Open Access Research

A new approach to BVPs with state-dependent impulses

Irena Rachůnková* and Jan Tomeček

Author Affiliations

Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 771 46, Czech Republic

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Boundary Value Problems 2013, 2013:22 doi:10.1186/1687-2770-2013-22

Published: 11 February 2013

Abstract

The paper deals with the second-order Dirichlet boundary value problem with one state-dependent impulse

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M1">View MathML</a>

Proofs 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2">View MathML</a>. 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.

Keywords:
impulsive differential equation; state-dependent impulses; Dirichlet problem; second-order ODE