content_image/316777.jpg content_image/316778.jpg content_image/316779.jpg
paper are true or can be easily adapted to the following systems, 

<img src='content_image/316777.jpg'>

that is, differential equations with piecewise constant argument, EP CAG, 

<img src='content_image/316778.jpg'>

differential equations with variable delay and piecewise constant argument, 

<img src='content_image/316779.jpg'>

In other words, what we have suggested are sufficiently general models, which can be easily specified for concrete applications. 

Let us introduce the initial condition for SICNN (2.3). Fix a number σ ∈ R and functions φ = { φ$_{ij}$ } , ψ = { ψ$_{ij}$ } ∈ C , i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n. In the case γ ( σ ) < σ, we say that a solution x ( t ) = { x$_{ij}$ ( t ) } of (2.3) satisfies the initial condition and write x ( t ) = x ( t, σ, φ, ψ ) , t ≥ σ, if x$_{σ}$ ( s ) = φ ( s ) , x$_{γ}$$_{(}$$_{σ}$$_{)}$ ( s ) = ψ ( s ) for s ∈ [ − τ, 0] . In what follows, we assume that if the set [ γ ( σ ) − τ, γ ( σ )] ∪ [ σ − τ, σ ] is connected, then the equation φ ( s ) = ψ ( s + σ − γ ( σ )) is true for all s ∈ [ − τ, γ ( σ ) − σ ] . If γ ( σ ) ≥ σ, then we look for a solution x ( t ) = x ( t, σ, φ ) , t ≥ σ, such that x$_{σ}$ ( s ) = φ ( s ) , s ∈ [ − τ, 0] . Thus, if θ$_{p}$ ≤ σ < θ$_{p}$$_{+1}$ for some p ∈ Z , then there are two cases of the initial condition: 

(IC$_{1}$) x$_{σ}$ ( s ) = φ ( s ) , φ ∈ C , s ∈ [ − τ, 0] if θ$_{p}$ ≤ σ ≤ ζ$_{p}$ < θ$_{p}$$_{+1}$ ; 

(IC$_{2}$) x$_{σ}$ ( s ) = φ ( s ) , x$_{γ}$$_{(}$$_{σ}$$_{)}$ ( s ) = ψ ( s ) , φ, ψ ∈ C , s ∈ [ − τ, 0] , if θ$_{p}$ ≤ ζ$_{p}$ < σ < θ$_{p}$$_{+1}$. 

Considering SICNN (2.3) with these conditions, we shall say about the initial value problem ( IV P ) for (2.3). To be short, we shall say only about IV P in the form x ( t, σ, φ, ψ ) , specifying x ( t, σ, φ ) for ( IC$_{1}$ ) , if needed. Thus, we can provide the following definitions now. 

Definition 2.1 A function x ( t ) = { x$_{ij}$ ( t ) } , i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n, is a solution of (2.3) with ( IC$_{1}$ ) or ( IC$_{2}$ ) on an interval [ σ, σ + a ) if: 

(i) it satisfies the initial condition; 

(ii) x ( t ) is continuous on [ σ, σ + a ); 

(iii) the derivative x $^{′}$( t ) exists for t ≥ σ with the possible exception of the points θ$_{p}$ , where one-sided derivatives exist; 

