Monotone method for second order periodic boundary value problems

Abstract

IN RECENT years, there has been an extensive study of the existence of periodic solutions [l, S-11, 14, 151. In [8, 111, the existence of solutions of first and second order PBVP (period boundary value problems) has been obtained by a novel approach of combining the classical method of lower and upper solutions and the method of alternative problems (LyapunovSchmidt method), which provide conditions that are easily verifiable and which covers previous known results of other authors. As a constructive method for obtaining extremal solutions of initial and boundary value problems, the monotone iterative procedure has been employed by several researchers [5-7, 11-13, 151. The objective of this paper is to employ this useful technique for second order PBVP to obtain the minimal and maximal solutions as limits of monotone iterates. Our method can be used to study semilinear parabolic initial boundary value problems and other problems at resonance.