Journal of Mathematical Sciences

, Volume 197, Issue 1, pp 45–65 | Cite as

Oscillation Criteria for Higher order Nonlinear Functional Differential Equations with Advanced Argument

  • R. Koplatadze
We consider a differential equation
$$ {u^{(n) }}(t)+p(t)\left| {u\left( {\sigma (t)} \right)\left| {^{{\mu (t)}}} \right.\mathrm{sign}\,u\left( {\sigma (t)} \right)=0,} \right. $$
where pL loc (R +; R +), μ ∈ C(R +; (0, +∞)), σ ∈ C(R +; R +), and σ(t) ≥ t for tR +. We say that the equation is almost linear if the condition lim t→+∞ μ(t) = 1 is satisfied. At the same time, if lim sup t→+∞ μ(t) ≠ 1 or lim inf t→+∞ μ(t) ≠ 1, then Eq. is called an essentially nonlinear differential equation. The oscillatory properties of almost linear differential equations have been extensively studied. In the paper, new sufficient (necessary and sufficient) conditions are established for a general class of essentially nonlinear functional differential equations to have Property A.


Differential Equation Satisfying Condition Nonlinear Differential Equation Linear Differential Equation Functional Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Graef, R. Koplatadze, G. Kvinikadze, “Nonlinear functional differential equations with Properties A and B,” J. Math. Anal. Appl., 306, 136–160 (2005).CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    R. Koplatadze, “Quasilinear functional differential equations with Property A,” J. Math. Anal. Appl., 330, 483–510 (2007).CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    R. Koplatadze, “On oscillatory properties of solutions of generalized Emden–Fowler-type differential equations,” Proc. A. Razmadze Math. Inst., 145, 117–121 (2007).MATHMathSciNetGoogle Scholar
  4. 4.
    R. Koplatadze, “On asymptotic behavior of solutions of almost linear and essentially nonlinear differential equations,” Nonlinear Anal., 71, No 12, 396–400 (2009).CrossRefMathSciNetGoogle Scholar
  5. 5.
    R. Koplatadze, E. Litsyn, “Oscillation criteria for higher order ‘almost linear’ functional differential equations,” Funct. Differ. Equ., 16, No 3, 387–434 (2009).MATHMathSciNetGoogle Scholar
  6. 6.
    R. Koplatadze, “On asymptotic behavior of solutions of nth order Emden–Fowler differential equations with advanced argument,” Czechoslovak Math. J., 60(135), 817–833 (2010).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    R. Koplatadze, “On oscillatory properties of solutions of functional differential equations,” Mem. Differential Equations Math. Phys., 3, 33–179 (1994).MathSciNetGoogle Scholar
  8. 8.
    I. Ličko, M. Švec, “Le caractere oscillatore des solutions de i’equation y (n) + f (x) y α = 0, n > 1,” Czechoslovak Math. J., 13, 481–489 (1963).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Tbilisi State UniversityTbilisiGeorgia

Personalised recommendations