content_image/325499.jpg content_image/325500.jpg content_image/325501.jpg content_image/325502.jpg
Theorem 5.1 Assume that the conditions ( C 1) , ( C 2) , ( C 4) − ( C 6) , ( C 8) and ( C 9) are fulfilled. Then, the SICNN (2.3) admits a unique almost periodic solution. If, additionally, the condition ( C 7) is valid, then the solution is exponentially stable with exponential convergence rate γ$_{0}$/ 2 . 

Proof. It follows from Theorem 4.1 that (2.3) admits a unique bounded on R solution u ( t ) = { u$_{ij}$ ( t ) } , i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , m, which is exponentially stable provided that the condition ( C 7) is valid. We will show that it is an almost periodic function. 

Consider the operator Π defined by equation (4.9) again. It is sufficient to verify that Π u ( t ) is almost periodic, if u ( t ) is. kl 2 kl 

Let us denote β = max ( i,j ) [ 1 a$_{ij}$ + ( M + 3 LH ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C ij a$_{ij}$ + 4 LH ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C ij 1 − e − a$_{ij}$θ ] . Fix an arbi- trary positive number ϵ. Because Π u is uniformly continuous, there exists a positive number η satisfying η < θ 5 and η ≤ ϵ 3 β such that if | t ′ − t $^{′′}$| < 4 η, then 

<img src='content_image/325499.jpg'>

Next, we take into account a number ν with 0 < ν < η such that ‖ u ( t $^{′}$) − u ( t $^{′′}$) ‖ < η whenever | t ′ − t $^{′′}$| < ν, and let α and q be numbers as mentioned in Lemma 5.1 such that α is an η − translation number for u ( t ) . 

Assume that t ∈ ( θ$_{p}$ + η, θ$_{p}$$_{+1}$ − η ) for some p ∈ Z . Making use of the equation 

we obtain that 

<img src='content_image/325500.jpg'>

<img src='content_image/325501.jpg'>

According to Lemma 5.1, ( i ) , the inequality 

<img src='content_image/325502.jpg'>

