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The Dirichlet problem for differential equations in a Banach space


Abstract

In the paper, we consider an abstract differential equation of the form $\left(\frac{\partial^{2}}{\partial t^{2}}- B \right)^{m}y(t) = 0$, where $B$ is a positive operator in a Banach space $\mathfrak{B}$. For solutions of this equation on $(0, \infty)$, it is established the analogue of the Phragmen-Lindelof principle on the basis of which we show that the Dirichlet problem for the above equation is uniquely solvable in the class of vector-valued functions admitting an exponential estimate at infinity. The Dirichlet data may be both usual and generalized with respect to the operator $-B^{1/2}$.The formula for the solution is given, and some applications to partial differential equations are adduced.


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Article Information

TitleThe Dirichlet problem for differential equations in a Banach space
SourceMethods Funct. Anal. Topology, Vol. 18 (2012), no. 2, 140-151
MathSciNet MR2978190
zbMATH 1265.34214
CopyrightThe Author(s) 2012 (CC BY-SA)

Authors Information

M. L. Gorbachuk
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

V. I. Gorbachuk
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine 


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Citation Example

M. L. Gorbachuk and V. I. Gorbachuk, The Dirichlet problem for differential equations in a Banach space, Methods Funct. Anal. Topology 18 (2012), no. 2, 140-151.


BibTex

@article {MFAT628,
    AUTHOR = {Gorbachuk, M. L. and Gorbachuk, V. I.},
     TITLE = {The Dirichlet problem for differential equations in a Banach space},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {18},
      YEAR = {2012},
    NUMBER = {2},
     PAGES = {140-151},
      ISSN = {1029-3531},
  MRNUMBER = {MR2978190},
 ZBLNUMBER = {1265.34214},
       URL = {http://mfat.imath.kiev.ua/article/?id=628},
}


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