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 alpha and beta violate the boundary conditions alpha(0) less than or equal to alpha(2 pi) and beta(0) greater than or equal to beta(2 pi). 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 alpha and beta without further restrictions to f(t, u). The monotone method is also discussed with some new results. (C) 1999 Elsevier Science Ltd. All rights reserved.