Monotonic solutions of a functional equation arising from simultaneous utility representations (Q1430555)

The system consisting of a functional equation (1) \(f[h(x)]-f[h(y)]+f(y)=f[h(x-y)+y]\) and an inequality (2) \(h(x)\leq x,\) originating from utility theory, has been solved by \textit{J. Aczél, R. D. Luce} and \textit{A. A. J. Marley} [Result. Math. 43, 193--197 (2003; Zbl 1041.39015)] under differentiability assumptions. The present author eliminates with considerable amount of work and skills the differentiabilty, even continuity assumptions, which are not natural for the underlying application, and the inequality (2), which is, in solving the equation (1) for \(f:]0,a[\to\mathbb{R},\;h:[0,b[\to\mathbb{R}\) on the open domain \(\{(x,y)\mid x\in]0,b[,\,x\geq y;\;h(x),\;h(y),\;h(x-y)+y\in]0,a[;\;y\in]0,a[\cap]0,b[ \}.\) Only strict increasing of \(f\) and \(h,\;h(0)\) and the existence of a limit of \(h\) at 0, smaller than \(a,\) is assumed, properties again natural for the utility problem.