resp. x < −R) implies F (x) > F (+∞) − ε (resp. F (x) < F (−∞)+ε). If for some solution u(t) of (3.6) and t0 ≥ 0 we have u(t0) ≥ R 212 L. SANCHEZ, J. G. SILVA EJDE/SI/01 (resp. u(t0) ≤ −R), then for some s ∈ [t0, t0 + T ] the inequality u(s) ≤ R (resp. u(s) ≥ −R) must hold. Since u′ is bounded the claim follows. A little strengthening of the assumptions would lead to the existence of an almost periodic solution which, in case F is increasing, must be unique. Figure 1 shows a graphic simulation for a solution of (3.6) with F (u) = arctanu and E(t) = − cos t− sin(πt). The figure was drawn using the open source software available at the web address [15]. Aknowledgements. L. Sanchez was supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project: UIDB/04561/2020. J. G. Silva was supported by Fundação Calouste Gulbenkian, Novos Talentos em Matemática 2019. The authors are indebted to the anonymous referees for their valuable remarks and suggestions. References [1] S. Ahmad; A nonstandard resonance problem for ordinary differential equations, Trans. Amer. Math. Soc., 323 (1991), 857-875 [2] L. Amerio; Soluzioni quasi-periodiche, o limitate, di sistemi differenziali non lineari quasi- periodici, o limitati, Ann. Mat. Pura Appl., 39 (1955), 97-119. [3] J. O. C. Ezeilo; On the existence of almost periodic solutions of some dissipative second order differential equations, Ann. Mat. Pura Appl. (4) 65 (1964), 389-405. [4] P. O. Frederickson, A. C. Lazer; Necessary and sufficient damping in a second order oscilla- tor, J. Differential Eqs. 5 (1969), 262–270 [5] A. Fonda, F. Zanolin; Bounded solutions of nonlinear second order ordinary differential equations, Discrete Contin. Dynam. Systems, 4 (1998), no. 1, 91–98. [6] P. Korman, Y. Li; Harmonic oscillators at resonance perturbed by a non-linear friction force, Acta Mathematica Scientia 2014,34B(4):1025–1028. [7] E. M. Landesman, A. C. Lazer; Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609–623. [8] A. C. Lazer; On Schauder’s fixed point theorem and forced second-order nonlinear oscilla- tions. J. Math. Anal. Appl., 21 (1968), 421–425. [9] A. C. Lazer; A second look at the first result of Landesman-Lazer type, Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 113–119 [10] A. C. Lazer, D. E. Leach; Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl. 82 (1969), 49–68. [11] J. Mawhin; An extension of a theorem of A. C. Lazer on forced nonlinear oscillations, J. Math. Anal. Appl., 40 (1972), 20–29. [12] J. Mawhin, J. R. Ward, Jr; Bounded solutions of some second order nonlinear differential equations. J. London Math. Soc., (2) 58 (1998), no. 3, 733–747. [13] R. Ortega; A boundedness result of Landesman-Lazer type. Differential Integral Equations 8 (1995), no. 4, 729–734. [14] R. Ortega, A. Tineo; Resonance and non-resonance in a problem of boundedness. Proc. Amer. Math. Soc. 124 (1996), no. 7, 2089–2096. [15] J. R. Sennings; First order differential equation solver, website. http://www.math- cs.gordon.edu/ senning/software.html [16] Wolfgang Walter; Ordinary Differential Equations, Springer 1998. Lúıs Sanchez CMAFcIO, Faculdade de Ciências da Universidade de Lisboa, Campo Grande 1749-016 Lisboa, Portugal Email address: lfrodrigues@ciencias.ulisboa.pt João G. Silva Faculdade de Ciências da Universidade de Lisboa, Campo Grande 1749-016 Lisboa, Por- tugal Email address: jgs1891@outlook.com 1. Introduction 2. Main result 3. Boundedness of solutions Aknowledgements References