LANDESMAN-LAZER CONDITION FOR NONLINEAR PROBLEMS WITH JUMPING NONLINEARITIES

This paper was motivated by the result of Iannacci and Nkashama (J. Differential Equations 69 (1987), 289-309) concerning the existence of at least one solution for the differential equation x″(t) + m2x(t) + g(t, x(t)) = e(t) with periodic boundary data x(0) - x(2π) = x′(0) - x′(2π) = 0, where m ≥ 0 is an integer, e is integrable, and g satisfies Carathéodory's conditions. We intend to prove that the Landesman-Lazer type condition is sufficient for solvability of this periodic problem under rather more general assumptions on the growth of the nonlinear term g. Particularly, our hypotheses cover the nonlinearities which may "jump over" the eigenvalues different from m2 while the assumptions of Iannacci and Nkashama allow g only to asymptotically "touch" the neighbour eigenvalues. Our approach may be used also for two-point boundary value problems. The proofs are based on the Leray-Schauder degree theory and on the shooting method for ordinary differential equations. © 1990.