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A real valued function ‖ y ( t ) − y ( t ) ‖ takes its maximum on the interval J 1 l at η$_{l}$, that is, 

for some η$_{l}$ ∈ J 1 l . Using (2.9), we obtain 

Hence, for t ∈ [ η$_{l}$, η$_{l}$ + Δ$_{l}$] ⊂ J$_{l}$ with 0 < Δ$_{l}$ ≤ Δ$_{l}$ / $_{4}$, we derive 

Consequently, by taking 

where 

one can attain that 

for any t ∈ [ η$_{l}$, η$_{l}$ + Δ$_{l}$] ⊂ $_{J$_{l}$}$. Note that 

So, by taking 

and 

the proof is finished. □ 

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