Hyers–Ulam Stability of the Quadratic Functional Equation

Elhoucien Elqorachi (), Youssef Manar () and Themistocles M. Rassias ()
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Elhoucien Elqorachi: University Ibn Zohr
Youssef Manar: University Ibn Zohr
Themistocles M. Rassias: National Technical University of Athens

Chapter Chapter 8 in Functional Equations in Mathematical Analysis, 2011, pp 97-105 from Springer

Abstract: Abstract We prove a stability theorem for the quadratic functional equation $$f(x + y) + f(x + \sigma (y)) = 2f(x) + 2f(y),\quad x,y \in G,$$ where G is an abelian group and σ is an involution of G. We also prove that for functions f from G to an inner product space E, the inequality $$\|2f(x) + 2f(y) - f(x + \sigma (y))\| \leq \| f(x + y)\|,\quad x,y \in G.$$ implies that f is a solution to the equation.

Keywords: Hyers–Ulam stability; Quadratic functional equation; Group homomorphisms; Unbounded Cauchy difference; Abelian group (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-1-4614-0055-4_8

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