IJPAM: Volume 92, No. 4 (2014)

ASYMPTOTIC AND OSCILLATORY BEHAVIOUR
OF CERTAIN NONLINEAR GENERALIZED
$\alpha$-DIFFERENCE EQUATIONS

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


Abstract. In this paper, the authors discuss the asymptotic and oscillatory behavior of solutions of the nonlinear generalized $\alpha-$difference equation

\begin{displaymath} \Delta_{\alpha(\ell)}(p(k)\Delta_{\alpha(\ell)} (u(k)+q(k)u(k-\rho)))+t(k)f(u(k-\tau))=0, k\in[a,\infty), \end{displaymath} (1)

where the functions $p,q$ and $t$ are real numbers with $t(k)\geq0$ and $\alpha,\ell,\rho,\tau$ are positive real. Further, $uf(u)>0$ for $u\neq0,p(k)>0$ and
\begin{displaymath} R(k)=\sum\limits_{r=0}^{\infty}\frac{1}{\alpha^rp(r\ell)}=\infty \end{displaymath} (2)

and $u(k)$ is defined for $k\geq-\max\{\rho,\tau\}$ for all $k\in[a,\infty)$ for some $a\in[0,\infty)$.

Received: January 6, 2014

AMS Subject Classification: 39A12

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

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DOI: 10.12732/ijpam.v92i4.9 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 92
Issue: 4
Pages: 549 - 563

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).