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Positive solutions to second-order differential equations with dependence on the first-order derivative and nonlocal boundary conditions

Tadeusz Jankowski

Author affiliations

Department of Differential Equations and Applied Mathematics, Gdańsk University of Technology, 11/12 G. Narutowicz Str., Gdańsk, 80-233, Poland

Citation and License

Boundary Value Problems 2013, 2013:8  doi:10.1186/1687-2770-2013-8

Published: 17 January 2013

Abstract

In this paper, we consider the existence of positive solutions for second-order differential equations with deviating arguments and nonlocal boundary conditions. By the fixed point theorem due to Avery and Peterson, we provide sufficient conditions under which such boundary value problems have at least three positive solutions. We discuss our problem both for delayed and advanced arguments α and also in the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M2">View MathML</a>. In all cases, the argument β can change the character on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M3">View MathML</a>, see problem (1). It means that β can be delayed in some set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M4">View MathML</a> and advanced in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M5">View MathML</a>. An example is added to illustrate the results.

MSC: 34B10.

Keywords:
boundary value problems with delayed and advanced arguments; nonlocal boundary conditions; cone; existence of positive solutions; a fixed point theorem