Impulsive system of ODEs with general linear boundary conditions

摘要

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 egin{gather} z'(t) = A(t)z(t) + f(t,z(t)) extrm{ for a.e. }t in [a,b] subset mathbb{R}, z(t_i+) - z(t_i) = J_i(z(t_i)), quad i = 1,ldots,p, ell(z) = c_0, quad c_0 in mathbb{R}^n. end{gather} Here $p,n in N$, $a < t_1 < ldots < t_p < b$, $A in L^1([a,b];mathbb{R}^{nimes n})$, $f in operatorname{Car}([a,b]imesmathbb{R}^n;mathbb{R}^n)$, $J_i in C(mathbb{R}^n;mathbb{R}^n)$, $i=1,ldots,p$, and $ell$ is a linear bounded operator on the space of left-continuous regulated functions on interval $[a,b]$. The operator $ell$ is expressed by means of the Kurzweil-Stieltjes 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.