Hyers–Ulam stability of norm-additive functional equations via ( δ , ϵ ) $(δ , ε )$ -isometries - Muhammad Sarfraz; Jiang Zhou; Yongjin Li (Journal of Inequalities and Applications)
Abstract This research examines the Hyers–Ulam stability of norm-additive functional equations expressed as ∥ ξ ( g h − 1 ) ∥ = ∥ ξ ( g ) − ξ ( h ) ∥ , ∥ ξ ( g h ) ∥ = ∥ ξ ( g ) + ξ ( h ) ∥ , $$\begin{aligned}& \|\xi (gh^{-1})\|=\|\xi (g)-\xi (h)\|,\\& \|\xi (gh)\|=\|\xi (g)+\xi (h)\|, \end{aligned}$$ through the utilization of ( δ , ϵ ) $(\delta , \epsilon )$ -isometries. In this context, ξ : G → X $\xi : G \to X$ represents a surjective (δ-surjective) mapping, where G denotes noncommutative group (arbitrary group) and X signifies a Banach space (or a real Banach space).