Subdifferentials of convex functions on time scales
type: Item
instance of: scholarly article
title: Subdifferentials of convex functions on time scales
zbMATH Open document ID: 1215.26014
author: Małgorzata Wyrwas
author: Dorota Mozyrska
author: Ewa Girejko
published in: Discrete and Continuous Dynamical Systems
publication date: 2011-03-15T00:00:00Z
review text: The first part of this paper is devoted to convex functions. The authors consider real-valued functions defined on a time scale interval, give the definition of convexity, and present several necessary and sufficient conditions for convexity. They investigate the properties of convex and concave functions such as the relation to monotonicity of \(f^{\Delta}\) and \(f^{\nabla}\), and to the sign of \(f^{\Delta\Delta}\) and \(f^{\nabla\nabla}\). In the second part, they introduce the notion of a subdifferential. Consider a function \(f:I\to\mathbb R\), where \(I\) is a time scale interval. The subdifferential of \(f\) at \(t_0\in I\) is denoted by \(\partial f(t_0)\), and it is the set of all real numbers \(s\) such that \[ f(t)\geq f(t_0)+(t-t_0)s\;\;\;\text{ for every }t\in I. \] The authors prove several theorems which relate subdifferentiability, convexity and the existence of \(\Delta\) and \(\nabla\)-derivatives, including the following statements: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}]A function is subdifferentiable if and only if it is convex. \item[{\(\bullet\)}] If \(f\) is convex, \(t\in I\) and both \(f^\Delta(t)\) and \(f^\nabla(t)\) exist, then \(\partial f(t)=\{\alpha f^\Delta(t)+(1-\alpha)f^\nabla(t),\;\alpha\in[0,1]\}\). \item[{\(\bullet\)}] If \(f\) is convex and both \(\Delta\) and \(\nabla\)-differentiable, then \(\int_a^b\varphi(t)\Delta t\leq f(b)-f(a)\) and \(\int_a^b\varphi(t)\nabla t\geq f(b)-f(a)\) for every \(\varphi\in \partial f\). \item[{\(\bullet\)}] If \(f\) is subdifferentiable and attains a minimum at \(t\), then \(0\in\partial f(t)\). \end{itemize}}
reviewed by: Antonín Slavík
Mathematics Subject Classification ID: 26E70
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 5866058
zbMATH Keywords: time scales
zbMATH Keywords: convex functions
zbMATH Keywords: subdifferential
zbMATH Keywords: subderivative
zbMATH Keywords: subgradient
zbMATH Keywords: diamond-\(\alpha\) derivative
MaRDI profile type: Publication
full work available at URL: dcds.2011.29.671
OpenAlex ID: W2335695729
DOI: 10.3934/DCDS.2011.29.671
description: scientific article; zbMATH DE number 5866058