Similarly, we derive by the mean value theorem Then one can confirm that Hence, we arrive at Next, the quadratic function has two positive roots 0 < r$_{1}$ < r$_{2}$ when 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. □ 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. 7 2.1 Almost Periodic Functions ‖ H$_{2}$ ( y$_{1}$, y$_{2}$, y$_{3}$, s ) ‖ ≤ ∫ 1 0 ∥ D$_{u}$f ( θy$_{1}$ + (1 − θ ) y$_{3}$ + y$_{2}$, s ) − D$_{u}$f ( y$_{2}$, s ) ∥ dθ ‖ y$_{1}$ − y$_{3}$ ‖ ≤ ∫ 1 0 ∫ 1 0 ‖ D$_{uu}$f ( τ ( θy$_{1}$ + (1 − θ ) y$_{3}$ ) + y$_{2}$, s ) ‖ dτ ( θ ‖ y$_{1}$ ‖ + (1 − θ ) ‖ y$_{3}$ ‖ ) dθ ‖ y$_{1}$ − y$_{3}$ ‖ ≤ ∫ 1 0 ∫ 1 0 ‖ D$_{uu}$f ( τ ( θy$_{1}$ + (1 − θ ) y$_{3}$ ) + y$_{2}$, s ) ‖ dτdθ max {‖ y$_{1}$ ‖ , ‖ y$_{3}$ ‖}‖ y$_{1}$ − y$_{3}$ ‖ ≤ P$_{2}$ max {‖ y$_{1}$ ‖ , ‖ y$_{3}$ ‖}‖ y$_{1}$ − y$_{3}$ ‖ . ‖ Π y$_{1}$ ( t ) − Π y$_{2}$ ( t ) ‖ ≤ 2 KP$_{2}$ α max {‖ y$_{1}$ ‖$_{0}$ , ‖ y$_{2}$ ‖$_{0}$} ‖ y$_{1}$ − y$_{2}$ ‖$_{0}$ . ‖ Π y$_{1}$ − Π y$_{2}$ ‖$_{0}$ ≤ 2 KP$_{2}$ α max {‖ y$_{1}$ ‖$_{0}$ , ‖ y$_{2}$ ‖$_{0}$} ‖ y$_{1}$ − y$_{2}$ ‖$_{0}$ . q ( r ) = 2 KP$_{2}$ α r 2 − r + 2 KP$_{2}$ α M$_{h}$ M$_{h}$ < α 2 16 K $^{2}$P 2 ,