IJPAM: Volume 78, No. 3 (2012)

OSCILLATION AND NONOSCILLATION FOR CERTAIN CLASS
OF FIRST AND SECOND ORDER GENERALIZED
$\alpha-$DIFFERENCE EQUATIONS

M. Maria Susai Manuel$^1$, G. Britto Antony Xavier$^2$,
D.S. Dilip$^3$, G. Dominic Babu$^4$
$^1$Department of Science and Humanities
R.M.D. Engineering College
Kavaraipettai, 601 206, Tamil Nadu, INDIA
$^{2,3,4}$Department of Mathematics
Sacred Heart College
Tirupattur, 635 601, Vellore District
Tamil Nadu, INDIA


Abstract. In this paper, the authors discuss the oscillation and nonoscillation of the solutions of the generalized $\alpha-$difference equations

\begin{displaymath} \Delta_{\alpha(\ell)}u(k)+\delta\sum\limits_{i=1}^mf_i(k)F_i(u(g_i(k)))=0,\ k\in[0,\infty),\ \delta=\pm1 \end{displaymath} (1)


\begin{displaymath} \text{and }\Delta_{\alpha(\ell)}^2u(k)=f(k,u(k),\Delta_{\beta(\ell)} u(k)),\ k\in[0,\infty) \end{displaymath} (2)

where $\alpha,\beta$ and $\ell$ are positive real fixed constants, $f_i,F_i$ are defined on $\mathbb{R}$, for each $i, 1\leq i\leq m$, $\displaystyle{\{g_i(k)\}\subseteq[0,\infty)}$ and $f$ is defined on $[0,\infty)\times\mathbb{R}^2$.

Received: April 19, 2012

AMS Subject Classification: 39A12, 39A70, 47B39, 39B60

Key Words and Phrases: generalized $\alpha-$difference equation, generalized $\alpha-$difference operator, oscillation and nonoscillation

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 78
Issue: 3