Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-07T00:18:29.291Z Has data issue: false hasContentIssue false
  • English
  • Français

On Approximations to Solutions of Nonlinear Integral Equations of the Urysohn Type

Published online by Cambridge University Press:  20 November 2018

K. A. Zischka
K. A. Zischka*
Affiliation:
University of Windsor, Windsor, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This note will derive a priori estimates of the errors due to replacing the given integral operator A by a similar operator A* of the same type when successive approximations are applied to the integral equation φ=Aφ.

The existence and uniqueness of solutions to this equation follow easily by applying a well known fixed point theorem in a Banach space to the above mapping [1, 2]. Moreover, sufficient conditions for the existence and uniqueness of a solution to Urysohn's equation are stated explicitly in a note by the author [3].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 1 , March 1973 , pp. 137 - 141
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Thielman, H. P., (Applications of the fixed point theorem by Russian mathematicians) Nonlinear integral equations, by P. M. Anselone, Univ. of Wisconsin Press, Madison, Wis. (1964), 35–68.Google Scholar
2. Krasnosels’kii, M. A., Topological methods in the theory of nonlinear integral equations, Gosudarstvennoe Izdatel’stvo Tekhniko-teoreticheskoi literatury, (Russian), Moscow, 1958.Google Scholar
3. Zischka, K. A., On the existence and uniqueness of solutions of nonlinear equations of the Urysohn type, Math. Note, Amer. Math. Monthly (to appear).Google Scholar
4. Urabe, M., Convergence of numerical iterations in solutions of equations, J. Sci. Hiroshima Univ. Ser. A-19 Math., (1957), 479–489.Google Scholar
5. Collatz, L., Functional analysis and numerical mathematics, Academic Press, New York (1966), 218–220.Google Scholar