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On the existence of solutions of some second order nonlinear difference equations. (English) Zbl 1122.39001

The authors study the second order difference equation \[ \Delta ^2 y_n = a_n y_{n+1} + f(n,y_n,y_{n+1}),\quad n\in N. \tag{E} \] They give two theorems ensuring the existence of a solution \(y\) of equation (E) such that \(y_{n+1} = \alpha _{n}u_n + \beta _{n}v_n,\) where \(\lim \alpha _n=\alpha \), \(\lim \beta _n=\beta \), \(\alpha , \beta \in \mathbb R\) and \(u\), \(v\) are solutions of the equation \(\Delta ^2 z_n = a_{n+1} z_{n+1}.\)

MSC:

39A10 Additive difference equations
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