Oscillation of higher-order half-linear neutral differential equations

Abstract

Abstract In this paper, we establish some new oscillation criteria for the higher-order half-linear neutral differential equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msup> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo> </m:mo> <m:mo> </m:mo> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>+</m:mo> <m:mi>p</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>σ</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>γ</m:mo> </m:msup> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>′</m:mo> </m:msup> <m:mo>+</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>x</m:mi> <m:mo>γ</m:mo> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>τ</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>.</m:mo> </m:math> $$\left[ {a(t)\,\,\left( {[x(t) + p(t)x(\sigma (t))]^{(n - 1)} } \right)^\gamma } \right]^\prime + q(t)x^\gamma (\tau (t)) = 0.$$ An example is given to illustrate the main results.