Advertisement

Positive Non-monotone Solutions of Second-Order Delay Differential Equations

  • George E. Chatzarakis
  • Lana Horvat-Dmitrović
  • Mervan PašićEmail author
Article
  • 78 Downloads

Abstract

In this paper, we study the non-monotonic behavior of solutions of the second-order delay differential equations. We establish the criteria for positive solutions to be non-monotonic on a bounded interval. The results are the first to be reported on the non-monotonic behavior of solutions of functional differential equations. In the final section, two examples are given which illustrate the significance of our results.

Keywords

Delay differential equations Non-monotonic behavior Second-order Nonlinear term Dual equation 

Mathematics Subject Classification

34A30 34B30 34C10 34C11 

References

  1. 1.
    Akca, H., Chatzarakis, G.E., Stavroulakis, I.P.: An oscillation criterion for delay differential equations with several non-monotone arguments. Appl. Math. Lett. 59, 101–108 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berezansky, L., Domoshnitsky, A., Gitman, M., Stolbov, V.: Exponential stability of a second order delay differential equation without damping term. Appl. Math. Comput. 258, 483–488 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Braverman, E., Chatzarakis, G.E., Stavroulakis, I.P.: Iterative oscillation tests for differential equations with several non-monotone arguments. Adv. Differ. Equ. 2016, 1–18 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York (1995)zbMATHGoogle Scholar
  5. 5.
    Feza Guvenilir, A.: Interval oscillation of second-order functional differential equations with oscillatory potentials. Nonlinear Anal. 71, e2849–e2854 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kwong, M.K.: Oscillation of first-order delay equations. J. Math. Anal. Appl. 156, 274–286 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Liu, Y., Zhang, J., Yan, J.: Existence of oscillatory solutions of second order delay differential equations. J. Comput. Appl. Math. 277, 17–22 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ladde, G.S., Lakshmikanthan, V., Vatsala, A.S.: Monotone Iteration Technique for Nonlinear Differential Equations. Pitman, Boston (1985)Google Scholar
  9. 9.
    Mohapatra, R.N., Vajravelu, K., Yin, Y.: Generalized quasilinearization method and rapid convergence for first order initial value problems. J. Math. Anal. Appl. 207, 206–219 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Opluštil, Z., Šremr, J.: Myshkis type oscillation criteria for second-order linear delay differential equations. Monatsh. Math. 178, 143–161 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pašić, M.: Sign-changing first derivative of positive solutions of forced second-order nonlinear differential equations. Appl. Math. Lett. 40, 40–44 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pašić, M.: Strong non-monotonic behavior of particle density of solitary waves of nonlinear Schrödinger equation in Bose–Einstein condensates. Commun. Nonlinear Sci. Numer. Simul. 29, 161–169 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pašić, M., Tanaka, S.: Non-monotone positive solutions of second-order linear differential equations: existence, nonexistence and criteria. Electron. J. Qual. Theory Differ. Equ. 93, 1–25 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Potter, R.L.: On self-adjoint differential equations of second order. Pac. J. Math. 3, 467–491 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sun, Y.G.: A note on Nasr’s and Wong’s papers. J. Math. Anal. Appl. 286, 363–367 (2003)Google Scholar
  16. 16.
    Zhang, G.B.: Non-monotone traveling waves and entire solutions for a delayed nonlocal dispersal equation. Appl. Anal. (2016). doi: 10.1080/00036811.2016.1197913
  17. 17.
    Walter, V.: Ordinary Differential Equations, Graduate Texts in Mathematics. Readings in Mathematics, vol. 182. Springer-Verlag, New York (1998)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of Electrical and Electronic Engineering EducatorsSchool of Pedagogical and Technological Education (ASPETE)AthensGreece
  2. 2.Faculty of Electrical Engineering and Computing Department of Applied MathematicsUniversity of ZagrebZagrebCroatia

Personalised recommendations