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(iv) equation (2 . 3) is satisfied by x ( t ) for all t > σ except possibly at the points of θ, and it holds for the right derivative of x ( t ) at the points θ$_{p}$. 

Definition 2.2 A function x ( t ) = { x$_{ij}$ ( t ) } , i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n, is a solution of (2.3) on R if: 

(i) x ( t ) is continuous; 

(ii) the derivative x $^{′}$( t ) exists for all t ∈ R with the possible exception of the points θ$_{p}$, p ∈ Z , where one-sided derivatives exist; 

(iv) equation (2 . 3) is satisfied by x ( t ) for all t ∈ R except at the points of θ, and it holds for the right derivative of x ( t ) at the points θ$_{p}$, p ∈ Z . 

The existence and uniqueness of solutions of (2.3) will be investigated in the next section. 

## 3 Existence and uniqueness 

Throughout the paper we suppose in SICNN (2.3) that γ$_{0}$ = min ( i,j ) a$_{ij}$ > 0 and C kl ij are non-negative numbers. 

The following assumptions are required. 

(C1) The functional f satisfies the Lipschitz condition 

for some positive constant L, where ( φ$_{1}$, ψ$_{1}$ ) and ( φ$_{2}$, ψ$_{2}$ ) are from C 0 × C $^{0}$; 

(C2) There exists a positive number M such that sup ( φ,ψ ) ∈C $^{0}$×C0 | f ( φ, ψ ) | ≤ M ; 

(C3) There exists a positive number ¯ θ such that θ$_{p}$$_{+1}$ − θ$_{p}$ ≤ ¯ θ for all p ∈ Z ; 

<img src='content_image/314893.jpg'>

(C4) | L$_{ij}$ ( t ) | ≤ L$_{ij}$ for all i, j and t ∈ R , where L$_{ij}$ are non-negative real constants. In the remaining parts of the paper, the notations µ = max ( i,j ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl ij , ¯ c = max ( i,j ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl ij a$_{ij}$ , ¯ d = max ( i,j ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl ij 2 a$_{ij}$ − γ$_{0}$ , ¯ L = max ( i,j ) L$_{ij}$ and ¯ l = max ( i,j ) L$_{ij}$ a$_{ij}$ will be used. We assume that µ ¯ θM < 1 and M ¯ c < 1 . Let us denote C H$_{0}$ = { φ ∈ C : ‖ φ ‖ 0 ≤ H$_{0}$ } , where H$_{0}$ is a positive number. 

Lemma 3.1 Suppose that the conditions ( C 1) − ( C 4) hold and fix an integer p. If H$_{0}$ is a positive number such that µ ¯ θ [ M + 2 L ( H$_{0}$ + ¯ θ ¯ L ) 1 − µ ¯ θM ] < 1 , then for every ( σ, φ, ψ ) ∈ [ θ$_{p}$, θ$_{p}$$_{+1}$ ] × C H$_{0}$ × C H$_{0}$ there exists a unique solution x ( t ) = x ( t, σ, φ, ψ ) of (2.3) on [ σ, θ$_{p}$$_{+1}$ ] . 

