Open Access

On generalized rezolvents and characteristic matrices of first-order symmetric systems

     Article (.pdf)

Abstract

We study general (not necessarily Hamiltonian) first-ordersymmetric system $J y'-B(t)y=\Delta(t) f(t)$ on an interval $\mathcal I=[a,b)$ with the regular endpoint $a$ and singular endpoint $b$. It isassumed that the deficiency indices $n_\pm(T_{\min})$ of thecorresponding minimal relation $T_{\min}$ in $L_\Delta^2(\mathcal I)$ satisfy$n_-(T_{\min})\leq n_+(T_{\min})$. We describe all generalized resolvents$y=R(\lambda)f, \; f\in L_\Delta^2(\mathcal I),$ of $T_{\min}$ in terms of boundary problemswith $\lambda$-depending boundary conditions imposed on regular andsingular boundary values of a function $y$ at the endpoints $a$and $b$ respectively. We also parametrize all characteristicmatrices $\Omega(\lambda)$ of the system immediately in terms of boundaryconditions. Such a parametrization is given both by the blockrepresentation of $\Omega(\lambda)$ and by the formula similar to thewell-known Krein formula for resolvents. These results develop the Straus' results on generalized resolvents and characteristicmatrices of differential operators.

Key words: First-order symmetric system, boundary problem with a spectral parameter, generalized resolvent, characteristic matrix.


Full Text





Article Information

TitleOn generalized rezolvents and characteristic matrices of first-order symmetric systems
SourceMethods Funct. Anal. Topology, Vol. 20 (2014), no. 4, 328-348
MathSciNet MR3309671
zbMATH 1324.47081
MilestonesReceived 13/01/2014
CopyrightThe Author(s) 2014 (CC BY-SA)

Authors Information

Vadim Mogilevskii
Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 R. Luxemburg, Donets'k, 83050, Ukraine


Citation Example

Tim Mogilevskii, On generalized rezolvents and characteristic matrices of first-order symmetric systems, Methods Funct. Anal. Topology 20 (2014), no. 4, 328-348.


BibTex

@article {MFAT733,
    AUTHOR = {Mogilevskii, Tim},
     TITLE = {On generalized rezolvents and characteristic matrices of first-order symmetric systems},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {20},
      YEAR = {2014},
    NUMBER = {4},
     PAGES = {328-348},
      ISSN = {1029-3531},
  MRNUMBER = {MR3309671},
 ZBLNUMBER = {1324.47081},
       URL = {http://mfat.imath.kiev.ua/article/?id=733},
}


Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar


Export article

Save to Mendeley


All Issues