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

Malgorzata Migda, Ewa Schmeidel, Malgorzata Zbaszyniak

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

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.