Inequalities for solutions of multipoint boundary value problems

The concept of concavity is generalized to functions, y, satisfying nth ord er differential inequalities, y((n)) (t) greater than or equal to 0, 0 less than or equal to t less than or equal to 1 and homogeneous multipoint boun dary conditions, y((j)) (a(i)) = 0, j = 0,..., n(i), i = 1,..., k, where 0 = a(1) < a(2) < ... < a(k) = 1 and Sigma(i)(k)=1 n(i) = n. A piecewise poly nomial, which bounds the function, y, below, is constructed and then is emp loyed to obtain that if (3a(i) + a(i)+1)/4 less than or equal to t less tha n or equal to (a(i) + 3a(i)+1)/4, then (-1)(alpha i)y(t) greater than or eq ual to parallel to y parallel to(a/4)(m), i = 1,..., k - 1, where a = min(i )(a(i)+1 - a(i)), parallel to .parallel to denotes the supremum norm, m = m ax(n - nl, n - nk), and alpha(i) = Sigma(j)(k)=i+1 n(j), i = 1,..., k - 1. An analogous inequality for a related Green's function is also obtained. Th ese inequalities are useful in applications of certain cone theoretic fixed point theorems.