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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 

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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. 

