Periodic solutions of Duffing's equations with superquadratic potential

Abstract

This paper is devoted to the study of harmonic and subharmonic solutions for the second order scalar nonlinear Duffing's equation x″ + g( x) = p( t, x, x′), where g and p are continuous functions with p bounded and periodic in the first variable and g satisfying the assumption g(x) sign(x) → + ∞, as ¦x¦ → + ∞ . Among other results, we prove the existence of infinitely many harmonic and subharmonic solutions (of any order) p = p( t) and if the potential G( x) of g( x) satisfies certain conditions of superquadratic growth at ∞. The new existence results can be applied to situations in which the more classical superlinear growth condition g(x) x → + ∞ , as ¦x¦ → + ∞, is not satisfied. In this manner, various preceding theorems are improved and sharpened (see the “Introduction” for more details). Proofs are based on a generalized version of the Poincaré-Birkhoff “twist” theorem due to W. Ding.