Automatic continuity of almost 3-homomorphisms and almost 3-Jordan homomorphisms

For a Banach algebra A and a semisimple commutative Banach algebra B, we prove that every 3-homomorphism φ:A⟶B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :A\longrightarrow B$$\end{document} is automatically continuous, and we generalize this result for almost 3-homomorphism. We also investigate the automatic continuity of almost 3-Jordan homomorphisms and show that if A is a commutative Banach algebra, then every almost 3-Jordan homomorphism φ:A⟶C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :A\longrightarrow \mathbb {C}$$\end{document} is automatically continuous.