Semi-Fredholm Operators and Periodic Solutions for Linear Functional Differential Equations

✍ Scribed by Jong Son Shin; Toshiki Naito

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 245 KB
• Volume: 153
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1998.3547

✦ Synopsis

dedicated to professor junji kato for his 60th birthday

We deal with the inhomogeneous linear periodic equation with infinite delay of the form dxÂdt=Ax(t)+B(t, x t )+F(t), where A is the generator of a C 0 -semigroup on a Banach space. Assuming that it has a bounded solution, we obtain several criteria on the existence and the uniqueness of periodic solutions for the equation in the general phase space B and in the concrete phase space B=UC g . The key of our approach is the employment of the perturbation theory of semi-Fredholm operators to show that the period map satisfies the condition of the fixed point theorem by Chow and Hale (Funkcial. Ekvac. 17 (1974), 31 38).

1999 Academic Press

Let B be a Banach space, consisting of functions : (& , 0] Ä E, which satisfies some axioms demonstrated in Section 2. We assume that Eq. (L) always satisfies the following hypothesis (H):

(H-1) A: D(A)/E Ä E is the infinitesimal generator of a C 0 -semigroup T(t) on E;