Remarks on First and Second Order Periodic Boundary Value Problems
Abstract
**Please note that the full text is embargoed** ABSTRACT: We consider the first and second order periodic boundary value problems (PBVP for brevity)[see pdf for notation] and [see pdf for notation]and obtain
the existence of extremal solutions as limits of monotone iterates. In [2] and [3], the monotone method was employed for the problems (1.1) and (1.2)
when the corresponding lower and upper solutions a(t),B(t) satisfy [see pdf for notation] in addition to other conditions. Furthermore, when f is
increasing, it is shown [2,3] that the conditions (1.3) and (1.4) are tantamount to assuming the existence of a periodic solution. Accordingly, the
problem of proving the existence of periodic solutions when the conditions (1.3) and (1.4) are violated becomes important and this question has been
open. Our main objective in this paper is to investigate this open problem.