Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space

Consider a linear impulsive equation in a Banach space $$\dot{x}(t)+A(t)x(t)=f(t),t\ge 0,$$ $$x({\tau }_i+0)=B_{i}x({\tau }_i-0)+{\alpha }_i,$$ with ${\lim }_{i\to \mathrm{\infty }}{\tau }_i=\mathrm{\infty }$. 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 $\{{\alpha }_i\}$ ; © ${\lim }_{t\to \mathrm{\infty }}x(t)=0$ for any $f,{\alpha }_i$ tending to zero; (d) exponential estimate of $f$ implies a similar estimate for $x$.