Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
oscillatory solution; neutral differential equation; asymptotic behaviour
oscillatory solution; neutral differential equation; asymptotic behaviour
Summary:
We obtain sufficient conditions for every solution of the differential equation $$ [y(t)-p(t)y(r(t))]^{(n)}+v(t)G(y(g(t)))-u(t)H(y(h(t)))=f(t) $$ to oscillate or to tend to zero as $t$ approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when $G$ has sub-linear growth at infinity. Our results also apply to the neutral equation $$ [y(t)-p(t)y(r(t))]^{(n)}+q(t)G(y(g(t)))=f(t) $$ when $q(t)$ has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.
References:
[1] Ming-Po, Chen, Wang, Z. C., Yu, J. S., Zhang, B. G.: Oscillation and asymptotic behaviour of higher order neutral differential equations. Bull. Inst. Math. Acad. Sinica 22 (1994), 203-217. MR 1297358
[2] Pitambar, Das, Misra, N.: A necessary and sufficient condition for the solution of a functional differential equation to be oscillatory or tend to zero. J. Math. Anal. Appl. 205 (1997), 78-87. DOI 10.1006/jmaa.1996.5143 | MR 1426981
[3] Dix, J. G., Misra, N., Padhy, L. N., Rath, R. N.: On oscillation and asymptotic behaviour of a neutral differential equations of first order with oscillating coefficients. Electron. J. Qual. Theory Differ. Equ. (2008), 1-10. DOI 10.14232/ejqtde.2008.1.19 | MR 2407546
[4] Gyori, I., Ladas, G.: Oscillation Theory of Delay-Differential Equations with Applications. Clarendon Press, Oxford (1991). MR 1168471
[5] Hildebrandt, T. H.: Introduction to the Theory of Integration. Academic Press, New York (1963). MR 0154957 | Zbl 0112.28302
[6] Karpuz, B., Rath, R. N., Padhy, L. N.: On oscillation and asymptotic behaviour of a higher order neutral differential equation with positive and negative coefficients. Electron. J. Differ. Equations 2008 (2008), 1-15. MR 2430910
[7] Kubiaczyk, I., Wan-Tong, Li, Saker, S. H.: Oscillation of higher order delay differential equations with applications to hyperbolic equations. Indian J. Pure. Appl. Math. 34 (2003), 1259-1271. MR 2007884
[8] Ladde, G. S., Lakshmikantham, V., Zhang, B. G.: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, New York (1987). MR 1017244 | Zbl 0832.34071
[9] Wantong, Li, Quan, Hongshun: Oscillation of higher order neutral differential equations with positive and negative coefficients. Ann. Differ. Equations 11 (1995), 70-76. MR 1341653
[10] Manojlovic, J., Shoukaku, Y., Tanigawa, T., Yoshida, N.: Oscillation criteria for second order differential equations with positive and negative coefficients. Appl. Math. Comput. 181 (2006), 853-863. DOI 10.1016/j.amc.2006.02.015 | MR 2269964 | Zbl 1110.34046
[11] Ozkan, Ocalan: Oscillation of neutral differential equation with positive and negative coefficients. J. Math. Anal. Appl. 331 (2007), 644-654. DOI 10.1016/j.jmaa.2006.09.016 | MR 2306029
[12] Parhi, N., Rath, R. N.: Oscillation criteria for forced first order neutral differential equations with variable coefficients. J. Math. Anal. Appl. 256 (2001), 525-541. DOI 10.1006/jmaa.2000.7315 | MR 1821755 | Zbl 0982.34057
[13] Parhi, N., Chand, S.: On forced first order neutral differential equations with positive and negative coefficients. Math. Slovaca 50 (2000), 81-94. MR 1764347 | Zbl 0959.34051
[14] Parhi, N., Chand, S.: Oscillation of second order neutral delay differential equations with positive and negative coefficients. J. Ind. Math. Soc. 66 (1999), 227-235. MR 1749649
[15] Parhi, N., Rath, R. N.: On oscillation and asymptotic behaviour of solutions of forced first order neutral differential equations. Proc. Indian. Acad. Sci., Math. Sci. 111 (2001), 337-350. DOI 10.1007/BF02829600 | MR 1851095 | Zbl 0995.34058
[16] Parhi, N., Rath, R. N.: Oscillatory behaviour of solutions of non linear higher order neutral differential equations. Math. Bohem. 129 (2004), 11-27. MR 2048783
[17] Parhi, N., Rath, R. N.: On oscillation of solutions of forced nonlinear neutral differential equations of higher order. Czech. Math. J. 53 (2003), 805-825. DOI 10.1007/s10587-004-0805-8 | MR 2018832 | Zbl 1080.34522
[18] Parhi, N., Rath, R. N.: On oscillation of solutions of forced nonlinear neutral differential equations of higher order II. Ann. Pol. Math. 81 (2003), 101-110. DOI 10.4064/ap81-2-1 | MR 1976190 | Zbl 1037.34058
[19] Rath, R. N.: Oscillatory and asymptotic behaviour of higher order neutral equations. Bull. Inst. Math. Acad. Sinica 30 (2002), 219-228. MR 1922656
[20] Rath, R. N., Misra, N.: Necessary and sufficient conditions for oscillatory behaviour of solutions of a forced non linear neutral equation of first order with positive and negative coefficients. Math. Slovaca 54 (2004), 255-266. MR 2076362
[21] Rath, R. N., Mishra, P. P., Padhy, L. N.: On oscillation and asymptotic behaviour of a neutral differential equation of first order with positive and negative coefficients. Electron. J. Differ. Equations 2007 (2007), 1-7. MR 2278415 | Zbl 1118.34054
[22] Rath, R. N., Misra, N., Mishra, P. P., Padhy, L. N.: Non-oscillatory behaviour of higher order functional differential equations of neutral type. Electron. J. Diff. Equations 2007 (2007), 1-14. MR 2366056 | Zbl 1138.34031
[23] Sahiner, Y., Zafer, A.: Bounded oscillation of non-linear neutral differential equations of arbitrary order. Czech. Math. J. 51 (2001), 185-195. DOI 10.1023/A:1013763409361 | MR 1814644
[24] Jianshe, Yu: Neutral delay differential equations with positive and negative coefficients. Acta Math. Sinica 34 (1991), 517-523. MR 1152147
[25] Jianshe, Yu, Zhicheng, Wang: Some further results on oscillation of neutral differential equations. Bull. Austral. Math. Soc. 46 (1992), 149-157. DOI 10.1017/S0004972700011758 | MR 1170449
[26] Zhicheng, Wang, Xianhua, Tang: On the oscillation of neutral differential equations with integrally small coefficients. Ann. Differ. Equations 17 (2001), 173-186. MR 1853530
Search
Partner of
