Remarks on First and Second Order Periodic Boundary Value Problems
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  and , 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.