In this article, we obtain the general solution and prove the Hyers-Ulam stability of the following quadratic-multiplicative functional equation: $$\varphi (st-uv)+\varphi (sv+tu)=[\varphi (s)+\varphi (u)][\varphi (t)+\varphi (v)]$$ by using the direct method and the fixed point method.

‎In this paper‎, ‎we introduce the new kind of a-type additive functional equation and investigate the Hyers-Ulam stability of a-type additive functional equations within the Banach spaces using the direct method‎. ‎This approach provides a straightforward and efficient way to establish the stability of functional equations without relying on more complex or indirect techniques‎. ‎We begin by defining the specific a-type additive functional equation under consideration and outlining the conditions required for its stability‎. ‎Through rigorous mathematical analysis‎, ‎we demonstrate that under certain constraints‎, ‎the functional equation exhibits Hyers-Ulam stability‎.

In this paper, we study the general solution of the functional equation, which is derived from additive–quartic mappings. In addition, we establish the generalized Hyers–Ulam stability of the additive–quartic functional equation in Banach spaces by using direct and fixed point methods.

In this present work, we obtain the solution of the generalized additive functional equation and also establish Hyers–Ulam stability results by using alternative fixed point for a generalized additive functional equation . where is a nonnegative integer with in Banach spaces.