Oscillation and nonoscillation theorems for a class of even-order quasilinear functional differential equations

Jelena Manojlović,Tomoyuki Tanigawa

doi:10.1155/jia/2006/42120

Abstract

We are concerned with the oscillatory and nonoscillatory behavior of solutions of even-order quasilinear functional differential equations of the type , where and are positive constants, and are positive continuous functions on , and is a continuously differentiable function such that , . We first give criteria for the existence of nonoscillatory solutions with specific asymptotic behavior, and then derive conditions (sufficient as well as necessary and sufficient) for all solutions to be oscillatory by comparing the above equation with the related differential equation without deviating argument.

Highlights

We consider even-order quasilinear functional differential equations of the form y(n)(t) α sgn y(n)(t) (n) + q(t) y g(t) β sgn y g(t) = 0, (A)where (a) α and β are positive constants;(b) q : [0, ∞) → (0, ∞) is a continuous function;(c) g : [0,∞) → (0,∞) is a continuously differentiable function such that g (t) > 0, t ≥ 0, and limt→∞ g(t) = ∞.By a solution of (A) we mean a function y : [Ty, ∞) → R which is n times continuously differentiable together with |y(n)|α sgn y(n) and satisfies (A) at all sufficiently large t
The objective of this paper is to study the oscillatory and nonoscillatory behavior of solutions of (A)
Our purpose here is to make a detailed analysis of the structure of the set P of all possible positive solutions of (A)

Summary

Introduction

We consider even-order quasilinear functional differential equations of the form y(n)(t) α sgn y(n)(t) (n) + q(t) y g(t) β sgn y g(t) = 0,. (c) g : [0,∞) → (0,∞) is a continuously differentiable function such that g (t) > 0, t ≥ 0, and limt→∞ g(t) = ∞. By a solution of (A) we mean a function y : [Ty, ∞) → R which is n times continuously differentiable together with |y(n)|α sgn y(n) and satisfies (A) at all sufficiently large t. Those solutions which vanish in a neighborhood of infinity will be excluded from our consideration.

Quasilinear functional differential equations

Classification and integral representations of positive solutions

Nonoscillation criteria

Comparison theorems

Oscillation criteria

Example