| Summary: | In this paper, we consider a certain class of third-order nonlinear delay differential equations <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mfenced separators="" open="(" close=")"> <mi>r</mi> <msup> <mfenced separators="" open="(" close=")"> <msup> <mi>w</mi> <mrow> <mo>″</mo> </mrow> </msup> </mfenced> <mi>α</mi> </msup> </mfenced> <mo>′</mo> </msup> <mfenced open="(" close=")"> <mi>v</mi> </mfenced> <mo>+</mo> <mi>q</mi> <mfenced open="(" close=")"> <mi>v</mi> </mfenced> <msup> <mi>x</mi> <mi>β</mi> </msup> <mfenced separators="" open="(" close=")"> <mi>ς</mi> <mfenced open="(" close=")"> <mi>v</mi> </mfenced> </mfenced> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>v</mi> <mo>≥</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>w</mi> <mfenced open="(" close=")"> <mi>v</mi> </mfenced> <mo>=</mo> <mi>x</mi> <mfenced open="(" close=")"> <mi>v</mi> </mfenced> <mo>+</mo> <mi>p</mi> <mfenced open="(" close=")"> <mi>v</mi> </mfenced> <mi>x</mi> <mfenced separators="" open="(" close=")"> <mi>ϑ</mi> <mfenced open="(" close=")"> <mi>v</mi> </mfenced> </mfenced> </mrow> </semantics> </math> </inline-formula>. We obtain new criteria for oscillation of all solutions of this nonlinear equation. Our results complement and improve some previous results in the literature. An example is considered to illustrate our main results.
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