### A CLASS OF EXPONENTIALLY STABLE SEMIGROUPS ON BANACH SPACES

#### Abstract

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.

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.

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