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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.

References [Enhancements On Off] (What's this?)

  • [1] Corduneanu, C, Integral Equations and Applications, Cambridge Univ. Press, Cambridge, 1991. MR 92h:45001
  • [2] Hartman, Philip, Ordinary Differential Equations, Wiley, New York, 1973. MR 49:9294
  • [3] Krasnoselskii, M. A., in Amer. Math. Soc. Transl. (2) 10 (1958), 345--409. MR 20:1243
  • [4] Kreyszig, Erwin, Introductory Functional Analysis with Applications, Wiley, New York, 1978. MR 57:7084
  • [5] Rudin, Walter, Principles of Mathematical Analysis, 2nd ed., McGraw-Hill, New York, 1964. MR 29:3587
  • [6] Schauder, J., Über den Zusammenhang zwischen der Eindeutigkeit und Lösbarkeit partieller Differentialgleichungen zweiter Ordnung von Elliptischen Typus, Math. Ann. 106 (1932), 661--721.
  • [7] Sine, Robert C., Fixed Points and Nonexpansive Mappings, Amer. Math. Soc. (Contemporary Mathematics Vol. 18), Providence, R.I., 1983.
  • [8] Smart, D. R., Fixed Point Theorems, Cambridge Univ. Press, Cambridge, 1980.
  • [9] Taylor, Angus E. and Mann, W. Robert, Advanced Calculus, Third ed., Wiley, New York, 1983. MR 83m:26001

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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

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