Lyapunov stability solutions of fractional integrodifferential equations

Abstract

Lyapunov stability and asymptotic stability conditions for the solutions of the fractional integrodiffrential equations x (α) s, x(s))ds, 0 < α ≤ 1, with the initial condition x (α−1) (t 0 ) = x 0 , have been investigated. Our methods are applications of Gronwall's lemma and Schwartz inequality.

Boundedness and power-type decay of solutions for a class of generalized fractional Langevin equations

“…In this paper, we consider a much wider class of nonlinearities than the ones considered in the previous papers [14,16,30]. Also, we improve the results in these papers by weakening the imposed conditions.…”

“…Clearly, this nonlinearity is much more general than the one in [14,30] and even more general than the ones in [6,7,15,31].…”

“…The long-time behavior of solutions of differential equations has attracted many researchers, see [6,7,14,16,19,27,30,31] . In many cases, the main idea is to establish sufficient reasonable conditions ensuring comparison or similarity with the long-time behavior of solutions of simpler differential equations.…”

“…In 2004, Momani et al [30] discussed the Lyapunov stability and asymptotic stability for solutions of the fractional integro-differential equation (2) for t ≥ a > 0. The assumptions:…”

Comparison of Adomian decomposition and Homotopy perturbation methods for higher-order linear fractional integrodifferential equations

“…In this paper, we considered the linear boundary value problems for higher-order fractional integro-differential equations with a Caputo fractional derivative of the type: , k x t are given and can be approximated by Taylor polynomials. The existence and stability of solutions for fractional integro-differential equations [12][13][14]. He [15][16][17][18][19] was the first to propose the Adomian decomposition method (ADM) and homotopy perturbation method (HPM) for finding the solutions of non-linear problems.…”