Aczél, János; Gronau, Detlef; Schwaiger, Jens Increasing solutions of the homogeneity equation and of similar equations. (English) Zbl 0806.39009 J. Math. Anal. Appl. 182, No. 2, 436-464 (1994). Let \(\mathbb{R}\) denote the set of real numbers, \(\mathbb{R}_{++} = \{t:0 < t < \infty\}\) and \(\mathbb{R}^ n_{++} = \mathbb{R}_{++} \times \mathbb{R}_{++} \times \cdots \times \mathbb{R}_{++}\). Let \(f\) be a real-valued function defined on \(\mathbb{R}^ n\). The authors consider the equations \[ f(rx+s1) = rf(x) + s, \quad x \in \mathbb{R}^ n, \quad s \in \mathbb{R}, \quad r>0, \quad 1 = (1,1,\dots,1), \]\[ f(rx) = rf(x), \quad x \in \mathbb{R}^ n_{++}, \quad r \in \mathbb{R}_{++} \quad \text{or} \quad x \in \mathbb{R}^ n, \quad r \in \mathbb{R}_{++} \quad \text{or} \quad x \in \mathbb{R}^ n, \quad r \in \mathbb{R}, \]\[ f(x + s1) = f(x) + s, \quad x \in \mathbb{R}^ n, \quad s \in \mathbb{R}, \] and provide necessary and sufficient conditions for a solution of either of these equations to be increasing. Moreover, the increasing solutions of equations, more general than listed above, are described.Necessary and sufficient conditions, both for differentiable solutions and without any regularity assumptions other than monotonicity, are given. Reviewer: H.K.L.Vasudeva (Chandigarh) Cited in 6 Documents MSC: 39B22 Functional equations for real functions Keywords:homogeneity equation; increasing solutions; differentiable solutions PDFBibTeX XMLCite \textit{J. Aczél} et al., J. Math. Anal. Appl. 182, No. 2, 436--464 (1994; Zbl 0806.39009) Full Text: DOI