2011/01/18 by Murat Adıvar, Adivar, Murat, Youssef N. Raffoul +1
Engineering · Mathematics · Medicine · #34K20 #39A13 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Differential Equations Analysis #Primary 34N05 #Secondary 39A12 #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1101.3475
openalex publication_date 2011/01/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper, we use what we call the shift operator so that general delay dynamic equations of the form xΔ(t)=a(t)x(t)+b(t)x(δ-(h,t))δ-Δ% (h,t), t∈\lbrack t0,∞)_\mathbbT% 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. Keywords: Delay dynamic equation, instability, shift operators, stability, time scales.