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

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 
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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Tbilisi State UniversityTbilisiGeorgia

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