Fixed points and stability for quartic mappings in β-Normed left Banach modules on Banach algebras

The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equation ∑k=2n (∑i1 =2k ∑i2 =i1 +1k+1…∑in −k+1=in −k+1n )f(∑i=1,i≠i1 ,…,in −k+1nxi −∑r=1n−k+1xir  ) + f(∑i=1nxi ) = 2n−2∑1≤i