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

The purpose of the paper is to show that the canonical operator L-3 given by L-3(.) = (r(2)(r(1)(.)')')' where the functions r(i)(t) is an element of C([t(0), infinity), [0, infinity)) satisfy integral(infinity )(t0)ds/r(i)(s) = infinity,(i = 1, 2), can be written in a certain strongly noncanonical form L-3(.) b(3 )(b(2)(b(1)(b(0)(.))')')', such that the functions b(i)(t) is an element of ([t(0), infinity), [0, infinity)) satisfy integral(infinity )(t0)ds/b(i)(s) < infinity, (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 L-3 in the study of oscillatory and asymptotic properties of third-order delay differential equations.