First-order nonlinear differential equations with state-dependent impulses
Department of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 77146, Czech Republic
Boundary Value Problems 2013, 2013:195 doi:10.1186/1687-2770-2013-195Published: 28 August 2013
The paper deals with the state-dependent impulsive problem
where , , f fulfils the Carathéodory conditions on , the impulse function is continuous on , the barrier function γ has a continuous first derivative on some subset of ℝ and ℓ 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 ℓ 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.