Hyers-Ulam stability of quadratic operators in locally convex cones

The stability problem in Ulam's sense has recently been explored in locally convex cone environments. In continuation of this research direction, our work examined the stability properties of the quadratic functional equation

$$2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right)=f(x)+f(y)$$

in such structures. We presented novel stability theorems that offered enhanced comprehension of operator behavior when subjected to perturbations. These results advanced the theoretical framework of Hyers-Ulam stability within locally convex cones while elucidating distinctive characteristics of quadratic operators in this context. Our investigation both strengthened the mathematical underpinnings of stability theory and provided new perspectives on interactions between certain operators and locally convex spaces.