4 Bounded solutions In this section, we will investigate the existence of a unique bounded solution of$_{SICNN}$ (2.3). Moreover, the exponential stability of the bounded solution will be considered. An auxiliary result is presented in the following lemma. Lemma 4.1 Assume that the conditions ( C 1) − ( C 4) are fulfilled. If H$_{0}$ is a positive number such that µ ¯ θ [ M + 2 L ( H$_{0}$ + ¯ θ ¯ L ) 1 − µ ¯ θM ] < 1 , then a function x ( t ) = { x$_{ij}$ ( t ) } , i = 1 , 2 , . . ., m, j = 1 , 2 , . . ., n, satisfying sup t ∈ $_{R}$‖ x ( t ) ‖ ≤ H$_{0}$ is a solution of (2.3) if and only if it satisfies the following integral equation Proof. We consider only sufficiency. The necessity can be proved by using (3.7) in a very similar way to the ordinary differential equations case. One can obtain that Therefore, the integral in (4.8) is convergent. Differentiate (4.8) to verify that it is a solution of (2.3). □ The following conditions are needed. The main result concerning the existence and exponential stability of bounded solutions of (2.3) is mentioned in the next theorem. Theorem 4.1 Suppose that the conditions ( C 1) − ( C 6) hold. Then, (2.3) admits a unique bounded on R solution, which satisfies (4.8). If, additionally, the condition ( C 7) is valid, then the solution is exponentially stable with exponential convergence rate γ$_{0}$/ 2 . Proof. Let C$_{0}$ ( R ) be the set of uniformly continuous functions defined on R such that if u ( t ) ∈ C$_{0}$ ( R ) , then ‖ u ‖ ∞ ≤ H, where ‖ u ‖ ∞ = sup t ∈ $_{R}$‖ u ( t ) ‖ . Define on C$_{0}$ ( R ) the operator Π as (Π u ( t ))$_{ij}$ ≡ − ∫ t −∞ e − a$_{ij}$ ( t − s ) [ ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl $_{ij}$f ( u$_{kls}$, u$_{klγ}$$_{(}$$_{s}$$_{)}$ ) u$_{ij}$ ( s ) − L$_{ij}$ ( s ) ] ds. (4.9) ∣ ∫ t −∞ e − a$_{ij}$ ( t − s ) [ ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl $_{ij}$f ( x$_{kls}$, x$_{klγ}$$_{(}$$_{s}$$_{)}$ ) x$_{ij}$ ( s ) − L$_{ij}$ ( s ) ] ds ∣ ≤ 1 a$_{ij}$ ( MH$_{0}$ ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl ij + L$_{ij}$ ) . x$_{ij}$ ( t ) = − ∫ t −∞ e − a$_{ij}$ ( t − s ) [ ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl $_{ij}$f ( x$_{kls}$, x$_{klγ}$$_{(}$$_{s}$$_{)}$ ) x$_{ij}$ ( s ) − L$_{ij}$ ( s ) ] ds. (4.8) (C5) µ ¯ θ [ M + 2 L ( H + ¯ θ ¯ L ) 1 − µ ¯ θM ] < 1 , where H = ¯ l 1 − M ¯ c ; (C6) ( M + 2 LH )¯ c < 1; (C7) 2 ¯ d [ M + LHe γ$_{0}$τ/ 2 ( 1 + e γ$_{0}$ ¯ θ/ 2 )] < 1 . 9