Solving a nonlinear functional inequality

We solve the inequality aligned \| f( x²+yf( z) ) -xf( x) -af( y) z\| ≤ ε aligned f x 2 + y f z - x f x - a f y z ≤ ϵ on some domains in a composition algebra with division ( A,\| · \| ) A , · , where f:A→ A f : A → A is the unknown, a≠ 0 a ≠ 0 is a fixed central element in A and ε ϵ is a fixed nonnegative number. The results facilitates the expanding of some known Ulam stability theorems in R R and has as a consequence the following trichotomy: the function f:A→ A f : A → A is either the null function, or f( x) =ax f x = a x for all x∈ A x ∈ A , or $${aligned} {\| x\| +\| y\| +\| z\| ≥ d}{ }\| f( x²+yf( z) ) -xf( x) -af( y) z\| =∞ , {aligned}$$ sup x + y + z ≥ d f x 2