In (J. Math. Anal. Appl. 293, 79–88, 2004), the authors investigated the Lipschitz stability of the classical quadratic function defined on an Abelian group. In the present paper, we study the Lipschitz stability of the following variant of the quadratic functional equation: \(\begin{aligned} f(xy) + f(\sigma (y)\hspace{0.55542pt}x)=2f(x) +2f(y),\quad x,y\in M, \end{aligned}\) using the property of a symmetric left-invariant mean. Here, the function f is defined on a monoid \((M,\hspace{0.55542pt}{\cdot }\hspace{1.111pt})\) (not necessarily Abelian) and \(\sigma :M\rightarrow M\) is a homomorphism or an anti-homomorphism satisfying \(\sigma \hspace{1.111pt}{\circ }\hspace{1.111pt}\sigma =\textrm{id}_M\) . As an application, we obtain an approximate solution of the above equation in Lipschitz norms. This study improves and extends the main results of Theorems 1, 4 and 5 in loc. cit. [8], as well as several other works in the context of Lipschitz spaces.
