content_image/304349.jpg content_image/304350.jpg content_image/304348.jpg content_image/304347.jpg
conditions ( C 1) and ( C 2) that for each i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n, there exist positive numbers K$_{ij}$ such that | u$_{ij}$ ( s, τ ) | ≤ K$_{ij}$e − λ$_{ij}$ ( s − τ $^{)}$, s ≥ τ. 

Suppose that M$_{f}$c < 1 , where c = max ( i,j ) ( K$_{ij}$ λ$_{ij}$ + pδK$_{ij}$ 1 − e − λ$_{ij}$ψ ( ω ) ) ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij and δ = max 1 ≤ k ≤ $_{p}$δ$_{k}$ . In the remaining parts of the paper, we will denote and M$_{F}$ = max η ∈ Λ ‖ F ( η ) ‖ , H$_{0}$ = M$_{F}$ 1 − M$_{f}$c max ( i,j ) ( K$_{ij}$ λ$_{ij}$ + pδK$_{ij}$ 1 − e − λ$_{ij}$ψ ( ω ) ) and d = ( M$_{f}$ + H$_{0}$L$_{f}$ ) max ( i,j ) K$_{ij}$ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij . The following conditions are also needed. 

<img src='content_image/304349.jpg'>

<img src='content_image/304350.jpg'>

In the next section, we will deal with the existence, uniqueness and attractiveness property of the bounded solutions of SICNN (1.2). 

## 3 Bounded solutions 

According to the results of [77, 78], a bounded on R function y ( s ) = { y$_{ij}$ ( s ) } , i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n, is a solution of the impulsive system (2.7) if and only if the equation 

is valid. 

<img src='content_image/304348.jpg'>

The existence of bounded on R solutions of (2.7) is considered in the next assertion. 

Lemma 3.1 Suppose that the conditions ( C 1) − ( C 5) are fulfilled. For each solution ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ of (1.3) , there exists a unique bounded on R solution φ$_{ζ}$ ( s ) = { φ ij ζ ( s ) } , i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n of (2.7) such that sup s ∈ R ‖ φ$_{ζ}$ ( s ) ‖ ≤ H$_{0}$. 

Proof. Fix a solution ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ of (1.3) and take into account the set D$_{0}$ ⊂ P C ( R ) consisting of functions of the form z ( s ) = { z$_{ij}$ ( s ) } which have discontinuities at the points s$_{k}$, k ∈ Z such that ‖ z ‖$_{∞}$ ≤ H$_{0}$, where ‖ z ‖$_{∞}$ = sup s ∈ R ‖ z ( s ) ‖ . The set D$_{0}$ is complete [77]. 

Define an operator Π on D$_{0}$ as 

<img src='content_image/304347.jpg'>

