A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations

We introduce a generalized upper and lower solutions method for the solvability of first-order ordinary differential equations u′(t) = ƒ(t, u(t)), u(0) = u(1) in order to cover the case when the function ƒ satisfies Carathéodory conditions. This method is then applied to get multiplicity results when the nonlinearity ƒ interacts with the real eigenvalue of the linearized problem. Our proofs are based on differential inequalities and classical Leray-Schauder degree.