Thus, for s ∈ ( $^{s$_{k}$$_{0}$}$− $_{1}$, s$_{k}$$_{0}$$_{+2}$$_{E}$ ] , we have that is valid. Therefore, Accordingly, the inequality It can be shown by applying the analogue of the Gronwall’s Lemma for piecewise continuous functions that One can confirm by means of (4.9) that and Let us define the functions u ( s ) = e λs ∥ φ$_{ζ}$ ( s ) − φ$_{˜}$ $_{ζ}$( s ) ∥ and v ( s ) = β$_{1}$ + β$_{2}$e $^{λs}$, where ∥ φ$_{ζ}$ ( s ) − φ$_{˜}$ $_{ζ}$( s ) ∥ ≤ R$_{1}$e − λ ( $^{s}$−s$_{k}$$_{0}$$_{−}$$_{1}$ ) + Nϵ γλ ( 1 − $_{e}$− λ ( $^{s}$−s$_{k}$$_{0}$$_{−}$$_{1}$ ) ) + N ¯ δpϵ γ (1 − e − λψ ( ω $^{)}$) ( 1 − e − λ ( $^{s}$−s$_{k}$$_{0}$$_{−}$$_{1}$ + ψ ( ω )) ) + ∫ s s$_{k}$$_{0}$$_{−}$$_{1}$ NL$_{f}$$_{e}$− λ ( $^{s}$−r ) ∥ φ$_{ζ}$ ( r ) − φ$_{˜}$ $_{ζ}$( r ) ∥ dr + ∑ s$_{k}$$_{0}$$_{−}$$_{1}$<s$_{k}$<s NL$_{f}$ ¯ δe − λ ( $^{s}$−s$_{k}$ ) ∥ φ$_{ζ}$ ( s$_{k}$ ) − φ$_{˜}$ $_{ζ}$( s$_{k}$ ) ∥ . (4.9) β$_{1}$ = R$_{1}$e λs$_{k}$$_{0}$$_{−}$$_{1}$ − Nϵ γλ e λs$_{k}$$_{0}$$_{−}$$_{1}$ − N ¯ δpϵ γ (1 − e − λψ ( ω $^{)}$) e λ ( $^{s$_{k}$$_{0}$$_{−}$$_{1}$}$−ψ ( ω )) β$_{2}$ = Nϵ γ ( 1 λ + ¯ δp 1 − e − λψ ( ω ) ) . u ( s ) ≤ v ( s ) + ∫ s s$_{k}$$_{0}$$_{−}$$_{1}$ NL$_{f}$u ( r ) dr + ∑ s$_{k}$$_{0}$$_{−}$$_{1}$<s$_{k}$<s NL$_{f}$ ¯ δu ( s$_{k}$ ) . u ( s ) ≤ v ( s ) + ∫ s s$_{k}$$_{0}$$_{−}$$_{1}$ NL$_{f}$ (1 + NL$_{f}$ ¯ δ ) i (( r,s $^{))}$e NL$_{f}$ ( $^{s}$−r $^{)}$v ( r ) dr + ∑ s$_{k}$$_{0}$$_{−}$$_{1}$<s$_{k}$<s NL$_{f}$ ¯ δ (1 + NL$_{f}$ ¯ δ ) i (( s$_{k}$,s $^{))}$e NL$_{f}$ ( $^{s}$−s$_{k}$ $^{)}$v ( s$_{k}$ ) . u ( s ) ≤ β$_{1}$ (1 + NL$_{f}$ ¯ δ ) $^{p}$e ( $^{λ}$−α )( $^{s}$−s$_{k}$$_{0}$$_{−}$$_{1}$ ) + β$_{2}$e λs + NL$_{f}$β$_{2}$ (1 + NL$_{f}$ ¯ δ ) p α e λs ( 1 − e − α ( $^{s}$−s$_{k}$$_{0}$$_{−}$$_{1}$ ) ) + NL$_{f}$ ¯ δpβ$_{2}$ (1 + NL$_{f}$ ¯ δ ) p 1 − e − αψ ( ω ) e λs ( 1 − $_{e}$− α ( $^{s}$−s$_{k}$$_{0}$$_{−}$$_{1}$ + ψ ( ω )) ) . ∥ φ$_{ζ}$ ( s ) − φ$_{˜}$ $_{ζ}$( s ) ∥ < R$_{1}$ (1 + NL$_{f}$ ¯ δ ) $^{p}$$_{e}$−α ( $^{s}$−s$_{k}$$_{0}$$_{−}$$_{1}$ ) + Nϵ γ ( 1 λ + ¯ δp 1 − e − λψ ( ω ) ) ( 1 + NL$_{f}$ (1 + NL$_{f}$ ¯ δ ) p α + NL$_{f}$ ¯ δp (1 + NL$_{f}$ ¯ δ ) p 1 − e − αψ ( ω ) ) 8