Hyers\u2013Ulam stability and hyperstability of a Jensen-type functional equation on 2-Banach spaces

Abstract

The main aim of this paper is to establish the Hyers–Ulam stability and hyperstability of a Jensen-type quadratic mapping in 2-Banach spaces. That is, we prove the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form g(x+y2+z)+g(x+y2−z)+g(x−y2+z)+g(x−y2−z)=g(x)+g(y)+4g(z),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ g \\biggl( \\frac{x+y}{2} + z \\biggr) + g \\biggl( \\frac{x+y}{2} - z \\biggr) + g \\biggl( \\frac{x-y}{2} + z \\biggr) + g \\biggl( \\frac{x-y}{2} - z \\biggr) = g(x) + g(y) + 4 g(z), $$\\end{document} in 2-Banach spaces by using the Hyers direct method.