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Summary:
The neutral differential equation (1.1) $$ \frac{{\mathrm{d}}^n}{{\mathrm{d}} t^n} [x(t)+x(t-\tau)] + \sigma F(t,x(g(t))) = 0, $$ is considered under the following conditions: $n\ge 2$, $\tau >0$, $\sigma = \pm 1$, $F(t,u)$ is nonnegative on $[t_0, \infty) \times (0,\infty)$ and is nondecreasing in $u\in (0,\infty)$, and $\lim g(t) = \infty$ as $t\rightarrow \infty$. It is shown that equation (1.1) has a solution $x(t)$ such that (1.2) $$ \lim_{t\rightarrow \infty} \frac{x(t)}{t^k}\ \text{exists and is a positive finite value if and only if} \int^{\infty}_{t_0} t^{n-k-1} F(t,c[g(t)]^k){\mathrm{d}} t < \infty\text{ for some }c > 0. $$ Here, $k$ is an integer with $0\le k \le n-1$. To prove the existence of a solution $x(t)$ satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.
References:
[1] Q. Chuanxi and G. Ladas: Oscillations of higher order neutral differential equations with variable coefficients. Math. Nachr. 150 (1991), 15–24. DOI 10.1002/mana.19911500103 | MR 1109642
[2] J. R. Graef, P. W. Spikes and M. K. Grammatikopoulos: Asymptotic behavior of nonoscillatory solutions of neutral delay differential equations of arbitrary order. Nonlinear Anal. 21 (1993), 23–42. DOI 10.1016/0362-546X(93)90175-R | MR 1231526
[3] M. K. Grammatikopoulos, G. Ladas and A. Meimaridou: Oscillation and asymptotic behavior of higher order neutral equations with variable coefficients. Chinese Ann. Math. Ser. B, 9 (1988), 322–338. MR 0968469
[4] J. Jaroš, Y. Kitamura and T. Kusano: On a class of functional differential equations of neutral type, in “Recent Trends in Differential Equations”. (R. P. Agarwal, Ed.), World Scientific, 1992, pp. 317–333. MR 1180120
[5] J. Jaroš and T. Kusano: Oscillation theory of higher order linear functional differential equations of neutral type. Hiroshima Math. J. 18 (1988), 509–531. MR 0991245
[6] J. Jaroš and T. Kusano: Asymptotic behavior of nonoscillatory solutions of nonlinear functional differential equations of neutral type. Funkcial. Ekvac. 32 (1989), 251–263. MR 1019433
[7] J. Jaroš and T. Kusano: On oscillation of linear neutral differential equations of higher order. Hiroshima Math. J. 20 (1990), 407–419. MR 1063374
[8] J. Jaroš and T. Kusano: Existence of oscillatory solutions for functional differential equations of neutral type. Acta Math. Univ. Comenian. 60 (1991), 185–194. MR 1155243
[9] Y. Kitamura and T. Kusano: Existence theorems for a neutral functional differential equation whose leading part contains a difference operator of higher degree. Hiroshima Math. J. 25 (1995), 53–82. MR 1322602
[10] Y. Kitamura, T. Kusano and B. S. Lalli: Existence theorems for nonlinear functional differential equations of neutral type. Proc. Georgian Acad. Sci. Math. 2 (1995), 79–92. MR 1310502
[11] G. Ladas and C. Qian: Linearized oscillations for even-order neutral differential equations. J. Math. Anal. Appl. 159 (1991), 237–250. DOI 10.1016/0022-247X(91)90233-P | MR 1119433
[12] G. Ladas and Y. G. Sficas: Oscillations of higher-order neutral equations. J. Austral. Math. Soc. Ser. B, 27 (1986), 502–511. DOI 10.1017/S0334270000005105 | MR 0836222
[13] W. Lu: The asymptotic and oscillatory behavior of the solutions of higher order neutral equations. J. Math. Anal. Appl. 148 (1990), 378–389. DOI 10.1016/0022-247X(90)90008-4 | MR 1052351 | Zbl 0704.34081
[14] P. Marušiak: On unbounded nonoscillatory solutions of systems of neutral differential equations. Czechoslovak Math. J. 42 (1992), 117–128. MR 1152175
[15] Y. Naito: Nonoscillatory solutions of neutral differential equations. Hiroshima Math. J. 20 (1990), 231–258. MR 1063362 | Zbl 0721.34091
[16] Y. Naito: Existence and asymptotic behavior of positive solutions of neutral differential equations. J. Math. Anal. Appl. 188 (1994), 227–244. DOI 10.1006/jmaa.1994.1424 | MR 1301729 | Zbl 0818.34036
[17] A. I. Zahariev and D. D. Bainov: On some oscillation criteria for a class of neutral type functional differential equations. J. Austral. Math. Soc. Ser. B, 28 (1986), 229–239. DOI 10.1017/S0334270000005324 | MR 0862572
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