Solving a nonlinear functional inequality

Abstract

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