Skip to main content
SpringerLink
Log in
Book cover

Constructive Methods for the Practical Treatment of Integral Equations pp 93–102Cite as

Inclusion of Regular and Singular Solutions of Certain Types of Integral Equations

  • Chapter

  • 144 Accesses

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 73))

Abstract

In the functional equation u=Tu for a function u (x) = u (x1, … , xn let the completely continuous linear or nonlinear operator T be “monotonically decomposible” in the sense of J. Schroder. We suppose, one has calculated, for instance with an iteration procedure an interval J = [v1,w1]which contains under certain conditions at least one solution u Schauders fixed point theorem. This is in many cases the only easily calculable possibility of an inclusion for u. This method is also applicable for calculation of solutions with certain types of singularities. Recently three-dimensional singuities in the three-dimensional space.

This method is also applicable for calculation of solutions with certain types of singularities. Recently three-dimensional singularities became more important. Numerical examples are given, also for distributed singularities in the three-dimensional space.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  • Bohl, E. [74] Monotonie, Lösbarkeit und Numerik bei Operatorgleigleichungen, Springer, 1974, 255p.

    Google Scholar 

  • Collatz, L. [66] Funktionalanalysis and Numerical mathematics, Academic Press, 1966, 473p.

    Google Scholar 

  • Collatz, L. [71] Some applications of Functional analysis to Analysis particularly to Nonlinear Integral equations, Prod. Nonlin. Funct. Anal. Applic. Acad. Press, 1971, 1–43.

    Google Scholar 

  • Collatz, L. [81] Anwendung von Monotniesotzen zur Einsechliebung der Losungen von Gkleichungen; Jahrbuch Uberblicke der Mathematik, 1981, 189–225

    Google Scholar 

  • Collatz, L. [82] Some numerical Test Problems with Singularities in: The Mathem of Finite Elements and Applications, MAFELAP 4 (1982), ed. Whiteman, 65–76.

    Google Scholar 

  • Collatz, L. [84] Approximation of solution of differential equations and of their derivatives, Proc. Construct. Th. of functions, Varna 1984, ed. Sendov.

    Google Scholar 

  • Dobrowolski, M. [83] Vortrag Tagung: Singularities and Constructive Methods for their treatment. Proc. Oberwolfach 21–26. November 1983.

    Google Scholar 

  • Kantorowitsch, L.W. - G.P. Akilow [64] Funktionalanalysis in nor mierten Rumen, Acad. Verl. Berlin, 1964, 622 p.

    Google Scholar 

  • Neta, B. [84] Adaptive Method for the Numerical Solution of Fred holm Integral Equations of the Second Kind;Part II: Singular Kernels. Report No. INTT-29, March 1984, l6p. Dep. Math. Texas Techn. Univ. Lubbock, Texas 79409.

    Google Scholar 

  • Reissig, R. [69] Anwendung von Fixpunktstzen auf das Problem der periodischen LSsungen bei nichtautonomen Systemen, DLR Forschungsbericht 69-53 (1969), 63 p.

    Google Scholar 

  • Schauder, J. [30] Der Fixpunktsatz inFunktionenrumen, Studia Math. 2 (1930) 171–182.

    Google Scholar 

  • Schnack, E. [84] Vortrag MAFELAP 1.-4. May 1984, Proc. ed. J.R. Whiteman

    Google Scholar 

  • Schröder, J. [59] Anwendung von Fixpunktsatzen bei der numerischen Behandlung nichtlinearer Gleichungen in halbgeordne- ten Raumen, Arch. Rat. Mech. Anal. 4 (1959) 177–192.

    Article  Google Scholar 

  • Schröder, J. [80] Operator inequalities, Acad. Press, (1980) 367 p.

    Google Scholar 

  • Tolksdorf, P. [83] Vortrag Tagung: Singularities and Constructive Methods for their treatment. Proc. Oberwolfach 21–26. Nov. 1983.

    Google Scholar 

  • Walter, W. [70] Differential and Integral Inequalities, Springer (1970), 352 p.

    Google Scholar 

  • Whiteman, J.R. [83] Vortrag Tagung: Singularities and Constructive Methods for their treatment. Proc. Oberwolfach 21.-26. Nov. 1983, see also: E. Stephan - J.R. Whiteman: Singularities of the Laplacian at Corners and edges of three dimensional domains. Bicon 81/1 Institute of Computational Mathematics, Brunel University, Uxbridge, U.K.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik der Universität Hamburg, Bundesstraße 55, D-2000, Hamburg 13, Germany

    Prof. Dr Lothar Collatz

Authors
  1. Prof. Dr Lothar Collatz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematisches Institut der, Ludwig-Maximilians-Universität, Theresienstraße 39, D-8000, München 2, Germany

    G. Hämmerlin

  2. Mathematisches Institut, der Universität Augsburg, Memminger Straße 6, D-8900, Augsburg, Germany

    K.-H. Hoffmann

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag

About this chapter

Cite this chapter

Collatz, L. (1985). Inclusion of Regular and Singular Solutions of Certain Types of Integral Equations. In: Hämmerlin, G., Hoffmann, KH. (eds) Constructive Methods for the Practical Treatment of Integral Equations. International Series of Numerical Mathematics, vol 73. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9317-6_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9317-6_7

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9993-2

  • Online ISBN: 978-3-0348-9317-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation