Maximal monotone differential inclusions with memory
type: Item
instance of: scholarly article
title: Maximal monotone differential inclusions with memory
zbMATH Open document ID: 0758.34012
DOI: 10.1007/BF02837180
author: Nikolaos S. Papageorgiou
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1993-01-16T00:00:00Z
review text: Let \(b\), \(r>0\), \(\hat T=[-r,b]\), \(T_ 0=[-r,0]\), \(T=[0,b]\). The paper is dealing with the following differential inclusion with memory, in \(\mathbb{R}^ N\): \((*)\) \(-\dot x(t)\in Ax(t)+F(t,x_ t)\) a.e. on \(T\); \(x(v)=\varphi(v)\), \(v\in T_ 0\), where \(x_ t\in C(T_ 0;\mathbb{R}^ N)\), \(x_ t(s)=x(t+s)\), \(A:D(A)\subset\mathbb{R}^ N\to 2^{\mathbb{R}^ N}\) is maximal monotone and \(F\) is a multifunction defined on \(T\times C(T_ 0,\mathbb{R}^ N)\) which satisfies some Carathéodory type conditions and \(| F(t,y)|=\sup\{\| v\|:v\in F(t,y)\}\leq a(t)+b(t)\| y\|_ \infty\) a.e. \((a,b\in L^ 1_ +)\). The author proves two theorems on the existence of a solution \(x\in C(\hat T;\mathbb{R}^ N)\); one involving convex-valued mappings \(F\) and the other nonconvex-valued ones. Then he considers a parametrized family of problems, where \(A\), \(F\) and \(\varphi\) depend on a parameter \(r\) and the continuous dependence on \(r\) of the solution set is proved. The last result says that under reasonable hypotheses on \(F\), the solution set of the nonconvex problem \((*)\) is dense in that of the convexified problem \((*)_ c\) with \(\overline{\text{conv}} F\) instead of \(F\).
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 92806
zbMATH Keywords: differential inclusion with memory
zbMATH Keywords: maximal monotone
zbMATH Keywords: multifunction
reviewed by: Radu Precup
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 92806