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 $$\varphi$$.

Mathematics Subject Classification: 39B22, 39B72, .

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