On the oscillation of solutions of third order linear difference equations of neutral type

Anna Andruch-Sobiło; Małgorzata Migda

Mathematica Bohemica (2005)

  • Volume: 130, Issue: 1, page 19-33
  • ISSN: 0862-7959

Abstract

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In this note we consider the third order linear difference equations of neutral type Δ 3 [ x ( n ) - p ( n ) x ( σ ( n ) ) ] + δ q ( n ) x ( τ ( n ) ) = 0 , n ∈ N ( n 0 ) , ( E ) where δ = ± 1 , p , q N ( n 0 ) → ℝ + ; σ , τ N ( n 0 ) → ℕ , lim n → ∞ σ ( n ) = lim n → ∞ τ ( n ) = ∞ . We examine the following two cases: { 0 < p ( n ) ≤ 1 , σ ( n ) = n + k , τ ( n ) = n + l } , { p ( n ) > 1 , σ ( n ) = n - k , τ ( n ) = n - l } , where k , l are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.

How to cite

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Andruch-Sobiło, Anna, and Migda, Małgorzata. "On the oscillation of solutions of third order linear difference equations of neutral type." Mathematica Bohemica 130.1 (2005): 19-33. <http://eudml.org/doc/249603>.

@article{Andruch2005,
abstract = {In this note we consider the third order linear difference equations of neutral type \[ \Delta ^\{3\}[x(n)-p(n)x(\sigma (n))]+\delta q(n)x(\tau (n))=0, \quad n \in N(n\_0), \qquad \mathrm \{(\{\mathrm \{E\}\})\}\] where $\delta =\pm 1$, $p,q\: N(n_0)\rightarrow \mathbb \{R\}_+;$$\sigma ,\tau \: N(n_0)\rightarrow \mathbb \{N\}$, $\lim _\{n \rightarrow \infty \}\sigma (n)= \lim \limits _\{n \rightarrow \infty \}\tau (n)= \infty .$ We examine the following two cases: \[ \BOF \begin\{@align\}\{1\}\{-1\}\lbrace 0<p(n)&\le 1, \ \sigma (n)=n+k,\ \tau (n)=n+l\rbrace , \lbrace p(n)&>1, \ \sigma (n)=n-k,\ \tau (n)=n-l\rbrace , \BOF \end\{@align\}\] where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.},
author = {Andruch-Sobiło, Anna, Migda, Małgorzata},
journal = {Mathematica Bohemica},
keywords = {neutral type difference equation; nonoscillatory solution; asymptotic behavior; oscillation; third order linear difference equations; neutral type difference equation; nonoscillatory solution; asymptotic behavior; oscillation; third order linear difference equations},
language = {eng},
number = {1},
pages = {19-33},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the oscillation of solutions of third order linear difference equations of neutral type},
url = {http://eudml.org/doc/249603},
volume = {130},
year = {2005},
}

TY - JOUR
AU - Andruch-Sobiło, Anna
AU - Migda, Małgorzata
TI - On the oscillation of solutions of third order linear difference equations of neutral type
JO - Mathematica Bohemica
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 130
IS - 1
SP - 19
EP - 33
AB - In this note we consider the third order linear difference equations of neutral type \[ \Delta ^{3}[x(n)-p(n)x(\sigma (n))]+\delta q(n)x(\tau (n))=0, \quad n \in N(n_0), \qquad \mathrm {({\mathrm {E}})}\] where $\delta =\pm 1$, $p,q\: N(n_0)\rightarrow \mathbb {R}_+;$$\sigma ,\tau \: N(n_0)\rightarrow \mathbb {N}$, $\lim _{n \rightarrow \infty }\sigma (n)= \lim \limits _{n \rightarrow \infty }\tau (n)= \infty .$ We examine the following two cases: \[ \BOF \begin{@align}{1}{-1}\lbrace 0<p(n)&\le 1, \ \sigma (n)=n+k,\ \tau (n)=n+l\rbrace , \lbrace p(n)&>1, \ \sigma (n)=n-k,\ \tau (n)=n-l\rbrace , \BOF \end{@align}\] where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.
LA - eng
KW - neutral type difference equation; nonoscillatory solution; asymptotic behavior; oscillation; third order linear difference equations; neutral type difference equation; nonoscillatory solution; asymptotic behavior; oscillation; third order linear difference equations
UR - http://eudml.org/doc/249603
ER -

References

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