Non-oscillation of second order linear self-adjoint nonhomogeneous difference equations

N. Parhi

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 3, page 241-258
  • ISSN: 0862-7959

Abstract

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In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form Δ ( p n - 1 Δ y n - 1 ) + q y n = 0 , n ≥ 1 , where q is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type Δ ( p n - 1 Δ y n - 1 ) + q n g ( y n ) = f n - 1 , n ≥ 1 , where, unlike earlier works, f n ≥ 0 or ≤ 0 (but ¬ ≡ 0 ) for large n . Further, these results are used to obtain sufficient conditions for non-oscillation of all solutions of forced linear third order difference equations of the form y n + 2 + a n y n + 1 + b n y n + c n y n - 1 = g n - 1 , n ≥ 1 .

How to cite

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Parhi, N.. "Non-oscillation of second order linear self-adjoint nonhomogeneous difference equations." Mathematica Bohemica 136.3 (2011): 241-258. <http://eudml.org/doc/197043>.

@article{Parhi2011,
abstract = {In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form \[ \Delta (p\_\{n-1\}\Delta y\_\{n-1\}) + q y\_\{n\} =0 , \quad n\ge 1, \] where $q$ is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type \[ \Delta (p\_\{n-1\}\Delta y\_\{n-1\}) + q\_\{n\}g( y\_\{n\}) = f\_\{n-1\}, \quad n\ge 1, \] where, unlike earlier works, $f_\{n\}\ge 0$ or $\le 0$ (but $\lnot \equiv 0)$ for large $n$. Further, these results are used to obtain sufficient conditions for non-oscillation of all solutions of forced linear third order difference equations of the form \[ y\_\{n+2\}+ a\_\{n\}y\_\{n+1\}+ b\_\{n\}y\_\{n\}+ c\_\{n\}y\_\{n-1\}= g\_\{n-1\}, \quad n\ge 1. \]},
author = {Parhi, N.},
journal = {Mathematica Bohemica},
keywords = {oscillation; non-oscillation; second order difference equation; third order difference equation; generalized zero; oscillation; non-oscillation; generalized zero; linear second order difference equation; nonlinear nonhomogeneous second order difference equation; linear third order difference equation; Fibonacci equation; Sturm type comparison theorem; Leighton-Wintner type criterion},
language = {eng},
number = {3},
pages = {241-258},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Non-oscillation of second order linear self-adjoint nonhomogeneous difference equations},
url = {http://eudml.org/doc/197043},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Parhi, N.
TI - Non-oscillation of second order linear self-adjoint nonhomogeneous difference equations
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 3
SP - 241
EP - 258
AB - In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form \[ \Delta (p_{n-1}\Delta y_{n-1}) + q y_{n} =0 , \quad n\ge 1, \] where $q$ is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type \[ \Delta (p_{n-1}\Delta y_{n-1}) + q_{n}g( y_{n}) = f_{n-1}, \quad n\ge 1, \] where, unlike earlier works, $f_{n}\ge 0$ or $\le 0$ (but $\lnot \equiv 0)$ for large $n$. Further, these results are used to obtain sufficient conditions for non-oscillation of all solutions of forced linear third order difference equations of the form \[ y_{n+2}+ a_{n}y_{n+1}+ b_{n}y_{n}+ c_{n}y_{n-1}= g_{n-1}, \quad n\ge 1. \]
LA - eng
KW - oscillation; non-oscillation; second order difference equation; third order difference equation; generalized zero; oscillation; non-oscillation; generalized zero; linear second order difference equation; nonlinear nonhomogeneous second order difference equation; linear third order difference equation; Fibonacci equation; Sturm type comparison theorem; Leighton-Wintner type criterion
UR - http://eudml.org/doc/197043
ER -

References

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  8. Parhi, N., Panda, A., Oscillation of solutions of forced nonlinear second order difference equations, Proc. Eighth Ramanujan Symposium on Recent Developments in Nonlinear Systems R. Sahadevan, M. Lakshmanan Narosa Pub. House, New Delhi (2002), 221-238. (2002) Zbl1056.39010MR2010625
  9. Parhi, N., Panda, A., Oscillatory and non-oscillatory behaviour of solutions of difference equations of the third order, Math. Bohem. 133 (2008), 99-112. (2008) MR2400154
  10. Parhi, N., Tripathy, A. K., Oscillatory behaviour of second order difference equations, Commun. Appl. Nonlin. Anal. 6 (1999), 79-100. (1999) MR1665966
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  12. Parhi, N., Tripathy, A. K., 10.1080/10236190290017423, J. Difference Eq. Appl. 8 (2002), 415-426. (2002) Zbl1037.39001MR1897066DOI10.1080/10236190290017423
  13. Patula, W., 10.1137/0510006, SIAM J. Math. Anal. 10 (1979), 55-61. (1979) Zbl0397.39001MR0516749DOI10.1137/0510006
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