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satisfied 

where 

and 

Now, we are ready to prove the following result. 

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Lemma 2.1 If conditions ( C 1) − ( C 6) hold, then for each x ( t ) ∈ A$_{x}$ there exists a unique solution φ β x ( t $_{)}$( t ) of (2.9) such that sup t ∈ R ∥ φ β x ( t $_{)}$( t ) ∥ ≤ r$_{1}$ for a constant r$_{1}$ = 4 KP$_{2}$ α M$_{h}$ 1 + √ 1 − 16 K $^{2}$P 2 2 α 2 M$_{h}$ . 

Proof. Consider the set C$_{0}$ ( R ) of continuous functions y ( t ) satisfying ‖ y ‖$_{0}$ ≤ r$_{1}$, where ‖ y ‖$_{0}$ = sup t ∈ R ‖ y ( t ) ‖ . Define the operator Π on C$_{0}$ as 

Using the mean value theorem we obtain that 

Hence, the inequality 

is valid so that 

Let 

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