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

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*Malgorzata Migda, Ewa Schmeidel, Malgorzata Zbaszyniak
*

**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.