On the stability of the translation equation and dynamical systems

Stability has many names

“…all functions in consideration are in this 986 Z. Moszner AEM class) is strongly stable with δ = ε 10 [18]. Moreover, the solution, which is the approximation, is continuous [18].…”

On the Normal Stability of Functional Equations

“…Przebieracz has proved in [22], by using the result of J. Chudziak [7], the following Ulam-Hyers stability theorem of equation (1) in the class of continuous functions.…”

“…1. For I = R the function Ψ(δ) = 9δ is "good" [22]. For I bounded the function Ψ(δ) = 10δ for δ ∈ (0, 2 5 ) and Ψ(δ) = |I| (the length of I) for δ ∈ [ 2 5 , +∞) is the measure of b-stability for which inf Ψ(0, +∞) = 0.…”

“…From the above table this dynamical system (i.e., the system of the equations which determines this dynamical system) is stable only for I = R (from [22] the measure Φ( ) = 9 , thus this stability is normal) and it is uniformly b-stable only for I bounded (here this b-stability is not normal by the proposition for dynamical systems analogous to Proposition 4) or I = R (from [22] the measure Ψ(δ) = 9δ, thus this b-stability is normal too). And vice versa.…”

Recent results on iteration theory: iteration groups and semigroups in the real case

“…More precisely, for every proper interval I = R there exists ε > 0 such that for every δ > 0 there exists H satisfying (33) with the property that for every continuous iteration group {F t , t ∈ R} with F 0 = id there exist t 0 and x 0 such that |H(t 0 , x 0 ) − F t0 (x 0 )| > ε. Earlier, in [116], she proved that in the class of continuous functions (with respect to each variable) equation (2) (31) such that (32) holds true for x ∈ cl (I\L) and t ∈ (0, +∞).…”