Opuscula Mathematica
Opuscula Math. 40, no. 5 (2020), 523-536

Oscillatory criteria via linearization of half-linear second order delay differential equations

Blanka Baculíková
Jozef Džurina
Abstract. In the paper, we study oscillation of the half-linear second order delay differential equations of the form \[\left(r(t)(y'(t))^{\alpha}\right)'+p(t)y^{\alpha}(\tau(t))=0.\] We introduce new monotonic properties of its nonoscillatory solutions and use them for linearization of considered equation which leads to new oscillatory criteria. The presented results essentially improve existing ones.
Keywords: second order differential equations, delay, monotonic properties, linearization, oscillation.
Mathematics Subject Classification: 34K11, 34C10.
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References ▼
  1. [1] R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations, Kluver Academic Publishers, Dotrecht, 2002.
  2. [2] B. Baculikova, Oscillation of second-order nonlinear noncanonical differential equations with deviating argument, Appl. Math. Lett. 91 (2019), 68-75.
  3. [3] B. Baculíková, Oscillatory behavior of the second order noncanonical differential equation, Electron. J. Qual. Theory Differ. Equ. 89 (2019), 1-17.
  4. [4] B. Baculíková, Oscillation for second-order differential equations with delay, Electron. J. Differential Equations 2018 (2018) 96, 1-9.
  5. [5] O. Došly, P. Řehák, Half-linear Differential Equations, vol. 202, North-Holland Mathematics Studies, 2005.
  6. [6] J. Džurina, Comparison theorems for nonlinear ODE's, Math. Slovaca 42 (1992), 299-315.
  7. [7] L.H. Erbe, Q. Kong, B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1994.
  8. [8] S.R. Grace, R.P. Agarwal, R. Pavani, E. Thandapani, On the oscillation of certain third order nonlinear functional differential equations, Appl. Math. Comput. 202 (2008), 102-112.
  9. [9] I. Györi, G. Ladas, Oscillation theory of delay differential equations, The Clarendon Press, Oxford, 1991.
  10. [10] I.T. Kiguradze, T.A. Chaturia, Asymptotic Properties of Solutions of Nonatunomous Ordinary Differential Equations, Kluwer Acad. Publ., Dordrecht 1993.
  11. [11] R. Koplatadze, G. Kvinkadze, I.P. Stavroulakis, Properties \(A\) and \(B\) of \(n\)-th order linear differential equations with deviating argument, Georgian Math. J. 6 (1999), 553-566.
  12. [12] T. Kusano, M. Naito, Comparison theorems for functional differential equations with deviating arguments, J. Math. Soc. Japan 3 (1981), 509-533.
  13. [13] G.S. Ladde, V. Lakshmikantham, B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987.
About the authors ▼
Blanka Baculíková (corresponding author)
https://orcid.org/0000-0002-8689-6308
Technical University of Košice, Faculty of Electrical Engineering and Informatics, Department of Mathematics, Letná 9, 042 00 Košice, Slovakia
Jozef Džurina
https://orcid.org/0000-0002-6872-5695
Technical University of Košice, Faculty of Electrical Engineering and Informatics, Department of Mathematics, Letná 9, 042 00 Košice, Slovakia
About the article ▼
Communicated by Josef Diblík.
Received: 2020-06-11.
Revised: 2020-09-11.
Accepted: 2020-09-12.
Published online: 2020-10-10.
Cite this article as ▼
Blanka Baculíková, Jozef Džurina, Oscillatory criteria via linearization of half-linear second order delay differential equations, Opuscula Math. 40, no. 5 (2020), 523-536, https://doi.org/10.7494/OpMath.2020.40.5.523.
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