Impulsive system of ODEs with general linear boundary conditions

The paper provides an operator representation for a problem which consists of a system of ordinary differential equations of the first order with impulses at fixed times and with general linear boundary conditions z (t) = A(t)z(t) + f(t, z(t)) for a.e. t ∈ [a, b] ⊂ R, z(ti+)− z(ti) = Ji(z(ti)), i = 1, . . . , p, l(z) = c0, c0 ∈ R n . Here p, n ∈ N, a < t1 < . . . < tp < b, A ∈ L ([a, b];R), f ∈ Car([a, b] × R;R), Ji ∈ C(R ;R), i = 1, . . . , p, and l is a linear bounded operator on the space of left-continuous regulated functions on interval [a, b]. The operator l is expressed by means of the KurzweilStieltjes integral and covers all linear boundary conditions for solutions of the above system subject to impulse conditions. The representation, which is based on the Green matrix to a corresponding linear homogeneous problem, leads to an existence principle for the original problem. A special case of the n-th order scalar differential equation is discussed. This approach can be also used for analogical problems with state-dependent impulses. Mathematics Subject Classification 2010: 34B37, 34B15, 34B27