content_image/317450.jpg content_image/317451.jpg content_image/317452.jpg content_image/317453.jpg content_image/317454.jpg
Similarly, we derive by the mean value theorem 

<img src='content_image/317450.jpg'>

Then one can confirm that 

Hence, we arrive at 

Next, the quadratic function 

has two positive roots 0 < r$_{1}$ < r$_{2}$ when 

<img src='content_image/317451.jpg'>

<img src='content_image/317452.jpg'>

<img src='content_image/317453.jpg'>

<img src='content_image/317454.jpg'>

which is satisfied by (C6). Moreover, q $^{′}$( r$_{1}$ ) < $_{0}$, i.e., 2 KP$_{2}$ α r$_{1}$ < 1 $_{2}$. This means that for the ball B$_{r}$$_{1}$ = { y ∈ C$_{0}$ ( R ) : ‖ y ‖$_{0}$ ≤ r$_{1}$ $_{}}$, the map Π : B$_{r}$$_{1}$ → B$_{r}$$_{1}$ is a contraction with coefficient 1 / $_{2}$. Note that r$_{1}$ ≤ 4 KP$_{2}$ α M$_{h}$, i.e., the smaller M$_{h}$ the smaller $_{B$_{r}$$_{1}$}$, as we can expect this. Thus, (2.9) admits a unique solution from C$_{0}$ ( R ) . This finalizes the proof of the lemma. □ 

## 2.1 Almost Periodic Functions 

A continuous function F ( t ) is said to be almost periodic, if for any ϵ > 0 there exists l > 0 such that for any interval with length l there exists a number ω in this interval satisfying ‖F ( t + ω ) − F ( t ) ‖ < ϵ for all t ∈ R [29, 34, 45]. Now, let us show that the bounded solution is almost periodic. 

Lemma 2.2 If in addition in Lemma 2.1, x ( t ) and ν$_{β}$ ( t ) are almost periodic then φ β x ( t $_{)}$( t ) of (2.9) is almost periodic as well. Moreover, when x ( t ) and ν$_{β}$ ( t ) are both i ¯ l -periodic for some i ∈ N , then φ β x ( t $_{)}$( t ) is also i ¯ l -periodic. 

