Integral equations, implicit functions, and fixed points
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- by T. A. Burton
- Proc. Amer. Math. Soc. 124 (1996), 2383-2390
- DOI: https://doi.org/10.1090/S0002-9939-96-03533-2
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Corrigendum: Proc. Amer. Math. Soc. 141 (2013), 4425-4426.
Abstract:
The problem is to show that (1) $V(t,x) = S(t, \int _0^t H(t, s, x(s)) ds )$ has a solution, where $V$ defines a contraction, $\tilde V$, and $S$ defines a compact map, $\tilde S$. A fixed point of $P \varphi = \tilde S \varphi + (I - \tilde V) \varphi$ would solve the problem. Such equations arise naturally in the search for a solution of $f(t, x) = 0$ where $f(0,0) = 0$, but $\partial f(0,0) / \partial x = 0$ so that the standard conditions of the implicit function theorem fail. Now $P \varphi = \tilde S \varphi + ( I - \tilde V) \varphi$ would be in the form for a classical fixed point theorem of Krasnoselskii if $I - \tilde V$ were a contraction. But $I - \tilde V$ fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that $I - \tilde V$ has enough properties that an extension of Krasnoselskii’s theorem still holds and, hence, (1) has a solution. This substantially improves the classical implicit function theorem and proves that a general class of integral equations has a solution.References
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Bibliographic Information
- T. A. Burton
- Affiliation: Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901
- Email: taburton@math.siu.edu
- Received by editor(s): February 6, 1995
- Communicated by: Hal L. Smith
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2383-2390
- MSC (1991): Primary 45D05, 26B10, 47H10
- DOI: https://doi.org/10.1090/S0002-9939-96-03533-2
- MathSciNet review: 1346965