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Homogeneous Differential-Operator Equations in Locally Convex Spaces

RUSSIAN MATHEMATICS 2017年第1期61卷 22-38页

作者： Mishin, S. N. Orel State Univ Ul Komsomolskaya 95 Oryol 302026 Russia

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.

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

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Vector-Valued Functions Generated by the Operator of Finite order and Their Application to Solving Operator Equations in Locally Convex Spaces

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RUSSIAN MATHEMATICS 2018年第3期62卷 34-44页

作者： Man'ko, S. N. Orel State Inst Culture Ul Leskova 15 Oryol 302020 Russia

This work is devoted to solving some classes of operator equations, based on the solution of auxiliary one-parameter family of equations, which is obtained fromthe original operator equation by formal replacement of the operator of the integrated parameter. Solutions are vector-valued functions represented by power series or integral. We investigate some properties of these solutions, namely, growth characteristics, the domain of analyticity. The investigation is realized by means of order and type of operator, operator order and operator type of the vector relative to the operator.

关键词： locally convex space order and type of operators vector-valued functions differential-operator equation

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