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

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We consider a differential equation
where

$$ {u^{(n) }}(t)+p(t)\left| {u\left( {\sigma (t)} \right)\left| {^{{\mu (t)}}} \right.\mathrm{sign}\,u\left( {\sigma (t)} \right)=0,} \right. $$

*p*∈*L*_{loc}(*R*_{+};*R*_{+}), μ ∈*C*(*R*_{+}; (0, +∞)), σ ∈*C*(*R*_{+};*R*_{+}), and σ(*t*) ≥*t*for*t*∈*R*_{+}. 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**.## Keywords

Differential Equation Satisfying Condition Nonlinear Differential Equation Linear Differential Equation Functional Differential Equation
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