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On the structure of solutions of operator-differential equations on the whole real axis

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Abstract

We consider differential equations of the form $\left(\frac{d^{2}}{dt^{2}} - B\right)^{m}y(t) = f(t)$, $m \in \mathbb{N}, \ t \in (-\infty, \infty)$, where $B$ is a positive operator in a Banach space $\mathfrak{B}, \ f(t)$ is a bounded continuous vector-valued function on $(-\infty, \infty)$ with values in $\mathfrak{B}$, and describe all their solutions. In the case, where $f(t) \equiv 0$, we prove that every solution of such an equation can be extended to an entire $\mathfrak{B}$-valued function for which the Phragmen-Lindel\"{o}f principle is fulfilled. It is also shown that there always exists a unique bounded on $\mathbb{R}^{1}$ solution, and if $f(t)$ is periodic or almost periodic, then this solution is the same as $f(t)$.

Key words: Positive operator, differential equation in a Banach space, classic and generalized solutions, C0-semigroup of linear operators, bounded analytic C0-semigroup, entire vector of a closed operator, entire vector-valued function, Phragmen-Lindel¨of princip


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Article Information

TitleOn the structure of solutions of operator-differential equations on the whole real axis
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 2, 170–178
MathSciNet 3407908
zbMATH 06533474
MilestonesReceived 29/01/2015
CopyrightThe Author(s) 2015 (CC BY-SA)

Authors Information

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


Citation Example

V. M. Gorbachuk, On the structure of solutions of operator-differential equations on the whole real axis, Methods Funct. Anal. Topology 21 (2015), no. 2, 170–178.


BibTex

@article {MFAT775,
    AUTHOR = {Gorbachuk, V. M.},
     TITLE = {On the structure of solutions of operator-differential equations on the whole real axis},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {2},
     PAGES = {170–178},
      ISSN = {1029-3531},
  MRNUMBER = {3407908},
 ZBLNUMBER = {06533474},
       URL = {http://mfat.imath.kiev.ua/article/?id=775},
}


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References

  1. Einar Hille, Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society, Providence, R. I., 1957.  MathSciNet 
  2. K. Yosida, Functional Analysis, Springer, Berlin-Gottingen-Heidelberg, 1965.
  3. V. V. Vasil′ev, S. G. Krein, S. I. Piskarev, Operator semigroups, cosine operator functions, and linear differential equations, in: Mathematical analysis, Vol.\ 28 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990.  MathSciNet 
  4. Klaus-Jochen Engel, Rainer Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, 2000.  MathSciNet 
  5. I. M. Gelfand and G. E. Shilov, Generalized Functions, Vol. 2: Spaces of Test and Generalized Functions, Fizmatgiz, Moscow, 1958. (Russian)
  6. M. Gorbachuk, V. Gorbachuk, On extensions and restrictions of semigroups of linear operators in a Banach space and their applications, Math. Nachr. 285 (2012), no. 14-15, 1860-1879.  MathSciNet  CrossRef
  7. Hikosaburo Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966), 285-346.  MathSciNet 
  8. S. G. Krein, Lineikhye differentsialnye uravneniya v Banakhovom prostranstve, Izdat. ``Nauka'', Moscow, 1967.  MathSciNet 
  9. P. D. Lax, A Phragmen-Lindelof theorem in harmonic analysis and its application to some questions in the theory of elliptic equations, Comm. Pure Appl. Math. 10 (1957), 361-389.  MathSciNet 


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