Solvability of a (p, n-p)-type multi-point boundary-value problem for higher-order differential equations

Abstract

In this article, we study the differential equation (-1)n-p x(n) (t) = f (t, x(t), x'(t), ..., x(n-1) (t)), 0 < t < 1, subject to the multi-point boundary conditions x(i) (0) = 0 for i = 0, 1, ..., p - 1, x(i) (1) = 0 for i = p + 1, ..., n - 1, m∑i=1 αix(p) (ξi) = 0, where 1 ≤ p ≤ n - 1. We establish sufficient conditions for the existence of at least one solution at resonance and another at non-resonance. The emphasis in this paper is that <i>f</i> depends on all higher-order derivatives. Examples are given to illustrate the main results of this article.