Fixed point approach: Ulam stability results of a quartic functional equation in non-Archimedean Banach spaces

In the field of functional equations and their solutions, Ulam stability in the context of functional equations and their solutions is crucial. Theoretically, this concept assesses whether a function's approximation to a specific functional equation is close to the exact solution. In extended Ulam stability, the idea of stability is developed. This makes solving, analyzing, and classifying functional equations easier across different spaces. In this work, we explore the Ulam stability results of the quartic functional equation \[\begin{aligned} \psi\bigg(\sum_{i=1}^{n}\flat_{i}\bigg)&=\sum_{1\leq i<j<k<l \leq n} \psi(\flat_{i}+\flat_{j}+\flat_{k}+\flat_{l})+(-n+4) \sum_{1\leq i<j<k \leq n} \psi(\flat_{i}+\flat_{j}+\flat_{k})\\ &\quad+\bigg(\frac{n^2-7n+12}{2}\bigg)\sum_{1=i;i\neq j}^{n}\psi(\flat_{i}+\flat_{j})-\sum_{i=1}^{n} \psi(2\flat_{i})\\ &\quad+\bigg(\frac{-n^3+9n^2-26n+120}{6}\bigg)\sum_{i=1}^{n}\bigg(\frac{\psi(\flat_{i})+\psi(-\flat_{i})}{2}\bigg), \end{aligned} \] where \(n>4\), in non-Archimedean Banach spaces over a field utilizing direct and fixed-point techniques.