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Open Access Research Article

Existence principle for higher-order nonlinear differential equations with state-dependent impulses via fixed point theorem

Irena Rachůnková* and Jan Tomeček

Author Affiliations

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

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Boundary Value Problems 2014, 2014:118  doi:10.1186/1687-2770-2014-118


Dedicated to Professor Ivan Kiguradze for his merits in mathematical sciences

Published: 14 May 2014

Abstract

The paper provides an existence principle for a general boundary value problem of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M1">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M4">View MathML</a>, with the state-dependent impulses <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M5">View MathML</a>, where the impulse points t are determined as solutions of the equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8">View MathML</a>. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M10">View MathML</a>, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8">View MathML</a>, are Lebesgue integrable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M14">View MathML</a> satisfies the Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M15">View MathML</a>. The impulse functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M8">View MathML</a>, and the barrier functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M7">View MathML</a>, are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M22">View MathML</a>, respectively. The functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M24">View MathML</a>, are linear and bounded on the space of left-continuous regulated (i.e. having finite one-sided limits at each point) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M13">View MathML</a> vector functions. Provided the data functions h and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M16">View MathML</a> are bounded, transversality conditions which guarantee that each possible solution of the problem in a given region crosses each barrier <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M27">View MathML</a> at the unique impulse point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/118/mathml/M28">View MathML</a> are presented, and consequently the existence of a solution to the problem is proved.

MSC: 34B37, 34B10, 34B15.

Keywords:
nonlinear higher-order ODE; state-dependent impulses; general linear boundary conditions; transversality conditions; fixed point