Electron. J. Diff. Equ.,
Vol. 2015 (2015), No. 42, pp. 121.
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
wellknown theorems for semigroups. Furthermore, we show the relationship
between these Favard spaces and the
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
,
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 finitetime and infinitetime
(resp. finitetime and uniform finitetime)
admissibility
coincide for exponentially stable resolvent families (reps. for reflexive
state space), extending wellknown 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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