Abstract
We prove a general stability theorem of an n-dimensional quadratic-additive type functional equation \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} Df(x_1, x_2, \\ldots , x_n) = \\sum _{i=1}^m c_i f \\big ( a_{i1}x_1 + a_{i2}x_2 + \\cdots + a_{in}x_n \\big ) = 0 \\end{aligned}$$\\end{document}Df(x1,x2,…,xn)=∑i=1mcif(ai1x1+ai2x2+⋯+ainxn)=0by applying the direct method.
Highlights
Throughout this paper, let V and W be real vector spaces, let X and Y be a real normed space and a real Banach space, respectively, and let N0 denote the set of all nonnegative integers
We prove a general stability theorem that can be applied to the Hyers-Ulam stability of a large class of functional equations of the form Df (x1, x2, . . . , xn) = 0, which includes quadratic-additive type functional equations
If we confine ourselves to the stability problems of the quadratic-additive type functional equations, the condition (12) is a direct consequence of (11)
Summary
Throughout this paper, let V and W be real vector spaces, let X and Y be a real normed space and a real Banach space, respectively, and let N0 denote the set of all nonnegative integers. When |a| < 1, in view of Lemma 1, there exists a unique mapping F : V → Y satisfying the equalities in (12) and the inequality (13), since the inequality
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