Introdiiction
In [l], D. C. ANGELOVA and D. D. BAINOV consider the problem of boundedness and asymptotic behavior of the oscillatory solutions of some forced functional differential equations of second order. Their general equation is stated below in Theorem 1, as equation (I). There is a wealth of papers dealing with related problems, and typical references can be found in [ 11. The deviation t ( t ) is given in [ 11 a functional form: t ( t ) = y ( A ( t , y ( t ) ) where A can be larger than t (advanced equation) or less than t (retarded equation) or of both types. In fact, the very assumptions on A do not seem to be of importance as soon as the problem here is not to prove existence results. We can distinguish two types of assumptions on A depending on the question to be answered: for boundedness, A is supposed to be less than t. The proof consists in starting with some estimate on a first interval and then propagating it along the solution. Now, if boundedness has been proved or is accepted as an assumption, then there is no condtion whatever on A , the deviated term is replaced by a function depending only on the bounds of the solution.
This is particularly true for Theorems 3.3 and 3.4 in [l], and i t is something we wanted to point out in our introductory remark. It explains why we are expressing the deviation with t ( t ) instead of A ( t , y(t)). Now, our main purpose in this short note is to give a slightly different formulation ofTheorem 3.3, that is: the assumption 3 of Theorem 3.3 changed into (Hz) in our Theorem 1, and prove Theorem 1 by a method which differs from that used in [I]. We compare the solutions of (I) to