zbMATH — the first resource for mathematics

Necessary and sufficient conditions for the nonoscillation of a first order neutral equation with several delays. (English) Zbl 1057.34081
Summary: Necessary and sufficient conditions are obtained so that every solution of the neutral delay differential equation (NDDE) \[ \biggl (y(t)-\sum \limits _{j = 1}^k p_j y({t - \tau _j})\biggr )' + Q(t)G\bigl (y({t-\sigma })\bigr ) = f(t) \] is oscillatory or tends to zero as \(t\to \infty \) for different ranges of \(\sum_{j = 1}^k p_j \). This paper improves and generalizes two recent works [P. Das and N. Misra, J. Math. Anal. Appl. 205, 78–87 (1997; Zbl 0874.34058)] and [N. Parhi and R. N. Rath, Bull. Inst. Math. Acad. Sin. 28, 59–70 (2000; Zbl 0961.34059)]. The results of this paper hold for linear, sublinear and superlinear equations. Also, they are valid for homogeneous equations. The results can be extended to NDDE with variable coefficients without assumption of any further condition on the coefficient functions.

34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
34K25 Asymptotic theory of functional-differential equations
Full Text: EuDML
[1] DAS P. : Oscillation and asymptotic behaviour of solutions for second order neutral delay differential equation. J. Indian Math. Soc. (N.S.) 60 (1994), 159-170. · Zbl 0886.34066
[2] DAS P.-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. 204 (1997), 78-87. · Zbl 0874.34058
[3] DETANG, ZHOU: Oscillation of neutral functional differential equations. Acta Math. Scientia 12 (1992), 42-50. · Zbl 0765.34055
[4] ERBE L. H.-KAIKONG, QING: Oscillation and nonoscillation properties of neutral differential equations. Canad. J. Math. 46 (1994), 284-297. · Zbl 0797.34072
[5] GYORI I.-LADAS G.: Oscillation Theory of Delay-Differential Equations with Applications. Clarendon, Oxford, 1991.
[6] KULENOVIC M. R. S.-LADAS G.-MEIMARIDOU A.: Necessary and sujficient condition for oscillation of neutral differential equations. J. Austral. Math. Soc. Ser. B 28 (1987), 362-375. · Zbl 0616.34064
[7] LADAS G.-SFICAS Y. G.-STAVROULAKIS I. P.: Necessary and sufficient condition for oscillation of higher order delay differential equations. Trans. Amer. Math. Soc. 285 (1984), 81-90. · Zbl 0528.34070
[8] LIU X. Z.-YU J. S.-ZHANG B. G.: Oscillation and nonoscillation for a class of neutral differential equations. Differential Equations Dynam. Systems 1 (1993), 197-204. · Zbl 0873.34056
[9] LU, WUDU: Nonoscillation and oscillation of first order neutral equations with variable coefficients. J. Math. Anal. Appl. 181 (1994), 803-815. · Zbl 0809.34083
[10] OLAH R.: Oscillation of differential equation of neutral type. Hiroshima Math. J. 25 (1995), 1-10. · Zbl 0831.34076
[11] PARHI N.-MOHANTY P. K.: Maintenance of oscillation of neutral differential equations under the effect of a forcing term. Indian J. Pure Appl. Math. 26 (1995), 909-919. · Zbl 0838.34082
[12] PARHI N.-MOHANTY P. K.: Oscillatory behaviour of solutions of forced neutral differential equations. Ann. Polon. Math. 65 (1996), 1-10. · Zbl 0874.34063
[13] PARHI N.-RATH R. N.: On oscillation criteria for a forced neutral differential equation. Bull. Inst. Math. Acad. Sinica 28 (2000), 59-70. · Zbl 0961.34059
[14] 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. · Zbl 0982.34057
[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. Ill (2001), 337-350. · Zbl 0995.34058
[16] PARHI N.-RATH R. N.: On oscillation of forced non-linear neutral differential equations of higher order. Czechoslovak Math. J. · Zbl 1080.34522
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.