A note on stability of impulsive differential equations

Abstract

In this note, we study a new class of ordinary differential equations with non-instantaneous impulses. Both existence and generalized Ulam-Hyers-Rassias stability results are established. Finally, an example is given to illustrate our theoretical results.

p-Moment exponential stability of Caputo fractional differential equations with noninstantaneous random impulses

“…Therefore, from (34) and (38) it follows the function v(t) satisfies the linear impulsive fractional differential inequalities with fixed points of impulses …”

“…Since the introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous processes, we can interpret the situation as an impulsive action which starts abruptly and stays active on a finite time interval. Recently results concerning noninstantaneous impulses are obtained for differential equations [4,14,38], delay integro-differential equations [16], abstract differential equations [24,39], and fractional differential equations, FrDEs [2,33,37].…”

Stability analysis of higher order nonlinear differential equations in β–normed spaces

“…To the best of our knowledge, the mathematicians who first investigated Ulam's type stability of impulsive ordinary differential equations were Wang et al Following their own work, in 2014, they proved the Hyers–Ulam–Rassias stability and generalized Hyers–Ulam–Rassias stability for impulsive evolution equations on a compact interval, which then they extended for infinite impulses in the same paper. For more details, the reader may see the previous studies . The problems with instantaneous impulses can not characterize processes in which the problem solutions has an interval discontinuity, for example, the introduction of drugs in the bloodstream and the consequent absorption for the body is gradual and a continuous process.…”

“…For more details, the reader may see the previous studies. [24][25][26][27][28][29] The problems with instantaneous impulses can not characterize processes in which the problem solutions has an interval discontinuity, for example, the introduction of drugs in the bloodstream and the consequent absorption for the body is gradual and a continuous process. These type of processes are characterized by noninstantaneous impulses, which starts from an arbitrary fixed point and stays alive on a finite interval.…”

Hyers-Ulam Stability of First-Order Non-Linear Delay Differential Equations with Fractional Integrable Impulses

“…In 2016, Zada et al [38] using fixed point method discussed Hyers-Ulam stability and Hyers-Ulam-Rassias stability of first order impulsive delay differential equations. For more details on impulsive differential equations, we recommend [5,15,18,23].…”