Increasing solutions of the homogeneity equation and of similar equations
type: Item
instance of: scholarly article
title: Increasing solutions of the homogeneity equation and of similar equations
zbMATH Open document ID: 0806.39009
author: János Aczél
author: Detlef Gronau
author: Jens Schwaiger
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-02-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 584066
zbMATH Keywords: homogeneity equation
zbMATH Keywords: increasing solutions
zbMATH Keywords: differentiable solutions
reviewed by: Harkrishan L. Vasudeva
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1097
OpenAlex ID: W2005181007
DOI: 10.1006/JMAA.1994.1097
Recommended article: Increasing solutions of simultaneous Poincaré equations
Recommended article: Q3705772
Recommended article: Multiple solutions of asymptotically homogeneous problems
Recommended article: Increasing solutions of the Korteweg-de Vries equation with given asymptotic behaviour
Recommended article: Q4953897
Recommended article: Q4780999
Recommended article: On the growth of solutions of higher order linear homogeneous and non-homogeneous differential equations
Recommended article: Uniform asymptotic expansions of solutions of an inhomogeneous equation
Recommended article: The growth of solutions of non-homogeneous linear differential equations
Recommended article: Q4728873
description: scientific article; zbMATH DE number 584066