Second-Order Differential Equation: Oscillation Theorems and Applications

Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form <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>ς</a:mi> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>y</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:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>y</a:mi> </a:mrow> </a:mfenced> </a:mrow> </a:mrow> </a:mfenced> </a:mrow> <a:mrow> <a:mi>a</a:mi> </a:mrow> </a:msup> </a:mrow> </a:mrow> </a:mfenced> </a:mrow> <a:mrow> <a:mo>′</a:mo> </a:mrow> </a:msup> <a:mo>+</a:mo> <a:mi>p</a:mi> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>y</a:mi> </a:mrow> </a:mfenced> <a:msup> <a:mrow> <a:mi>u</a:mi> </a:mrow> <a:mrow> <a:mi>c</a:mi> </a:mrow> </a:msup> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>ϑ</a:mi> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>y</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:mtext> for </a:mtext> <a:mi>y</a:mi> <a:mo>≥</a:mo> <a:msub> <a:mrow> <a:mi>y</a:mi> </a:mrow> <a:mrow> <a:mn>0</a:mn> </a:mrow> </a:msub> <a:mo>,</a:mo> </a:math> under the assumption <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M2"> <x:msup> <x:mrow> <x:mstyle displaystyle="true"> <x:mo stretchy="false">∫</x:mo> </x:mstyle> </x:mrow> <x:mrow> <x:mi>∞</x:mi> </x:mrow> </x:msup> <x:msup> <x:mrow> <x:mfenced open="(" close=")" separators="|"> <x:mrow> <x:mi>ς</x:mi> <x:mrow> <x:mfenced open="(" close=")" separators="|"> <x:mrow> <x:mi>η</x:mi> </x:mrow> </x:mfenced> </x:mrow> </x:mrow> </x:mfenced> </x:mrow> <x:mrow> <x:mo>−</x:mo> <x:mfenced open="(" close=")" separators="|"> <x:mrow> <x:mn>1</x:mn> <x:mo>/</x:mo> <x:mi>a</x:mi> </x:mrow> </x:mfenced> </x:mrow> </x:msup> <x:mo>=</x:mo> <x:mi>∞</x:mi> </x:math> . Two cases are considered for <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M3"> <kb:mi>a</kb:mi> <kb:mo><</kb:mo> <kb:mi>c</kb:mi> </kb:math> and <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M4"> <mb:mi>a</mb:mi> <mb:mo>></mb:mo> <mb:mi>c</mb:mi> </mb:math> , where <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M5"> <ob:mi>a</ob:mi> </ob:math> and <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" id="M6"> <qb:mi>c</qb:mi> </qb:math> are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.