Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On the Stability of a Higher Functional Equation in Banach Algebras

  • Received : 2018.07.25
  • Accepted : 2019.05.10
  • Published : 2019.12.23
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Abstract

Let 𝓐 and 𝓑 be real (or complex) algebras. We investigate the stability of a sequence F = {f0, f1, ⋯, fn, ⋯ } of mappings from 𝓐 into 𝓑 satisfying the higher functional equation: $$f_n(x+y+zw)=f_n(x)+f_n(y)+\;{\normalsize\sum\limits_{\tiny{i+j=n \atop i{\leq}j}}}\;[f_i(z)f_j(w)+c_{ij}f_i(w)f_j(z)]$$ for each n = 0, 1, ⋯ and all x, y, z, w ∈ 𝓐, where $$c_{ij}=\left\{1{\text{ if }}i{\neq}j,\\0{\text{ if }}i=j.\right.$$.

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