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Homogeneous differential-operator equations in locally convex spaces

Authors
  • Mishin, S. N.1
  • 1 Orel State University named after I. S. Turgenev, ul. Komsomol’skaya 95, Orel, 302026, Russia , Orel (Russia)
Type
Published Article
Journal
Russian Mathematics
Publisher
Allerton Press
Publication Date
Jan 01, 2017
Volume
61
Issue
1
Pages
22–38
Identifiers
DOI: 10.3103/S1066369X17010042
Source
Springer Nature
Keywords
License
Yellow

Abstract

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.

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