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2002 | 22 | 2 | 185-212
Tytuł artykułu

Oscillation of nonlinear neutral delay differential equations of second order

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Oscillation criteria, extended Kamenev and Philos-type oscillation theorems for the nonlinear second order neutral delay differential equation with and without the forced term are given. These results extend and improve the well known results of Grammatikopoulos et. al., Graef et. al., Tanaka for the nonlinear neutral case and the recent results of Dzurina and Mihalikova for the neutral linear case. Some examples are considered to illustrate our main results.
Twórcy
  • Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
  • Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt
Bibliografia
  • [1] R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publisheres, Drdrechet 2000.
  • [2] R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation Theory for Second Order Dynamic Equations, to appear.
  • [3] D.D. Bainov and D.P. Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, New York 1991.
  • [4] E. Boe and H.C. Chang, Dynamics of delayed systems under feedback control, Chem. Engng. Sci. 44 (1989), 1281-1294.
  • [5] S.J. Bilchev, M.K. Grammatikopoulos and I.P. Stavroulakis, Oscillation of second order neutral differential equations with deviating arguments, Contemporary Math. 129 (1992), 1-21.
  • [6] J. Dzurina and B. Mihalikova, Oscillation criteria for second order neutral differential equations, Math. Boh. 125 (2000), 145-153.
  • [7] L.H. Erbe, Q. King and B.Z. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York 1995.
  • [8] J.R. Graef, M.K. Grammatikopoulos and P.W. Spikes, Asymptotic properties of solutions of nonlinear neutral delay differential equations of the second order, Radovi Mat. 4 (1988), 133-149.
  • [9] J.R. Graef, M.K. Grammatikopoulos and P.W. Spikes, On the Asymptotic behavior of solutions of the second order nonlinear neutral delay differential equations, J. Math. Anal. Appl. 156 (1991), 23-39.
  • [10] J.R. Graef, M.K. Grammatikopoulos and P.W. Spikes, Some results on the asymptotic behavior of the solutions of a second order nonlinear neutral delay differential equations, Contemporary Mathematics 129 (1992), 105-114.
  • [11] M.K. Grammatikopoulos, G. Ladas and A. Meimaridou, Oscillation of second order neutral delay differential equations, Radovi Mat. 1 (1985), 267-274.
  • [12] M.K. Grammatikopoulos, G. Ladas and A. Meimaridou, Oscillation and asymptotic behavior of second order neutral differential equations, Annali di Matematica Pura ed Applicata CXL, VIII (1987), 20-40.
  • [13] M.K. Grammatikopoulos and P. Marusiak, Oscillatory properties of second order nonlinear neutral differential inequalities with oscillating coefficients, Arch. Math. 31 (1995), 29-36.
  • [14] I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations With Applications, Clarendon Press, Oxford 1991.
  • [15] J. K.Hale, Theory of Functional Differential Equations, Springer-Verlag, New York 1977.
  • [16] P. Hartman, On nonoscillatory linear differential equations of second order, Amer. J. Math. 74 (1952), 389-400.
  • [17] I.V. Kamenev, Integral criterion for oscillation of linear differential equations of second order, Math. Zemetki (1978), 249-251.
  • [18] I.T. Kiguradze, On the oscillation of solutions of the equation [(d^{m}u)/(dt^{m})] + a(t)|u|^{n} sign u = 0, Math. Sb. 65 (1964), 172-187, (in Russian).
  • [19] G.S. Ladde, V. Lakshmikantham and B.Z. Zhang, Oscillation theory of differential equations with deviating arguments, Marcel Dekker, New York 1987.
  • [20] W. Leighton, The detection of the oscillation of solutions of a second order linear differential equation, Duke J. Math. 17 (1950), 57-62.
  • [21] W.T. Li, Classification and existence of nonoscillatory solutions of second order nonlinear neutral differential equations, Ann. Polon Math. LXV. 3 (1997), 283-302.
  • [22] Ch.G. Philos, Oscillation theorems for linear differential equation of second order, Arch. Math. 53 (1989), 483-492.
  • [23] E.P. Popove, Automatic Regulation and Control, Nauka, Moscow 1966, (in Russian).
  • [24] S. Tanaka, Oscillation properties of solutions of second order neutral differential equations with deviating arguments, Analysis 17 (1997), 99-111.
  • [25] C.C. Travis, Oscillation theorems for second order differential equations with functional arguments, Proc. Amer. Math. Soc. 31 (1972), 199-202.
  • [26] P. Waltman, A note on an oscillation criterion for an equation with function argument, Canad. Math. Bull. 11 (1968), 593-595.
  • [27] A. Wintner, A cireterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115-117.
  • [28] J. Yan, Oscillation theorems for second order linear differential equations with damping, Proc. Amer. Math. Soc. 98 (1986), 276-282, 482-492.
  • [29] N. Yoshida and K. Takeuchi, Oscillation properties of solutions of second order nonlinear differential equations with delay, Math. J. Toyama Univ. 17 (1994), 167-173.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1037
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