Asymptotic Behavior of Solutions of Generalized Nonlinear $\alpha$-difference Equation of Second Order

M. Maria Susai Manuel, G. Britto Antony Xavier, D. S. Dilip, G. Dominic Babu

Abstract


In this paper, the authors discuss the asymptotic behavior of solutions of the generalized nonlinear $\alpha$-difference equation
\begin{equation}
\Delta_{\alpha(\ell)}(p(k)\Delta_{\alpha(\ell)} u(k))+f(k)F(u(k))=g(k),
\end{equation}
$k\in[a,\infty),$ where the functions $p$, $f$, $F$ and $g$ are defined in their domain of definition and $\alpha>1$, $\ell$ is positive real.\ Further, $uF(u)>0$ for $u\neq0$, $p(k)>0$ for all $k\in[a,\infty)$ for some $a\in[0,\infty)$ and for all $0\leq j<\ell$, $R_{a+j,k}\to\infty$, where $R_{t+j,k}=\sum\limits_{r=0}^{\frac{k-\ell-t-j}{\ell}}\frac{1}{p(t+j+r\ell)}$,
$t\in[a,\infty)$ and $ k\in\mathbb{N}_\ell(t+j+\ell)$.


Keywords


Generalized difference equation; Generalized difference operator; Oscillation and nonoscillation

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eISSN 0975-5748; pISSN 0974-875X