Weighted Paley-Wiener theorem, with applications to stability of the linear multi-step methods for Volterra equations in Hilbert spaces
The author studies the weighted \(l^{1}\) Paley-Wiener Theorem, with applications to stability of the linear multi-step time discretization for a linear Volterra-type equation of scalar type in a Hilbert space: \[ \frac{\partial u}{\partial t}+\int\limits_{0}^{t}\beta(t-s)Au(s)ds=0, \quad t>0,\quad u(0)=u_0. \] Here \(A\) is a self-adjoint positive definite linear operator in a real Hilbert space \(\mathbb{H}\) with dense domain. The positive nonincreasing kernel \(\beta\) is assumed to be the real-valued and satisfies \(\beta\in L^1(0,1)\cap C(0,\infty)\). The results extend to weighted \(l^{1}\) stability some recent \(l^{1}\) stability theorem due to the author [Numer. Math. 109, No. 4, 571--595 (2008; Zbl 1166.65401)]. Theoretical results are then applied to investigate the asymptotic behavior as \(t_n \to \infty\) of the numerical solutions of certain integro-differential equations in Hilbert space.
scientific article; zbMATH DE number 6015263
scientific article