  Consider a linear impulsive equation in a Banach space
  $$\dot{x}(t)+A(t)x(t) = f(t), ~t \geq 0,$$
  $$x(\tau_i +0)= B_i x(\tau_i -0) + \alpha_i,$$
  with $\lim_{i \rightarrow \infty} \tau_i = \infty $. Suppose each solution of
the corresponding semi-homogeneous equation
  $$\dot{x}(t)+A(t)x(t) = 0,$$
  (2) is bounded for any bounded sequence $\{ \alpha_i \}$.
  The conditions are determined ensuring
  (a) the solution of the corresponding homogeneous equation has an exponential
estimate;
  (b) each solution of (1),(2) is bounded on the half-line for any bounded $f$
and bounded sequence $\{ \alpha_i \}$ ;
  (c) $\lim_{t \rightarrow \infty}x(t)=0$ for any $f, \alpha_i$ tending to
zero;
  (d) exponential estimate of $f$ implies a similar estimate for $x$.
