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

Archivum Mathematicum (2005)

• Volume: 041, Issue: 4, page 379-388
• ISSN: 0044-8753

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## Abstract

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We consider a second order nonlinear difference equation ${\Delta }^{2}{y}_{n}={a}_{n}{y}_{n+1}+f\left(n,{y}_{n},{y}_{n+1}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}n\in N\phantom{\rule{0.166667em}{0ex}}.\phantom{\rule{2.0em}{0ex}}\left(\text{E}\right)$ The necessary conditions under which there exists a solution of equation (E) which can be written in the form ${y}_{n+1}={\alpha }_{n}{u}_{n}+{\beta }_{n}{v}_{n}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}\text{are}\phantom{\rule{4.0pt}{0ex}}\text{given.}$ Here $u$ and $v$ are two linearly independent solutions of equation ${\Delta }^{2}{y}_{n}={a}_{n+1}{y}_{n+1}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}\left(\underset{n\to \infty }{lim}{\alpha }_{n}=\alpha <\infty \phantom{\rule{1.0em}{0ex}}\mathrm{and}\phantom{\rule{1.0em}{0ex}}\underset{n\to \infty }{lim}{\beta }_{n}=\beta <\infty \right)\phantom{\rule{0.166667em}{0ex}}.$ A special case of equation (E) is also considered.

## How to cite

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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 -

## References

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11. Schmeidel E., Asymptotic behaviour of solutions of the second order difference equations, Demonstratio Math. 25 (1993), 811–819. (1993) Zbl0799.39001MR1265844
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