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

Abstract

<p>This article concerns the existence of positive solutions to the differential equation</p> <pre>(-1)^m x^(2m)(t) = ƒ(t, x(t), x'(t),...,x^(m)(t)), 0 < t < π,</pre> <p>subject to boundary condition</p> <pre>x⁽²ⁱ⁾(0) = x⁽²ⁱ⁾ (π) = 0,</pre> <p>or to the boundary condition</p> <pre>x⁽²ⁱ⁾(0) = x^{(2i + 1)} (π) = 0,</pre> <p>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.</p>