Cauchy–Rassias Stability of Cauchy–Jensen Additive Mappings in Banach Spaces

Let X, Y be vector spaces. It is shown that if a mapping f : X → Y satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f{\left( {\frac{{x + y}} {2} + z} \right)} + f{\left( {\frac{{x - y}} {2} + z} \right)} = f{\left( x \right)} + 2f{\left( z \right)}, $$\end{document}(0.1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f{\left( {\frac{{x + y}} {2} + z} \right)} - f{\left( {\frac{{x - y}} {2} + z} \right)} = f{\left( y \right)}, $$\end{document}(0.2) or\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 2f{\left( {\frac{{x + y}} {2} + z} \right)} = f{\left( x \right)} + f{\left( y \right)} + 2f{\left( z \right)} $$\end{document} (0.3) for all x, y, z ∈ X, then the mapping f : X → Y is Cauchy additive. Furthermore, we prove the Cauchy–Rassias stability of the functional equations (0.1), (0.2) and (0.3) in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.