Geodesic
On the existence of solutions of some second order nonlinear difference equations
Archivum mathematicum, Tome 41 (2005) no. 4, pp. 379-388 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 39A10, 39A11
Keywords: nonlinear difference equation; nonoscillatory solution; second order 
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Migda, Małgorzata; Schmeidel, Ewa; Zbąszyniak, Małgorzata. On the existence of solutions of some second order nonlinear difference equations. Archivum mathematicum, Tome 41 (2005) no. 4, pp. 379-388. http://geodesic.mathdoc.fr/item/ARM_2005_41_4_a2/

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