A generalized Green operator for a boundary value problem with impulse action
The author considers a linear nonhomogeneous differential equation with impulses of the form \[ {dz\over dt}= A(t) z+ f(t),\quad t\neq\tau_i,\tag{1} \] \[ \ell_i z(\cdot)= a_i,\quad i= 1,\dots, P,\tag{2} \] where \(A(t)\) is a continuous \(n\times n\)-matrix on \([a,b]\); \(f(t)\) is an \(n\)-vector function, continuous for \(t\neq \tau_i\) and has jump discontinuities at the points \(\tau_i\); \(a_i\) and \(c\) are constants. Necessary and sufficient conditions for the solvability of (1), (2) in the case \(\ell_i= \ell\), \(i= 1,\dots, P\), and \(a_i= \alpha\), \(\alpha\in \mathbb{R}^m\), \(m\geq n\), \(i= 1,\dots, P\), are derived.
scientific article; zbMATH DE number 1758743