Geodesic
Evolution systems for differential equations with variable time lags
Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 4, Tome 48 (2013), pp. 5-26.

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

We consider nonautonomous retarded functional differential equations under hypotheses which are designed for the application to equations with variable time lags, which may be unbounded, and construct an evolution system of solution operators which are continuously differentiable. These operators are defined on manifolds of continuously differentiable functions. The results apply to pantograph equations and to nonlinear Volterra integro-differential equations, for example. For linear equations we also provide a simpler evolution system with solution operators on a Banach space of continuous functions. 
@article{CMFD_2013_48_a0,
     author = {H.-O. Walther},
     title = {Evolution systems for differential equations with variable time lags},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {5--26},
     publisher = {mathdoc},
     volume = {48},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2013_48_a0/}
}
TY  - JOUR
AU  - H.-O. Walther
TI  - Evolution systems for differential equations with variable time lags
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2013
SP  - 5
EP  - 26
VL  - 48
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2013_48_a0/
LA  - ru
ID  - CMFD_2013_48_a0
ER  - 
%0 Journal Article
%A H.-O. Walther
%T Evolution systems for differential equations with variable time lags
%J Contemporary Mathematics. Fundamental Directions
%D 2013
%P 5-26
%V 48
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2013_48_a0/
%G ru
%F CMFD_2013_48_a0
H.-O. Walther. Evolution systems for differential equations with variable time lags. Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 4, Tome 48 (2013), pp. 5-26. http://geodesic.mathdoc.fr/item/CMFD_2013_48_a0/

[1] Valter Kh.-O., “Gladkost polupotokov dlya differentsialnykh uravnenii s zapazdyvaniem, zavisyaschim ot resheniya”, Sovrem. mat. Fundam. napravl., 1, 2003, 40–55 | MR | Zbl

[2] Diekmann O., van Gils S. A., Verduyn Lunel S. M., Walther H.-O., Delay equations: functional-, complex- and nonlinear analysis, Springer, N.Y., 1995 | MR | Zbl

[3] Hartung F., Krisztin T., Walther H.-O., Wu J., “Functional differential equations with state-dependent delays: theory and applications”, Handbook of Differential Equations, Ordinary Differential Equations, v. 3, 2006, 435–545 | DOI | MR

[4] Iserles A., “On the generalized pantograph functional-differential equation”, European J. Appl. Math., 4 (1993), 1–38 | DOI | MR | Zbl

[5] Kato T., McLeod J. B., “The functional-differential equation $y'(x)=ay(\lambda x)+by(x)$”, Bull. Amer. Math. Soc., 77 (1971), 891–937 | DOI | MR | Zbl

[6] Krisztin T., [lichnaya beseda], 2011

[7] Mallet-Paret J., Nussbaum R. D., Paraskevopoulos P., “Periodic solutions for functional differential equations with multiple state-dependent time lags”, Topol. Methods Nonlinear Anal., 3 (1994), 101–162 | MR | Zbl

[8] Walther H.-O., “The solution manifold and $C^1$-smoothness of solution operators for differential equations with state dependent delay”, J. Differential Equations, 195 (2003), 46–65 | DOI | MR | Zbl

[9] Walther H.-O., “Differential equations with locally bounded delay”, J. Differential Equations, 252:4 (2012), 3001–3039 | DOI | MR | Zbl