Abstract and Applied Analysis

Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

Gabriele Gühring and Frank Räbiger

Full-text: Open access

Abstract

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t)=Au(t)+B(t)u(t)+f(t),t, where (A,D(A)) is a Hille-Yosida operator on a Banach space X,B(t),t, is a family of operators in (D(A)¯,X) satisfying certain boundedness and measurability conditions and fLloc1(,X). The solutions of the corresponding homogeneous equations are represented by an evolution family (UB(t,s))ts. For various function spaces we show conditions on (UB(t,s))ts and f which ensure the existence of a unique solution contained in . In particular, if (UB(t,s))ts is p-periodic there exists a unique bounded solution u subject to certain spectral assumptions on UB(p,0),f and u. We apply the results to nonautonomous semilinear retarded differential equations. For certain p-periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of (UB(t,s))ts.

Article information

Source
Abstr. Appl. Anal., Volume 4, Number 3 (1999), 169-194.

Dates
First available in Project Euclid: 9 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1049907202

Digital Object Identifier
doi:10.1155/S1085337599000214

Mathematical Reviews number (MathSciNet)
MR1811234

Zentralblatt MATH identifier
0987.34062

Subjects
Primary: 34C25: Periodic solutions 34C27
Secondary: 34C28 34G10 47D06 47H15

Citation

Gühring, Gabriele; Räbiger, Frank. Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations. Abstr. Appl. Anal. 4 (1999), no. 3, 169--194. doi:10.1155/S1085337599000214. https://projecteuclid.org/euclid.aaa/1049907202


Export citation