Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line
Let u be a bounded, uniformly continuous, mild solution of an inhomogeneous Cauchy problem on R+: u′(t) = Au(t) + (Latin small letter o with stroke)(t) (t ≥ 0). Suppose that u has uniformly convergent means, σ(A) ∩ i R is countable, and (Latin small letter o with stroke) is asymptotically almost per...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
1999
|
| Summary: | Let u be a bounded, uniformly continuous, mild solution of an inhomogeneous Cauchy problem on R+: u′(t) = Au(t) + (Latin small letter o with stroke)(t) (t ≥ 0). Suppose that u has uniformly convergent means, σ(A) ∩ i R is countable, and (Latin small letter o with stroke) is asymptotically almost periodic. Then u asymptotically almost periodic. Related results have been obtained by Ruess and Vũ, and by Basit, using different methods. A direct proof is given of a Tauberian theorem of Batty, van Neerven and Räbiger, and applications to Volterra equations are discussed. |
|---|