In this paper we will consider the functional inequalities (1) u(t) ^ f(t) + g(t)G~l(§ G(u(s))h(s)ds\ a^x^t^b, (2) u(t) ^ u(x)-g(t)G~l(f G(u(s))h(s)ds\ where the functions u, f, g and h are nonnegative and continuous on the interval [a, b]. The function G(u) is continuous and strictly increasing for ra^O, G(0)=0, lim,,..,, G(u)= oo and G_1 denotes the inverse function of G. If G(u) =u, we have the well-known Gronwall inequality and a case similar to the Langenhop inequality [3]. If G(u)=up, p^l, then Willett [4] has studied (1) in connection with a singular perturbation problem. Our purpose here is to obtain new estimates for u(t) if G is a convex or concave function. Before giving the main results, we will state two important preliminary lemmas. Lemma 1. Let u(t),f(t), g(t) and h(t) be nonnegative, continuous functions on the interval [a, b]; and

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This paper is concerned with the problem of specifying growth conditions on the positive function q(t) which imply that all solutions of the nonlinear second order ordinary differential equation y" +q(t) | y \ ° sgn y = 0, a > 0, are nonoscillatory on a half line. Several different results are given, and the usual explicit monotonicity condition on q has been avoided to a certain degree.

Abstract.This paper is concerned with the problem of specifying growth conditions on the positive function q(t) which imply that all solutions of the nonlinear second order ordinary differential equation y" +q(t) | y \ ° sgn y = 0, a > 0, are nonoscillatory on a half line. Several different results are given, and the usual explicit monotonicity condition on q has been avoided to a certain degree.In the real, nonlinear differential equation (1) y' + î(0|y|"sgny = 0, C = d/dt), let q(t) be positive, continuous and locally of bounded variation on a half line (a, °o), and suppose that (Xa^l. Our purpose here is to give several sets of conditions which imply that all solutions of (1) are nonoscillatory.These conditions are of interest since the usual explicit monotonicity condition on q can be avoided in some sense. We begin with some preliminary facts and definitions.

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