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neutral type difference equation; nonoscillatory solution; asymptotic behavior; oscillation; third order linear difference equations
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: \[ \BOF\align \lbrace 0<p(n)&\le 1, \ \sigma (n)=n+k,\ \tau (n)=n+l\rbrace , \lbrace p(n)&>1, \ \sigma (n)=n-k,\ \tau (n)=n-l\rbrace , \BOF\endalign \] where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.
[1] R. P. Agarwal: Difference Equations and Inequalities. 2nd edition, Pure Appl. Math. 228, Marcel Dekker, New York, 2000. MR 1740241 | Zbl 0952.39001
[2] R. P. Agarwal, S. R. Grace, D. O’Regan: Oscillation Theory of Difference and Functional Differential Equations. Kluwer Academic Publishers, Dordrecht, 2000. MR 1774732
[3] B. Bačová: Oscillation of third order linear neutral differential equations. Studies of University in Žilina 13 (2001), 7–15. MR 1873438 | Zbl 1040.34078
[4] B. Bačová: Some oscillatory properties of third order linear neutral differential equations. Folia FSN Universitatis Masarykianae Brunensis, Mathematica 13 (2003), 17–23. MR 2030018 | Zbl 1111.34332
[5] M. P. Chen, B. S. Lalli, J. S. Yu: Oscillation in neutral delay difference equations with variable coefficients. Computers Math. Appl. 29 (1995), 5–11. DOI 10.1016/0898-1221(94)00223-8 | MR 1326277
[6] Y. Gao, G. Zhang: Oscillation of nonlinear first order neutral difference equations. Appl. Math. E-Notes 1 (2001), 5–10. MR 1833830
[7] S. R. Grace, G. G. Hamedani: On the oscillation of certain neutral difference equations. Math. Bohem. 125 (2000), 307–321. MR 1790122
[8] J. R. Graef, R. Savithri, E. Thandapani: Oscillatory properties of third order neutral delay differential equations. Dynamical systems and differential equations (Wilmington, NC, 2002). Discrete Contin. Dyn. Syst. (2003), suppl., 342–350. MR 2018134
[9] B. S. Lalli: Oscillation theorems for neutral difference equations. Comput. Math. Appl. 28 (1994), 191–202. DOI 10.1016/0898-1221(94)00107-3 | MR 1284234 | Zbl 0807.39004
[10] B. S. Lalli, B. G. Zhang, J. Z. Li: 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
[11] B. S. Lalli, B. G. Zhang: Oscillation and comparison theorems for certain neutral difference equations. J. Austral. Math. Soc. Ser. B 34 (1992), 245–256. DOI 10.1017/S0334270000008754 | MR 1181576
[12] M. Migda, J. Migda: On a class of first order nonlinear difference equations of neutral type. (to appear).
[13] X. Tang, J. Yan: Oscillation and nonoscillation of an odd-order nonlinear difference equation. Funct. Differ. Equ. 7 (2000), 1–2, 157–166. MR 1941865
[14] E. Thandapani, R. Arul, P. S. Raja: Oscillation of first order neutral delay difference equations. Appl. Math. E-Notes 3 (2003), 88–94. MR 1980570
[15] E. Thandapani, K. Mahalingam: Oscillatory properties of third order neutral delay difference equations. Demonstratio Math. 35 (2002), 325–337. MR 1907305
[16] E. Thandapani, P. Sundaram: Oscillation properties of first order nonlinear functional difference equations of neutral type. Indian J. Math. 36 (1994), 59–71. MR 1315896
[17] E. Thandapani, E. Sundaram: Asymptotic and oscillatory behavior of solutions of first order nonlinear neutral difference equations. Rivista Math. Pura Appl. 18 (1996), 93–105. MR 1600048
[18] A. Zafer, R. S. Dahiya: Oscillation of a neutral difference equation. Appl. Math. Lett. 6 (1993), 71–74. DOI 10.1016/0893-9659(93)90015-F | MR 1347777
[19] G. Zhang: Oscillation of nonlinear neutral difference equations. Appl. Math. E-Notes 2 (2002), 22–24. MR 1979405
[20] F. Zhou: Oscillation for nonlinear difference equation of higher order. J. Math. Study 34 (2001), 282–286. MR 1864910
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