Positive solutions of a boundary value problem (Q1174939)

Let \(K\) be a cone in \(\mathbb{R}^ n\) and \(u\) a function in \(C^ 2([a,b],R^ n)\). It is known that if \(u(a)\) and \(u(b)\) belong to \(K\), and \(-u''(t)\in K\) for all \(a\leq t\leq b\), then \(u(t)\in K\) for all \(a\leq t\leq b\). The present paper provides a generalization of this result to (solid) cones in a Banach space \(X\) and to functions in \(C^ n([a,b],X)\), \(n\geq 2\).