Asymptotic conditions at the two first eigenvalues for the periodic solutions of Liénard differential equations and an inequality of E. Schmidt

Abstract

We study the periodic boundary problem $$ x’’(t) + f(x(t)) x’(t) + g(t, x(t)) = e(t), $$ $$ x(0) — x(2 \\pi) = x’(0) —x’(2 \\pi) = 0 $$ under some non-resonance conditions on the asymptotic behavior of $x^{-1}g(t, x)$ for $|x| \\to \\infty$.