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Approximation of solutions to impulsive functional differential equations. (English) Zbl 1214.34069

The authors consider the impulsive semilinear functional differential equation \[ u'(t)+ Au(t)=f(t,u_t),\quad t\in (0,T), \;t\neq t_k, \]
\[ \Delta u(t_k)=I_k(u(t_k)), \quad k=1,2,\dots, p,\tag{1} \]
\[ u(t)=h(t), \quad t\in [-\tau,0], \]
where \(-A\) is the infinitesimal generator of an analytic semigroup on a separable Hilbert space \(H\), \(I_k:H\to H\), \(\Delta u(t_k)=u(t_k^+)-u(t_k^-)\), \(f:[0,T]\to C_0\) and \(h\in C_0\). Here \(C_0\) consists of all piecewise continuous functions from \([-\tau,0]\) to \(H\). Using fixed point arguments, they prove the existence, uniqueness and convergence of a suitably defined sequence of approximate solutions of problem (1) to the solution of that problem. Some results concerning finite-dimensional approximation of the solutions of (1) – Theorems 4.1 and 4.2 – as well as an illustrating example referring to an impulsive semilinear functional differential equation of parabolic type are also included.

MSC:

34K30 Functional-differential equations in abstract spaces
34K05 General theory of functional-differential equations
34K45 Functional-differential equations with impulses
47D06 One-parameter semigroups and linear evolution equations
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