Proximal (sub-)differential inequalities
type: Item
instance of: scholarly article
title: Proximal (sub-)differential inequalities
zbMATH Open document ID: 0939.34029
DOI: 10.1016/S0362-546X(98)00136-9
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2000-01-27T00:00:00Z
review text: The authors study the interesting question on a characterization of comparison properties of nonsmooth vector functions in terms of proximal subderivatives. For a function \(\phi: \mathbb{R}\to (-\infty,+\infty]\), the number \(\zeta\) is called a proximal subderivative of \(\phi\) at \(t\) if \(\exists\sigma>0\) such that for all \(\tau\) near \(t\), \[ \phi(\tau)\geq \phi(t)+ \zeta(\tau- t)- \sigma|\tau- t|^2. \] The set of all proximal subderivatives of \(\phi\) at \(t\) is denoted by \(\partial_p\phi(t)\). If \(\phi(t)= +\infty\), then by convention \(\partial_p\phi(t)= \emptyset\). Let \(\rho\) be a vector function with lower-semicontinuous components \(\rho^i: \mathbb{R}\to (-\infty,+\infty]\) \((i= 1,\dots, m)\), and \(f: (a,b)\times \mathbb{R}^m\to \mathbb{R}^m\) is a continuous function. Definition: The function \(\rho\) is said to satisfy the comparison principle with respect to solutions to the ODE \(\dot r= f(t,r)\) if for every \[ t_0\in \text{dom }\rho:= \bigcap^m_{i=1} \{t:\rho^i(t)< +\infty\}, \] and every \(r_0\in \mathbb{R}^m\) with \(r_0\geq \rho(t_0)\), there exists a solution \(r(\cdot)\) to this ODE satisfying \(\rho(t)\leq r(t)\) on the maximal existent interval of \(r(\cdot)\). The main result of the paper is embodied in the following theorem: Let the functions \(\rho\), \(f\) be as defined above. Then \(\rho(t)\) satisfies the comparison principle iff \[ \partial_p\rho^i(t)\leq f^i(t,r),\quad \forall i\in\{i: r^i= \rho^i(t)\},\quad \forall r\geq \rho(t),\quad \forall t\in \text{dom }\rho,\tag{1} \] and one of the following conditions holds: \(\forall t\in\text{dom }\rho\), \[ \partial_p\Biggl( \sum^m_{i=1} \rho^i\Biggr)(t)\leq \sum^m_{i= 1}f^i(t, \rho(t))\quad\text{or }\lim_{\tau\downarrow t} \rho(\tau)= \rho(t). \] Note, inequality (1) means that every element of the left set is less or equal to the right number. Related problems on invariance, weak and strong comparison principles are also discussed in detail.
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34A40
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 1383335
zbMATH Keywords: comparison principle
zbMATH Keywords: nonsmooth vector function
zbMATH Keywords: proximal subderivative
author: V. Lakshmikantham
author: Yu S. Ledyaev
MaRDI profile type: Publication
full work available at URL: s0362-546x(98)00136-9
OpenAlex ID: W1981934628
description: scientific article; zbMATH DE number 1383335