An abstract spectral approximation theorem from the theory of semigroups (Q1184814)

Using the theory of semigroups, under some conditions, the author establishes convergence of the sequence of approximations to a solution to the initial value problem \((du/dt)=Au(t)+f(t,u(t))\), \(t\in[0,T_ 0]\), \(u(0)=x_ 0\), \(x_ 0\in B\), in a Banach space B. Each of the approximations satisfies the initial value problem \((du_ n/dt)=P_ nAP_ nu_ n(t)+P_ nf(t,u_ n(t))\), \(u_ n(0)=P_ nx_ 0\), where \(P_ n\) is a projection onto a subspace of \(B\). There are examples of application of the established results to certain equations.