Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Regularization of singular systems of integral equations with kernels of finite double-norm on $ L\sb{\infty }$


Author: Guillermo Miranda
Journal: Proc. Amer. Math. Soc. 26 (1970), 423-427
MSC: Primary 45.15
DOI: https://doi.org/10.1090/S0002-9939-1970-0264349-9
MathSciNet review: 0264349
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: There are known examples of linear integral transformations $ T$ of finite double-norm on $ {L_\infty }$ such that neither the transformation nor any of its iterates is compact, so that Fredholm's alternative does not hold unrestrictedly for the equation $ (I - \lambda T)g = f$ ($ \lambda $ a complex number, $ g,f \in {L_\infty }$. It is also known that the alternative holds true for $ \vert\lambda \vert$ less than the Fredholm radius of $ T$. Using a kernel decomposition, a quantity $ \omega $ is introduced and the equivalence of an integral transformation system with components of finite double-norm on $ {L_\infty }$, to a similar system that satisfies the Fredholm alternative for $ \vert\lambda \vert < \omega $ is proved. In contrast to the Fredholm radius, an easy computation for $ \omega $ is available.


References [Enhancements On Off] (What's this?)

  • [1] Adriaan Cornelis Zaanen, Linear analysis. Measure and integral, Banach and Hilbert space, linear integral equations, Interscience Publishers Inc., New York; North-Holland Publishing Co., Amsterdam; P. Noordhoff N.V., Groningen, 1953. MR 0061752
  • [2] W. J. Trjitzinsky, Singular non-linear integral equations, Duke Math. J. 11 (1944), 517–564. MR 0011392
  • [3] D. Willett, Nonlinear vector integral equations as contraction mappings, Arch. Rational Mech. Anal. 15 (1964), 79–86. MR 0159200, https://doi.org/10.1007/BF00257405
  • [4] Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
  • [5] G. Miranda, Application of singular integral equation methods to static problems of non-smooth elastic bodies, Thesis, Purdue University, 1969; Notices Amer. Math. Soc. 16 (1969), 646. Abstract #665-73.
  • [6] -, Integral equation solution of the first initial-boundary value problem for the heat equation in domains with nonsmooth boundary, Comm. Pure and Appl. Math. 23 (1970); Notices Amer. Math. Soc. 17 (1970), 171. Abstract #672-312.
  • [7] T. Carleman, Über das Neumann-Poincaresche problem für ein gebiet mit ecken, Thesis, Uppsala, 1916.
  • [8] -, La théorie des équations intégrales singulières et ses applications, Ann. Inst. Henri Poincaré 1 (1930), 401-430.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 45.15

Retrieve articles in all journals with MSC: 45.15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0264349-9
Keywords: Singular systems, integral equations, Fredholm alternative, Fredholm radius, finite double-norm transformations, Neumann series, space of bounded functions
Article copyright: © Copyright 1970 American Mathematical Society