Opuscula Math. 28, no. 4 (2008), 517-527

Opuscula Mathematica

Remarks on the stability of some quadratic functional equations

Zygfryd Kominek

Abstract. Stability problems concerning the functional equations of the form \[f(2x+y)=4f(x)+f(y)+f(x+y)-f(x-y),\tag{1}\] and \[f(2x+y)+f(2x-y)=8f(x)+2f(y)\tag{2}\] are investigated. We prove that if the norm of the difference between the LHS and the RHS of one of equations \((1)\) or \((2)\), calculated for a function \(g\) is say, dominated by a function \(\varphi\) in two variables having some standard properties then there exists a unique solution \(f\) of this equation and the norm of the difference between \(g\) and \(f\) is controlled by a function depending on \(\varphi\).

Keywords: quadratic functional equations, stability.

Mathematics Subject Classification: 39B22, 39B72, .

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  • Zygfryd Kominek
  • Silesian University, Institute of Mathematics, ul. Bankowa 14, 40-007 Katowice, Poland
  • Received: 2008-01-24.
  • Revised: 2008-10-14.
  • Accepted: 2008-07-04.
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Zygfryd Kominek, Remarks on the stability of some quadratic functional equations, Opuscula Math. 28, no. 4 (2008), 517-527

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