Second-Order Neutral Differential Equations: Improved Criteria for Testing the Oscillation

Abstract

The main purpose of this study is to establish new improved conditions for testing the oscillation of solutions of second-order neutral differential equation <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:msup> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>r</a:mi> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>l</a:mi> </a:mrow> </a:mfenced> <a:msup> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:msup> <a:mrow> <a:mi>u</a:mi> </a:mrow> <a:mrow> <a:mo>′</a:mo> </a:mrow> </a:msup> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>l</a:mi> </a:mrow> </a:mfenced> </a:mrow> </a:mfenced> </a:mrow> <a:mrow> <a:mi>γ</a:mi> </a:mrow> </a:msup> </a:mrow> </a:mfenced> </a:mrow> <a:mrow> <a:mo>′</a:mo> </a:mrow> </a:msup> <a:mo>+</a:mo> <a:mi>q</a:mi> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>l</a:mi> </a:mrow> </a:mfenced> <a:msup> <a:mrow> <a:mi>x</a:mi> </a:mrow> <a:mrow> <a:mi>β</a:mi> </a:mrow> </a:msup> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>σ</a:mi> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>l</a:mi> </a:mrow> </a:mfenced> </a:mrow> </a:mrow> </a:mfenced> <a:mo>=</a:mo> <a:mn>0</a:mn> <a:mo>,</a:mo> </a:math> where <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M2"> <x:mi>l</x:mi> <x:mo>≥</x:mo> <x:msub> <x:mrow> <x:mi>l</x:mi> </x:mrow> <x:mrow> <x:mn>0</x:mn> </x:mrow> </x:msub> </x:math> and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M3"> <z:mi>u</z:mi> <z:mfenced open="(" close=")" separators="|"> <z:mrow> <z:mi>l</z:mi> </z:mrow> </z:mfenced> <z:mo>≔</z:mo> <z:mi>x</z:mi> <z:mfenced open="(" close=")" separators="|"> <z:mrow> <z:mi>l</z:mi> </z:mrow> </z:mfenced> <z:mo>+</z:mo> <z:mi>p</z:mi> <z:mi>x</z:mi> <z:mfenced open="(" close=")" separators="|"> <z:mrow> <z:mi mathvariant="normal">ϱ</z:mi> <z:mrow> <z:mfenced open="(" close=")" separators="|"> <z:mrow> <z:mi>l</z:mi> </z:mrow> </z:mfenced> </z:mrow> </z:mrow> </z:mfenced> </z:math> . By optimizing the commonly used relationship <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M4"> <ob:mi>x</ob:mi> <ob:mo>&gt;</ob:mo> <ob:mfenced open="(" close=")" separators="|"> <ob:mrow> <ob:mn>1</ob:mn> <ob:mo>−</ob:mo> <ob:mi>p</ob:mi> </ob:mrow> </ob:mfenced> <ob:mi>u</ob:mi> </ob:math> , we obtain new criteria that give sharper results for oscillation than the previous related results. Moreover, we obtain criteria of an iterative nature. Our new results are illustrated by an example.

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