Asymptotic equivalence of impulsive differential equations in a Banach space (Q805856)

In this paper the authors consider two impulsive equations in a Banach space X: \(x'(t)=A(t)x(t)\), \(x(t^+_ i)=L_ ix(t^-_ i)\), and \(y'(t)=A(t)y(t)+f(t,y(t))\), \(y(t^+_ i)=(L_ i+H_ i)y(t^-_ i)\), where t varies in an interval \([T,+\infty)\) with the exception of a set of points \(t_ 1<t_ 2<...\), A(t) and \(L_ i\) are bounded and linear operators in X, and f is a nonlinear operator from \([T,+\infty)\times X\) to X. By using the Schauder's fixed point theorem, it is proved that the two problems are asymptotically equivalent under suitable conditions involving the evolutionary operator of the first equation.