Advanced Search

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