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of ( i ) . Let x ( t ) be an eventually positive solution of inequality (3) such that x ( t ) > 0 , x ( t − τ ) > 0 , x ([ t − 1]) > 0 for t > T ≥ t$_{0}$ +max { τ, 2 } , where T is sufficiently large. 

Set y ( t ) = ∏ T <t$_{j}$ ≤ t (1 − b$_{j}$ ) x ( t ) . Since 1 − b$_{j}$ > 0 , it is clear that y ( t ) > 0 , y ( t − τ ) > 0 , and y ([ t − 1]) > 0 for t > T. Now, we will show that y ( t ) is a solution of inequality (6). From (8), (9), and (3) we obtain that 

<img src='content_image/297001.jpg'>

<img src='content_image/297000.jpg'>

So, y ( t ) is an eventually positive solution of inequality (6). On the other hand, from (2), we have 

and 

<img src='content_image/296999.jpg'>

<img src='content_image/296998.jpg'>

