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Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces

Authors
  • Mishin, S. N.1
  • 1 Turgenev Orel State University, Orel, Russia , Orel (Russia)
Type
Published Article
Journal
Mathematical Notes
Publisher
Pleiades Publishing
Publication Date
Jan 01, 2018
Volume
103
Issue
1-2
Pages
75–88
Identifiers
DOI: 10.1134/S0001434618010091
Source
Springer Nature
Keywords
License
Yellow

Abstract

The well-known Lagrange method for linear inhomogeneous differential equations is generalized to the case of second-order equations with constant operator coefficients in locally convex spaces. The solutions are expressed in terms of uniformly convergent functional vector-valued series generated by a pair of elements of a locally convex space. Sufficient conditions for the continuous dependence of solutions on the generating pair are obtained. The solution of the Cauchy problem for the equations under consideration is also obtained and conditions for its existence and uniqueness are given. In addition, under certain conditions, the so-called general solution of the equations (a function of most general form from which any particular solution can be derived) is obtained. The study is carried out using the characteristics (order and type) of an operator and of a sequence of operators. Also, the convergence of operator series with respect to equicontinuous bornology is used.

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