Kamenev-Type Asymptotic Criterion of Fourth-Order Delay Differential Equation

• Main Author: Omar Bazighifan
• Format: Article
• Language: English
• Published: MDPI AG 2020-03-01
• Series: Fractal and Fractional
• Subjects: oscillatory solutions; fourth-order; delay differential equations
• Online Access: https://www.mdpi.com/2504-3110/4/1/7

In this paper, we obtain necessary and sufficient conditions for a Kamenev-type oscillation criterion of a fourth order differential equation of the form <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mfenced separators="" open="(" close=")"> <msub> <mi>r</mi> <mn>3</mn> </msub> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> <msup> <mfenced separators="" open="(" close=")"> <msub> <mi>r</mi> <mn>2</mn> </msub> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> <msup> <mfenced separators="" open="(" close=")"> <msub> <mi>r</mi> <mn>1</mn> </msub> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> <msup> <mi>y</mi> <mo>&#8242;</mo> </msup> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mfenced> <mo>&#8242;</mo> </msup> </mfenced> <mo>&#8242;</mo> </msup> </mfenced> <mo>&#8242;</mo> </msup> <mo>+</mo> <mi>q</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> <mi>f</mi> <mfenced separators="" open="(" close=")"> <mi>y</mi> <mfenced separators="" open="(" close=")"> <mi>&#963;</mi> <mfenced open="(" close=")"> <mi>t</mi> </mfenced> </mfenced> </mfenced> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>&#8805;</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>.</mo> <mspace width="4pt"></mspace> </mrow> </semantics> </math> </inline-formula>The results presented here complement some of the known results reported in the literature. Moreover, the importance of the obtained conditions is illustrated via some examples.