for s ∈ ( s$_{k}$$_{1}$$_{−}$$_{1}$, s$_{k}$$_{1}$$_{+2}$$_{E}$ ] . The bounded solutions φ$_{ζ}$ ( s ) = { φ ij ζ ( s ) } , φ$_{˜}$ $_{ζ}$( s ) = { φ ij ˜ ζ ( s ) } of (2.7) satisfy the relation Therefore, for s ∈ ( s$_{k}$$_{1}$$_{−}$$_{1}$, s$_{k}$$_{1}$$_{+2}$$_{E}$ ] , we have that Define the function v ( s ) = e λs ∥ φ$_{ζ}$ ( s ) − φ$_{˜}$ $_{ζ}$( s ) ∥ . The inequality (4.9) implies that v ( s ) ≤ β$_{1}$e λs$_{k}$$_{1}$$_{−}$$_{1}$ + β$_{2}$ϵ γ e λs + ∫ s s$_{k}$$_{1}$$_{−}$$_{1}$ dv ( τ ) dτ + ∑ s$_{k}$$_{1}$$_{−}$$_{1}$<s$_{k}$<s δdv ( s$_{k}$ ) , s ∈ ( s$_{k}$$_{1}$$_{−}$$_{1}$, s$_{k}$$_{1}$$_{−}$$_{1+2}$$_{E}$ ] . Making use of the Gronwall’s Lemma for piecewise continuous functions, one can obtain that Since the inequality (1 + δd ) i (( a$_{0}$,b$_{0}$ $^{))}$e d ( b$_{0}$ − a$_{0}$ ) ≤ (1 + δd ) $^{p}$e ( λ − α )( b$_{0}$ − a$_{0}$ ) is valid for any real numbers a$_{0}$ and b$_{0}$ with b$_{0}$ > a$_{0}$, we have 11 v ( s ) ≤ β$_{1}$e $^{λs$_{k}$$_{1}$$_{−}$$_{1}$}$(1 + δd ) $^{p}$e ( λ − α )( s − s$_{k}$$_{1}$$_{−}$$_{1}$ ) + β$_{2}$ϵ γ e λs + β$_{2}$d (1 + δd ) $^{p}$ϵ αγ e λs ( 1 − e − α ( s − s$_{k}$$_{1}$$_{−}$$_{1}$ ) ) + β$_{2}$pδd (1 + δd ) $^{p}$ϵ γ (1 − e − αψ ( ω $^{)}$) e λs ( 1 − e − α ( s − s$_{k}$$_{1}$$_{−}$$_{1}$ + ψ ( ω )) ) . v ( s ) ≤ β$_{1}$e $^{λs$_{k}$$_{1}$$_{−}$$_{1}$}$(1 + δd ) i (( s$_{k}$$_{1}$$_{−}$$_{1}$,s $^{))}$e d ( s − s$_{k}$$_{1}$$_{−}$$_{1}$ ) + β$_{2}$ϵ γ e λs + β$_{2}$dϵ γ ∫ s s$_{k}$$_{1}$$_{−}$$_{1}$ (1 + δd ) i (( τ,s $^{))}$e d ( s − τ $^{)}$e $^{λτ}$dτ + β$_{2}$δdϵ γ ∑ s$_{k}$$_{1}$$_{−}$$_{1}$<s$_{k}$<s (1 + δd ) i (( s$_{k}$,s $^{))}$e d ( s − s$_{k}$ $^{)}$e $^{λs$_{k}$}$. ∣ φ ij ζ ( s ) − φ ij ˜ ζ ( s ) ∣ ≤ 2 ( H$_{0}$M$_{f}$ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij + M$_{F}$ ) ( K$_{ij}$ λ$_{ij}$ + pδK$_{ij}$ 1 − e − λ$_{ij}$ψ ( ω ) ) e − λ$_{ij}$ ( s − s$_{k}$$_{1}$$_{−}$$_{1}$ ) + ϵK$_{ij}$ γλ$_{ij}$ ( 1 − e − λ$_{ij}$ ( s − s$_{k}$$_{1}$$_{−}$$_{1}$ ) ) + pδϵK$_{ij}$ γ (1 − e − λ$_{ij}$ψ ( ω $^{)}$) ( 1 − e − λ$_{ij}$ ( s − s$_{k}$$_{1}$$_{−}$$_{1}$ + ψ ( ω )) ) + ∫ s s$_{k}$$_{1}$$_{−}$$_{1}$ K$_{ij}$ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij ( M$_{f}$ + H$_{0}$L$_{f}$ ) e − λ$_{ij}$ ( s − τ ) ∥ φ$_{ζ}$ ( τ ) − φ$_{˜}$ $_{ζ}$( τ ) ∥ dτ + ∑ s$_{k}$$_{1}$$_{−}$$_{1}$<s$_{k}$<s K$_{ij}$ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij δ ( M$_{f}$ + H$_{0}$L$_{f}$ ) e − λ$_{ij}$ ( s − s$_{k}$ ) ∥ φ$_{ζ}$ ( s$_{k}$ ) − φ$_{˜}$ $_{ζ}$( s$_{k}$ ) ∥ . (4.9) φ ij ζ ( s ) − φ ij ˜ ζ ( s ) = − ∫ s −∞ u$_{ij}$ ( s, τ ) [ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij f ( φ hl ζ ( τ ) φ ij ζ ( τ ) − L$_{ij}$ ( ψ − $^{1}$( τ ) , ζ ) − ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij f ( φ hl ˜ ζ ( τ ) φ ij ˜ ζ ( τ ) + L$_{ij}$ ( ψ − $^{1}$( τ ) , ˜ ζ ) ] dτ − −∞ <s$_{k}$<s u$_{ij}$ ( s, s$_{k}$ +) [ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij f ( φ hl ζ ( s$_{k}$ ) φ ij ζ ( s$_{k}$ ) − ζ ij k − ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij f ( φ hl ˜ ζ ( s$_{k}$ ) φ ij ˜ ζ ( s$_{k}$ ) + ˜ ζ ij k ] δ$_{k}$.