Some qualitative properties of linear dynamic equations with multiple delays

Jan Čermák

doi:10.1186/1687-1847-2013-155

Abstract

This paper discusses stability and asymptotic properties of the delay dynamic equation y ∇ (t)= ∑ j = 0 k a j y ( τ j ( t ) ) ,t∈T, where a j ∈R are scalars, τ j are iterates of a function τ:T→T and T is a time scale unbounded above. Under some specific choices of τ, this dynamic equation involves several significant particular cases such as linear autonomous differential equations with several delays or linear autonomous higher-order difference equations. For proportional τ, we formulate an asymptotic result joint for two different time scales, including a joint form of its proof. For a general τ, we investigate stability and asymptotic properties of solutions on the continuous and discrete time scales separately. Besides a related character of the relevant results, we discuss also a possible related character of their proofs.MSC:34N05, 34K25, 39A12, 39A30.

Highlights

We consider the delay dynamic equation k y∇ (t) = ajy τ j(t), t ∈ T, ( . ) j=where aj are real scalars, τ j are the jth iterates of an increasing function τ : T → T with τ ≡ id, τ ≡ τ, τ (t) < t for t ∈ T and T is a time scale unbounded above
Qualitative analysis of delay dynamic equations on time scales has already been the subject of several investigations
Besides the methods developed directly for delay dynamic equations, there are some proof procedures utilised originally either for delay differential or difference equations, but they seem to be applicable without any extra difficulties to a general dynamic case

Summary

Introduction

Qualitative analysis of delay dynamic equations on time scales has already been the subject of several investigations These papers present techniques which enable a joint analysis of delay differential and difference equations. ) can be taken for a basic type of delay dynamic equations, its qualitative analysis has not been described yet. It might be surprising because delay dynamic equations studied in the above mentioned papers mostly have more complicated forms. Considering ( . ) with unbounded lags, we describe the cases when a joint investigation of ( . ) is possible

Objectives

Conclusion