Periodic solutions of systems with asymptotically even nonlinearities (Q1566583)

First, the \(2\pi\)-periodic problem for the equation \[ x''+ n^2 x= g(|x|) + f(t,x) + b(t) \] is considered with bounded \(g(u)\) and \(f(t,x) \to 0\) as \(|x|\to 0\). The existence of at least one \(2\pi\)-periodic solution is proved under certain assumptions. The proof is based on a general theorem on the calculation of the index at infinity for vector fields having degenerate principal part as well as degenerate ``next order'' terms. The abstract result is also applied to the solvability of the two-point boundary value problem and to resonance problems for equations arising in control theory.