Abstract
In this paper, we have investigated that initial time difference boundedness criteria and Lagrange stability for fractional order differential equation in Caputo's sense are unified with Lyapunov-like functions to establish comparison result. The qualitative behavior of a perturbed fractional order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional order differential equation with Caputo's derivative has been investigated. We present a comparison result that again gives the null solution a central role in the comparison fractional order differential equation when establishing initial time difference boundedness criteria and Lagrange stability of the perturbed fractional order differential equation with respect to the unperturbed fractional order differential equation in Caputo's sense.
Highlights
The concept of noninteger-order derivative, popularly known as fractional derivative, goes back to the 17th century [1,2]
Lyapunov’ s direct method is a standard technique used in the study of the qualitative behavior of differential systems along with a comparison result [4,8,9,10,11] that allows the prediction of behavior of a differential system when the behavior of the null solution of a comparison system is known
The classical notions of boundedness and Lagrange stability [5,7,8,9,10,21] are with respect to the null solution, but initial time difference (ITD) boundedness and Lagrange stability [2,12,13,14,15,16,17,18,19,20] are with respect to the unperturbed fractional order differential system where the perturbed fractional order differential system and the unperturbed fractional order differential system differ both in initial position and in initial time [2,12,13,14,15,16,17,18,19,20]
Summary
The concept of noninteger-order derivative, popularly known as fractional derivative, goes back to the 17th century [1,2]. We have dissipated this complexity and have a new comparison result that again gives the null solution a central role in the comparison fractional order differential system This result creates many paths for continuing research by direct application and generalization [15,19,20,22].
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