content_image/303531.jpg content_image/303530.jpg content_image/303529.jpg
## 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 

<img src='content_image/303531.jpg'>

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 

<img src='content_image/303530.jpg'>

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. 

(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 . 

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 

<img src='content_image/303529.jpg'>

