A fixed point approach to the stability of the mixed type functional equation (Q2911454)

The article deals with the Ulam stability property for the following functional equation NEWLINE\[NEWLINEf(x + y + z) - f(x + y) - f(y + z) - f(x + z) + f(x) + f(y) + f(z) = 0.NEWLINE\]NEWLINE More exactly, the authors study the conditions for a function \(\varphi(x,y,z)\) under which the inequality NEWLINE\[NEWLINE\|f(x + y + z) - f(x + y) - f(y + z) - f(x + z) + f(x) + f(y) + f(z)\| \leq \varphi(x,y,z)NEWLINE\]NEWLINE implies the representation \(f(x)\) in the form \(f(x) = F(x) + \omega(x)\), where \(F(x)\) is a homogeneous quadratic operator, and \(\omega\) is a `small' operator. This `smallness' is determined by the inequality NEWLINE\[NEWLINE\|\omega(x)\| \leq \frac{c(\varphi(x,x,-x) + \varphi(-x,-x,x))}{1 - L}NEWLINE\]NEWLINE (\(c\), \(L\) are constants, \(0 < L < 1\)). Under these conditions, the operator \(F\) if defined as a limit of some explicit approximations. As corollaries are the partial cases considered when NEWLINE\[NEWLINE\varphi(x,y,z) = \theta(\|x\|^p + \|y\|^p + \|z\|^p), \quad \varphi(x,y,z) = \theta\|x\|^p\|y\|^q\|z\|^r,NEWLINE\]NEWLINE and also the Ulam stability property for the classical functional equation NEWLINE\[NEWLINEf(x + y) - f(x) - f(y) = 0, \quad f(x + y) + f(x - y) - 2f(x) - 2f(y) = 0.NEWLINE\]