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Oscillation criteria for nonlinear neutral functional dynamic equations on time scales

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Mathematica Slovaca

Abstract

In this paper, we establish some new sufficient conditions for oscillation of the second-order neutral functional dynamic equation

$$\left[ {r\left( t \right)\left[ {m\left( t \right)y\left( t \right) + p\left( t \right)y\left( {\tau \left( t \right)} \right)} \right]^\Delta } \right]^\Delta + q\left( t \right)f\left( {y\left( {\delta \left( t \right)} \right)} \right) = 0$$

on a time scale \(\mathbb{T}\) which is unbounded above, where m, p, q, r, T and δ are real valued rd-continuous positive functions defined on \(\mathbb{T}\). The main investigation of the results depends on the Riccati substitutions and the analysis of the associated Riccati dynamic inequality. The results complement the oscillation results for neutral delay dynamic equations and improve some oscillation results for neutral delay differential and difference equations. Some examples illustrating our main results are given.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, PL-61-614, Poznan, Poland

    I. Kubiaczyk & A. Sikorska-Nowak

  2. Department of Mathematics Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    S. H. Saker

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  1. I. Kubiaczyk
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  2. S. H. Saker
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  3. A. Sikorska-Nowak
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Correspondence to I. Kubiaczyk.

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Communicated by Michal Fečkan

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Kubiaczyk, I., Saker, S.H. & Sikorska-Nowak, A. Oscillation criteria for nonlinear neutral functional dynamic equations on time scales. Math. Slovaca 63, 263–290 (2013). https://doi.org/10.2478/s12175-012-0097-7

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  • DOI: https://doi.org/10.2478/s12175-012-0097-7

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