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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.

### Article Information

 Title On approximation of solutions of operator-differential equations with their entire solutions of exponential type Source Methods Funct. Anal. Topology, Vol. 22 (2016), no. 3, 245-255 MathSciNet MR3554651 zbMATH 06742109 Milestones Received 01/04/2016; Revised 12/04/2016 Copyright The 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},
MRNUMBER = {MR3554651},
ZBLNUMBER = {06742109},
URL = {http://mfat.imath.kiev.ua/article/?id=892},
}

### References

1. J. M. Ball, Continuity properties of nonlinear semigroups, J. Functional Analysis 17 (1974), 91-103.  MathSciNet
2. Yu. M. Berezanskiĭ, Selfadjoint operators in spaces of functions of infinitely many variables, Translations of Mathematical Monographs, vol. 63, American Mathematical Society, Providence, RI, 1986.  MathSciNet
3. M. L. Gorbachuk, On analytic solutions of operator-differential equations, Ukrainian Math. J. 52 (2000), no. 5, 680-693.  MathSciNet CrossRef
4. V. I. Gorbachuk and M. L. Gorbachuk, Operator approach to approximation problems, St. Petersburg Math. J. 9 (1998), no. 6, 1097-1110.  MathSciNet
5. M. L. Gorbachuk, Ya. I. Grushka, and S. M. Torba, Direct and inverse theorems in the theory of approximations by the Ritz method, Ukrainian Math. J. 57 (2005), no. 5, 751-764.  MathSciNet CrossRef
6. V. I. Gorbachuk, On summability of expansions in eigenfunctions of self-adjoint operators, Soviet Math. Dokl. 35 (1987), no. 1, 11-15.
7. N. P. Kupcov, Direct and inverse theorems of approximation theory and semigroups of operators, Uspehi Mat. Nauk 23 (1968), no. 4 (142), 117-178.  MathSciNet
8. S. G. Mihlin, Lineinye uravneniya v chastnykh proizvodnykh, Izdat. Vysš. Škola'', Moscow, 1977.  MathSciNet