Affiliations: Facultad de Ciencias, Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Santiago, Chile. E-mail: carlos.lizama@usach.cl | Departamento de Matemáticas & I.U.M.A., Universidad de Zaragoza, Zaragoza, Spain. E-mail: pjmiana@unizar.es | Facultad de Ciencias, Universidad de Chile, Las Palmeras, Santiago, Chile. E-mail: fepobletg@ug.uchile.cl
Note: [] Corresponding author: Carlos Lizama, Facultad de Ciencias, Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile. E-mail: carlos.lizama@usach.cl
Abstract: In this article we study the uniform stability of an (a,k)-regularized family {S(t)}t≥0 generated by a closed operator A. We give sufficient conditions, on the scalar kernels a, k and the operator A, to ensure the uniform stability of the family {S(t)}t≥0 in Hilbert spaces. Our main result is a generalization of Theorem 1 in [Proc. Amer. Math. Soc. 132(1) (2004), 175–181], concerning the stability of resolvent families, and can be seen as a substantial generalization of the Gearhart–Greiner–Prüss characterization of exponential stability for strongly continuous semigroups.
