Previous |  Up |  Next


oscillation; positive solutions; neutral equation
The paper is concerned with oscillation properties of $n$-th order neutral differential equations of the form \[ [x(t)+cx(\tau (t))]^{(n)}+q(t)f\bigl (x(\sigma (t))\bigr )=0,\quad t\ge t_0>0, \] where $c$ is a real number with $|c|\ne 1$, $q\in C([t_0,\infty ),\mathbb R)$, $f\in C(\mathbb R,\mathbb R)$, $\tau ,\sigma \in C([t_0,\infty ),\mathbb R_+)$ with $\tau (t)<t$ and $\lim _{t\rightarrow \infty }\tau (t)=\lim _{t\rightarrow \infty }\sigma (t)=\infty $. Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which $c$ is a function of $t$ and a certain type of a forcing term is present.
[1] D. D.  Bainov and D. P.  Mishev: Oscillation Theory for Neutral Differential Equations with Delay. IOP Publishing Ltd., Bristol, UK, 1992. MR 1147908
[2] M. P.  Chen, J. S.  Yu and B. G. Zhang: The existence of positive solutions for even order neutral delay differential equations. Panamer. Math. J. 3 (1993), 61–77. MR 1234191
[3] Q. Chuanxi and G.  Ladas: Oscillation of neutral differential equations with variable coefficients. Appl. Anal. 32 (1989), 215–228. DOI 10.1080/00036818908839850 | MR 1030096
[4] R. S.  Dahiya and O.  Akinyele: Oscillation theorems of $n$th order functional differential equations with forcing terms. J.  Math. Anal. Appl. 109 (1985), 325–332. DOI 10.1016/0022-247X(85)90153-2 | MR 0802898
[5 R. S.  Dahiya and A.  Zafer] Asymptotic behavior and oscillation in higher order nonlinear differential equations with retarded arguments. Acta Math. Hungar. 76(3) (1997), 257–266. DOI 10.1023/A:1006577321359 | MR 1459234
[6] E. A. Grove, M. R. S.  Kulenovic and G.  Ladas: Sufficient conditions for oscillation and nonoscillation of neutral equations. J.  Differential Equations 68 (1987), 673–682. MR 0891334
[7] I. T.  Kiguradze: On the oscillation of solutions of equation $^m u/t^m+a(t)u^m \mathop {\mathrm {sgn}} u=0$. Mat. Sb. 65 (1964), 172–187. MR 0173060
[8] V.  Lakshmikantham, L.  Wen and B. G.  Zhang: Theory of Differential Equations with Unbounded Delay. MIA Kluwer Academic Publishers, London, 1994. MR 1319339
[9] A.  Zafer and R. S.  Dahiya: Oscillation of bounded solutions of neutral differential equations. Appl. Math. Lett. 6 (1993), 43–46. DOI 10.1016/0893-9659(93)90010-K | MR 1347773
[10] A.  Zafer: Oscillation criteria for even order neutral differential equations. Appl. Math. Lett. 11 (1998), 21–25. DOI 10.1016/S0893-9659(98)00028-7 | MR 1628987 | Zbl 0933.34075
[11] B. G.  Zhang and J. S.  Yu: On the existence of asymptotically decaying positive solutions of second order neutral differential equations. J.  Math. Anal. Appl. 166 (1992), 1–11. DOI 10.1016/0022-247X(92)90322-5 | MR 1159633
Partner of
EuDML logo