OSCILLATION THEOREMS FOR n-TH ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH FORCING TERMS AND DEVIATING ARGUMENTS DEPENDING ON THE UNKNOWN FUNCTION

TitleOSCILLATION THEOREMS FOR n-TH ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH FORCING TERMS AND DEVIATING ARGUMENTS DEPENDING ON THE UNKNOWN FUNCTION
Publication TypeJournal Article
Year of Publication2005
AuthorsMarkova, NT, Simeonov, PS
Volume9
Issue3
Start Page417
Pagination12
Date Published2005
ISSN1083-2564
AMS34K15
Abstract

In this paper differential equations of the type $${ x^{(n)} (t) + a(t)F(x(δ(t, x(t)))) + b(t)G(x(∆(t, x(t)))) = q(t) \ \ \ \ \ \ (E) }$$are considered, where ${n ≥ 2}$ and the deviating arguments ${δ}$ and ${∆}$ depend on the independent variable t as well as on the unknown function ${x}$. Sufficient conditions are found under which every solution ${x(t)}$ of equation ${(E)}$ is either oscillatory or such that ${ \lim_{t→+∞} x^{(i)} (t) = 0,\  i = 0, 1, . . . , n − 1.}$
URLhttp://www.acadsol.eu/en/articles/9/3/12.pdf
Refereed DesignationRefereed
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REFERENCES
[1] D.C. Angelova, D.D. Bainov, On the oscillation of solutions to second order functional differential
equations, Math. Rep. Toyama Univ., 5 (1982), 1-13.
[2] D.C. Angelova, D.D. Bainov, On oscillation of solutions of forced functional differential equations
of second order, Math. Nachr., 122 (1985), 289-300.
[3] F.V. Atkinson, On second-order differential inequalities, Proc. Roy. Soc. Edinburgh, 72 (1974), 109-127.
[4] D.D. Bainov, N.T. Markova, P.S. Simeonov, Asymptotic behaviour of the nonoscillatory solutions
of differential equations of second order with delay depending on the unknown function,
Journal of Computational and Applied Mathematics, 91 (1998), 87-96.
[5] D.D. Bainov, N.T. Markova, P.S. Simeonov, Oscillation of second order differential equations
with retarded argument depending on the unknown function, Communications in Applied
Analysis, 7 (2003), no. 4, 593-604.
[6] J.S. Bradley, Oscillation theorems for second order delay equations, Journal of Differential
Equations, 8 (1970), 397-403.
[7] R.S. Dahiya, Oscillation and asymptotic behavior of bounded solutions of functional differential
equations with delay, Jour. Math. Anal. Appl., 91 (1983), 276-286.
[8] R.S. Dahiya, O. Akinyele, Oscillation theorems of n-th order funtional differential equations
with forcing terms, Jour. Math. Anal. Appl., 109 (1985), no. 2, 325-332.
[9] J. Dˇzurina, Oscillation of second order delay differential equations, Archivum Mathematicum
(Brno), Tomus 33 (1997), 309-314.
[10] S.R. Grace, B.S. Lalli, Asymptotic and oscillatory behavior of n-th order forced functional
differential equations, Jour. Math. Anal. Appl., 140 (1989), 10-25.
[11] I.T. Kiguradze, On the oscillation of solutions of the equation d mu dtm + a(t)|u| n signu = 0, Mat.
Sb., 65 (1964), no. 2, 172-187, In Russian.
[12] G.S. Ladde, V. Lakshmikantham, B.G. Zhang, Oscillation Theory of Differential Equations
with Deviating Arguments, Pure and Applied Mathematics, Volume 110, Marcel Dekker, 1987.
[13] N.T. Markova, P.S. Simeonov, Asymptotic and oscillatory properties of the solutions of differential
equations with delays depending on the unknown function, In: Invited Lectures Delivered
at the VII-th Int. Colloquium on Differential Equations, August 18-23, 1996, Plovdiv, Bulgaria, II (1996), 71-78.
[14] N.T. Markova, P.S. Simeonov, On the asymptotic behaviour of the solutions of a class of
differential equations with delays depending on the unknown function, In: Invited Lectures
Delivered at the VII-th Int. Colloquium on Differential Equations, August 18-23, 1996, Plovdiv, Bulgaria, I (1996), 89-100.
[15] H. Onose, A comparison theorem and the forced oscillation, Bull. Austral. Math. Soc., 13 (1975), 13-19.
[16] Ch.G. Philos, Y.G. Sficas, Oscillation and asymptotic behavior of second and third order
retarded differential equations, Czechoslovak Mathematical Journal, 32 (1982), no. 2, 169-182.
[17] B.N. Shevelo, Oscillations of the Solutions of Differential Equation, Naukova Dumka Press,
Kiev, 1978, In Russian.
[18] P.S. Simeonov, Oscillation of the solution of second order retarded impulsive differential equations,
Communications in Applied Analysis, To Appear.