Existence Results for Nonoscillatory Solutions of Third Order Nonlinear Neutral Difference Equations
DOI:
https://doi.org/10.5644/SJM.05.1.07Keywords:
Nonoscillation, third order, nonlinear, difference equationsAbstract
In this paper the authors consider the third order neutral difference equation
\begin{equation*}
\Delta ^{3}\left( x_{n}+p_{n}x_{n-k}\right) +q_{n}f\left(
x_{n-\ell }\right) =h_{n}
\end{equation*}
where $\left\{ p_{n}\right\} ,\left\{ q_{n}\right\} ,\left\{
h_{n}\right\} \ $are real sequences. They use Krasnoselskii's fixed point theorem to establish the existence of nonoscillatory solutions. The results are illustrated with examples.
2000 Mathematics Subject Classification. 39A10
Downloads
References
R. P. Agarwal,Difference Equations and Inequalities , 2nd Edition, Marcel Dekker, New York, 2000.
R. P. Agarwal, M. Bohner, S. R. Grace and D. O Regan, Discrete Oscillation Theory, Hindawi publishing corporation, New York, 2005.
R. P. Agarwal and S. R. Grace, The oscillation of higher - order nonlinear difference equations of neutral type, Appl. Math. Lett., 12 (1999), 77–83.
R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, Kluwer Publications, Dordrecht, 1997.
R. P. Agarwal and S. R. Grace, Oscillation of certain third order difference equations, Comput.Math.Applic.,42 (2001), 379–384.
Y. Bolat and O. Akin, ¨ Oscillatory behaviour of a higher-order nonlinear neutral type functional difference equation with oscillating coefficients, Appl. Math. Lett., 17 (2004), 1073–1078.
L. H. Erbe, Q. K. Kong, and B. G. Zang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.
J. R. Graef and E. Thandapani, Oscillatory and asymptotic behaviour of solutions of third order delay difference equations, Funk.Ekv., 42 (1999), 355–369.
V. Lakshmikantham and D. Trigiante, Theory of Difference Equations, Numerical Methods and Applications, Academic Press, New York, 1998.
W. T. Li, Classifications and existence of positive solutions of higher-order nonlinear neutral difference equations, Appl. Math. Comp., 152 (2004), 351–366.
M. Migda, E. Schmeidel and A. Drozdowicz, Nonoscillation results for some third order nonlinear difference equations, Colloquium on Differential and Difference Equations, Brno (2002), 185–192.
S. H. Saker, Oscillation of third order difference equations, Portugal. Math., 61 (2004), 249–257.
B. Smith, Oscillatory and asymptotic behaviour in certain third order difference equations, Rocky Mountain J.Math., 17 (1987), 597–606.
B. Smith and W. E. Taylor, Nonlinear third order difference equation: Oscillatory and asymptotic behaviour, Tamk. J. Math., 19 (1988), 91–95.
B. Smith and W. E. Taylor, Asymptotic behaviour of third order difference equations, Portugal. Math., 44 (1987), 113–117.
B. Smith and W. E. Taylor, Oscillation and nonoscillation in nonlinear third order difference equations, Int. J. Math. Math. Sci., 13 (1990), 281–286.
S. H. Saker, Oscillation and asymptotic behavior of third order nonlinear neutral delay difference equations, Dynamic Systems and Applications, 15 (2006), 549–568.
