Geodesic
Integral resolvent for Volterra equations and Favard spaces
Problemy analiza, Tome 11 (2022) no. 1, pp. 67-80.

Voir la notice de l'article provenant de la source Math-Net.Ru

The objective of this work is to give a characterization of the domain $D\left(A\right)$ of $A$ in terms of the integral resolvent family of the equation $x\left(t\right)=x_{0}+\int\limits_{0}^{t}a\left(t-s\right)Ax(s)ds$, $t\geq0$, where $A$ is a linear closed densely defined operator, $a\in L_{loc}^{1}\left(\mathbb{R}^{+}\right)$ in a general Banach space $X$ and $ x_{0}\in X $. Furthermore, we give a relationship between the Favard classes (temporal and frequency) for integral resolvents.
Keywords: semigroups, scalar Volterra integral equations, integral resolvent families
Mots-clés : Favard spaces. 
@article{PA_2022_11_1_a5,
     author = {A. Fadili and F. Maragh},
     title = {Integral resolvent for {Volterra} equations and {Favard} spaces},
     journal = {Problemy analiza},
     pages = {67--80},
     publisher = {mathdoc},
     volume = {11},
     number = {1},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2022_11_1_a5/}
}
TY  - JOUR
AU  - A. Fadili
AU  - F. Maragh
TI  - Integral resolvent for Volterra equations and Favard spaces
JO  - Problemy analiza
PY  - 2022
SP  - 67
EP  - 80
VL  - 11
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2022_11_1_a5/
LA  - en
ID  - PA_2022_11_1_a5
ER  - 
%0 Journal Article
%A A. Fadili
%A F. Maragh
%T Integral resolvent for Volterra equations and Favard spaces
%J Problemy analiza
%D 2022
%P 67-80
%V 11
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2022_11_1_a5/
%G en
%F PA_2022_11_1_a5
A. Fadili; F. Maragh. Integral resolvent for Volterra equations and Favard spaces. Problemy analiza, Tome 11 (2022) no. 1, pp. 67-80. http://geodesic.mathdoc.fr/item/PA_2022_11_1_a5/

[1] H. Bounit, A. Fadili, “On the Favard spaces and the admissibility for Volterra systems with scalar kernel”, Electronic Journal of Differential Equations, 42 (2015), 1–21

[2] P. L. Butzer, H. Berens, Semi-Groups of Operators and Approximation, Springer-Verlag, New York, 1967

[3] W. Desch, J. Pruss, “Counterexamples for abstract linear Volterra equations”, J. Integral Equations Applications, 5:1 (1993), 29–45 | DOI

[4] K. J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, New York–Berlin–Heidelberg, 2000 | DOI

[5] R. Grimmer, J. Prüss, “On linear Volterra equations in Banach spaces”, Comp Maths with Appls., 11:1 (1985), 189–205

[6] M. Jung, “Duality theory for solutions to Volterra integral equations”, J.M.A.A, 230:1 (1999), 112–134 | DOI

[7] C. Lizama, V. Poblete, “On multiplicative perturbation of integral resolvent families”, Journal of Mathematical Analysis and Applications, 327:2 (2007), 1335–1359 | DOI

[8] C. Lizama, J. Sanchez, “On perturbation of k-regularized resolvent families”, Taiwanese Journal of Mathematics, 7:2 (2003), 217–227 | DOI

[9] F. Maragh, H. Bounit, A. Fadili, H. Hammouri, “On the admissible control operators for linear and bilinear systems and the Favard spaces”, Bulletin of the Belgian Mathematical Society, 21:4 (2014), 711–732 | DOI

[10] J.M.A.M. van Neerven, The adjoint of a Semigroup of Linear Operators, Lecture Notes Math., 1529, Springer-Verlag, Berlin–Heidelberg–New York, 1992

[11] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser-Verlag, Basel, 1993

[12] Michiaki Watanabe, “A new proof of the generation theorem of cosine families in Banach spaces”, Houston J. Math., 10:2 (1984), 285–290