Abstract

New conditions of solvability based on a general theorem on the calculation of the index at infinity for vector fields that have degenerate principal linear part as well as degenerate “next order” terms are obtained for the 2π-periodic problem for the scalar equation x″+n2x=g(|x|)+f(t,x)+b(t) with bounded g(u) and f(t,x)→0 as |x|→0. The result is also applied to the solvability of a two-point boundary value problem and to resonant problems for equations arising in control theory.