Uniformly Lipschitz Stability and Asymptotic Property in Perturbed Nonlinear Differential Systems

Abstract

Abstract. This paper shows that the solutions to the perturbed differential system g(s, y(s), T y(s))ds + h(t, y(t))have asymptotic property and uniform Lipschitz stability. To show these properties, we impose conditions on the perturbed part g(s, y(s), T y(s))ds, h(t, y(t)), and on the fundamental matrix of the unperturbed system y = f (t, y).

Lipschitz Stability for Impulsive Riemann–Liouville Fractional Differential Equations

“…Many stability concepts exist, describing various behavior of the solutions, e.g., Lipschitz stability defined for ODEs [15]. Later, this type of stability has been studied for various types of differential equations and problems such as, e.g., nonlinear differential systems [14,19,28], impulsive differential equations with delays [9], fractional differential systems [29], Caputo fractional differential equations with noninstantaneous impulses [4], a piecewise linear Schrödinger potential [7], a hyperbolic inverse problem [10], the electrical impedance tomography problem [11], and the radiative transport equation [25]. See also [2,3,8,14,19,20,27,28] for related references.…”

Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations

“…There are various types of stability that describe different properties of the solutions. One of them is the Lipschitz stability, introduced in [4] and later studied for nonlinear differential equations [5,6], for functional differential equations [7,8], for impulsive functional differential equations [9], for Caputo fractional differential equations [10], for partial differential equations [11,12], and applied to some models such as neural networks [13], electrical impedance tomography [14], and the radiate transport problem [15].…”

Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

“…One of them is Lipschitz stability, defined and studied for ordinary differential equations in [5]. Later, this type of stability was studied for various types of differential equations and problems, such as nonlinear differential systems [6][7][8], impulsive differential equations with delays [9], fractional differential systems [10], Caputo fractional differential equations with non-instantaneous impulses [11], a piecewise linear Schrödinger potential [12], a hyperbolic inverse problem [13], the electrical impedance tomography problem [14], the radiative transport equation [15] and neural networks with non-instantaneous impulses [16].…”