期刊名称：International Journal of Differential Equations and Applications
印刷版ISSN：1311-2872
出版年度：2003
卷号：7
期号：1
DOI：10.12732/ijdea.v7i1.284
语种：English
出版社：International Journal of Differential Equations and Applications
摘要：We consider a mild solution $u_f$ of a well-posed inhomogeneous Cauchy problem$$\dot u(t)=Au(t)+f(t), \quad u(0)=0\eqno{(A, f, 0)}$$on a complex Banach space $X$, where $A$ is the infinitesimal generator of a strongly continuous semigroup ${\bf T}.$It is proved that if ${\bf T}$ is exponentially stable then for each$f$ belonging to a certain subspace ${\bf{\cal X}_0}$ of $\text{BUC}\,(\bf{R}_+, X)$ the solution $u_f$ of $(A, f, 0)$ lies in ${\bf{\cal X}_0}.$ The converse statement is also true if ${\bf {\cal X}_0}$ and ${\bf T}$ verify supplementaryconditions as in Theorem 4, below.
Loading...
