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 Mat. Sb. (N.S.), 1978, Volume 105(147), Number 4, Pages 467–484 (Mi msb2535)

An expression for the solution of a differential equation in terms of iterates of differential operators

A. V. Babin

Abstract: We obtain theorems on the expression of $A^{-1}$ in terms of iterates of the operator $A$, which is the reproducing operator of a $1$-parameter group of linear transformations of a Banach space, and whose spectrum does not surround $0$. These results are applied to first order differential equations with analytic coefficients and right-hand sides (symmetric first order systems on a compact manifold without boundary), and to second order elliptic equations (equations with a real principal part on a manifold without boundary, selfadjoint equations degenerate on the boundary of the domain, and the Dirichlet problem for a selfadjoint equation in a domain with an analytic boundary). We obtain formulas expressing the value of the solution at a point in terms of the derivatives of the coefficients and the right-hand side at this point.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1978, 34:4, 411–424

Bibliographic databases:

UDC: 517.944
MSC: Primary 47A10, 47F05, 35J20; Secondary 47G05

Citation: A. V. Babin, “An expression for the solution of a differential equation in terms of iterates of differential operators”, Mat. Sb. (N.S.), 105(147):4 (1978), 467–484; Math. USSR-Sb., 34:4 (1978), 411–424

Citation in format AMSBIB
\Bibitem{Bab78} \by A.~V.~Babin \paper An expression for the solution of a~differential equation in terms of iterates of differential operators \jour Mat. Sb. (N.S.) \yr 1978 \vol 105(147) \issue 4 \pages 467--484 \mathnet{http://mi.mathnet.ru/msb2535} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=496590} \zmath{https://zbmath.org/?q=an:0382.35049|0442.35056} \transl \jour Math. USSR-Sb. \yr 1978 \vol 34 \issue 4 \pages 411--424 \crossref{https://doi.org/10.1070/SM1978v034n04ABEH001214} 

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This publication is cited in the following articles:
1. A. V. Babin, “Fractional powers of a nonlinear analytic differential operator”, Math. USSR-Sb., 37:1 (1980), 9–38
2. A. V. Babin, “Solution of the cauchy problem with the help of weighted approximations of exponents by polynomials”, Funct. Anal. Appl., 17:4 (1983), 305–307
3. A. V. Babin, “Construction and investigation of solutions of differential equations by methods in the theory of approximation of functions”, Math. USSR-Sb., 51:1 (1985), 141–167
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