lim t $_{→∞}$sup t ∫ t − l   ∏ s − τ<t$_{j}$ ≤ s (1 − b$_{j}$ $_{)}$  b ( s $_{) exp}$  s ∫ s − τ a ( u ) $_{du}$  ds > 1 , (10) lim n $_{→∞}$sup n +1 ∫ n +1 − l   ∏ n − 1 <t$_{j}$ ≤ s (1 − b$_{j}$ $_{)}$  c ( s $_{) exp}$  s ∫ n − 1 a ( u ) $_{du}$  ds > 1 , (11) where l = min { τ, 1 } . Proof. Let conditions (10) or (11) is satisfied. We shall prove that the existence of eventually positive (or negative) solutions leads to a contradiction. Let x ( t ) be an eventually positive solution of equation (1)-(2). Then y ( t ) = ∏ T <t$_{j}$ ≤ t (1 − b$_{j}$ ) x ( t ) is an eventually positive solution of equation (5) such that y ( t ) > 0 , y ( t − τ ) > 0 , y ([ t − 1]) > 0 for n + 1 > t ≥ n > T. Taking it is obtained from equation (5) that for n + 1 > t ≥ n > T. Since B ( t ) , C ( t ) ≥ 0 for t ∈ R and z ( t − τ ) , z ([ t − 1]) ≥ 0 for n + 1 > t ≥ n > T, we get z ( t ) is nonincreasing for t > T. Now, we consider two cases: Case 1. τ > 1 . Then it is clear that z ( t − τ ) ≥ z ( t − 1) and z ([ t − 1]) ≥ z ( t − 1) for t > T. 7 z $^{′}$( t ) = −    $^{B}$$^{(}$$^{t}$$^{)}$$^{z}$$^{(}$$^{t}$ − τ $_{) exp}$  t ∫ t − τ a ( s ) $_{ds}$  + C ( t ) z ([ t − $_{1]) exp}$   t ∫ [ t − 1] a ( s ) $_{ds}$       (13) z ( t ) = y ( t $_{) exp}$  t ∫ T a ( s ) $_{ds}$  , t > T, (12)