Comparison theorems for functional differential equations with deviating arguments

Lₙ±, F, g)$ do possess proper solutions.A proper solution of $(Lₙ⁺, F, g)$ or $(Lₙ⁻, F, g)$ is called oscillatory if it has arbitrarily large zeros; otherwise it is called nonoscillatory.Equation $(Lₙ⁺, F, g)$ is said to be oscillatory if all of its proper solutions are oscillatory.We say that the operator $Lₙ$ is in canonical form if(3)It is known that any differential operator of the form (1) can always be repre- sented in canonical form in an essentially unique way (see Trench [30]).Let $iₖ∈\{1, ⋯ , n-1\},$ $1 k n-1$ , and $t,$ $ s∈[a, ∞$ ).We define$Iₖ(t, s;p_{iₖ}, ⋯ p_{i₁})=∫ₛᵗp_{iₖ}(r)Iₖ₋₁(r, s;p_{iₖ₋₁}, ⋯ p_{i₁})dr$ .In case (3) holds, the functions $p₀(t)Iₖ(t, a;p₁, ⋯ , pₖ)$ , $0 k n-1$ , form a fundamental set of solutions of the differential equation $Lₙx=0$ .DEFINITION 1.Let $Lₙ$ be in canonical form.Equation $(Lₙ⁺, F, g)$ is said to have property (A) if (i) for $n$ even, equation $(Lₙ⁺, F, g)$ is oscillatory, and (ii) for $n$ odd, every nonoscillatory solution $x(t)$ of $(Lₙ⁺, F, g)$ is strongly decreasing in the sense that(5)$|{x(t)}{p₀(t)}|↓ 0$ as $ t↑∞$ .Equation $(Lₙ⁻, F, g)$ is said to have property (B) if (i) for $n$ odd, every nonoscillatory solution $x(t)$ of $(Lₙ⁻, F, g)$ is strongly- increasing in the sense that (6)$|{x(t)}{p₀(t)Iₙ₋₁(t,a;p₁,⋯,pₙ₋₁)}|↑∞$ as $ t↑∞$ , and (ii) for $n$ even, every nonoscillatory solution is either strongly decreasing or strongly increasing.We are interested in comparing the oscillatory and asymptotic properties of equations $(Lₙ⁺, F, g),$ $(Lₙ⁻, F, g)$ with those of the equations $(Mₙ⁺, G, h)$ $Mₙy(t)+G(t, y(h(t)))=0$ ,