Research Article Characterization and Stability of Multi-Euler-Lagrange Quadratic Functional Equations Abasalt Bodaghi , 1 Hossein Moshtagh, 2 and Amir Mousivand 3 1 Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran 2 Department of Computer Science, University of Garmsar, Garmsar, Iran 3 Department of Mathematics, West Tehran Branch, Islamic Azad University, Tehran, Iran Correspondence should be addressed to Abasalt Bodaghi; abasalt.bodaghi@gmail.com Received 22 March 2022; Revised 22 August 2022; Accepted 22 September 2022; Published 0 October 2022 Academic Editor: Cristian Chifu Copyright © 2022 Abasalt Bodaghi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aim of the current article is to characterize and to prove the stability of multi-Euler-Lagrange quadratic mappings. In other words, it reduces a system of equations defining the multi-Euler-Lagrange quadratic mappings to an equation, say, the multi-Euler-Lagrange quadratic functional equation. Moreover, some results corresponding to known stability (Hyers, Rassias, and Gӑvruta) outcomes regarding the multi-Euler-Lagrange quadratic functional equation are presented in quasi- β-normed and Banach spaces by using the fixed point methods. Lastly, an example for the nonstable multi-Euler- Lagrange quadratic functional equation is indicated. 1. Introduction The celebrated Ulam challenge [1] arises from this question that how we can find an exact solution near to an approxi- mate solution of an equation. This phenomenon of mathe- matics is called the stability of functional equations which has many applications in nonlinear analysis. The mentioned question has been partially solved by Hyers [2], Aoki [3], and Rassias [4] for the linear, additive, and linear (unbounded Cauchy difference) mappings, respectively. Next, many Hyers-Ulam stability problems for miscellaneous functional equations were studied by authors in the spirit of Rassias approach (see for instance [5–14] and other resources). During the last two decades, stability problems for multivariable mappings were studied and extended by a number of authors. One of the mappings is the multiquadra- tic mapping, studied, for example, in [15–17]. Recall that a multivariable mapping f : V n ⟶ W is said to be multiqua- dratic [11] if it fulfills the famous quadratic equation Qu + v ð Þ + Qu − v ð Þ =2Qu ðÞ +2Qv ðÞ, ð1Þ in each component. Note that equation (1) is a suitable tool for obtaining some characterizations in the setting of inner product spaces and in fact plays a prominent role. In other words, any square norm on an inner product space fulfills u + v k k 2 + u − v k k 2 =2 u kk 2 + v kk 2 , ð2Þ which is called the parallelogram equality. However, some functional equations have been applied to characterize inner product spaces and are available in [18, 19] and references therein. In addition, the quadratic functional equation was used to characterize inner product spaces in [20, 21]. A lot of information about equation (1) and some equa- tions which are equivalent to it (in particular, about their solutions and stability) and more applications can be found for instance in [22–24]. Park was the first author who stud- ied the stability of multiquadratic in the setting of Banach algebras [16]. After that, some authors introduced various versions of multiquadratic mappings and investigated the Hyers-Ulam stability of such mappings in Banach spaces and non-Archimedean spaces; these results are available for instance in [15, 25–29]. As for an unification of the Hindawi Journal of Function Spaces Volume 2022, Article ID 3021457, 9 pages https://doi.org/10.1155/2022/3021457