η$_{k}$ = s$_{k}$ − τ/ 2 . One can find an integer j$_{k}$ with 1 ≤ j$_{k}$ ≤ n such that Using (7.22) one can obtain for t ∈ [ η$_{k}$, η$_{k}$ + τ ] that Moreover, inequality (7.23) yields for all t ∈ [ η$_{k}$, η$_{k}$ + τ ] . Since there exist numbers r$_{1}$, r$_{2}$, . . . , r$_{n}$ ∈ [ η$_{k}$, η$_{k}$ + τ ] satisfying we have that Let ζ ∈ Θ be an arbitrary sequence. Making use of the relations and it can be deduced that Hence, 16 | h$_{j}$$_{k}$ ( x ( s$_{k}$ )) − h$_{j}$$_{k}$ $^{(}$˜ x ( s$_{k}$ )) | ≥ L$_{2}$ √$_{n}$ ‖ x ( s$_{k}$ ) − ˜ x ( s$_{k}$ ) ‖ > L$_{2}$ϵ$_{0}$ √$_{n}$ . (7.23) | h$_{j}$$_{k}$ ( x ( s$_{k}$ )) − h$_{j}$$_{k}$ $^{(}$˜ x ( s$_{k}$ )) | − | h$_{j}$$_{k}$ ( x ( t )) − h$_{j}$$_{k}$ $^{(}$˜ x ( t )) | < L$_{2}$ϵ$_{0}$ $_{2}$√$_{n}$ . | h$_{j}$$_{k}$ ( x ( t )) − h$_{j}$$_{k}$ $^{(}$˜ x ( t )) | > | h$_{j}$$_{k}$ ( x ( s$_{k}$ )) − h$_{j}$$_{k}$ $^{(}$˜ x ( s$_{k}$ )) | − L$_{2}$ϵ$_{0}$ $_{2}$√$_{n}$ > L$_{2}$ϵ$_{0}$ $_{2}$√$_{n}$ ∥ ∫ η$_{k}$ + τ η$_{k}$ [ h ( x ( s )) − h $^{(}$˜ x ( s ))] ds ∥ = τ ( n ∑ k =1 [ h$_{k}$ ( x ( r$_{k}$ )) − h$_{k}$ $^{(}$˜ x ( r$_{k}$ ))] ) 1 / 2 , ∥ ∫ η$_{k}$ + τ η$_{k}$ [ h ( x ( s )) − h $^{(}$˜ x ( s ))] ds ∥ ≥ τ | h$_{j}$$_{k}$ ( x ( r$_{j}$$_{k}$ )) − h$_{j}$$_{k}$ $^{(}$˜ x ( r$_{j}$$_{k}$ )) | > τL$_{2}$ϵ$_{0}$ $_{2}$√$_{n}$ . φ$_{x,ζ}$ ( t ) = φ$_{x,ζ}$ ( η$_{k}$ ) + ∫ t η$_{k}$ [ Aφ$_{x,ζ}$ ( s ) + f ( φ$_{x,ζ}$ ( s ) , s ) + ν ( s, ζ ) + h ( x ( s ))] ds φ$_{˜}$ $_{x,ζ}$( t ) = φ$_{˜}$ $_{x,ζ}$( η$_{k}$ ) + ∫ t η$_{k}$ [ Aφ$_{˜}$ $_{x,ζ}$( s ) + f ( φ$_{˜}$ $_{x,ζ}$( s ) , s ) + ν ( s, ζ ) + h $^{(}$˜ x ( s ))] ds, ‖ φ$_{x,ζ}$ ( η$_{k}$ + τ ) − φ$_{˜}$ $_{x,ζ}$( η$_{k}$ + τ ) ‖ ≥ ∥ ∫ η$_{k}$ + τ η$_{k}$ [ h ( x ( s )) − h $^{(}$˜ x ( s ))] ds ∥ − ‖ φ$_{x,ζ}$ ( η$_{k}$ ) − φ$_{˜}$ $_{x,ζ}$( η$_{k}$ ) ‖ − ( ‖ A ‖ + L$_{f}$ ) ∫ η$_{k}$ + τ η$_{k}$ ‖ φ$_{x,ζ}$ ( s ) − φ$_{˜}$ $_{x,ζ}$( s ) ‖ ds. max t ∈ [ η$_{k}$,η$_{k}$ + τ ] ‖ φ$_{x,ζ}$ ( t ) − φ$_{˜}$ $_{x,ζ}$( t ) ‖ > τL$_{2}$ϵ$_{0}$ 2[2 + τ ( ‖ A ‖ + L$_{f}$ $_{)]}$√$_{n}$ .