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aequationes mathematicae

, Volume 67, Issue 3, pp 216–224 | Cite as

On quadratic differences that depend on the product of arguments - revisited

  • J. K. Chung
  • Prasanna K. Sahoo
  • László Székelyhidi
Research paper
  • 26 Downloads

Summary.

Let \( \mathbb{K} \) be a field of real or complex numbers and \( \mathbb{K}_0 \) denote the set of nonzero elements of \( \mathbb{K} \). Let \( \mathbb{G} \) be an abelian group. In this paper, we solve the functional equation f 1 (x + y) + f 2 (x - y) = f 3 (x) + f 4 (y) + g(xy) by modifying the domain of the unknown functions f 3, f 4, and g from \( \mathbb{K} \) to \( \mathbb{K}_0 \) and using a method different from [3]. Using this result, we determine all functions f defined on \( \mathbb{K}_0 \) and taking values on \( \mathbb{G} \) such that the difference f(x + y) + f (x - y) - 2 f(x) - 2 f(y) depends only on the product xy for all x and y in \( \mathbb{K}_0 \)

Mathematics Subject Classification (2000).

Primary 39B22. 

Keywords.

Additive function Biadditive function Functional equation Local polynomial n-additive map Quadratic map Quadratic difference. 

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Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  • J. K. Chung
    • 1
  • Prasanna K. Sahoo
    • 2
  • László Székelyhidi
    • 3
  1. 1.Department of Applied MathematicsSouth China University of TechnologyGuangzhouPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  3. 3.Institute of Mathematics and InformaticsKossuth Lajos UniversityDebrecenHungary

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