Practical stability in terms of two measures with initial time difference

h-Stability for Differential Systems Relative to Initial Time Difference

“…For the pioneering works in this area we can refer to the papers [11,12]. After that, there are many stability results for various of differential and difference systems; see [13][14][15][16][17][18][19][20]. However, the above results were obtained by using comparison principle and differential inequalities; there are few stability criteria by using the method of variation of parameters; see [21][22][23][24].…”

Boundedness and Lagrange stability of fractional order perturbed system related to unperturbed systems with initial time difference in Caputo's sense

“…The application of Lyapunov's direct method in boundedness theory [4,9,10,12,13] has the advantage of not requiring knowledge of solutions. However, there has been difficulty with this approach when trying to apply it to unperturbed fractional differential systems [14,15] and associated perturbed fractional differential systems with an ITD. The difficulty arises because there is a significant difference between ITD boundedness and Lagrange stability [2,[12][13][14][15][16][17][18][19][20] and the classical notion of boundedness and Lagrange stability for fractional order differential systems [4,7].…”

“…The difficulty arises because there is a significant difference between ITD boundedness and Lagrange stability [2,[12][13][14][15][16][17][18][19][20] and the classical notion of boundedness and Lagrange stability for fractional order differential systems [4,7]. The classical notions of boundedness and Lagrange stability [5,[7][8][9][10]21] are with respect to the null solution, but 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].…”

“…This result creates many paths for continuing research by direct application and generalization [15,19,20,22].…”

Fractional Differential Equations in Terms of Comparison Results and Lyapunov Stability with Initial Time Difference

“…However, there has been difficulty with this approach when trying to apply it to unperturbed fractional differential systems 2, 6, 8 and associated perturbed fractional differential systems with an initial time difference. The difficulty arises because there is a significant difference between initial time difference ITD stability 7, [9][10][11][12][13][14][15][16] and the classical notion of stability for fractional differential systems 2, 6 . The classical 2 Abstract and Applied Analysis notions of stability 2-6, 8, 17 are with respect to the null solution, but ITD stability 7, [9][10][11][12][13][14][15][16] is 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 initial time 7, 9-16 . In this paper, we have dissipated this complexity and have a new comparison result which again gives the null solution a central role in the comparison fractionalorder differential system.…”