content_image/299506.jpg content_image/299505.jpg content_image/299503.jpg content_image/299502.jpg content_image/299501.jpg content_image/299500.jpg
Proof. Using the technique for quasi-linear equations [13], one can confirm under the conditions ( C 1) − ( C 3) that system (4.2) possesses a unique bounded on R solution φ ( t ) which satisfies the relation 

<img src='content_image/299506.jpg'>

Moreover, sup t ∈ R ‖ φ ( t ) ‖ ≤ M$_{φ}$, where M$_{φ}$ = K ( M$_{f}$ + M$_{g}$ ) ω and M$_{g}$ = sup t ∈ R ‖ g ( t ) ‖ . The solution φ ( t ) is uniformly continuous on R since sup t ∈ R ‖ φ $^{′}$( t ) ‖ ≤ ‖ A ‖ M$_{φ}$ + M$_{f}$ + M$_{g}$. 

Suppose that x ( t ) is a solution of (4.2) such that x ( t$_{0}$ ) = x$_{0}$ for some t$_{0}$ ∈ R and x$_{0}$ ∈ R $^{m}$. It can be verified that 

and therefore, φ ( t ) is uniformly exponentially stable. 

<img src='content_image/299505.jpg'>

Since the function g ( t ) is unpredictable, there exist a positive number ϵ$_{0}$ ≤ 1 and sequences { t$_{n}$ } , { τ$_{n}$ } , both of which diverge to infinity, such that d ( g$_{t}$$_{n}$, g ) → 0 as n → ∞ and d ( g$_{t}$$_{n}$$_{+}$$_{τ}$$_{n}$, g$_{τ}$$_{n}$ ) ≥ ϵ$_{0}$ for all n ∈ N , where the distance function d is given by (3.1). 

First of all, we shall show that d ( φ$_{t}$$_{n}$, φ ) → 0 as n → ∞ . Fix an arbitrary small positive number ϵ < 1 and suppose that α is a positive number satisfying α ≤ ω − KL$_{f}$ 2 ω + K − 2 KL$_{f}$ . Let k$_{0}$ be a sufficiently large natural number such that 

<img src='content_image/299503.jpg'>

There exists a natural number n$_{0}$ such that if n ≥ n$_{0}$ then d ( g$_{t}$$_{n}$, g ) < 2 − 2 k 0 αϵ. Therefore, for n ≥ n$_{0}$, the inequality ρ$_{2}$$_{k}$$_{0}$ ( g$_{t}$$_{n}$, g ) < αϵ is valid. Since αϵ < 1 , we have that ‖ g ( t$_{n}$ + s ) − g ( s ) ‖ < αϵ for s ∈ [ − 2 k$_{0}$, 2 k$_{0}$ ] . 

Making use of the relation (4.3), one can obtain that 

Thus, if s belongs to the interval [ − 2 k$_{0}$, 2 k$_{0}$ ] , then it can be verified that 

<img src='content_image/299502.jpg'>

<img src='content_image/299501.jpg'>

Now, let us define the functions ψ$_{n}$ ( s ) = e ωs ‖ φ ( t$_{n}$ + s ) − φ ( s ) ‖ , n ≥ n$_{0}$. Inequality (4.5) implies that 

<img src='content_image/299500.jpg'>

