Inequalities for solutions of multipoint boundary value problems (Q1974419)

The concept of concavity is generalized to functions \(y\) satisfying \(y^{(n)}\geq 0\), and homogeneous multipoint boundary conditions, \[ y^{(j)}(a_i)=0,\;j=0,\cdots,n_i,i=1,\cdots,k, \] with \(0=a_1<\cdots<a_k=1\), and \(\sum_{i=1}^k n_i=n\). A piecewise polynomial, which bounds the function \(y\) below, is constructed and is employed to obtain \((-1)^{\alpha_i}y(t)\geq ||y||a^m\) for some constants \(a, m, \alpha_i,i=1,\cdots,k-1\). This inequality is useful in applications of certain cone theoretic fixed point theorems.