By Gobel R., Hill P., Liebert W. (eds.)

ISBN-10: 0821851780

ISBN-13: 9780821851784

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According to (a) there is d > 0 with Γ (c) > d Then c > 0. Γ kn +1 (t) = Γ Γ kn (t) ≥ d n ∈ N, which is a contradiction to (7). Thus we have proved (b). Finally, we prove (c). So, fix t ∈ R+ and k ∈ N. First, suppose that Γ is not continuous at 0. Then, according to (b), there is m ∈ N such that Γ n (t) = 0 n > m. Next, by subadditivity of Γ , we get ∞ ∞ m Γ n (t) Γ ≤ n=k Γ n+1 (t) + Γ n=k Γ n (t) n=m+1 m = Γ n+1 (t) + Γ (0) , n=k whence, by (a), ∞ ∞ m Γ n (t) ≤ Γ Γ n+1 (t) = n=k n=k Γ n (t).

Tabor in [54, pp. 67–68], several authors (see [2, 3, 9, 19, 30, 42]) studied stability of various particular cases of (38). Corollary 2 Let E := A + B = 1, A = −B, S ⊂ X be nonempty, s : X → X be given by s(x) := (α + β)x + γ x ∈ X, ϕ : S 2 → R, and f : S → Y satisfy f (αx + βy + γ ) − Af (x) − Bf (y) − C ≤ ϕ(x, y) (39) 54 A. Bahyrycz and J. Brzd¸ek for every x, y ∈ S with αx + βy + γ ∈ S. Suppose that there is ε ∈ {−1, 1} such that s ε (S) ⊂ S and, for every x, y ∈ S, ∞ H (x) := |E|−iε ϕ(s iε (x), s iε (x)) < ∞, i=0 lim inf |E −nε ϕ(s nε (x), s nε (y))| = 0.

Appl. 4, 581–588 (2002) 43. : Functional inclusions on square-symmetric groupoids and Hyers-Ulam stability. Math. Inequal. Appl. 7, 419–428 (2004) 44. : Selections of set-valued maps satisfying functional inclusions on square-symmetric grupoids. , Brzd¸ek, J. ) Functional Equations in MathematicalAnalysis. Springer Optimization and Its Applications, vol. 52, pp. 261–272. Springer, New York (2012) 45. : On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–130 (1982) 46.

### Abelian Group Theory and Related Topics by Gobel R., Hill P., Liebert W. (eds.)

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