A new approach to BVPs with state-dependent impulses

Abstract

The paper deals with the second-order Dirichlet boundary value problem with one state-dependent impulseProofs of the main results contain a new approach to boundary value problems with state-dependent impulses which is based on a transformation to a fixed point problem of an appropriate operator in the space. Sufficient conditions for the existence of solutions to the problem are given here. The presented approach can be extended to more impulses and to other boundary conditions. MSC: 34B37; 34B15

Second order BVPs with state dependent impulses via lower and upper functions

“…The following three lemmas and their proofs are simple generalizations of those in the paper [22]. For the sake of completeness we state them with their full proofs.…”

“…Let us show that [22,Theorem 7] cannot be applied in this case. The basic assumption needed there has the form: there…”

“…Our main existence result (Theorem 3.1), which is based on assumption (7) and which deals with ∈ N, can be applied to problems which are not covered by [22,Theorem 7] even in the case = 1. See Examples 4.1-4.3.…”

“…We construct an auxiliary problem (21)- (23) and transform it to a fixed point problem for a proper operator in the space X introduced in (12). To this end we use the approach of [22], where such transformation was done for = 1. Let us consider K from (11) and define a set…”

“…In our previous paper [22] we investigated the solvability of problem (1)- (3) with = 1 and the main existence result there (Theorem 7) has been reached by means of the transformation of the problem to a fixed point problem for a proper operator in the space…”