Electron. J. Diff. Equ., Vol. 2015 (2015), No. 42, pp. 1-21.

Favard spaces and admissibility for Volterra systems with scalar kernel

Hamid Bounit, Ahmed Fadili

Abstract:
We introduce the Favard spaces for resolvent families, extending some well-known theorems for semigroups. Furthermore, we show the relationship between these Favard spaces and the $L^p$-admissibility of control operators for scalar Volterra linear systems in Banach spaces, extending some results in [22]. Assuming that the kernel a(t) is a creep function which satisfies $a(0^+)>0$, we prove an analogue version of the Weiss conjecture for scalar Volterra linear systems when p=1. To this end, we also show that the finite-time and infinite-time (resp. finite-time and uniform finite-time) $L^{1}$-admissibility coincide for exponentially stable resolvent families (reps. for reflexive state space), extending well-known results for semigroups.

Submitted March 22, 2014. Published February 12, 2015.
Math Subject Classifications: 45D05, 45E05, 45E10, 47D06.
Key Words: Semigroups; Volterra integral equations; resolvent family; Favard space; admissibility.

Show me the PDF file (327 KB), TEX file, and other files for this article.

  Hamid Bounit
Department of Mathematics, Faculty of Sciences
Ibn Zohr University, BP 8106
Agadir 80 000, Morocco
email: h.bounit@uiz.ac.ma
Ahmed Fadili
Department of Mathematics, Faculty of Sciences
Ibn Zohr University, BP 8106
Agadir 80 000, Morocco
email: ahmed.fadili@edu.uiz.ac.ma

Return to the EJDE web page