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

Abstract

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