Oscillatory and asymptotic behavior of delay differential equations with a nonlinear damping term. (English) Zbl 0769.34049

Grace, S. R.

J. Math. Anal. Appl. 168, No. 2, 306-318 (1992).

The author studies the oscillatory and asymptotic behavior of the solutions of some delay differential equations with a nonlinear damping term by comparing with certain differential equations of the same or lower order whose behavior is known.

Reviewer: W.M.Oliva (São Paulo)

MSC:

34K99	Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20	Stability theory of functional-differential equations
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

oscillatory and asymptotic behavior; delay differential equations; nonlinear damping term

References:

[1]	Foster, K. E.; Grimmer, R. C., Nonoscillatory solutions of higher order delay equations, J. Math. Anal. Appl., 77, 150-164 (1980) · Zbl 0455.34053
[2]	Grace, S. R.; Lalli, B. S., An oscillation criterion for certain second order strongly sublinear differential equations, J. Math. Anal. Appl., 123, 584-586 (1987) · Zbl 0641.34031
[3]	Grace, S. R.; Lalli, B. S., Oscillation theorems for nonlinear second order functional differential equations with damping, Bull. Inst. Math. Acad. Sinica, 13, 183-192 (1985) · Zbl 0595.34069
[4]	Grace, S. R.; Lalli, B. S., Oscillation theorems for \(n\) th order delay differential equations, J. Math. Anal. Appl., 91, 352-366 (1983) · Zbl 0546.34055
[5]	Grace, S. R.; Lalli, B. S., Oscillation theorems for certain delay differential inequalities, J. Math. Anal. Appl., 106, 414-426 (1985) · Zbl 0605.34056
[6]	Grammatikopoulos, M. K.; Sficas, Y. G.; Staikos, V. A., Oscillatory property of strongly superlinear differential equations with deviating arguments, J. Math. Anal. Appl., 67, 171-187 (1979) · Zbl 0405.34062
[7]	Kartsatos, A. G., Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order, (Stability of Dynamical Systems: Theory and Applications. Stability of Dynamical Systems: Theory and Applications, Lecture Notes in Pure and Applied Math., No. 28 (1977), Dekker: Dekker New York), 17-72
[8]	Kartsatos, A. G., On \(n\) th order differential inequalities, J. Math. Anal. Appl., 52, 1-9 (1975) · Zbl 0327.34012
[9]	Kartsatos, A. G.; Onose, H., A comparison theorem for functional differential equations, Bull. Austral. Math. Soc., 14, 343-347 (1976) · Zbl 0318.34044
[10]	Kiguradze, I. T., On the oscillation of solutions of the equation \(d^mudt^m\) + a(t) ¦u¦\(^n\) sgn u = 0\), Mat. Sb., 65, 172-187 (1964), [Russian] · Zbl 0135.14302
[11]	Kitamura, Y.; Kusano, T., Oscillation of first order nonlinear differential equations with deviating arguments, (Proc. Amer. Math. Soc., 78 (1980)), 64-68 · Zbl 0433.34051
[12]	Ladas, G.; Stavroulakis, I. P., Oscillation caused by several retarded and advanced arguments, J. Differential Equations, 44, 134-152 (1982) · Zbl 0452.34058
[13]	Lalli, B. S.; Grace, S. R., Some oscillation criteria for delay differential equations of even order, J. Math. Anal. Appl., 119, 164-170 (1986) · Zbl 0608.34067
[14]	Lovelady, D. L., Oscillation and a class of linear delay differential equations, Trans. Amer. Math. Soc., 226, 345-364 (1977) · Zbl 0355.34059
[15]	Philos, Ch. G., A new criterion for the oscillatory and asymptotic behavior of delay differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Mat., 39, 61-64 (1981)
[16]	Philos, Ch. G., On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays, Arch. Math., 36, 168-178 (1981) · Zbl 0463.34050
[17]	Philos, Ch. G., Bounded oscillations generated by retardations for differential equations of arbitrary order, Utilitas Math., 15, 161-182 (1979) · Zbl 0399.34061
[18]	Sficas, Y. G.; Staikos, V. A., Oscillation of differential equations with deviating arguments, Funckcial. Ekvac., 19, 35-43 (1976) · Zbl 0351.34045
[19]	Staikos, V. A., Basic results on oscillation for differential equations with deviating arguments, Hiroshima Math. J., 10, 495-516 (1980) · Zbl 0453.34055
[20]	Wong, J. S.W; Burton, T. A., Some properties of \(u^″ + a(t) f(u) g(u^′) = 0\), Monatsh. Math., 69, 364-374 (1965) · Zbl 0142.06402