Remarks on the stability of some quadratic functional equations

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 \(\varphi\).