# A Variant of a Generalized Quadratic Functional Equation on Groups

A Variant of a Generalized Quadratic Functional Equation on Groups Let G be a group and $$\mathbb {C}$$ C the field of complex numbers. Suppose $$\sigma : G \rightarrow G$$ σ : G → G is an involutive endomorphism, that is, $$\sigma$$ σ is an endomorphism of G and it satisfies the condition $$\sigma (\sigma (x)) = x$$ σ ( σ ( x ) ) = x for all x in G. In this paper, we find the solutions $$f, g, h, k : G\rightarrow \mathbb {C}$$ f , g , h , k : G → C of the equation $$f(xy) + g(\sigma (y) x) = h(x) + k(y)$$ f ( x y ) + g ( σ ( y ) x ) = h ( x ) + k ( y ) $$\text {for all } x, y \in G$$ for all x , y ∈ 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 $$f, g, h , k : G \times G \rightarrow \mathbb {C}$$ f , g , h , k : G × G → C of the equation $$f(pr, qs)+g(sp,rq) = h(p,q) + k(r,s)$$ f ( p r , q s ) + g ( s p , r q ) = h ( p , q ) + k ( r , s ) for all $$p, q, r, s \in G$$ p , q , r , s ∈ G assuming f and g to be central functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# A Variant of a Generalized Quadratic Functional Equation on Groups

, Volume 72 (2) – Feb 28, 2017
17 pages

Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-017-0662-z
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Results in MathematicsSpringer Journals

Published: Feb 28, 2017

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