Planar systems with critical points: multiple solutions of two-point nonlinear boundary value problems

Abstract

Two-point boundary value problems for the second-order ordinary nonlinear differential equations are considered. First, we consider the planar systems equivalent to equation x ″ = f ( x ) , where f ( x ) has multiple zeros and the respective system has centers and saddle points in various combinations. Estimations of the number of solutions are given. Then results are extended to nonautonomous equations which have superlinear behavior at infinity.