Solution of the Ulam stability problem for nonlinear five-dimensional Euler quadratic mappings (Q5929672)

Let \(X,Y\) be linear spaces and \(f:X\longrightarrow Y\). Consider the equation NEWLINE\[NEWLINE \begin{aligned} &\bigl[ f(x_0-x_1)+f(x_1-x_2)+f(x_2-x_3)+f(x_3-x_4)+f(x_4-x_0) \bigr]\\ -&\Bigl\{ \bigl[ f(x_0-x_2)+f(x_0-x_3)+f(x_1-x_3)+f(x_1-x_4)+f(x_2-x_4) \bigr]\\ +&\bigl[ f(x_0-x_1+x_2-x_3)+f(x_0-x_1+x_2-x_4)+f(x_0-x_1+x_3-x_4)\\ +&f(x_0-x_2+x_3-x_4)+f(x_1-x_2+x_3-x_4) \bigr] \Bigr\}=0. \end{aligned} \tag{1}NEWLINE\]NEWLINE for all \((x_0,x_1,x_2,x_3,x_4)\in X^5\). NEWLINENEWLINENEWLINENow let \(X\) be a normed linear space and let \(Y\) be a real complete normed linear space. It is proved that if the left hand side of (1) is bounded, then close to \(f\) there exists a unique mapping \(Q:X\rightarrow Y\) satisfying equation (1).