First-order nonlinear differential equations with state-dependent impulses

The paper deals with the state-dependent impulsive problem z'(t) = f(t,z(t)) for a.e. t epsilon [a,b], z(tau+) - z(tau) I (tau,z(tau)), gamma(z(tau)) = tau, l(z) = c(0), where , , f fulfils the Carath,odory conditions on , the impulse function is continuous on , the barrier function gamma has a continuous first derivative on some subset of a"e and a"" is a linear bounded functional which is defined on the Banach space of left-continuous regulated functions on equipped with the sup-norm. The functional a"" is represented by means of the Kurzweil-Stieltjes integral and covers all linear boundary conditions for solutions of first-order differential equations subject to state-dependent impulse conditions. Here, sufficient and effective conditions guaranteeing the solvability of the above problem are presented for the first time.