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## 3 Unpredictable sequences 

The following theorem is concerned with the existence of an unpredictable solution of the discrete equa- tion (2.1). 

Theorem 3.1 The bounded solution { φ$_{n}$ } , n ∈ Z , of equation (2.1) is unpredictable under the conditions ( C 1) − ( C 3) . 

Proof. Fix an arbitrary positive number ϵ, and suppose that γ is a positive number satisfying 

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Let n$_{1}$ and n$_{2}$ be integers such that n$_{2}$ > n$_{1}$ , and take a natural number E with 

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Since { ψ$_{n}$ } , 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 ‖ ψ$_{n}$$_{+}$$_{ζ}$$_{k}$ − ψ$_{n}$ ‖ → 0 uniformly as k → ∞ for each n with n$_{1}$ − E − 1 ≤ n ≤ n$_{2}$ − 1 and ‖ ψ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − ψ$_{η}$$_{k}$ ‖ ≥ ϵ$_{0}$ for each k ∈ N . 

First of all, we will show that ‖ φ$_{n}$$_{+}$$_{ζ}$$_{k}$ − φ$_{n}$ ‖ → 0 uniformly as k → ∞ for each n with n$_{1}$ ≤ n ≤ n$_{2}$ . There exists a natural number k$_{0}$, independent of n, such that for each k ≥ k$_{0}$ the inequality ‖ ψ$_{n}$$_{+}$$_{ζ}$$_{k}$ − ψ$_{n}$ ‖ < γϵ is valid whenever n$_{1}$ − E − 1 ≤ n ≤ n$_{2}$ − 1 . 

Fix an arbitrary integer k ≥ k$_{0}$ . One can obtain using the relation (2.2) that 

Therefore, for n$_{1}$ − E ≤ n ≤ n$_{2}$, we have 

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Let us denote 

and 

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