A note on a functional inequality

Abstract

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 seems interesting since the known transformations for changing (3) to the normal form (1) do not preserve the square integrability of solutions (see §3); it is also easier to find examples of the limit circle case for the self-adjoint form (3). In addition, the proof given here is more straightforward since a Priifer type transformation is not needed.…”

“…Proofs of the theorems. The proofs of Theorems 1 and 2 rely on the following lemma which is a corollary to a version of the Gronwall inequality recently proved by H. E. Gollwitzer [3].…”

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“…These results are extensions of those proved by Ibrahim in [10]. Our approach is based on an extension of Gronwall's inequality used by Bradley and due to Gollwitzer [6], on a technical lemma from Goldberg's book [5] and on an appropriate formulation of the variation of parameters formula.…”

“…The next lemma is a special case of an extension of the well-known Gronwall inequality due to Gollwitzer [6] (See also Willett [12] and Willet-Wong [13]). …”

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