Existence of multiple solutions for some nonlinear boundary value problems

Abstract

By imposing one-sided conditions on the nonlinearity, where neither regularity nor uniformity is required, we prove the existence of either a nonnegative or a nonpositive solution for first and second order ordinary differential equations with periodic, Neumann, or Dirichlet boundary conditions. Positone and nonpositone problems are considered. Some nonexistence results are also obtained. When using generalized Ambrosetti-Prodi type conditions we get the existence of nonnegative and nonpositive solutions for first and second order periodic or Neumann boundary value problems. Our method of proof makes use of topological degree arguments in Cones.