Abstract
A stability theory of nonlinear impulsive delay differential equations (IDDEs) is established. Existing algorithm may not converge when the impulses are variable. A convergent numerical scheme is established for nonlinear delay differential equations with variable impulses. Some stability conditions of analytical and numerical solutions to IDDEs are given by the properties of delay differential equations without impulsive perturbations.
Highlights
We study the stability of nonlinear impulsive differential equations: x (t) = f (t, x (t), x (t − τ)), t ≥ 0, t ≠ kτ, Δx = rkx (t), t = kτ, k = 0, 1, . . . , (1)
Impulsive differential equations are widely used in actual modeling such as epidemic, optimal control and population dynamics
Extensive work has been done in the area of qualitative theories of impulsive delay differential equations (IDDEs)
Summary
We study the stability of nonlinear impulsive differential equations: x (t) = f (t, x (t) , x (t − τ)) , t ≥ 0, t ≠ kτ, Δx = rkx (t) , t = kτ, k = 0, 1, . In [8], the convergence of Euler method for linear IDDEs is studied. The numerical scheme may not converge when the impulses are variable. It is necessary to find a convergent stable method for IDDEs with variable pulses.
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