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. Let us denote β = max ( i,j ) [ 1 a$_{ij}$ + ( M + 3 LH ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl ij a$_{ij}$ + 4 LH 2 ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl 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 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}$$_{+1}$ − η ) for some p ∈ Z . Making use of the equation we obtain that According to Lemma 5.1, ( i ) , the inequality ‖ Π u ( t $^{′}$) − Π u ( t $^{′′}$) ‖ < ϵ 3 . (5.11) (Π u ( t + α ))$_{ij}$ − (Π u ( t ))$_{ij}$ = − ∫ t −∞ e − a$_{ij}$ ( t − s ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl ij × [ f ( u$_{kl}$$_{(}$$_{s}$$_{+}$$_{α}$$_{)}$, u$_{klγ}$$_{(}$$_{s}$$_{+}$$_{α}$$_{)}$ ) u$_{ij}$ ( s + α ) − f ( u$_{kls}$, u$_{klγ}$$_{(}$$_{s}$$_{)}$ ) u$_{ij}$ ( s ) ] ds + ∫ t −∞ e − a$_{ij}$ ( t − s $^{)}$[ L$_{ij}$ ( s + α ) − L$_{ij}$ ( s )] ds, | (Π u ( t + α ))$_{ij}$ − (Π u ( t ))$_{ij}$ | ≤ ∫ t −∞ e − a$_{ij}$ ( t − s ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl ij M | u$_{ij}$ ( s + α ) − u$_{ij}$ ( s ) | ds + ∫ t −∞ e − a$_{ij}$ ( t − s ) ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl $_{ij}$LH ∥ u$_{kl}$$_{(}$$_{s}$$_{+}$$_{α}$$_{)}$ − u$_{kls}$ ∥ $_{0}$ds + t −∞ e − a$_{ij}$ ( t − s ) C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl $_{ij}$LH u$_{klγ}$$_{(}$$_{s}$$_{+}$$_{α}$$_{)}$ − u$_{klγ}$$_{(}$$_{s}$$_{)}$ ∥ $_{0}$ds + ∫ t −∞ e − a$_{ij}$ ( t − s ) | L$_{ij}$ ( s + α ) − L$_{ij}$ ( s ) | ds. (5.12) ∫ t −∞ e − a$_{ij}$ ( t − s ) | L$_{ij}$ ( s + α ) − L$_{ij}$ ( s ) | ds < η a$_{ij}$ 14