# On the existence of solutions of some second order nonlinear difference equations

Archivum Mathematicum (2005)

• Volume: 041, Issue: 4, page 379-388
• ISSN: 0044-8753

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## Abstract

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We consider a second order nonlinear difference equation ${\Delta }^{2}{y}_{n}={a}_{n}{y}_{n+1}+f\left(n,{y}_{n},{y}_{n+1}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}n\in N\phantom{\rule{0.166667em}{0ex}}.\phantom{\rule{2.0em}{0ex}}\left(\text{E}\right)$ The necessary conditions under which there exists a solution of equation (E) which can be written in the form ${y}_{n+1}={\alpha }_{n}{u}_{n}+{\beta }_{n}{v}_{n}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}\text{are}\phantom{\rule{4.0pt}{0ex}}\text{given.}$ Here $u$ and $v$ are two linearly independent solutions of equation ${\Delta }^{2}{y}_{n}={a}_{n+1}{y}_{n+1}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}\left(\underset{n\to \infty }{lim}{\alpha }_{n}=\alpha <\infty \phantom{\rule{1.0em}{0ex}}\mathrm{and}\phantom{\rule{1.0em}{0ex}}\underset{n\to \infty }{lim}{\beta }_{n}=\beta <\infty \right)\phantom{\rule{0.166667em}{0ex}}.$ A special case of equation (E) is also considered.

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