Existence of solutions for some higher order boundary value problems

Abstract

In this paper, we are concerned with the existence of solutions for the higher order boundary value problem in the formu(2 m + 2)(x) = f (x, u (x), u″(x), ..., u(2 m)(x)), x ∈ (0, 1),u(2 i)(0) = u(2 i)(1) = 0, 0 ≤ i ≤ m, where m is a given positive integer and f : [0, 1] × Rm + 1→ R is continuous. We introduce a new maximum principle of higher order equations and develop a monotone method in the presence of lower and upper solutions for this problem. © 2006 Elsevier Inc. All rights reserved.