Hence, it can be verified that For s ∈ ( s$_{k}$$_{1}$$_{−}$$_{1+}$$_{E}$ , s$_{k}$$_{1}$$_{+2}$$_{E}$ ] , utilizing the inequality β$_{1}$ (1 + δd ) $^{p}$e − α ( s − s$_{k}$$_{1}$$_{−}$$_{1}$ ) < β$_{1}$ (1 + δd ) $^{p}$e − αEη ≤ ϵ γ we attain that ∥ φ$_{ζ}$ ( s ) − φ$_{˜}$ $_{ζ}$( s ) ∥ < ϵ γ [ 1 + β$_{2}$ ( 1 + d (1 + δd ) p α + pδd (1 + δd ) p 1 − e − αψ ( ω ) )] ≤ ϵ. Thus, ∥ ϕ$_{ζ}$ ( t ) − ϕ$_{˜}$ $_{ζ}$( t ) ∥ < ϵ for t ∈ [ θ$_{2(}$$_{k}$$_{1}$$_{+}$$_{E}$$_{)}$$_{−}$$_{1}$, θ$_{2(}$$_{k}$$_{1}$$_{+2}$$_{E}$$_{)}$ ] ∩ T $_{0}$. Consequently, the pair ϕ$_{ζ}$ ( t ) , ϕ$_{˜}$ $_{ζ}$( t ) is proximal. □ The next lemma is devoted to the frequent separation feature in SICNN (1.2). Lemma 4.2 Under the conditions ( C 1) − ( C 5) , if a pair of sequences ζ, ˜ ζ ∈ Θ is frequently separated, then the pair of bounded solutions ϕ$_{ζ}$ ( t ) , ϕ$_{˜}$ $_{ζ}$( t ) ∈ A is frequently ( ϵ$_{0}$, ϵ$_{1}$ ) -separated for some positive numbers ϵ$_{0}$ and ϵ$_{1}$. Proof. Since the pair ζ, ˜ ζ is frequently separated, there exists a positive number ϵ$_{0}$ and a sequence { k$_{q}$ } of integers satisfying k$_{q}$ → ∞ as q → ∞ such that ∥ ζ$_{k}$$_{q}$ − ˜ ζ$_{q}$ ∥ > ϵ$_{0}$, q ∈ N . Fix a natural number q. If s ∈ ( s$_{k}$$_{q}$$_{−}$$_{1}$, s$_{k}$$_{q}$ ] , then the bounded solutions φ$_{ζ}$ ( s ) = { φ ij ζ ( s ) } and φ$_{˜}$ $_{ζ}$( s ) = { φ ij ˜ ζ ( s ) } of (2.7) satisfy the relation Thus, it can be verified that 12 ∣ φ ij ζ ( s$_{k}$$_{q}$ ) − φ ij ˜ ζ ( s$_{k}$$_{q}$ ) ∣ ≥ ∣ ζ ij k$_{q}$ − ˜ ζ ij k$_{q}$ ∣ ( s$_{k}$$_{q}$ − s$_{k}$$_{q}$$_{−}$$_{1}$ ) − ∣ φ ij ζ ( s$_{k}$$_{q}$$_{−}$$_{1}$ +) − φ ij ˜ ζ ( s$_{k}$$_{q}$$_{−}$$_{1}$ +) ∣ − ∫ s$_{k}$$_{q}$ s$_{k}$$_{q}$$_{−}$$_{1}$ a$_{ij}$ ∣ φ ij ζ ( τ ) − φ ij ˜ ζ ( τ ) ∣ dτ − ∫ s$_{k}$$_{q}$ s$_{k}$$_{q}$$_{−}$$_{1}$ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij ∣ f ( φ hl ζ ( τ ) φ ij ζ ( τ ) − f ( φ hl ˜ ζ ( τ ) φ ij ˜ ζ ( τ ) ∣ dτ ≥ ∣ ζ ij k$_{q}$ − ˜ ζ ij k$_{q}$ ∣ η − [ 1 + a$_{ij}$ η + ( M$_{f}$ + H$_{0}$L$_{f}$ ) η ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij ] × sup s ∈ ( s$_{k}$$_{q}$$_{−}$$_{1}$,s$_{k}$$_{q}$ ] ∥ φ$_{ζ}$ ( s ) − φ$_{˜}$ $_{ζ}$( s ) ∥ . φ ij ζ ( s ) − φ ij ˜ ζ ( s ) = φ ij ζ ( s$_{k}$$_{q}$$_{−}$$_{1}$ +) − φ ij ˜ ζ ( s$_{k}$$_{q}$$_{−}$$_{1}$ +) − ∫ s s$_{k}$$_{q}$$_{−}$$_{1}$ a$_{ij}$ ( φ ij ζ ( τ ) − φ ij ˜ ζ ( τ ) dτ − ∫ s s$_{k}$$_{q}$$_{−}$$_{1}$ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij [ f ( φ hl ζ ( τ ) φ ij ζ ( τ ) − f ( φ hl ˜ ζ ( τ ) φ ij ˜ ζ ( τ ) ] dτ + ∫ s s$_{k}$$_{q}$$_{−}$$_{1}$ ( ζ ij k$_{q}$ − ˜ ζ ij k$_{q}$ ) dτ. ∥ φ$_{ζ}$ ( s ) − φ$_{˜}$ $_{ζ}$( s ) ∥ < β$_{1}$ (1 + δd ) $^{p}$e − α ( s − s$_{k}$$_{1}$$_{−}$$_{1}$ ) + β$_{2}$ϵ γ ( 1 + d (1 + δd ) p α + pδd (1 + δd ) p 1 − e − αψ ( ω ) ) .