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neutral differential equations; nonoscillatory solutions
Consider the forced higher-order nonlinear neutral functional differential equation \[ \frac{{\mathrm d}^n}{{\mathrm d}t^n}[x(t)+C(t) x(t-\tau )]+\sum ^m_{i=1} Q_i(t)f_i(x(t-\sigma _i))=g(t), \quad t\ge t_0, \] where $n, m \ge 1$ are integers, $\tau , \sigma _i\in {\mathbb{R}}^+ =[0, \infty )$, $C, Q_i, g\in C([t_0, \infty ), {\mathbb{R}})$, $f_i\in C(\mathbb{R}, \mathbb{R})$, $(i=1,2,\dots ,m)$. Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general $Q_i(t)$ $(i=1,2,\dots ,m)$ and $g(t)$ which means that we allow oscillatory $Q_i(t)$ $(i=1,2,\dots ,m)$ and $g(t)$. Our results improve essentially some known results in the references.
[1] R. P. Agarwal, S. R.  Grace and D.  O’Regan: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic Publishers, , 2000. MR 1774732
[2] R. P.  Agarwal, S. R.  Grace and D.  O’Regan: Oscillation criteria for certain $n$th  order differential equations with deviating arguments. J.  Math. Anal. Appl. 262 (2001), 601–622. DOI 10.1006/jmaa.2001.7571 | MR 1859327
[3] R. P.  Agarwal and S. R.  Grace: The oscillation of higher-order differential equations with deviating arguments. Computers Math. Applied 38 (1999), 185–190. DOI 10.1016/S0898-1221(99)00193-5 | MR 1703416
[4] M. P.  Chen, J. S.  Yu and Z. C.  Wang: Nonoscillatory solutions of neutral delay differential equations. Bull. Austral. Math. Soc. 48 (1993), 475–483. DOI 10.1017/S0004972700015938 | MR 1248051
[5] L. H.  Erbe, Q. K.  Kong and B. G.  Zhang: Oscillation Theory for Functional Differential equations. Marcel Dekker, New York, 1995. MR 1309905
[6] I.  Gyori and G.  Ladas: Oscillation Theory of Delay Differential Equations with Applications. Oxford Univ. Press, London, 1991. MR 1168471
[7] J. R.  Graef, B.  Yang and B. G.  Zhang: Existence of nonoscillatory and oscillatory solutions of neutral differential equations with positive and negative coefficients. Math. Bohemica 124 (1999), 87–102. MR 1687484
[8] M. R. S.  Kulenovic and S.  Hadziomerspahic: Existence of nonoscillatory solution of second order linear neutral delay equation. J.  Math. Anal. Appl. 228 (1998), 436–448. DOI 10.1006/jmaa.1997.6156 | MR 1663585
[9] H. A. El-Morshedy and K.  Gopalsamy: Nonoscillation, oscillation and convergence of a class of neutral equations. Nonlinear Anal. 40 (2000), 173–183. DOI 10.1016/S0362-546X(00)85010-5 | MR 1768408
[10] C. H.  Ou and J. S. W.  Wong: Forced oscillation of $n$th-order functional differential equations. J.  Math. Anal. Appl. 262 (2001), 722–732. DOI 10.1006/jmaa.2001.7614 | MR 1859335
[11] S.  Tanaka: Existence of positive solutions for a class of higher order neutral differential equations. Czechoslovak Math.  J. 51 (2001), 573–583. DOI 10.1023/A:1013736122991 | MR 1851548
[12] N.  Parhi and R. N.  Rath: Oscillation criteria for forced first order neutral differential equations with variable coefficients. J.  Math. Anal. Appl. 256 (2001), 525–241. DOI 10.1006/jmaa.2000.7315 | MR 1821755
[13] B. G.  Zhang and B.  Yang: New approach of studying the oscillation of neutral differential equations. Funkcial Ekvac. 41 (1998), 79–89. MR 1627357
[14] Yong Zhou: Oscillation of neutral functional differential equations. Acta Math. Hungar. 86 (2000), 205–212. DOI 10.1023/A:1006716411558 | MR 1756173
[15] Yong Zhou and B. G.  Zhang: Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients. Appl. Math. Lett. 15 (2002), 867–874. DOI 10.1016/S0893-9659(02)00055-1 | MR 1920988
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