On stability of almost surjective functional equations of uniformly convex Banach spaces

Abstract

Abstract Let Y be a uniformly convex space with power type p, and let ( G , + ) $(G,+)$ be an abelian group, δ , ε ≥ 0 $\delta ,\varepsilon \geq 0$ , 0 < r < 1 $0< r<1$ . We first show a stability result for approximate isometries from an arbitrary Banach space into Y. This is a generalization of Dolinar’ results for ( δ , r ) $(\delta ,r)$ -isometries of Hilbert spaces and L p $L_{p}$ ( 1 < p < ∞ $1< p<\infty $ ) spaces. As a result, we prove that if a standard mapping F : G → Y $F:G\rightarrow Y$ satisfies d ( u , F ( G ) ) ≤ δ ∥ u ∥ r $d(u,F(G))\leq \delta \|u\|^{r}$ for every u ∈ Y $u\in Y$ and | ∥ F ( x ) − F ( y ) ∥ − ∥ F ( x − y ) ∥ | ≤ ε , x , y ∈ G , $$ \bigl\vert \bigl\Vert F(x)-F(y) \bigr\Vert - \bigl\Vert F(x-y) \bigr\Vert \bigr\vert \leq \varepsilon , \quad x,y \in G, $$ then there is an additive operator A : G → Y $A:G\rightarrow Y$ such that ∥ F ( x ) − A x ∥ = o ( ∥ F ( x ) ∥ ) as ∥ F ( x ) ∥ → ∞ . $$ \bigl\Vert F(x)-Ax \bigr\Vert =o\bigl( \bigl\Vert F(x) \bigr\Vert \bigr) \quad \text{as } \bigl\Vert F(x) \bigr\Vert \rightarrow \infty . $$

Keywords

• Functional equation
• ( δ , r ) $(\delta ,r)$ -surjective
• Additive function
• Linear isometry
• Uniformly convex space