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. □ 12 Δ = min { Δ / 4 , L$_{1}$ Δ ϵ$_{0}$ 4 (8 + ( P$_{1}$ + P$_{2}$r$_{1}$ )Δ) P$_{3}$ } , ϵ$_{1}$ = L$_{1}$ Δ ϵ$_{0}$ 2 (8 + ( P$_{1}$ + P$_{2}$r$_{1}$ )Δ) , (3.17) Δ$_{l}$ ≥ min { Δ / 4 , L$_{1}$ Δ ϵ$_{0}$ 4 (8 + ( P$_{1}$ + P$_{2}$r$_{1}$ )Δ) P$_{3}$ } . ‖ y ( t ) − y ( t ) ‖ ≥ L$_{1}$ Δ$_{l}$ ϵ$_{0}$ 2 (8 + ( P$_{1}$ + P$_{2}$r$_{1}$ )Δ$_{l}$) ≥ L$_{1}$ Δ ϵ$_{0}$ 2 (8 + ( P$_{1}$ + P$_{2}$r$_{1}$ )Δ) P$_{3}$ = ( P$_{1}$ + P$_{2}$r$_{1}$ ) r$_{1}$ + L$_{2}$H + ‖ h (0) ‖ , Δ$_{l}$ = min { Δ$_{l}$ / 4 , L$_{1}$ Δ$_{l}$ ϵ$_{0}$ 4 (8 + ( P$_{1}$ + P$_{2}$r$_{1}$ )Δ$_{l}$) P$_{3}$ } , ‖ y ( t ) − y ( t ) ‖ ≥ ‖ y ( η$_{l}$ ) − y ( η$_{l}$ ) ‖ − ‖ y ( η$_{l}$ ) − y ( t ) ‖ − ‖ y ( η$_{l}$ ) − y ( t ) ‖ ≥ L$_{1}$ Δ$_{l}$ ϵ$_{0}$ 8 + ( P$_{1}$ + P$_{2}$r$_{1}$ )Δ$_{l}$ − 2 (( P$_{1}$ + P$_{2}$r$_{1}$ ) r$_{1}$ + L$_{2}$H + ‖ h (0) ‖ ) Δ$_{l}$ . ‖ y $^{′}$‖$_{0}$ + ‖ y $^{′}$‖$_{0}$ ≤ 2 (( P$_{1}$ + P$_{2}$r$_{1}$ ) r$_{1}$ + L$_{2}$H + ‖ h (0) ‖ ) . (3.16) max t ∈ J 1 l ‖ y ( t ) − y ( t ) ‖ = ‖ y ( η$_{l}$ ) − y ( η$_{l}$ ) ‖