to t, it is obtained that where P ( t ) is defined in (15). Since e x ≥ ex for x ∈ R , we obtain that Let u ( t ) = z ( t − 1) z ( t ) . Since z ( t ) is nonincreasing for t > T , lim inf t →∞ u ( t ) ≥ 1 . Assume that lim inf t →∞ u ( t ) = + ∞ . Then integrating inequality (14) from t − 1 2 to t, we have Since z ( t ) is nonincreasing, from the above inequality, we obtain that Dividing inequality (25) by z ( t ) and z ( t − 1 $_{2}$) , we get and 13 z ( t ) z ( t − 1 $_{2}$) − 1 + z ( t − 1) z ( t − 1 $_{2}$) t ∫ t − 1 2 P ( s ) ds ≤ 0 , (27) 1 − z ( t − 1 $_{2}$) z ( t ) + z ( t − 1) z ( t ) t ∫ t − 1 2 P ( s ) ds ≤ 0 , (26) z ( t ) − z ( t − 1 2 ) + z ( t − 1) t ∫ t − 1 2 P ( s ) ds ≤ 0 . (25) z ( t ) − z ( t − 1 2 ) + t ∫ t − 1 2 P ( s ) z ( s − 1) ds ≤ 0 . z ( t − 1) z ( t ) ≥ $_{exp}$  t ∫ t − 1 P ( s ) z ( s − 1) z ( s ) $_{ds}$  ≥ $_{e}$  t ∫ t − 1 P ( s ) z ( s − 1) z ( s ) $_{ds}$  . (24) ln z ( t − 1) z ( t ) ≥ t ∫ t − 1 P ( s ) z ( s − 1) z ( s ) ds, (23)