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 Izv. Vyssh. Uchebn. Zaved. Mat., 2017, Number 1, Pages 26–43 (Mi ivm9194)

Homogeneous differential-operator equations in locally convex spaces

S. N. Mishin

Orel State University, 95 Komsomol'skaya str., Oryol, 302026 Russia

Abstract: We describe a general method that allows to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is interpreted not in terms of null right-hand side of an equation but in terms that the left-hand side is homogeneous function of operators appearing in an equation. Solutions are presented by a uniformly convergent functional vector-valued series, generated by a set of solutions to some ordinary differential equation of $k$th degree, roots of characteristical polynomial, and some set of elements of locally convex space. We find sufficient conditions of continuous dependence of solution on generating set, and a solution to Cauchy's problem for considered equations. We specify conditions of its existence and uniqueness. Besides, under certain conditions we find a general solution to considered equation (a function of most general form from which any particular solution can be found). The investigation is realized by means of characteristics (order and type) of operator and operator characteristics (operator order and operator type) of vector relative to an operator. Also we use a convergence of operator series relative to equicontinuous bornology.

Keywords: locally convex space, order and type of operators, differential-operator equation, equicontinuous bornology, convergence by bornology, vector-valued function.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2017, 61:1, 22–38

Bibliographic databases:

UDC: 517.983

Citation: S. N. Mishin, “Homogeneous differential-operator equations in locally convex spaces”, Izv. Vyssh. Uchebn. Zaved. Mat., 2017, no. 1, 26–43; Russian Math. (Iz. VUZ), 61:1 (2017), 22–38

Citation in format AMSBIB
\Bibitem{Mis17} \by S.~N.~Mishin \paper Homogeneous differential-operator equations in locally convex spaces \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2017 \issue 1 \pages 26--43 \mathnet{http://mi.mathnet.ru/ivm9194} \transl \jour Russian Math. (Iz. VUZ) \yr 2017 \vol 61 \issue 1 \pages 22--38 \crossref{https://doi.org/10.3103/S1066369X17010042} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000408827200004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85013923919} 

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• http://mi.mathnet.ru/eng/ivm/y2017/i1/p26

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. N. Mishin, “Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces”, Math. Notes, 103:1 (2018), 75–88
2. S. N. Mishin, “Ob odnom klasse operatornykh uravnenii v lokalno vypuklykh prostranstvakh”, Izv. vuzov. Matem., 2018, no. 11, 33–50
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