Analog of the Cauchy function for a generalized equation with deviating argument
type: Item
instance of: scholarly article
title: Analog of the Cauchy function for a generalized equation with deviating argument
zbMATH Open document ID: 1161.45001
DOI: 10.1134/S0012266107100060
published in: Differential Equations
publication date: 2008-09-22T00:00:00Z
review text: The author studies the properties of the Cauchy function for the Volterra-type integral equation with deviating argument \[ x(t)-\lambda\int_\alpha^t x(F(\cdot)) dq = f(t),\qquad \alpha, t\in D:=[a,b], \eqno{(1)} \] where \(x,f\in C(D)\) and \(q\in C(D)\cap BV(D)\), \(F : D\to D\) is a continuous function, and \(\lambda\) plays the role of a spectral parameter (actually all presented results refer to the case \(|\lambda|<<1\)). Defining the multiplication \(*\) generated by the deviation \(F\), the author introduces first the algebra of formal power series \(\sum_{k=0}^\infty \lambda^k x_k\) with function coefficients \(x_k\in C(D^n)\). This algebra is a unital associative algebra over \(\mathbb{R}\) with the Riemann-Stieltjes integral being defined in it. Then the original equation (1) is represented in an abstract form \[ x(t) - \int_\alpha^t (dQ * x) = f(t). \eqno{(2)} \] Further analysis demonstrates that the scheme of the classical theory of linear differential equations is transferred to equation (2) in full extent. In particular, there exists an analog of the Cauchy function, which allows to express every solution to equation (2) in the integral form.
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 5345603
zbMATH Keywords: Volterra integral equation
zbMATH Keywords: deviating argument
zbMATH Keywords: Cauchy function
Wikidata QID: Q115253308
author: Vitaliĭ Ivanovich Rodionov
reviewed by: Oleh Omel'chenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5345603