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

Migda, Małgorzata, Schmeidel, Ewa, and Zbąszyniak, Małgorzata. "On the existence of solutions of some second order nonlinear difference equations." Archivum Mathematicum 041.4 (2005): 379-388. <http://eudml.org/doc/249499>.

@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 -