An axiomatic theory of Volterra integral equations
type: Item
instance of: scholarly article
title: An axiomatic theory of Volterra integral equations
zbMATH Open document ID: 0846.45002
published in: Differential and Integral Equations
publication date: 1996-03-11T00:00:00Z
review text: The authors are concerned with an axiomatic theory for Volterra integral equations of the form \[ x(t)= f(t)+ \int^t_0 k(t, s, x(s)) ds, \] with \(x\) taking values in a finite-dimensional vector space. The basic axioms are formulated in accordance with the basic properties of the set of solutions of the integral equation, and they relate to the possibility of extending (continuing) solutions to larger intervals, the uniqueness and the compactness of the set of solutions with graph in a compact set. Two more requirements (axioms) are formulated, one known as the fixed initial value property, and another concerning the possibility of constructing saturated solutions by means of continuable ones. Under adequate properties imposed on the kernel \(k(t, s, x)\), various properties are derived for the set of all solutions of the class of equations of the form indicated above. For instance, one shows that any solution whose graph belongs to a compact set, can be extended to a saturated solution. A Kneser type property is also investigated. That theory is a generalization of Filippov's theory for ordinary differential equations [cf. \textit{V. V. Fedorchuk} and \textit{V. V. Filippov}, ``General topology; basic constructions'' (1988; Zbl 0658.54001)].
Mathematics Subject Classification ID: 45G15
zbMATH DE Number: 844436
zbMATH Keywords: compact solution set
zbMATH Keywords: systems
zbMATH Keywords: axiomatic theory
zbMATH Keywords: Volterra integral equations
zbMATH Keywords: saturated solutions
author: Boris S. Klebanov
author: B. D. Sleeman
reviewed by: Constantin Corduneanu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 844436