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.

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  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

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