×

zbMATH — the first resource for mathematics

On oscillation properties of delay differential equations with positive and negative coefficients. (English) Zbl 1056.34063
J. Math. Anal. Appl. 274, No. 1, 81-101 (2002); corrigendum ibid. 351, No. 2, 819-820 (2009).
Summary: For a scalar delay differential equation
\[ x^{\prime}(t) +a(t)x(h(t))-b(t)x(g(t))=0,\quad t\geq t_{0},\tag{E} \]
where \(a(t)\geq 0\), \(b(t)\geq 0\), \(h(t)\leq t\), \(g(t)\leq t\), a connection between the following properties is established: nonoscillation of the differential equation and the corresponding differential inequalities, positiveness of the fundamental function and existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality. A comparison theorem and explicit nonoscillation and oscillation results are presented.

MSC:
34K11 Oscillation theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ladas, G.; Sficas, Y.G., Oscillation of delay differential equations with positive and negative coefficients, (), 232-240
[2] Chuanxi, Q.; Ladas, G., Oscillation in differential equations with positive and negative coefficients, Canad. math. bull., 33, 442-451, (1990) · Zbl 0723.34068
[3] Kreith, K.; Ladas, G., Allowable delays for positive diffusion processes, Hiroshima math. J., 15, 437-443, (1985) · Zbl 0591.35025
[4] Farell, K.; Grove, E.A.; Ladas, G., Neutral delay differential equations with positive and negative coefficients, Appl. anal., 27, 181-197, (1988) · Zbl 0618.34063
[5] Yu, J.S., Neutral differential equations with positive and negative coefficients, Acta math. sinica, 34, 517-523, (1991) · Zbl 0738.34040
[6] Yu, J.S.; Yan, J., Oscillation in first order differential equations with “integral smaller” coefficients, J. math. anal. appl., 187, 361-370, (1994) · Zbl 0814.34062
[7] Li, W.; Quan, H.; Wu, J., Oscillation of first order differential equations with variable coefficients, Comm. appl. anal., 3, 1-13, (1999) · Zbl 0924.34069
[8] Zhang, X.; Yan, J., Oscillation criteria for first order neutral differential equations with positive and negative coefficients, J. math. anal. appl., 253, 204-214, (2001) · Zbl 0980.34066
[9] Li, W.; Quan, H.S., Oscillation of higher order neutral differential equations with positive and negative coefficients, Ann. differential equations, 11, 70-76, (1995) · Zbl 0918.34068
[10] Li, W.; Jan, J., Oscillation of first order neutral differential equations with positive and negative coefficients, Collect. math., 50, 199-209, (1999) · Zbl 0935.34055
[11] Cheng, S.S.; Guan, X.-P.; Yang, J., Positive solutions of a nonlinear equation with positive and negative coefficients, Acta math. hungar., 86, 169-192, (2000) · Zbl 0980.34002
[12] Erbe, L.N.; Kong, Q.; Zhang, B.G., Oscillation theory for functional differential equations, (1995), Dekker New York
[13] Domshlak, Yu.I.; Aliev, A.I., On oscillatory properties of the first order differential equations with one or two arguments, Hiroshima math. J., 18, 31-46, (1988) · Zbl 0714.34105
[14] Berezansky, L.; Braverman, E., On non-oscillation of a scalar delay differential equation, Dynamic systems appl., 6, 567-580, (1997) · Zbl 0890.34059
[15] Györi, I.; Ladas, G., Oscillation theory of delay differential equations, (1991), Clarendon Oxford · Zbl 0780.34048
[16] Domshlak, Y., Sturmian comparison method in investigation of the behavior of solutions for differential-operator equations, (1986), Elm, Baku, in Russian
[17] Domshlak, Y., Properties of delay differential equations with oscillating coefficients, Funct. differential equations, 2, 59-68, (1994) · Zbl 0860.34036
[18] Ladas, G.; Sficas, Y.G.; Stavroulakis, I.P., Functional differential inequalities and equations with oscillating coefficients, (), 277-284 · Zbl 0531.34051
[19] Yu, J.S.; Wang, Z.C.; Zhang, B.G.; Qian, X.Z., Oscillation of differential equations with deviating arguments, Panamer. math. J., 2, 59-72, (1992) · Zbl 0845.34082
[20] Ladde, G.S.; Lakshmikantham, V.; Zhang, B.G., Oscillation theory of differential equations with deviating arguments, (1987), Dekker New York · Zbl 0832.34071
[21] Berezansky, L.; Domshlak, Y., Differential equations with several deviating arguments: Sturmian comparison method in oscillation theory, I, Electron. J. differential equations, 40, 1-19, (2001) · Zbl 0982.34056
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.