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On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales

We study oscillatory and asymptotic properties of the third-order nonlinear dynamic equation

$$ {{\left[ {{{{\left( {\frac{1}{{{r_2}(t)}}{{{\left( {{{{\left( {\frac{1}{{{r_1}(t)}}{x^{\varDelta }}(t)} \right)}}^{{{\gamma_1}}}}} \right)}}^{\varDelta }}} \right)}}^{{{\gamma_2}}}}} \right]}^{\varDelta }}+f\left( {t,{x^{\sigma }}(t)} \right)=0,\quad t\in \mathbb{T}. $$

By using the Riccati transformation, we present new criteria for the oscillation or certain asymptotic behavior of solutions of this equation. It is supposed that the time scale T is unbounded above.

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Author information

Correspondence to M. T. Şenel.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 7, pp. 996–1004, July, 2013.

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Şenel, M.T. On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales. Ukr Math J 65, 1111–1121 (2013). https://doi.org/10.1007/s11253-013-0845-z

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Keywords

  • Dynamic Equation
  • Nonoscillatory Solution
  • Riccati Transformation
  • Nonoscillation Criterion
  • Dynamic Functional Equation