content_image/325713.jpg content_image/325712.jpg content_image/325711.jpg
Hence, it can be verified that 

<img src='content_image/325713.jpg'>

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 

<img src='content_image/325712.jpg'>

Thus, it can be verified that 

<img src='content_image/325711.jpg'>

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