we obtain that Therefore, it can be verified for each k ∈ $^{N}$that Let us denote It can be confirmed by means of inequality (5.8) that ρ$_{k}$ ( φ$_{i}$$_{n}$$_{+}$$_{j}$$_{n}$, φ$_{j}$$_{n}$ ) ≥ ϵ$_{0}$, k ∈ $^{N}$. Thus, ρ ( φ$_{i}$$_{n}$$_{+}$$_{j}$$_{n}$, φ$_{j}$$_{n}$ ) ≥ ϵ$_{0}$ for each n ∈ $^{N}$. Consequently, the function φ ( t ) is unpredictable. □ One of the possible ways useful for applications to generate unpredictable functions from a given one is provided in the next theorem. Theorem 5.2 Let φ : R → H be an unpredictable function, where H is a bounded subset of R $^{p}$. If h : H → R q is a function such that there exist positive numbers L$_{1}$ and L$_{2}$ satisfying L$_{1}$ ‖ u − u ‖ ≤ ‖ h ( u ) − h ( u ) ‖ ≤ L$_{2}$ ‖ u − u ‖ for all u, u ∈ H , then the function ψ : $^{R}$→ R q defined as ψ ( t ) = h ( φ ( t )) is also unpredictable. Proof. Since φ ( t ) is an unpredictable function, there exist a positive number ϵ$_{0}$ and sequences { t$_{n}$ } and { τ$_{n}$ } , both of which diverge to infinity, such that lim n $_{→∞}$ρ ( φ$_{t}$$_{n}$, φ ) = 0 and ρ ( φ$_{t}$$_{n}$$_{+}$$_{τ}$$_{n}$, φ$_{τ}$$_{n}$ ) ≥ ϵ$_{0}$ for each n ∈ $^{N}$. Firstly, we will show that lim n →∞ ρ ( ψ$_{t}$$_{n}$, ψ ) = 0 . Fix an arbitrary positive number ϵ, and let us denote α = max { 1 , L$_{2}$ } . There exists a natural number n$_{0}$ such that for all n ≥ n$_{0}$ the inequality ρ ( φ$_{t}$$_{n}$ , φ ) < ϵ/α is valid. For each k ∈ $^{N}$, one can confirm that Therefore, it can be verified for each n ≥ n$_{0}$ that 10 ρ ( ψ$_{t}$$_{n}$, ψ ) ≤ αρ ( φ$_{t}$$_{n}$ , φ ) < ϵ. ρ$_{k}$ ( ψ$_{t}$$_{n}$ , ψ ) = min { 1 , sup s ∈ [ − k,k ] ‖ h ( φ ( t$_{n}$ + s )) − h ( φ ( s )) ‖ } ≤ min 1 , L$_{2}$ sup s ∈ [ − k,k ] ‖ φ ( t$_{n}$ + s ) − φ ( s ) ‖ } ≤ αρ$_{k}$ ( φ$_{t}$$_{n}$ , φ ) . ϵ$_{0}$ = min { 1 , ( e γϵ$_{0}$ − 1) ‖ d$_{1}$ − d$_{2}$ ‖ γ (1 + e $^{γ}$) } . sup s ∈ [ − k,k ] ‖ φ ( i$_{n}$ + j$_{n}$ + s ) − φ ( j$_{n}$ + s ) ‖ ≥ ( e γϵ$_{0}$ − 1) ‖ d$_{1}$ − d$_{2}$ ‖ γ (1 + e $^{γ}$) . (5.8) ‖ φ ( i$_{n}$ + j$_{n}$ + 1) − φ ( j$_{n}$ + 1) ‖ ≥ ∥ ζ$_{j}$$_{n}$ − j$_{n}$ ∫ ζ$_{i}$$_{n}$$_{+}$$_{j}$$_{n}$ − i$_{n}$ − j$_{n}$ e − γ (1 − u $^{)}$( d$_{1}$ − d$_{2}$ ) du ∥ − e − γ ‖ φ ( i$_{n}$ + j$_{n}$ ) − φ ( j$_{n}$ ) ‖ ≥ e γϵ$_{0}$ − 1 γe γ ‖ d$_{1}$ − d$_{2}$ ‖ − e − γ ‖ φ ( i$_{n}$ + j$_{n}$ ) − φ ( j$_{n}$ ) ‖ .