Periodic solutions of systems with asymptotically evennonlinearities

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 2-periodic problem for the scalar equation x" + n2x g( x I)+ f(t, x)+ b(t) with bounded g(u) and f(t,x)O as Ix I-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.