RESONANCE PROBLEM FOR A CLASS OF DUFFING’S EQUATIONS

Abstract:

Consider the Duffing’s equation

where g∈C(R, R) and f ∈ P≡{f ∈ C(R, R); f is ω-periodic for some ω > 0}. The function g is said to be resonant if there exists f ∈ P such that eq. (1) has no bounded solutions on [0, ∞). Using a generalized version of the Poincare-Birkhoff fixed point theorem, the authors establish conditions on g which guarantee the following result holds: for any f ∈ P with period ω, there exists $\overline{k}⩾0$ such that eq. (1) has infinitely many kω-periodic solutions for every integer $k⩾\overline{k}$. In such a case, g is clearly non-resonant.