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Proceedings of the American Mathematical Society

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



Integral equations, implicit functions,
and fixed points

Author: T. A. Burton
Journal: Proc. Amer. Math. Soc. 124 (1996), 2383-2390
MSC (1991): Primary 45D05, 26B10, 47H10
Corrigendum: Proc. Amer. Math. Soc. 141 (2013), 4425-4426
MathSciNet review: 1346965
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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.

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Additional Information

T. A. Burton
Affiliation: Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901

Keywords: Integral equations, implicit functions, fixed points
Received by editor(s): February 6, 1995
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society