where ϵ$_{0}$ = ϵ$_{0}$ 1 + ‖ A ‖ + L$_{f}$ . For each k ∈ N , let us take ˜ η$_{k}$ = η$_{k}$ + 1 if ‖ φ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$$_{+1}$ − φ$_{η}$$_{k}$$_{+1}$ ‖ ≥ ‖ φ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − φ$_{η}$$_{k}$ ‖ , and we set ˜ η$_{k}$ = η$_{k}$ otherwise. Clearly, ˜ η$_{k}$ → ∞ as k → ∞ . According to inequality (3.5), we have ‖ φ$_{ζ}$$_{k}$$_{+}$$_{˜}$ η$_{k}$ − φ$_{˜}$ $_{η$_{k}$}$‖ ≥ ϵ$_{0}$ for each k ∈ N . Consequently, the bounded solution { φ$_{n}$ } , n ∈ Z , of (2.1) is unpredictable. □ A possible way to obtain a different unpredictable sequence from a given one is mentioned in the following theorem. Theorem 3.2 Suppose that { ϕ$_{n}$ } , n ∈ Z , is an unpredictable sequence such that ϕ$_{n}$ ∈ Λ for each n , where Λ is a bounded subset of R $^{p}$. If h : Λ → R q is a function such that there exist positive numbers L$_{1}$ and L$_{2}$ with L$_{1}$ ‖ s$_{1}$ − s$_{2}$ ‖ ≤ ‖ h ( s$_{1}$ ) − h ( s$_{2}$ ) ‖ ≤ L$_{2}$ ‖ s$_{1}$ − s$_{2}$ ‖ for all s$_{1}$, s$_{2}$ ∈ Λ , then the sequence { ϕ$_{n}$ } defined through the equation ϕ$_{n}$ = h ( ϕ$_{n}$ ) , n ∈ Z , is also unpredictable. Proof. Since { ϕ$_{n}$ } , n ∈ Z , is an unpredictable sequence, there exist a positive number ϵ$_{0}$ and sequences { ζ$_{k}$ } , { η$_{k}$ } , k ∈ N , of positive integers both of which diverge to infinity such that ‖ ϕ$_{n}$$_{+}$$_{ζ}$$_{k}$ − ϕ$_{n}$ ‖ → 0 uniformly as k → ∞ for each n in bounded intervals of integers and ‖ ϕ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − ϕ$_{η}$$_{k}$ ‖ ≥ ϵ$_{0}$ for each k ∈ N . Fix an arbitrary positive number ϵ, and let n$_{1}$ and n$_{2}$ be any two integers such that n$_{2}$ > n$_{1}$ . One can find a natural number k$_{0}$ , which does not depend on n , such that for each k ≥ k$_{0}$ we have ‖ ϕ$_{n}$$_{+}$$_{ζ}$$_{k}$ − ϕ$_{n}$ ‖ < ϵ/L$_{2}$ whenever n$_{1}$ ≤ n ≤ n$_{2}$ . Therefore, the inequality is satisfied for each k ≥ k$_{0}$ and each n with n$_{1}$ ≤ n ≤ n$_{2}$ . This shows that ∥ ϕ$_{n}$$_{+}$$_{ζ}$$_{k}$ − ϕ$_{n}$ ∥ → 0 uniformly as k → ∞ on bounded intervals of integers. On the other hand, for each k ∈ N , we have that In the next section, an example which supports the result of Theorem 3 is provided. Consequently, { ϕ$_{n}$ } , n ∈ Z , is an unpredictable sequence. 4 An example Consider the logistic map where n ∈ Z and λ is a parameter. Based on the result of the papers [12] and [18], it was demonstrated in paper [13] that for λ ∈ [3 + (2 / 3) 1 / $^{2}$, 4] the map (4.6) possesses an unpredictable solution. For such values of the parameter, the unit interval [0 , 1] is invariant under the iterations of the map [19]. 6 u$_{n}$$_{+1}$ = λu$_{n}$ (1 − u$_{n}$ ) , (4.6) ∥ ϕ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − ϕ$_{η}$$_{k}$ ∥ ≥ L$_{1}$ ‖ ϕ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − ϕ$_{η}$$_{k}$ ‖ ≥ L$_{1}$ϵ$_{0}$. ∥ ϕ$_{n}$$_{+}$$_{ζ}$$_{k}$ − ϕ$_{n}$ ∥ ≤ L$_{2}$ ‖ ϕ$_{n}$$_{+}$$_{ζ}$$_{k}$ − ϕ$_{n}$ ‖ < ϵ