First-order nonlinear differential equations with state-dependent impulses: Doc 450 - Rachnek, Lukás; Rachnková, Irena (Boundary value problems, 2013)
The paper deals with the state-dependent impulsive problem [Equation not available: see fulltext.] where [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], f fulfils the Carathéodory conditions on [InlineEquation not available: see fulltext.], the impulse function [InlineEquation not available: see fulltext.] is continuous on [InlineEquation not available: see fulltext.], the barrier function γ has a continuous first derivative on some subset of and is a linear bounded functional which is defined on the Banach space of left-continuous regulated functions on [InlineEquation not available: see fulltext.] equipped with the sup-norm. The functional 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. MSC: 34B37, 34B15.[PUBLICATION ABSTRACT]