Article

Full entry | PDF   (0.2 MB)
Keywords:
neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior
Summary:
In the paper we consider the difference equation of neutral type $$\Delta ^{3}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \Bbb N (n_0),$$ where $p,q\colon\Bbb N(n_0)\rightarrow \Bbb R_+$; $\sigma , \tau \colon\Bbb N\rightarrow \Bbb Z$, $\sigma$ is strictly increasing and $\lim \limits _{n \rightarrow \infty }\sigma (n)=\infty ;$ $\tau$ is nondecreasing and $\lim \limits _{n \rightarrow \infty }\tau (n)=\infty$, $f\colon\Bbb R\rightarrow {\Bbb R}$, $xf(x)>0$. We examine the following two cases: $0<p(n)\leq \lambda ^*< 1,\quad \sigma (n)=n-k,\quad \tau (n)=n-l,$ and $1<\lambda _*\leq p(n),\quad \sigma (n)=n+k,\quad \tau (n)=n+l,$ 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.
References:
[1] Agarwal, R. P.: Difference Equations and Inequalities, 2nd edition. Pure Appl. Math. 228, Marcel Dekker, New York (2000). MR 1740241
[2] Dorociaková, B.: Asymptotic behaviour of third order linear neutral differential equations. Studies of University in Žilina 13 (2001), 57-64. MR 1874004 | Zbl 1040.34098
[3] Dorociaková, B.: Asymptotic criteria for third order linear neutral differential equations. Folia FSN Universitatis Masarykianae Brunensis, Mathematica 13 (2003), 107-111. MR 2030427 | Zbl 1111.34342
[4] Grace, S. R., Hamedani, G. G.: On the oscillation of certain neutral difference equations. Math. Bohem. 125 (2000), 307-321. MR 1790122 | Zbl 0969.39006
[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. DOI 10.1016/S0377-0427(00)00264-8 | MR 1835720 | Zbl 0984.39006
[6] Luo, J., Yu, Y.: Asymptotic behavior of solutions of second order neutral difference equations with "maxima''. Demonstratio Math. 34 (2001), 83-89. MR 1823087
[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. DOI 10.1016/0022-247X(92)90342-B | MR 1159653 | Zbl 0763.39002
[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. DOI 10.1016/0022-247X(91)90278-8 | MR 1113411 | Zbl 0732.39002
[9] Migda, M., Migda, J.: On a class of first order nonlinear difference equations of neutral type. Math. Comput. Modelling 40 (2004), 297-306. DOI 10.1016/j.mcm.2003.12.006 | MR 2091062
[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. DOI 10.1016/S0022-247X(03)00298-1 | MR 1998666 | Zbl 1037.39002
[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. MR 1480399 | Zbl 0887.39004
[12] Thandapani, E., Arul, R., Raja, P. S.: Oscillation of first order neutral delay difference equations. Appl. Math. E-Notes 3 (2003), 88-94. MR 1980570 | Zbl 1027.39003
[13] Thandapani, E., Sundaram, P.: Asymptotic and oscillatory behavior of solutions of nonlinear neutral delay difference equations. Utilitas Math. 45 (1994), 237-244. MR 1284034
[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. MR 1600048 | Zbl 0901.39004
[15] Zafer, A., Dahiya, R. S.: Oscillation of a neutral difference equation. Appl. Math. Lett. 6 (1993), 71-74. DOI 10.1016/0893-9659(93)90015-F | MR 1347777 | Zbl 0772.39001

Partner of