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First-order nonlinear differential equations with state-dependent impulses

Lukáš Rachůnek and Irena Rachůnková*

Author Affiliations

Department of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 77146, Czech Republic

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Boundary Value Problems 2013, 2013:195  doi:10.1186/1687-2770-2013-195

Published: 28 August 2013

Abstract

The paper deals with the state-dependent impulsive problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M3">View MathML</a>, f fulfils the Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M4">View MathML</a>, the impulse function is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M4">View MathML</a>, the barrier function γ has a continuous first derivative on some subset of ℝ and is a linear bounded functional which is defined on the Banach space of left-continuous regulated functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/195/mathml/M7">View MathML</a> 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.

Keywords:
first-order ODE; state-dependent impulses; transversality conditions; general linear boundary conditions; existence; Kurzweil-Stieltjes integral