Oscillatory and asymptotic behavior of fourth order quasilinear difference equations

E. Thandapani, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, India
M. Vijaya, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, India

E. J. Qualitative Theory of Diff. Equ., No. 64. (2009), pp. 1-15.

Abstract: The authors consider the fourth order quasilinear difference equation $$\Delta^{2}\left(p_{n}|\Delta^{2}x_n|^{\alpha-1}\Delta^{2}x_n\right)+q_{n}|x_{n+3}|^{\beta -1}x_{n+3}=0,$$ where $\alpha$ and $\beta$ are positive constants, and ${\{p_{n}\}}$ and ${\{q_{n}\}}$ are positive real sequences. They obtain sufficient conditions for oscillation of all solutions when $\sum\limits_{n=n_{0}}^{\infty}\left(\frac{n}{p_{n}}\right)^\frac{1}{\alpha}<\infty $ and $\sum\limits_{n=n_{0}}^{\infty}\left(\frac{n}{{p_{n}}^{\frac{1}{\alpha}}}\right)<\infty.$ The results are illustrated with examples.

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