Properties of third-order half-linear dynamic equations with an unbounded neutral coefficient

Tongxing Li,Chenghui Zhang

doi:10.1186/1687-1847-2013-333

Abstract

We study oscillation and asymptotic behavior of a class of third-order half-linear functional dynamic equations with an unbounded neutral coefficient. Several comparison theorems are presented that are essentially new.

Highlights

1 Introduction Neutral differential equations appear in modeling of the networks containing lossless transmission lines, in the study of vibrating masses attached to an elastic bar, as the Euler equation in some variational problems, in the theory of automatic control and in neuromechanical systems in which inertia plays an important role; see Hale [ ]
A time scale T is an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the theories of differential and of difference equations
Does the new theory of the so-called dynamic equations unify the theories of differential equations and difference equations, and extends these classical cases to cases ‘in between’, e.g., to the so-called q-difference equations, when T = qN := {qt : t ∈ N for q > } (which has important applications in quantum theory)

Summary

Introduction

Neutral differential equations appear in modeling of the networks containing lossless transmission lines, in the study of vibrating masses attached to an elastic bar, as the Euler equation in some variational problems, in the theory of automatic control and in neuromechanical systems in which inertia plays an important role; see Hale [ ].A time scale T is an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the theories of differential and of difference equations. Where t ∈ [t , ∞) T := [t , ∞) ∩ T, z := x + p · x ◦ τ , and we assume that the following conditions are satisfied: (H ) γ ≤ is a quotient of odd positive integers; (H ) p ∈ Crd( [t , ∞) T , [ , ∞)) and q ∈ Crd( [t , ∞) T , ( , ∞)); (H ) r ∈ C rd(T, R), τ , δ, δ– ∈ C rd(T, T), r(t) > , and limt→∞ τ (t) = limt→∞ δ(t) = ∞, where δ– denotes the inverse function of δ; (H ) τ ( [t , ∞) T) = [τ (t ), ∞) T, δ– ( [t , ∞) T) = [δ– (t ), ∞) T, τ > , and (δ– ) > .

Results

Conclusion