  The main focus of this paper is to define the notion of
quasi-$(2,\beta)$-Banach space and show some properties in this new space, by
help of it and under some natural assumptions, we prove that the fixed point
theorem [16, Theorem 2.1] is still valid in the setting of
quasi-$(2,\beta)$-Banach spaces, this is also an extension of the fixed point
result of Brzd\k{e}k et al. [12, Theorem 1] in $2$-Banach spaces to
quasi-$(2,\beta)$-Banach spaces. In the next part, we give a general solution
of the radical-type functional equation (1.2). In addition, we study the
hyperstability results for these functional equation by applying the
aforementioned fixed point theorem, and at the end of this paper we will derive
some consequences.
