Boundedness and dichotomies for impulse equations (Q810718)

The authors consider the impulse equation \(\dot y(t)=A(t)y(t)+f(t),\quad t\neq t_ i,\) \(y(t_{i+})-y(t_{i-})=B_ iy(t_{i-})+I_ i(f(t_ i),y(t_{i-})),\) where A(t) is a continuous \(n\times n\) matrix valued function on \([0,\infty)\), \(\{t_ i\}\) is an increasing sequence, \(t_ i\to \infty\) as \(i\to \infty\). They establish that, under certain hypothesis, the existence of bounded solutions of the problem for each f from families of functions implies a dichotomy for the corresponding linear equation. The main tool in the paper is a closed graph theorem for multivalued maps from a Banach space into another.