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Convergence of the solution of an impulsive differential equation with piecewise constant arguments. (English) Zbl 1299.34247

Summary: We prove the existence of the unique solution of the impulsive differential equation \[ \begin{aligned} &x' (t)= a(t) (x(t)-x (\lfloor t-1 \rfloor ))+ f (t),\;t\neq n \in \mathbb Z^+ =\{1,2,\dots\},\;t\geq 0,\\ &\Delta x(t)= c_t x(t)+ d_t,\;t = n \in \mathbb Z^+, \end{aligned} \] with the initial conditions \[ x (-1)=x_{-1},\;x (0)= x_0, \] where \(\lfloor \, .\,\rfloor\) denotes the floor integer function. Moreover, we obtain sufficient conditions for the asymptotic constancy of this equation and we compute, as \(t\to \infty\), the limits of the solutions of the impulsive equation with \(c_n= 0\) in terms of the initial conditions, a special solution of the corresponding adjoint equation and a solution of the corresponding difference equation.

MSC:

34K25 Asymptotic theory of functional-differential equations
34K45 Functional-differential equations with impulses
34K06 Linear functional-differential equations
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