Abstract
We study first order linear impulsive delay differential equations with periodic coefficients and constant delays. This study presents some new results on the asymptotic behavior and stability. Thus, a proper real root was used for a representative characteristic equation. Applications to special cases, such as linear impulsive delay differential equations with constant coefficients, were also presented. In this study, we gave three different cases (stable, asymptotic stable and unstable) in one example. The findings suggest that an equation that is in a way that characteristic equation plays a crucial role in establishing the results in this study.
Highlights
Our aim in this paper is to present some new results on the asymptotic behavior and stability for linear impulsive delay differential equations with periodic coefficients
Note that the results in Reference [45] are that sufficient conditions are provided for the oscillatory and non-oscillatory solutions of linear impulsive delay differential equations to tend to zero
We determined the stability of the trivial solution by converting the constructed equation into two integral equations
Summary
The impulsive delay differential equation is considered as:. where I is the initial segment of natural numbers, a and bi for i ∈ I the continuous real-valued functions on the interval [0, ∞), and τi for i ∈ I positive real numbers with τi1 6= τi for i1 , i2 ∈ I such that i1 6= i2. The authors conducted the first to study the oscillation behavior of solutions of linear impulsive delay differential equations. Some significant results were obtained behavior of solutions of the linear impulsive delay differential equations with variable delays [48,49]. The same authors of Reference [52] used the center manifold theory for the impulsive delay differential equations [50,51] to obtain information about the orbit structure in a particular pulsed SIR vaccination model involving delay. Our aim in this paper is to present some new results on the asymptotic behavior and stability for linear impulsive delay differential equations with periodic coefficients. Note that the results in Reference [45] are that sufficient conditions are provided for the oscillatory and non-oscillatory solutions of linear impulsive delay differential equations to tend to zero. X is the single solution of the initial value problem (1)–(3) if and only if the function y defined by hλ0 (u)du t ≥ −τ for is the solution of the integral Equation (10) which gives the initial condition t ∈ [−τ, 0]
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