Exponential stability of the exact solutions and the numerical solutions for a class of linear impulsive delay differential equations

Stability of Stochastic Delay Differential Systems With Variable Impulses Due to Logic Choice

“…In [4], G. Zhang et al have studied the exponential stability of the following linear impulsive delay differential equation:…”

Existence Theorems for Fractional Semilinear Integrodifferential Equations with Noninstantaneous Impulses and Delay

“…Much attention has been paid to the existence of solutions for the differential equations with impulses in abstract space. For details, see [7,[20][21][22][23][24][25][26][27][28].…”

“…In [7], the authors discussed the existence of solutions for the fractional ordinary differential equation with a generalized impulsive term. In [20][21][22][23], the authors discussed the integer or fractional differential equations with instantaneous impulses and the linear operator A is independent of t. In [26,29,30], the authors discussed the integer-order differential equations with noninstantaneous impulses and the linear operator A is independent of t. In [31][32][33], the authors discussed the fractional differential equations with noninstantaneous impulses and the linear operator A is also independent of t. In this paper, we consider the fractional semilinear integrodifferential equations with noninstantaneous impulses and delay, and the linear operator AðtÞ is assumed to be dependent on t. Therefore, the mentioned results above are special cases of the problem investigated in this paper. Our results improve and generalize the results in References [7, 20-23, 26, 29-33].…”

Stability Analysis of Analytical and Numerical Solutions to Nonlinear Delay Differential Equations with Variable Impulses

“…In [8], the convergence of Euler method for linear IDDEs is studied. Zhang et al studied the numerical stability of Runge-Kutta methods to IDDEs in [9,10] and proved that backward Euler method and 2-stage Lobatto IIIC method can preserve the stability of nonlinear IDDEs with fixed impulses. Unfortunately, the numerical scheme may not converge when the impulses are variable.…”

“…Discrete Dynamics in Nature and Society (see [9][10][11][12]). But the equivalent equations may not work when the impulses are variable.…”