Hyers-Ulam stability of Davison functional equation on restricted domains (Q6622672)

Let \(A\) be a normed algebra with the unit and \(B\) a Banach space. A mapping \(f\colon A\to B\) satisfies the Davison functional equation if \[ f(xy)+f(x+y)-f(xy+x)-f(y)=0,\qquad x,y\in A. \] The authors consider the Hyers-Ulam stability of the above equation on a restricted domain. Namely, they show that whenever \(f\) satisfies, with \(\varepsilon\geq 0\) and \(d>0\), the functional inequality \[ \|f(xy)+f(x+y)-f(xy+x)-f(y)\|\leq\varepsilon,\qquad x,y\in A,\ \|x\|\geq d, \] then it has to be close to a unique affine mapping \(\varphi\colon A\to B\). Moreover, it is proved that if a mapping \(f\colon [0,\infty)\to B\) satisfies \[ \|f(xy)+f(x+y)-f(xy+x)-f(y)\|\leq\varepsilon,\qquad x,y\geq 0, \] then \(f\) is close to an affine mapping \(\psi\colon\mathbb{R}\to B\). A similar result is obtained for a generalized Davison equation, with four unknown functions \(f,g,h,k \colon [0,\infty)\to B\): \[ f(xy)+g(x+y)-h(xy+x)-k(y)=0,\qquad x,y\geq 0. \]