In this paper, we investigate the Hyers-Ulam stability problem of the following functional equation

$$f(x+y)+g(x-y)=h(x)+k(y),$$

on an unbounded restricted domain, which generalizes some of the results already obtained by other authors (for example [9,Theorem 2], [11,Theorem 5] and [21,Theorem 2]). Particular cases of this functional equation are Cauchy, Jensen, quadratic and Drygas functional equations. As a consequence, we obtain asymptotic behaviors of this functional equation.

In this paper, we investigate the Hyers-Ulam stability of additive and Fréchet functional equations on restricted domains. We improve the bounds and thus the results obtained by S. M. Jung and J. M. Rassias. As a consequence, we obtain asymptotic behaviors of functional equations of different types. One of the objectives of this paper is to bring out the involvement of functional equations in various characterizations of inner product spaces.

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.