Increasing solutions of the homogeneity equation and of similar equations
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.
scientific article; zbMATH DE number 584066