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

Migda, Małgorzata ; Schmeidel, Ewa ; Zbąszyniak, Małgorzata

Archivum mathematicum, Tome 41 (2005) no. 4, pp. 379-388.

Résumé

We consider a second order nonlinear difference equation \[ \Delta ^2 y_n = a_n y_{n+1} + f(n,y_n,y_{n+1})\,,\quad n\in N\,. \qquad \mathrm {(\mbox{E})}\] The necessary conditions under which there exists a solution of equation (E) which can be written in the form \[ y_{n+1} = \alpha _{n}{u_n} + \beta _{n}{v_n}\,,\quad \mbox{are given.} \] 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.

Détail

Classification : 39A10, 39A11
Keywords: nonlinear difference equation; nonoscillatory solution; second order