Address.
Institute of Mathematics, Poznan University of Technology, Piotrowo 3a, 60-965 Poznan, Poland
E-mail. mmigda@math.put.poznan.pl
E-mail. eschmeid@math.put.poznan.pl
E-mail. mmielesz@math.put.poznan.pl
Abstract.
We consider a second order nonlinear difference equation
$$
\Delta^2 y_n = a_n y_{n+1} + f(n,y_n,y_{n+1})\,,\quad n\in N\,.
\eqno(\mbox{E})
$$
The necessary conditions under which there exists a solution of equation
{\rm (E)} which can be written in
the form
$$
y_{n+1} = \alpha_{n}{u_n} + \beta_{n}{v_n}\,,\quad \mbox{are given.}
$$
Here $u$ and $v$ are two linearly independent solutions of equation
$$
\Delta^2 y_n = a_{n+1} y_{n+1}\,, \quad ({\lim\limits_{n \rightarrow \infty}
\alpha_{n} =
\alpha<\infty } \quad {\rm and} \quad {\lim\limits_{n \rightarrow \infty}
\beta_{n} = \beta<\infty })\,.
$$
A special case of equation (E) is also considered.
AMSclassification. 39A10.
Keywords. Nonlinear difference equation, nonoscillatory solution, second order.