Shift operators and stability in delayed dynamic equations

TL;DR

This paper introduces shift operators to analyze the stability and solutions of general delay dynamic equations on time scales, unifying various delay equations through Lyapunov methods.

Contribution

It develops a new framework using shift operators and Lyapunov functionals to unify stability analysis of diverse delay dynamic equations on time scales.

Findings

Unified stability criteria for delay differential and difference equations.

Extension of Lyapunov's direct method to general delay dynamic equations.

Framework applicable to various types of delay equations on time scales.

Abstract

In this paper, we use what we call the shift operator so that general delay dynamic equations of the form \[ x^{\Delta}(t)=a(t)x(t)+b(t)x(\delta_{-}(h,t))\delta_{-}^{\Delta}% (h,t),\ \ \ t\in\lbrack t_{0},\infty)_{\mathbb{T}}% \] can be analyzed with respect to stability and existence of solutions. By means of the shift operators we define a general delay function opening an avenue for the construction of Lyapunov functional on time scales. Thus, we use the Lyapunov's direct method to obtain inequalities that lead to stability and instability. Therefore, we extend and unify stability analysis of delay differential, delay difference, delay $h-$difference, and delay $q-$difference equations which are the most important particular cases of our delay dynamic equation. \textbf{Keywords}: Delay dynamic equation, instability, shift operators, stability, time scales.