On Cauchy-differences that are also quasisums. (English) Zbl 1071.39026

Let \(I=]0,K[\) where \(0<K\leq \infty\) and denote by \(\Delta\) the set of all \((x,y)\in {\mathbb R}^2\) for which \(x,y,x+y\in I.\) The authors solve the functional equation \[ f(x)+f(y)-f(x+y)=a\big(b(x)+b(y)\big)\quad ((x,y)\in \Delta) \] for the unknown functions \(f,b:I\to{\mathbb R}\) and \(a:\left\{\,b(x)+b(y)\,| \,(x,y)\in \Delta\,\right\}\to{\mathbb R}\) under the only regularity assumption that \(a,b\) are strictly monotone functions. First it is proved that \(f\) is of the form \(g+A\) where \(g\) is a convex/concave function and \(A\) is an additive function.
Then utilizing and extending previous ideas of the authors concerning composite equations, differentiability properties of the unknown functions \(g,b,a\) are proved. By differentiation the composite term is eliminated and using the regularity theory developed by A. Járai [Regularity properties of functional equations in several variables, Adv. Math. 8, New York (2005; Zbl 1081.39022)] for the equation so obtained, \(C^\infty\) properties are derived for the unknown functions.
Finally the nine families of solutions are obtained by help of differential equations. The results are applied to solve a functional equation arising in utilization theory.


39B22 Functional equations for real functions
91B16 Utility theory


Zbl 1081.39022