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This course is a follow-up of the course in finite dimensions (by E. Ait Dads).
It deals with linear delay differential equations in infinite dimensions. Two
classes of equations will be presented : so-called « ordinary » delay
differential equations in infinite dimensions, which are just an extension of
the previous ones in the frame of infinite dimensions ; partial differential
equations with delay. Two approaches will be presented.
The first one will be restricted to ordinary dde in infinite dimensions,
and follows essentially the same steps as in finite dimensions. A Riesz-type
representation formula for the delay functional will be presented, as well as
a variation of constants formula and a possible extension of the concept of
formal adjoint. See References for articles dealing with this approach.

The other approach uses the semigroup theory and extensions such as the
theory to integrated semigroups or extrapolation theory. This part of the
course will rather be introductory and serve as a support for the course by
M. Adimy and E. Ezzinbi and the one by L. Maniar.

- Linear theory for abstract differential equations of retarded
type, O. Arino, E. Sanchez, J. Math. Anal. Appl., 1995,
**191**: 547-571. Link to the paper on the JMAA site. - A variation of constants formula for abstract differential equations of retarded type, O. Arino, E. Sanchez, Differential and Integral Equations, 1996, Vol.9, No.6, pp.1305-1320.

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