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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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
  • C. Corduneanu, Integral equations and applications, Cambridge University Press, Cambridge, 1991. MR 1109491, DOI 10.1017/CBO9780511569395
  • Philip Hartman, Ordinary differential equations, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. MR 0344555
  • M. A. Krasnosel′skiĭ, Some problems of nonlinear analysis, American Mathematical Society Translations, Ser. 2, Vol. 10, American Mathematical Society, Providence, R.I., 1958, pp. 345–409. MR 0094731
  • Erwin Kreyszig, Introductory functional analysis with applications, John Wiley & Sons, New York-London-Sydney, 1978. MR 0467220
  • Walter Rudin, Principles of mathematical analysis, 2nd ed., McGraw-Hill Book Co., New York, 1964. MR 0166310
  • Schauder, J., Über den Zusammenhang zwischen der Eindeutigkeit und Lösbarkeit partieller Differentialgleichungen zweiter Ordnung von Elliptischen Typus, Math. Ann. 106 (1932), 661–721.
  • Sine, Robert C., Fixed Points and Nonexpansive Mappings, Amer. Math. Soc. (Contemporary Mathematics Vol. 18), Providence, R.I., 1983.
  • Smart, D. R., Fixed Point Theorems, Cambridge Univ. Press, Cambridge, 1980.
  • Angus E. Taylor and W. Robert Mann, Advanced calculus, 3rd ed., John Wiley & Sons, Inc., New York, 1983. MR 674807
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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