Nonlinear Analysis 70 (2009) 4027–4038 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A note on a class of problems for a higher-order fully nonlinear equation under one-sided Nagumo-type condition M.R. Grossinho a,∗ , F. Minhós b , A.I. Santos b a Departamento de Matemática e Centro de Matemática e Aplicações à Previsão e Decisão Económica. Instituto Superior de Economia e Gestão. Technical University of Lisbon. Rua do Quelhas, n ◦ 6, 1200-781 Lisboa, Portugal b Departamento de Matemática e Centro de Investigação em Matemática e Aplicações da U.E. (CIMA-UE). Universidade de Évora. R. Romão Ramalho, n ◦ 59, 7000-671 Évora, Portugal article info Article history: Received 15 October 2007 Accepted 28 August 2008 MSC: 34B10 34B15 34L30 Keywords: Higher-order BVP One-sided Nagumo-type conditions Lower and upper solutions A priori estimates Leray–Schauder degree abstract The purpose of this work is to establish existence and location results for the higher-order fully nonlinear differential equation u (n) (t ) = f (t , u(t ), u ′ (t ),..., u (n−1) (t )), n ≥ 2, with the boundary conditions u (i) (a) = A i , for i = 0,..., n − 3, u (n−1) (a) = B, u (n−1) (b) = C or u (i) (a) = A i , for i = 0,..., n − 3, c 1 u (n−2) (a) − c 2 u (n−1) (a) = B, c 3 u (n−2) (b) + c 4 u (n−1) (b) = C , with A i , B, C ∈ R, for i = 0,..., n − 3, and c 1 , c 2 , c 3 , c 4 real positive constants. It is assumed that f : [a, b] × R n−1 → R is a continuous function satisfying one- sided Nagumo-type conditions which allows an asymmetric unbounded behaviour on the nonlinearity. The arguments are based on the Leray–Schauder topological degree and lower and upper solutions method. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Let us consider the nth-order differential equation u (n) (t ) = f (t , u(t ),..., u (n−1) (t )), (1) for n ≥ 2, with the following boundary conditions u (i) (a) = A i , (2) u (n−1) (a) = B, u (n−1) (b) = C , ∗ Corresponding author. E-mail addresses: mrg@iseg.utl.pt (M.R. Grossinho), fminhos@dmat.uevora.pt (F. Minhós), aims@dmat.uevora.pt (A.I. Santos). 0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.08.011