On the nonoscillatory phase function for Legendre's differential equation

Effect of nonlinear perturbations on second order linear nonoscillatory differential equations

Abstract. The aim of this paper is to show that any second order nonoscillatory linear differential equation can be converted into an oscillating system by applying a “sufficiently large”nonlinear perturbation. This can be achieved through a detailed analysis of possible nonoscillatory solutions of the perturbed differential equation which may exist when the perturbation is “sufficiently small”...

Research Article Legendre's Differential Equation and Its Hyers-Ulam Stability

Comparison Theorems for Fourth Order Differential Equations Garret

This paper establishes an apparently overlooked relationship between the iv yiV p'air of fourth order line,r equations y p(x)y 0 and + p(x)y 0, where p is , positive, continuous function defined on [O,m). It is shown that if all solutions of the first equation are nonoscillatory, then all solutions of the second equation must be nonoscillatory as well. An oscillation criterion for these equatio...

Oscillation and Nonoscillation Theorems for a Class of Even-order Quasilinear Functional Differential Equations

We are concerned with the oscillatory and nonoscillatory behavior of solutions of evenorder quasilinear functional differential equations of the type (|y(n)(t)|α sgn y(n)(t))(n) + q(t)|y(g(t))|β sgn y(g(t))= 0, where α and β are positive constants, g(t) and q(t) are positive continuous functions on [0,∞), and g(t) is a continuously differentiable function such that g′(t) > 0, limt→∞ g(t)=∞. We ...

Oscillation Criteria for Second-order Linear Differential Equations^)

where p(x) is a continuous positive function for 0<x< oo. Equation (1) is said to be nonoscillatory in (a, oo) if no solution of (1) vanishes more than once in this interval. Because of the Sturm separation theorem, this is equivalent to the existence of a solution which does not vanish at all in (a, oo). The equation will be called nonoscillatory—without the interval being mentioned —if there ...