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- W3037432904 abstract "Abstract The main objective of this article is to improve and complement some of the oscillation criteria published recently in the literature for third order differential equation of the form $$ bigl( r(t) bigl( z^{prime prime }(t) bigr) ^{alpha } bigr) ^{prime }+q(t)f bigl(x bigl(sigma (t) bigr) bigr)=0,quad tgeq t_{0}>0, $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo>″</mml:mo></mml:mrow></mml:msup><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mi>q</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>σ</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace /><mml:mi>t</mml:mi><mml:mo>≥</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math> where $z(t)=x(t)+p(t)x(tau (t))$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>z</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mi>x</mml:mi><mml:mo>(</mml:mo><mml:mi>τ</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:math> and α is a ratio of odd positive integers in the two cases $int _{t_{0}}^{infty }r^{frac{-1}{alpha } }(s),mathrm {d}s<infty $ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msubsup><mml:mo>∫</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>∞</mml:mi></mml:msubsup><mml:msup><mml:mi>r</mml:mi><mml:mfrac><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>α</mml:mi></mml:mfrac></mml:msup><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mspace /><mml:mi>d</mml:mi><mml:mi>s</mml:mi><mml:mo><</mml:mo><mml:mi>∞</mml:mi></mml:math> and $int _{t_{0}}^{infty }r^{frac{-1}{alpha } }(s),mathrm {d}s=infty $ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msubsup><mml:mo>∫</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>∞</mml:mi></mml:msubsup><mml:msup><mml:mi>r</mml:mi><mml:mfrac><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>α</mml:mi></mml:mfrac></mml:msup><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mspace /><mml:mi>d</mml:mi><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mi>∞</mml:mi></mml:math> . Some illustrative examples are presented." @default.
- W3037432904 created "2020-07-02" @default.
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- W3037432904 date "2020-06-24" @default.
- W3037432904 modified "2023-09-25" @default.
- W3037432904 title "Oscillation of solutions of third order nonlinear neutral differential equations" @default.
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- W3037432904 doi "https://doi.org/10.1186/s13662-020-02777-9" @default.
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