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On approximation of solutions of operator-differential equations with their entire solutions of exponential type


Abstract

We consider an equation of the form $y'(t) + Ay(t) = 0, \ t \in [0, \infty)$, where $A$ is a nonnegative self-adjoint operator in a Hilbert space. We give direct and inverse theorems on approximation of solutions of this equation with its entire solutions of exponential type. This establishes a one-to-one correspondence between the order of convergence to $0$ of the best approximation of a solution and its smoothness degree. The results are illustrated with an example, where the operator $A$ is generated by a second order elliptic differential expression in the space $L_{2}(\Omega)$ (the domain $\Omega \subset \mathbb{R}^{n}$ is bounded with smooth boundary) and a certain boundary condition.

Key words: Hilbert and Banach spaces, differential-operator equation, weak solution, $C_{0}$-semigroup of linear operators, entire vector-valued function, entire vector-valued function of exponential type, the best approximation, direct and inverse theorems of the approximation theory.


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Article Information

TitleOn approximation of solutions of operator-differential equations with their entire solutions of exponential type
SourceMethods Funct. Anal. Topology, Vol. 22 (2016), no. 3, 245-255
MathSciNet MR3554651
MilestonesReceived 01/04/2016; Revised 12/04/2016
CopyrightThe Author(s) 2016 (CC BY-SA)

Authors Information

V. M. Gorbachuk
National Technical University "KPI", 37 Peremogy Prosp., Kyiv, 06256, Ukraine


Citation Example

V. M. Gorbachuk, On approximation of solutions of operator-differential equations with their entire solutions of exponential type, Methods Funct. Anal. Topology 22 (2016), no. 3, 245-255.


BibTex

@article {MFAT892,
    AUTHOR = {Gorbachuk, V. M.},
     TITLE = {On  approximation of solutions of operator-differential
equations with their entire solutions of exponential type},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {3},
     PAGES = {245-255},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=892},
}


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