Proof. Consider the set B AP r$_{1}$ of all almost periodic functions of $_{B$_{r}$$_{1}$}$. By (2.9), the function z ( t ) = Π y ( t ) from the above proof is defined as the bounded solution of where and If y ( t ) is almost periodic, by [29], A$_{β}$ ( t ) and f$_{β}$ ( t ) are almost periodic. Then using results of [20, pp. 72], we know that z ( t ) is also almost periodic. This means that Π : B AP r$_{1}$ → B AP r$_{1}$ . Since Π is contractive, its fixed point φ β x ( t $_{)}$( t ) is almost periodic. Next, when x ( t ) and ν$_{β}$ ( t ) are both i ¯ $_{l}$-periodic for some i ∈ N , then y ( t ) = φ β x ( t $_{)}$( t + i ¯ l ) satisfies (2.9) and ‖ y ‖ ≤ $_{r$_{1}$}$. The uniqueness of such a solution implies y ( t ) = y ( t $_{)}$, so φ β x ( t $_{)}$( t ) is also i ¯ $_{l}$-periodic. The proof is finished. □ In the original paper of Li and Yorke [35], infinitely many periodic solutions, which are separated from the elements of a scrambled set, is introduced. In the present study, we make use of the definition defined in [10], where Akhmet et al. modified the definition of Li-Yorke chaos by considering infinitely many almost periodic solutions as a basis of the developed continuous Li-Yorke chaos instead of periodic solutions, which are separated from the elements of the scrambled set, to generate the Li-Yorke chaos. Let us denote by a collection of functions, where K ⊂ R p is a bounded region. Since the concept of chaotic set of functions is used in the theoretical discussions, let us explain briefly the ingredients of Li-Yorke chaos for the set B , which are introduced in paper [10]. The proofs indicated in Section 3 are predicated on the definitions of these ingredients. For more information about Devaney and Li-Yorke chaos, one can see [12, 19, 21, 31, 35, 40]. Let us introduce the following ingredients of Li-Yorke chaos for the set B . 8 (LY1) A couple of functions ( ψ ( t ) , ψ ( t ) ∈ B × B is called proximal if for arbitrary small ϵ > 0 and arbitrary large E > 0 , there exists an interval J of length not less than E such that ∥ ψ ( t ) − ψ ( t ) ∥ < B = { ψ ( t ) | ψ : R → K is continuous$_{}}$ (2.12) 2.2 Li-Yorke Chaos f$_{β}$ ( t ) = f ( y ( t ) + ν$_{β}$ ( t ) , t ) − f ( ν$_{β}$ ( t ) , t ) − D$_{u}$f ( ν$_{β}$ ( t ) , t ) y ( t ) + h ( x ( t )) . A$_{β}$ ( t ) = D$_{u}$f ( ν$_{β}$ ( t ) , t ) z ′ = A$_{β}$ ( t ) z + f$_{β}$ ( t ) , (2.11)