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Yayın Different convergences in approximation of evolution equations(Maltepe Üniversitesi, 2009) Piskarev, S.Consider the semilinear equation in Banach space E ? u 0 (t) = Au(t) + f(u(t)), t ? 0, u(0) = u 0 ? E ?, (0.1) where f(·) : E ? ? E ? E, 0 ? ? < 1, is assumed to be continuous, bounded and continuously Fr´echet differentiable function. The problem (0.1) in the neighborhood of the hyperbolic equilibrium can be written in the form, where Au? = A+f 0 (u ? ), Fu? (v(t)) = f(v(t)+u ? )?f(u ? )?f 0 (u ? )v(t). We consider approximation of (0.2) by the following scheme, with initial data Vn(0) = v 0 n. The solution of such problem is given by formula Vn(t + ?n) = (In ? ?nAu? n,n) ?1Vn(t) + ?n(In ? ?nAu? n,n) ?1Fu? n,n(Vn(t)) = = (In ? ?nAu? n,n) ?kVn(0) + ?n? k j=0(In ? ?nAu? n,n) ?(k?j+1)Fu? n,n(Vn(j?n)), t = k?n, where Vn(0) = v 0 n. We consider different kind of consistency of generators under which one can get convergence of solutions in the vicinity of hyperbolic stationary point.
