Inequalities Based on a Generalization of Concavity

The concept of concavity is generalized to functions, y, satisfying nth order differential inequalities, (−1)n−ky(n)(t) ≥ 0, 0 ≤ t ≤ 1, and homogeneous two-point boundary conditions, y(0) = . . . = y(k−1)(0) = 0, y(1) = . . . = y(n−k−1)(1) = 0, for some k ∈ {1, . . . , n− 1}. A piecewise polynomial, which bounds the function, y, below, is constructed, and then is employed to obtain that y(t) ≥ ||y||/4m, 1/4 ≤ t ≤ 3/4, where m = max{k, n − k} and || · || denotes the supremum norm. An analogous inequality for a related Green’s function is also obtained. These inequalities are useful in applications of certain cone theoretic fixed point theorems. In recent applications of cone theoretic fixed point theorems to boundary value problems (BVPs), inequalities that provide lower bounds for positive functions as a function of the supremum norm have been applied. This type of inequality has been useful in applications to both regular two-point BVPs ([5], [4]) on annular like regions, and singular two-point BVPs ([6], [3]). The particular inequality to which we refer is as follows: if y′′(t) ≤ 0, 0 ≤ t ≤ 1, and y(t) ≥ 0, 0 ≤ t ≤ 1, then for 1/4 ≤ t ≤ 3/4, y(t) ≥ ||y||/4, (1) where ||y|| = sup0≤t≤1|y(t)|. An analogous inequality for a Green’s function has been employed for regular two-point BVPs [5], [4]. The purpose of this short paper is to obtain generalizations of (1) and the analogous inequalities for Green’s functions. These inequalities will play analogous roles in the study of BVPs for nth order ordinary differential equations. In particular, we shall show that if n ≥ 2 is an integer, k ∈ {1, . . . , n− 1}, and if (−1)(n−k)y(n) ≥ 0, 0 ≤ t ≤ 1, (2) y(0) = 0, j = 0, . . . , k − 1, y(1) = 0, j = 0, . . . , n− k − 1, (3) then for 1/4 ≤ t ≤ 3/4, y(t) ≥ ||y||/4m, (4) where m = max{k, n− k}. Received by the editors July 12, 1995 and, in revised form, February 6, 1996. 1991 Mathematics Subject Classification. Primary 34A40; Secondary 34B27.