Unbounded Solutions to Systems of Differential Equations at Resonance
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Unbounded Solutions to Systems of Differential Equations at Resonance A. Boscaggin1 · W. Dambrosio1 · D. Papini2 Dedicated to the memory of Russell Johnson Received: 25 June 2020 / Revised: 11 August 2020 © The Author(s) 2020
Abstract We deal with a weakly coupled system of ODEs of the type x j + n 2j x j + h j (x1 , . . . , xd ) = p j (t),
j = 1, . . . , d,
with h j locally Lipschitz continuous and bounded, p j continuous and 2π-periodic, n j ∈ N (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms h 1 , . . . , h d are assumed. Keywords Systems of ODEs · Unbounded solutions · Resonance · Lyapunov function Mathematics Subject Classification 34C11 · 34C15
1 Introduction In this paper, we deal with the existence of unbounded solutions for weakly coupled systems of ODEs of the type
Under the auspices of GNAMPA-I.N.d.A.M., Italy.
B
D. Papini [email protected] A. Boscaggin [email protected] W. Dambrosio [email protected]
1
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
2
Dipartimento di Matematica, Informatica e Fisica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy
123
Journal of Dynamics and Differential Equations
⎧ 2 ⎪ ⎪ x1 + n 1 x1 + h 1 (x1 , . . . , xd ) = p1 (t), ⎪ ⎪ ⎨ x + n 2 x2 + h 2 (x1 , . . . , xd ) = p2 (t), 2 2 .. ⎪ ⎪ . ⎪ ⎪ ⎩ xd + n 2d xd + h d (x1 , . . . , xd ) = pd (t),
(1.1)
where the functions h 1 , . . . , h d : Rd → R are locally Lipschitz continuous and bounded and the functions p1 , . . . , pd : R → R are continuous and periodic with the same period, say 2π for simplicity. We will also assume that nj ∈ N
for every j ∈ {1, . . . , d},
(1.2)
implying, as well-known, that the scalar equation x j +n 2j x j = 0 has a nontrivial 2π-periodic solution (in fact, all its nontrivial solutions are 2π-periodic). Following a popular terminology (cf. [21]), system (1.1) is thus said to be at resonance. In the scalar case (that is, d = 1), the problem of the existence of unbounded solutions has been considered since the nineties. Indeed, the first result can be essentially traced back to Seifert [22], establishing the existence of unbounded solutions to the equation x + n 2 x + h(x) = p(t),
x ∈ R,
(1.3)
as a consequence of a non-existence result for 2π-periodic solutions by Lazer and Leach [17] together with the classical Massera’s theorem. Later on, sharper conclusions were obtained by Alonso and Ortega in [1]. In particular, according to [1, Proposition 3.1], any solution of (1.3) is unbounded both in the past and in the future whenever 2π (1.4) p(t)eint dt ; 2 (sup h − inf h) ≤ 0
moreover, due to [1, Proposition 3.4], any sufficiently large solution is unbounded either in the past or in the future when the global condition (1.4) is replaced by the (weaker) asymptotic assumption 2 max lim sup
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