On asymptotic behaviors of solutions of “almost linear” and essential nonlinear functional differential equations

In the paper the following differential equation equation(0.1)u(n)(t)+p(t)|u(σ(t))|μ(t)signu(σ(t))=0 is considered, where p∈Lloc(R+;R−), μ∈C(R+;(0,+∞))μ∈C(R+;(0,+∞)), σ∈C(R+;R+)σ∈C(R+;R+) and limt→+∞σ(t)=+∞limt→+∞σ(t)=+∞.We say that the equation is “almost linear” if the condition limt→+∞μ(t)=1limt→+∞μ(t)=1 is fulfilled, while if there exists λ∈(0,1)(λ∈(1,+∞)) such that μ(t)≤λ(μ(t)≥λ), then we say that the equation is an essentially nonlinear differential equation.“Almost linear” and essentially nonlinear differential equations are considered and sufficient (necessary and sufficient) conditions for the equation to have Property B are established.