Positive solutions of boundary-value problems for 2m-order differential equations

Abstract

This article concerns the existence of positive solutions to the differential equation (-1)m x(2m)(t) = f(t,x(t), x'(t),...,x(m)(t)), 0 < t < π, subject to boundary condition x(2i)(0) = x(2i) (π) = 0, or to the boundary condition x(2i)(0) = x(2i + 1) (π) = 0, for i = 0,1,...,m - 1. Sufficient conditions for the existence of at least one positive solution of each boundary-value problem are established. Motivated by references [7, 17, 21], the emphasis in this paper is that f depends on all higher-order derivatives.