Let Y be a uniformly convex space with power type p, and let (G,+) be an abelian group, delta ,varepsilon geq 0, 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)-isometries of Hilbert spaces and L_{p} (1< p<infty ) spaces. As a result, we prove that if a standard mapping F:Grightarrow Y satisfies d(u,F(G))leq delta |u|^{r} for every uin Y and |∥F(x)−F(y)∥−∥F(x−y)∥|≤ε,x,y∈G,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\bigl\\vert \\bigl\\Vert F(x)-F(y) \\bigr\\Vert - \\bigl\\Vert F(x-y) \\bigr\\Vert \\bigr\\vert \\leq \\varepsilon , \\quad x,y \\in G, $$\\end{document} then there is an additive operator A:Grightarrow Y such that ∥F(x)−Ax∥=o(∥F(x)∥)as ∥F(x)∥→∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\bigl\\Vert F(x)-Ax \\bigr\\Vert =o\\bigl( \\bigl\\Vert F(x) \\bigr\\Vert \\bigr) \\quad \ ext{as } \\bigl\\Vert F(x) \\bigr\\Vert \\rightarrow \\infty . $$\\end{document}