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Oscillation criteria for neutral second-order half-linear differential equations with applications to Euler type equations

Simona Fišnarová and Robert Mařík*

Author Affiliations

Department of Mathematics, Mendel University in Brno, Zemědělská 1, Brno, 613 00, Czech Republic

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Boundary Value Problems 2014, 2014:83  doi:10.1186/1687-2770-2014-83

Published: 10 April 2014

Abstract

We study the second-order neutral delay half-linear differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M4">View MathML</a>. We use the method of Riccati type substitution and derive oscillation criteria for this equation. By an example of the neutral Euler type equation we show that the obtained results are sharp and improve the results of previous authors. Among others, we improve the results of Sun et al. (Abstr. Appl. Anal. 2012:819342, 2012) and discuss also the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M5">View MathML</a>.

MSC: 34K11, 34K40.

Keywords:
half-linear differential equation; oscillation criteria; Riccati technique; delay equation; neutral equation; Euler type equation