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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.

##### MSC:
 39B22 Functional equations for real functions 91B16 Utility theory