Oscillation and nonoscillation in first order neutral differential equations. (English) Zbl 0725.34088
The neutral differential equation \(\dot x(t)+C\dot x(t-\tau)+Px(t- \sigma)=0\) is considered. Sufficient conditions for oscillation and nonoscillation of this equation are given.
Reviewer: R.G.Koplatadze (Tbilisi)
MSC:
| 34K40 | Neutral functional-differential equations |
| 34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
References:
| [1] | Arino, O.; Györi, I.; Jawahari, A., Oscillation criteria in delay equations, J. Differential Equations, 53, 115-123 (1984) · Zbl 0547.34060 |
| [2] | Brumley, W. E., On the asymptotic behaviour of solutions of differential-difference equations of neutral type, J. Differential Equations, 7, 175-188 (1970) · Zbl 0215.15405 |
| [3] | Fukagai, N.; Kusano, T., Oscillation theory of first order functional differential equations with deviating arguments, Ann. Mat., 95-117 (1983) · Zbl 0552.34062 |
| [4] | Grammatikopoulos, M. K.; Grove, E. A.; Ladas, G., Oscillations of first order neutral delay differential equations, J. Math. Anal. Appl., 120, 510-520 (1986) · Zbl 0566.34056 |
| [5] | Grammatikopoulos, M. K.; Grove, E. A.; Ladas, G., Oscillation and Asymptotic Behaviour of First Order Neutral Differential Equations with Deviating Arguments, URI. T.R. No. 86 (1985) · Zbl 0631.34073 |
| [6] | M. K. Grammatikopoulos, G. Ladas, and Y. G. Sficas; M. K. Grammatikopoulos, G. Ladas, and Y. G. Sficas · Zbl 0617.34067 |
| [7] | Grammatikopoulos, M. K.; Sficas, Y. G.; Stavroulakis, I. P., Necessary and sufficient conditions for oscillations of neutral equations with several coefficients, J. Differential Equations, 76, 294-311 (1988) · Zbl 0669.34069 |
| [8] | Gromova, P. S., Stability of solutions of nonlinear equations of the neutral type in the asymptotically critical case, Math. Notes, 1, 472-479 (1967) · Zbl 0162.13203 |
| [9] | Gromova, P. S.; Zverkin, A. M., On trigonometric series whose sums are continuous unbounded functions on the real axis—solutions of equations with retarded arguments, Differential Equations, 4, 1774-1784 (1968) · Zbl 0212.43602 |
| [10] | Hunt, B. R.; Yorke, J. A., When all solutions of \(x\)′ = −∑\(q_ix (t\) −\(τ_i (t))\) oscillate, J. Differential Equations, 53, 139-145 (1984) · Zbl 0571.34057 |
| [11] | Jiono, R., On the oscillation of neutral differential difference equations with several delays, Sci. Sinica, 5, 467-477 (1986) |
| [12] | Jiong, R., Types and criteria of nonoscillatory solutions for second order linear neutral differential difference equations, Chinese Ann. Math. Ser. A, 8, 114-124 (1987) · Zbl 0632.34071 |
| [13] | Koplatadze, R. G.; Canturija, T. A., On oscillatory and monotone solutions of first order differential equations with deviating arguments, Differentsial’nye Uravneniya, 18, 1463-1465 (1982) · Zbl 0496.34044 |
| [14] | Kulenovic, M. R.S; Ladas, G.; Meimaridou, A., Necessary and sufficient condition for oscillations of neutral differential equations, J. Austral. Math. Soc. Ser. B, 28, 362-375 (1987) · Zbl 0616.34064 |
| [15] | Kusano, T., On even order functional differential equations with advanced and retarded arguments, J. Differential Equations, 45, 75-84 (1982) · Zbl 0512.34059 |
| [16] | Ladas, G., Sharp conditions for oscillations caused by delays, Appl. Anal., 9, 93-98 (1979) · Zbl 0407.34055 |
| [17] | Ladas, G.; Sficas, Y. G., Oscillations of neutral delay differential equations, Canad. Math. Bull., 29, 438-445 (1986) · Zbl 0566.34054 |
| [18] | Ladas, G.; Sficas, Y. G., Oscillations of higher order neutral equations, J. Austral. Math. Soc. Ser. B, 27, 502-511 (1986) · Zbl 0566.34055 |
| [19] | V. Lakshmikantham, G. S. Ladde, and B. G. Zhang; V. Lakshmikantham, G. S. Ladde, and B. G. Zhang · Zbl 0832.34071 |
| [20] | Nashed, M.; Wong, J. S.W, Some variants of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations, J. Math. Mech., 18, 767-777 (1969) · Zbl 0181.42301 |
| [21] | Onose, H., Oscillatory properties of first order differential inequalities with deviating arguments, Funkcial. Ekvac., 26, 189-195 (1983) · Zbl 0525.34051 |
| [22] | Y. G. Sficas and I. P. StavroulakisJ. Math. Anal. Appl.; Y. G. Sficas and I. P. StavroulakisJ. Math. Anal. Appl. |
| [23] | Stavroulakis, I. P., Nonlinear delay differential inequalities, Nonlinear Anal. Theory Meth. Appl., 6, 389-396 (1982) · Zbl 0488.34062 |
| [24] | Zahariev, A. I.; Bainov, D. D., Oscillating properties of the solutions of a class of neutral type functional differential equations, Bull. Austral. Math. Soc., 22, 365-372 (1980) · Zbl 0465.34042 |
| [25] | B. G. Zhang; B. G. Zhang · Zbl 0683.34037 |
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