PERIODIC SOLUTIONS AND ATTRACTIVENESS FOR SOME PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH LACK OF COMPACTNESS

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semi-group, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution implies the existence of a periodic solution. Several results regarding uniqueness and global attractiveness of periodic solutions are also established. The analysis relies on a fixed point theorem of Chow and Hale's type and uses some arguments of weak topology. Our theorems extend in a broad sense some new and classical related results. An application to a transport equation with delay is also presented.

Automated summary

This paper discusses the existence and attractiveness of periodic solutions for some partial functional differential equations in Banach spaces. By assuming a first linear part generates a strongly continuous semi-group and the delayed part is periodic, it is proved that the existence of a bounded solution implies the existence of a periodic solution. The analysis relies on a fixed point theorem and weak topology arguments, extending both new and classical results in a broad sense. An application to a transport equation with delay is also presented.