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

@article{Migda2005,
abstract = {We consider a second order nonlinear difference equation \[ \Delta ^2 y\_n = a\_n y\_\{n+1\} + f(n,y\_n,y\_\{n+1\})\,,\quad n\in N\,. \qquad \mathrm \{(\mbox\{E\})\}\] 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.},
author = {Migda, Małgorzata, Schmeidel, Ewa, Zbąszyniak, Małgorzata},
journal = {Archivum Mathematicum},
keywords = {nonlinear difference equation; nonoscillatory solution; second order; nonoscillatory solution},
language = {eng},
number = {4},
pages = {379-388},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the existence of solutions of some second order nonlinear difference equations},
url = {http://eudml.org/doc/249499},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Migda, Małgorzata
AU - Schmeidel, Ewa
AU - Zbąszyniak, Małgorzata
TI - On the existence of solutions of some second order nonlinear difference equations
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 4
SP - 379
EP - 388
AB - 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.
LA - eng
KW - nonlinear difference equation; nonoscillatory solution; second order; nonoscillatory solution
UR - http://eudml.org/doc/249499
ER -