Abstract: This paper is concerned with the following n-th ordinary differential equation: {u(n)(t) = ...=0, where a, c ∈ R, ≥, such that a²+b² >0 and c²+d²>0, n ≥ 2, f: [0,1] × R → R is a continuous function. Assume that f satisfies one-sided Nagumo condition, the existence theorems of solutions of the boundary value problem for the n-th-order nonlinear differential equations above are established by using Leray-Schauder degree theory, lower and upper solutions, a priori estimate technique. 

Key words: n-th order boundary value problems, one-sided Nagumo condition, lower and upper solutions, a priori estimates, Leray-Schauder degree