On the Hyers–Ulam stability of the linear differential equation

Abstract

We obtain some results on generalized Hyers-Ulam stability of the linear differential equation in a Banach space. As a consequence we improve some known estimates of the difference between the perturbed and the exact solutions.

Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives

“…Popa and Raşa proved the generalized Hyers-Ulam stability of linear differential equations in a Banach space (see [10]). András and Mészáros presented Hyers-Ulam stability of dynamic equations on time scales via Picard operators (see [1]).…”

An integrating factor approach to the Hyers-Ulam stability of a class of exact differential equations of second order

“…Soon afterwards, such stability results of the differential equation y = λy in various abstract spaces have been obtained by Miura and Takahasi et al [12,13,22]. Since then, many interesting results on the Ulam stability of different types of differential equations have been established by various authors [1,3,4,5,7,8,9,10,11,14,15,17,18,19,20,21,23].…”

Ulam's Stabilities of Fractional Integro-Differential Equations

“…Thereafter, Alsina and Ger published their paper [14], which handles the Hyers-Ulam stability of the linear differential equation y ′ (t) = y(t): If a differentiable function y(t) is a solution of the inequality |y ′ (t) − y(t)| ≤ ε for any t ∈ (a, ∞), then there exists a constant c such that |y(t) − ce t | ≤ 3ε for all t ∈ (a, ∞).Recently, the Hyers-Ulam stability problems of linear differential equations of first order and second order with constant coefficients were studied by using the method of integral factors (see [15,16]). The results given in [17][18][19] have been generalized by Popa and Rus [20,21] for the linear differential equations of nth order with constant coefficients. For more details on Hyers-Ulam stability and the generalized Hyers-Ulam stability, we refer the reader to the papers [22][23][24][25][26][27][28].…”