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A Variant of a Generalized Quadratic Functional Equation on Groups

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Abstract

Let G be a group and \(\mathbb {C}\) the field of complex numbers. Suppose \(\sigma : G \rightarrow G\) is an involutive endomorphism, that is, \(\sigma \) is an endomorphism of G and it satisfies the condition \(\sigma (\sigma (x)) = x\) for all x in G. In this paper, we find the solutions \(f, g, h, k : G\rightarrow \mathbb {C}\) of the equation \(f(xy) + g(\sigma (y) x) = h(x) + k(y)\) \(\text {for all } x, y \in G\) assuming f and g to be central functions. This equation is a variant of a generalized quadratic functional equation on groups with an involutive endomorphism. As an application, using the solutions of this equation, we find the solutions \(f, g, h , k : G \times G \rightarrow \mathbb {C}\) of the equation \(f(pr, qs)+g(sp,rq) = h(p,q) + k(r,s)\) for all \(p, q, r, s \in G\) assuming f and g to be central functions.

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Authors and Affiliations

  1. Department of Mathematics, Robert Morris University, Pittsburgh, PA, 15108, USA

    Heather Hunt Elfen

  2. Department of Mathematics, University of Louisville, Louisville, KY, 40292, USA

    Thomas Riedel & Prasanna K. Sahoo

Authors
  1. Heather Hunt Elfen
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  2. Thomas Riedel
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  3. Prasanna K. Sahoo
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Correspondence to Thomas Riedel.

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Elfen, H.H., Riedel, T. & Sahoo, P.K. A Variant of a Generalized Quadratic Functional Equation on Groups. Results Math 72, 555–571 (2017). https://doi.org/10.1007/s00025-017-0662-z

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  • DOI: https://doi.org/10.1007/s00025-017-0662-z

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