Nonoscillation Theorems for a Nonlinear Differential Equation

Abstract

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.

Remarks on Nonoscillation Theorems for a Second Order Nonlinear Differential Equation

“…Returning to (9), we deduce therefore that lim^^, t,l+xa\t) exists and is nonnegative. However, if such a limit is positive, then iMax(r) > e/t for some e > 0 and for all sufficiently large t, which would contradict assumption (5), proving (6). We now assume (7) holds.…”

“…The function a(t) is said to be locally of bounded variation on [0, oo) if it is of bounded variation on each compact subinterval of [0, oo). If a(t) is continuous and locally of bounded variation on [0, oo), then a(t) admits the Jordan representation a(t) = a+(t) -a_(t) where a+, a_ are continuous nondecreasing functions of t. Gollwitzer [6] showed that the nonincreasing assumption on a(t) can be weakened to that of…”

Nonoscillation Theorems for a Second Order Sublinear Ordinary Differential Equation

“…•'n°c da+(t) i s < oo, a(t) due to Gollwitzer [5]. Under this condition, he proved the following three nonoscillation criteria for (1):…”

“…Equation (1) is said to be nonoscillatory if every solution has only a finite number of zeros. For details, we refer to [1,2,5,9].Since a(t) is assumed to be locally of bounded variation it admits a Jordan decomposition a(t) = a+(t) -a_(t), where a+ and a_ are continuous nondecreasing functions. A number of nonoscillation results for the nonlinear equation (1), y ¥= 1, require some sort of restriction on the growth of the function a(t), typically that a(t) be nonincreasing.…”

“…Equation (1) is said to be nonoscillatory if every solution has only a finite number of zeros. For details, we refer to [1,2,5,9].…”