Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type

Anna Andruch-Sobiło; Andrzej Drozdowicz

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 3, page 247-258
  • ISSN: 0862-7959

Abstract

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In the paper we consider the difference equation of neutral type Δ 3 [ x ( n ) - p ( n ) x ( σ ( n ) ) ] + q ( n ) f ( x ( τ ( n ) ) ) = 0 , n ( n 0 ) , where p , q : ( n 0 ) + ; σ , τ : , σ is strictly increasing and lim n σ ( n ) = ; τ is nondecreasing and lim n τ ( n ) = , f : , x f ( x ) > 0 . We examine the following two cases: 0 < p ( n ) λ * < 1 , σ ( n ) = n - k , τ ( n ) = n - l , and 1 < λ * p ( n ) , σ ( n ) = n + k , τ ( n ) = n + l , where k , l are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as n with a weaker assumption on q than the usual assumption i = n 0 q ( i ) = that is used in literature.

How to cite

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Andruch-Sobiło, Anna, and Drozdowicz, Andrzej. "Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type." Mathematica Bohemica 133.3 (2008): 247-258. <http://eudml.org/doc/250534>.

@article{Andruch2008,
abstract = {In the paper we consider the difference equation of neutral type \[ \Delta ^\{3\}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \mathbb \{N\} (n\_0), \] where $p,q\colon \mathbb \{N\}(n_0)\rightarrow \mathbb \{R\}_+$; $\sigma , \tau \colon \mathbb \{N\}\rightarrow \mathbb \{Z\}$, $\sigma $ is strictly increasing and $\lim \limits _\{n \rightarrow \infty \}\sigma (n)=\infty ;$$\tau $ is nondecreasing and $\lim \limits _\{n \rightarrow \infty \}\tau (n)=\infty $, $f\colon \mathbb \{R\}\rightarrow \{\mathbb \{R\}\}$, $xf(x)>0$. We examine the following two cases: \[ 0<p(n)\le \lambda ^*< 1,\quad \sigma (n)=n-k,\quad \tau (n)=n-l, \] and \[1<\lambda \_*\le p(n),\quad \sigma (n)=n+k,\quad \tau (n)=n+l,\] where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\rightarrow \infty $ with a weaker assumption on $q$ than the usual assumption $\sum \limits _\{i=n_0\}^\{\infty \}q(i)=\infty $ that is used in literature.},
author = {Andruch-Sobiło, Anna, Drozdowicz, Andrzej},
journal = {Mathematica Bohemica},
keywords = {neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior; neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior},
language = {eng},
number = {3},
pages = {247-258},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type},
url = {http://eudml.org/doc/250534},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Andruch-Sobiło, Anna
AU - Drozdowicz, Andrzej
TI - Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 3
SP - 247
EP - 258
AB - In the paper we consider the difference equation of neutral type \[ \Delta ^{3}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \mathbb {N} (n_0), \] where $p,q\colon \mathbb {N}(n_0)\rightarrow \mathbb {R}_+$; $\sigma , \tau \colon \mathbb {N}\rightarrow \mathbb {Z}$, $\sigma $ is strictly increasing and $\lim \limits _{n \rightarrow \infty }\sigma (n)=\infty ;$$\tau $ is nondecreasing and $\lim \limits _{n \rightarrow \infty }\tau (n)=\infty $, $f\colon \mathbb {R}\rightarrow {\mathbb {R}}$, $xf(x)>0$. We examine the following two cases: \[ 0<p(n)\le \lambda ^*< 1,\quad \sigma (n)=n-k,\quad \tau (n)=n-l, \] and \[1<\lambda _*\le p(n),\quad \sigma (n)=n+k,\quad \tau (n)=n+l,\] where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\rightarrow \infty $ with a weaker assumption on $q$ than the usual assumption $\sum \limits _{i=n_0}^{\infty }q(i)=\infty $ that is used in literature.
LA - eng
KW - neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior; neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior
UR - http://eudml.org/doc/250534
ER -

References

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  1. Agarwal, R. P., Difference Equations and Inequalities, 2nd edition, Pure Appl. Math. 228, Marcel Dekker, New York (2000). (2000) MR1740241
  2. Dorociaková, B., Asymptotic behaviour of third order linear neutral differential equations, Studies of University in Žilina 13 (2001), 57-64. (2001) Zbl1040.34098MR1874004
  3. Dorociaková, B., Asymptotic criteria for third order linear neutral differential equations, Folia FSN Universitatis Masarykianae Brunensis, Mathematica 13 (2003), 107-111. (2003) Zbl1111.34342MR2030427
  4. Grace, S. R., Hamedani, G. G., On the oscillation of certain neutral difference equations, Math. Bohem. 125 (2000), 307-321. (2000) Zbl0969.39006MR1790122
  5. Luo, J. W., Bainov, D. D., Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima, J. Comp. Appl. Math. 131 (2001), 333-341. (2001) Zbl0984.39006MR1835720
  6. Luo, J., Yu, Y., Asymptotic behavior of solutions of second order neutral difference equations with "maxima'', Demonstratio Math. 34 (2001), 83-89. (2001) MR1823087
  7. Lalli, B. S., Zhang, B. G., On existence of positive solutions and bounded oscillations for neutral difference equations, J. Math. Anal. Appl. 166 (1992), 272-287. (1992) Zbl0763.39002MR1159653
  8. Lalli, B. S., Zhang, B. G., Li, J. Z., On the oscillation of solutions and existence of positive solutions of neutral difference equations, J. Math. Anal. Appl. 158 (1991), 213-233. (1991) Zbl0732.39002MR1113411
  9. Migda, M., Migda, J., On a class of first order nonlinear difference equations of neutral type, Math. Comput. Modelling 40 (2004), 297-306. (2004) MR2091062
  10. Parhi, N., Tripathy, A. K., Oscillation of a class of nonlinear neutral difference equations of higher order, J. Math. Anal. 284 (2003), 756-774. (2003) Zbl1037.39002MR1998666
  11. Szmanda, B., Note on the behavior of solutions of difference equations of arbitrary order, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8 (1997), 52-59. (1997) Zbl0887.39004MR1480399
  12. Thandapani, E., Arul, R., Raja, P. S., Oscillation of first order neutral delay difference equations, Appl. Math. E-Notes 3 (2003), 88-94. (2003) Zbl1027.39003MR1980570
  13. Thandapani, E., Sundaram, P., Asymptotic and oscillatory behavior of solutions of nonlinear neutral delay difference equations, Utilitas Math. 45 (1994), 237-244. (1994) MR1284034
  14. Thandapani, E., Sundaram, E., Asymptotic and oscillatory behavior of solutions of first order nonlinear neutral difference equations, Rivista Math.Pura Appl. 18 (1996), 93-105. (1996) Zbl0901.39004MR1600048
  15. Zafer, A., Dahiya, R. S., Oscillation of a neutral difference equation, Appl. Math. Lett. 6 (1993), 71-74. (1993) Zbl0772.39001MR1347777

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