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

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)$.