THE METHOD OF LOWER AND UPPER SOLUTIONS FOR 2ND, 3RD, 4TH, AND HIGHER-ORDER BOUNDARY-VALUE-PROBLEMS

In this paper we develop the monotone method in the presence of lower and upper solutions for the problem u(n)(t) = f(t, u(t)); u(i)(a) - u(i)(b) = lambda(i) is-an-element-of R; i = 0, ..., n - 1 Where f is a Caratheodory function. We obtain necessary and sufficient conditions in f to guarantee the existence of solutions between a lower solution alpha and an upper solution beta for n = 2 (if alpha greater-than-or-equal-to beta), n = 3 (either alpha less-than-or-equal-to beta or a greater-than-or-equal-to beta) and n = 4 (if alpha less-than-or-equal-to beta). Furthermore, we obtain sufficient conditions in f for n = 2k greater-than-or-equal-to 6 when alpha less-than-or-equal-to beta. (C) 1994 Academic Press, Inc.