A second-order impulsive Cauchy problem. (English) Zbl 1013.34061
Summary: The author studies the existence of mild and classical solutions to an abstract second-order impulsive Cauchy problem modeled in the form
\[
\ddot u(t)=A u(t) + f(t,u(t),\dot u(t)),\;t\in (-T_{0},T_{1}),\;t\neq t_{i};\quad u(0) = x_{0},\;\dot u(0) = y_{0};
\]
\[ \Delta u(t_{i})= I_i^1(u(t_{i})), \quad \Delta \dot u(t_{i})= I_i^2 (\dot u(t_i^+)), \] where \(A\) is the infinitesimal generator of a strongly continuous cosine family of linear operators on a Banach space \(X\) and \(f,I_i^1,I_i^2\) are appropriate continuous functions.
\[ \Delta u(t_{i})= I_i^1(u(t_{i})), \quad \Delta \dot u(t_{i})= I_i^2 (\dot u(t_i^+)), \] where \(A\) is the infinitesimal generator of a strongly continuous cosine family of linear operators on a Banach space \(X\) and \(f,I_i^1,I_i^2\) are appropriate continuous functions.
MSC:
34G20 | Nonlinear differential equations in abstract spaces |
34A37 | Ordinary differential equations with impulses |
47D09 | Operator sine and cosine functions and higher-order Cauchy problems |