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Stability of the Quadratic Functional Equation

  • Donald H. Hyers
  • George Isac
  • Themistocles M. Rassias
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)

Abstract

The quadratic functional equation
$$ f\left( {x + y} \right) + f\left( {x - y} \right) - 2f\left( x \right) - 2f\left( y \right) = 0$$
(3.1)
clearly has f(x) = cx2 as a solution with c an arbitrary constant when f is a real function of a real variable. We define any solution of (3.1) to be a quadratic function, even in more general contexts. We shall be interested in functions f: E1→ E2 where both E1 and E2 are real vector spaces, and we need a few facts concerning the relation between a quadratic function and a biadditive function sometimes called its polar. This relation is explained in Proposition 1, p. 166, of the book by J. Aczél and J. Dhombres (1989) for the case where E2 = R, but the same proof holds for functions f: E1→ E2. It follows then that f: E1→ E2 is quadratic if and only if there exists a unique symmetric function B: E1 × E1 E2, additive in x for fixed y, such that f (x) = B(x, x). The biadditive function B, the polar of f, is given by
$$B\left( {x,y} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ - \\ 4 \end{array}} \right)\left( {f\left( {x + y} \right) - f\left( {x - y} \right)} \right)$$

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Donald H. Hyers
  • George Isac
    • 1
  • Themistocles M. Rassias
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

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