Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-20T12:36:54.687Z Has data issue: false hasContentIssue false
  • English
  • Français

4.—Perturbations of Non-linear Integral Operators

Published online by Cambridge University Press:  14 February 2012

Friedrich Stummel
Friedrich Stummel
Affiliation:
Department of Mathematics, Johann Wolfgang Goethe- Universität, Frankfurt am Main.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Non-linear integral operators are studied under perturbations of the integrals, the domains of integration, the kernels and the inhomogeneous terms. The main theorem establishes the local solubility of the perturbed integral equations, the biconvergence of their solutions and associated two-sided error estimates. This theorem is of interest even for the special class of linear Fredholm integral equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 74 , 1976 , pp. 55 - 70
Copyright
Copyright © Royal Society of Edinburgh 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Anselone, P. M.. Convergence and error bounds for approximate solutions of intergral and operator equations. Proc. Symp. Wisconsin, Madison, 231–252 (New York: Wiley, 1965).Google Scholar
2Anselone, P. M.. Collectively compact operator approximation theory and applications to integral equations. (Englewood Cliffs: Prentice Hall, 1971).Google Scholar
3Atkinson, K. E.. The numerical evaluation of fixed points for completely continuous operators. SIAMJ. Numer. Anal. 10 (1973), 799–807.CrossRefGoogle Scholar
4Bourbaki, N.. Intégration, Ch. I-IV (Paris: Hermann, 1965).Google Scholar
5Dieudonné, J.. Foundations of modern analysis (New York: Academic, 1960).Google Scholar
6Hausdorff, F.. Grundzüge der Mengenlehre (New York: Chelsea,1965).Google Scholar
7Kantorovich, L. and Akilov, G.. Functional analysis is normed spaces(Oxford: Pergamon, 1964).Google Scholar
8Krasnosel'skii, M. A. and Vainikko, G. M., et al. Approximate solution of operator equations. (Groningen: Wolters-Noordhoff, 1972).CrossRefGoogle Scholar
9Moore, R. H.. Differentiability and convergence for compact nonlinear operators. J. Math.Anal. Appl. 16 (1966), 65–72.CrossRefGoogle Scholar
10Rinow, W.. Die innere Geometric der metrischen Räume. (Berlin: Springer, 1961).CrossRefGoogle Scholar
11Stummel, F.. Diskrete Konvergenz linearer Operatoren. I. Math. Ann. 190 (1970), 45–92; II. Math. Z. 120 (1971), 231–264;III. Proc. Oberwolfach Conf. Linear Operators and Approximation 1971, Int. Series of Numer. Math. 20,196–216 (Basel: Birkhäuser, 1972).CrossRefGoogle Scholar
12Stummel, F.. Approximation methods in analysis (Lecture Notes, Spring Semester, Aarhus Univ., 1973).Google Scholar
13Stummel, F.. Discretely uniform approximation of continuous functions. J. Approximation Theory 13 (1975), 178–191.CrossRefGoogle Scholar
14Stummel, F.. Perturbation theory for Sobolev spaces. Proc. Royal Soc. Edinburgh Sect. A, 73 (1975), 5–49.CrossRefGoogle Scholar
15Stummel, F.. Stability and discrete convergence of differentiable mappings. Rev. Roumaine Math. Pures Appl. submitted for publication.Google Scholar
16Vainikko, G. M.. On the convergence of the collocation method for non-linear differential equations. USSR Computational Math, and Math. Phys. 6 (1966), 47–58.CrossRefGoogle Scholar
17Vainikko, G. M.. Galerkin's perturbation method and the general theory of approximate method for non-linear equations. USSR Computational Math, and Math. Phys. 7 (1967), 1–41.CrossRefGoogle Scholar