Electron. J. Differential Equations, Vol. 2017 (2017), No. 90, pp. 1-13.

Improved oscillation conditions for third-order neutral type difference equations

Srinivasan Selvarangam, Mayakrishnan Madhan, Ethiraju Thandapani, Sandra Pinelas

Abstract:
In this article, we study the oscillatory behavior of the third-order neutral type difference equation
$$ \Delta(a_n(\Delta^2(x_n+p_nx_{n-k}))^{\alpha})+q_nf(x_{n-l})=0 $$
where $\alpha>0$, $a_n>0$, $q_n\geq 0$ and $0\leq p_n\leq p<\infty$. By using generalized Ricatti type transformation we present some new criteria which ensure that every solution is oscillatory. Also we provide examples that illustrate the importance of our results.

Submitted October 25, 2016. Published March 29, 2017.
Math Subject Classifications: 39A10.
Key Words: Asymptotic behavior; oscillation; third order; neutral difference equation}

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Srinivasan Selvarangam
Department of Mathematics
Presidency College
Chennai - 600 005, India
email: selvarangam.9962@gmail.com
Mayakrishnan Madhan
Department of Mathematics
Presidency College
Chennai - 600 005, India
email: mcmadhan24@gmail.com
Ethiraju Thandapani
Ramanujan Institute for Advanced Study in Mathematics
University of Madras
Chennai - 600 005, India
email: ethandapani@yahoo.co.in
Sandra Pinelas
Academia Militar
Departamento de Cincias Exactas e Naturais
2720-113 Amadora, Portugal
email: sandra.pinelas@gmail.com

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