Boundary value problems for systems of implicit differential equations
type: Item
instance of: scholarly article
title: Boundary value problems for systems of implicit differential equations
zbMATH Open document ID: 0799.34023
author: Marlène Frigon
author: Tomasz Kaczynski
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-11-13T00:00:00Z
review text: The aim of the authors is to give some sufficient conditions which guarantee the existence of solutions to the nonlinear systems of differential equations of the form \(F(t, y(t), y'(t), \dots, y^{(k)} (t))=0\), \(t\in [0,T]\), \(y\in \mathbb{R}^ n\), under various boundary conditions. Their approach consists in reducing first the given problem to a boundary-value problem for a differential inclusion of the type \(y^{(k)} (t)\in \psi(t, y(t), y'(t),\dots, y^{(k-1)} (t))\), \(t\in [0,T]\), with an appropriate multifunction \(\psi\) and then looking for solutions in the Sobolev space \(W^{k,p} ([0,T], \mathbb{R}^ n)\). To show the necessity of Sobolev space they give an example which shows that even the smoothness of \(F\) does not imply the existence of \(C^ k\)-solutions. In addition to a main theorem of general character the paper includes also four theorems related to a series of special cases such as i) \(\sum_{j=0}^ m a_ j (z, z(t), z'(t),\dots, z^{(k-1)} (t)) (z^{(k)}(t)- \varepsilon z(t))^ j =0\), \(\varepsilon\geq 0\); ii) \(F(a(t,y(t), y'(t),\dots, y^{(k-1)} (t)), y^{(k)} (t))=0\), a.e. \(t\) with \(y^{(j)}(t_ j)= r_ j\) \((j=0,1,\dots, k-1)\), where \(t_ j\in [0,\infty)\) and \(r_ j\in \mathbb{R}^ n\) and iii) \(F(a(t,y(t),y'(t)), y''(t))=0\), a.e. \(t\in [0,1]\) with \(y(0)= r_ 0\), \(y(1)= r_ 1\) or \(y(0)= y(1)\), \(y'(0)= y'(1)\).
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34A09
zbMATH DE Number: 566454
zbMATH Keywords: boundary-value problem
zbMATH Keywords: differential inclusion
reviewed by: Mithat Idemen
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1353
OpenAlex ID: W2058985708
DOI: 10.1006/JMAA.1993.1353
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description: scientific article; zbMATH DE number 566454