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<p>The subject of this discussion is third-order nonlinear dynamic equations with deviating arguments on time scales. The goal of this paper is to establish the oscillatory behavior of the aforementioned dynamic equations via some <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2025_766_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(\limsup \)</EquationSource> <EquationSource Format="MATHML"><math> <mo movablelimits="true">lim sup</mo> </math></EquationSource> </InlineEquation> type inequalities and comparison with first-order equations whose oscillatory behavior is known. Our results guarantee that all solutions to the said problem are only oscillatory, different from any other result in the literature. The results obtained are applied to three examples. We point out that the results are new even for the case <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2025_766_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {T}}={\mathbb {R}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40590_2025_766_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {T}}={\mathbb {Z}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Some new oscillation theorems for third-order nonlinear dynamic equations with deviating arguments

  • 期刊论文

  • 发表日期：

    2025年6月9日

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  • S. Kaleeswari,
  • Said R. Grace,
  • G. N. Chhatria

摘要

The subject of this discussion is third-order nonlinear dynamic equations with deviating arguments on time scales. The goal of this paper is to establish the oscillatory behavior of the aforementioned dynamic equations via some \(\limsup \) lim sup type inequalities and comparison with first-order equations whose oscillatory behavior is known. Our results guarantee that all solutions to the said problem are only oscillatory, different from any other result in the literature. The results obtained are applied to three examples. We point out that the results are new even for the case \({\mathbb {T}}={\mathbb {R}}\) T = R or \({\mathbb {T}}={\mathbb {Z}}\) T = Z .