Let V and W be vector spaces over rational numbers, n be an integer with \(n\ge 2\) . In this paper, we first define a multi-quadratic Euler-Lagrange type mapping \(f:V^n\longrightarrow W\) by means of a general system of n quadratic Euler-Lagrange equations. Then, we represent such mappings as a single unified equation (the first kind). In continuation, we consider a special case of this equation (the second kind) and moreover in this case, we show that such equation describes a multi-quadratic Euler-Lagrange mapping. We also establish the (Hyers-Ulam, Rassias and Găvruţa) stability and hyperstability of the mentioned equations, by applying the so-called direct (Hyers) method and a fixed point approach in the setting of Banach spaces. Using a characterization result, we present an example to illustrate that a multi-quadratic mapping (the second kind) in the singularity case cannot be stable.
