A fixed point theorem in locally convex spaces
type: Item
instance of: scholarly article
title: A fixed point theorem in locally convex spaces
zbMATH Open document ID: 1227.47035
DOI: 10.1007/BF03191243
author: B. E. Eshmatov
published in: Collectanea Mathematica
publication date: 2010-05-25T00:00:00Z
review text: In this interesting paper, the authors study the existence and uniqueness of fixed points for a mapping \(\mathcal K : \mathcal D_{\mathcal K} \rightarrow \mathcal D_{\mathcal K}\) defined on some set \(\mathcal D_{\mathcal K} \subset \mathcal X\), where \(\mathcal X\) is a locally convex space with the topology given by a family \(\{p(\cdot;\alpha)\}_{\alpha \in \Omega}\) of seminorms. By assuming that there exists a linear and positive operator \(K\), acting on functions defined on the index set \(\Omega\), such that for every \(u,v \in \mathcal D_{\mathcal K}\), \[ p( {\mathcal K}(u) - {\mathcal K }(v);\alpha ) \leq K (p(u-v;\cdot))(\alpha) , \quad \alpha \in \Omega, \] and some additional assumptions including the existence of a fixed point for the operator \(K + p({\mathcal K}(0);\cdot)\), they prove that there exists a fixed point of \(\mathcal K\). For a class of elements satisfying \(K^{n} (p(u;\cdot))(\alpha) \rightarrow 0\) as \(n\rightarrow \infty\), including, in particular, the class for which they prove the existence of fixed points, the authors show that fixed points are unique. In addition, the authors present several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. They also consider pseudodifferential equations with nonlinear terms.
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 46N20
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 46A03
zbMATH DE Number: 5711740
zbMATH Keywords: fixed point theorem
zbMATH Keywords: locally convex spaces
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: pseudodifferential operators
MaRDI profile type: Publication
cites work: Q4421936
cites work: Q3992443
cites work: Q3244654
cites work: An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities
cites work: Riesz potential equations in localLp-spaces
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description: scientific article; zbMATH DE number 5711740