Asymptotic behavior of solutions of nonlinear difference equations

Abstract: The nonlinear difference equation
\label{E} x_{n+1}-x_n=a_n\varphi_n(x_{\sigma(n)})+b_n, \tag{$\text E$}
where $(a_n), (b_n)$ are real sequences, $\varphi_n \Bbb R\longrightarrow\Bbb R$, $(\sigma(n))$ is a sequence of integers and $\lim_{n\longrightarrow\infty}\sigma(n)=\infty$, is investigated. Sufficient conditions for the existence of solutions of this equation asymptotically equivalent to the solutions of the equation $y_{n+1}-y_n=b_n$ are given. Sufficient conditions under which for every real constant there exists a solution of equation (\ref{E}) convergent to this constant are also obtained.

Keywords: difference equation, asymptotic behavior

Classification (MSC2000): 39A10

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