Marrakech 2002

The theory of linear delay differential equations in infinite dimensional space



O. Arino

I.R.D. Bondy
France


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.

References

  1. 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.
  2. 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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