Remarks on the periodic boundary value problems for first-order differential equations

This paper is concerned with the existence of solutions and the monotone method of first-order periodic boundary value problems when the lower and upper solutions α and β violate the boundary conditions α(0) ≤ α(2π) and β(0) ≥ β(2π). Using the topological degree theory, two existence theorems are established under weaker conditions than the one-side Lipschitz conditions. An example is given, which illustrates that PBVP may not have solutions between α and β without further restrictions to f(t, u). The monotone method is also discussed with some new results.