The third order non linear difference equations of the form
for n ≥ n0 are considered. Here {cn}, {dn}, {pn} and {qn} are sequences of positive real numbers for n0Nf is a continuous function such that
for u ≠ 0. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results. Where n0N is a fixed integer, Δ denotes to forward difference operator Δxn = xn+1xn and σ is a nonnegative integer.
1.
Agarwal
,
R.P.
,
Bohner
,
M.
,
Grace
,
S.R.
,
ORegan
,
D.
:
Discrete Oscillation Theory
,
Hindawi Publishing Corporation
.
New York
.
2005
.
2.
Agarwal
,
R.P.
:
Difference Equations and Inequalities, Theory, Methods and Applications
, Second Edition, Revised and Expanded.
New York
.
Marcel Dekker
2000
.
3.
Agarwal
,
R.P.
,
Wong
,
P.J.Y.
:
Advanced Topics in Difference Equations
,
Kluwer Academic Publishers
,
Drodrecht
.
1997
.
4.
R. P.
Agarwal
,
S. R.
Grace
,
D.
ORegan
,
Oscillation Theory for Difference and Functional Differential Equations
,
Kluwer Academic
,
Dordrecht
,
2000
.
5.
T.
Asaela
,
H.
YoshidaH
,
Stability, instability and complex behavior in macrodynamic models with policy lag
,
Discrete Dynamics in Nature and Society
,
5
, (
2001
),
281
295
.
6.
Andruch-Sobilo
,
A.
,
Migda
,
M.
:
On the oscillation of solutions of third order linear difference equations of neutral type
.
Math. Bohem.
130
,
19
33
(
2005
).
7.
Artzroumi
,
M.
:
Generalized stable population theory
.
J. Math. Biology.
21
,
363
381
(
1985
).
8.
B.
Baculkova
,
J.
Dzurina
,
Oscillation of third order neutral differential equations
,
Math. Comput. Modelling
,
52
, (
2010
),
215
226
.
9.
L.
Berenzansky
,
E.
Braverman
,
Some oscillation problems for a second order linear delay differential equations
,
J. Math. Anal. Appl.
220
, (
1998
),
719
740
.
10.
C.
Cesarano
and
O.
Bazighifan
,
Qualitative behavior of solutions of second order differential equations
,
Symmetry (MDPI)
,
11
(
777
) (
2019
)
8
pages.
11.
Dosla
,
Z.
,
Kobza
,
A.
:
Global asymptotic properties of third-order difference equations
.
Comput. Math. Appl.
48
,
191
200
(
2004
).
12.
Dosla
,
Z.
,
Kobza
,
A.
:
On third-order linear difference equations involving quasi-differences
.
Adv. Difference Equ.
1
13
(
2006
).
13.
O.
Dosly
,
J.
Graef
,
J.
Jaros
,
Forced oscillation of second order linear and half linear difference equations
,
Proc. Amer. Math. Soc.
131
(
2002
),
2859
2867
.
14.
D. M.
DuboisX
,
Extension of the Kaldor-Kalecki models of business cycle with a computational anticipated capital stock
,
Journal of Organisational Transformation and Social Change
,
1
, (
2004
),
63
80
.
15.
L. H.
Erbe
,
Qingkai
Kong
,
B.G.
Zhang
,
Oscillation Theory for Functional Differential Equations
,
Marcel Dekker, New York
,
1995
.
16.
J. M.
Ferreira
,
S.
Pinelas
,
Oscillatory mixed difference systems, Hindawi publishing corporation, Advanced in Difference Equations ID
(
2006
),
1
18
.
17.
G.
Gandolfo
,
Economic dynamics
, Third Edition,
Berlin
Springer-verlag
,
1996
.
18.
Gepreel
,
K
,
Shehata
,
AR
,
Rational Jacobi elliptic solutions for nonlinear difference differential lattice equation
,
Appl. Math. Lett.
25
(
2012
)
1173
1178
.
19.
S.R.
Grace
,
On the oscillations of mixed neutral equations
,
J. Math. Anal. Appl.
,
194
, (
1995
),
377
388
.
20.
Grace
,
S.R.
,
Lalli
,
B.S.
:
Oscillation theorems for forced neutral difference equations
.
J. Math. Anal. Appl.
187
,
91
106
(
1994
).
21.
Grace
,
S.R.
,
Lalli
,
B.S.
:
Oscillation theorems for second order delay and neutral difference equations
.
Utilitas Math.
45
,
197
211
(
1994
).
22.
Graef
,
J.
,
Thandapani
,
E.
:
Oscillatory and asymptotic behavior of solutions of third order delay difference equations
.
Funk. Ekvac.
42
,
355
369
(
1999
).
23.
John R.
Graef
and
Mary
Hill
,
Non-oscillation of all solutions of a higher order nonlinear delay dynamic equation on time scales
,
J. Math. Anal. Appl.
,
423
(
2015
)
1693
1703
.
24.
I.
Gyri
and
G.
Ladas
,
Oscillation Theory of Delay Differential Equations
,
Clarendon Press
,
New York
,
1991
.
25.
Z. L.
Han
,
T. X.
Li
,
S. R.
Sun
,
W. S.
Chen
,
On the oscillation of second order neutral delay differential equations
,
Adv. Diff. Eqn.
2010, (2010),
26.
Hooker
,
J.W.
,
Patula
,
W.T.
:
A second-order nonlinear difference equation: Oscillation and asymptotic behavior
.
J. Math. Anal. Appl.
91
,
9
29
(
1983
).
27.
V.
Iakoveleva
and
C. J.
Vanegas
,
On the oscillation of differential equations with delayed and advanced arguements
,
Elec. J. Diff. Equation
,
13
, (
2005
),
57
63
.
28.
R. W.
James
and
M. H.
Belz
,
The significance of the characteristic solutions of mixed difference and differential equations
,
Econometrica
,
6
, (
1938
),
326
343
.
29.
R.
Jankowski
and
E.
Schmeidel
,
Almost oscillation criteria for second order neutral difference equations with quasi-differences
,
International Journal of Difference Equations
,
9
(
1
) (
2014
),
77
86
.
30.
J.
Jiang
,
Oscillation criteria for second order quasilinear neutral delay difference equations
,
Appl. Math. Comput.
,
125
(
2002
),
287
293
.
31.
Jiang
,
J.
:
Oscillation of second order nonlinear neutral delay difference equations
.
Appl. Math. Comput.
146
,
791
801
(
2003
).
32.
Jiang
,
J.
,
Li
,
X.
:
Oscillation of second order nonlinear neutral differential equations
.
Appl. Math. Comput.
135
,
531
540
(
2003
).
This content is only available via PDF.
You do not currently have access to this content.