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The purpose of the paper is to show that the canonical operator L3 given by L3(·) = ( r2 ( r1(·) )′)′ where the functions ri(t) ∈ C([t0, ∞), [0, ∞)) satisfy ∫ ∞ t0 ds ri(s) = ∞, (i = 1, 2), can be written in a certain strongly noncanonical form L3(·) ≡ b3 ( b2 ( b1 (b0(·)) )′)′ , such that the functions bi(t) ∈ C([t0, ∞), [0, ∞)) satisfy ∫ ∞ t0 ds bi(s) < ∞, (i = 1, 2). We study some relations between canonical and strongly noncanonical operators, showing the advantage of this reverse approach based on the use of a noncanonical representation of L3 in the study of oscillatory and asymptotic properties of third-order delay differential equations.

On asymptotic properties of solutions to third-order delay differential equations