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Integral equations, implicit functions, and fixed points. (English) Zbl 0873.45003

The author is concerned with a generalized integral equation of the form \[ V(t,x)= S\Bigl(t, \int^t_0 H(t,s,x(s))\,ds\Bigr) \] in the space of continuous functions. The operator \(V\) will define a contraction \(\widetilde V\), while the operator \(S\) is assumed to be compact. The existence problem is then reduced to finding fixed points for the mapping \(Px=\widetilde Sx+ (I-\widetilde V)x\). The implicit function theorem and classical fixed point theorems (Schauder, Krasnoselskii) are the tools used by the author in obtaining two fixed point theorems, the first of the type of contraction mapping and the second of Krasnoselskii type (for the sum of two operators). He then applies them to the generalized integral equation considered. Many known results can be derived from the Theorem 3 in this paper. Some examples are provided.

MSC:

45G10 Other nonlinear integral equations
47H10 Fixed-point theorems
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References:

[1] C. Corduneanu, Integral equations and applications, Cambridge University Press, Cambridge, 1991. · Zbl 0714.45002
[2] Philip Hartman, Ordinary differential equations, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. · Zbl 0281.34001
[3] M. A. Krasnosel\(^{\prime}\)skiĭ, Some problems of nonlinear analysis, American Mathematical Society Translations, Ser. 2, Vol. 10, American Mathematical Society, Providence, R.I., 1958, pp. 345 – 409.
[4] Erwin Kreyszig, Introductory functional analysis with applications, John Wiley & Sons, New York-London-Sydney, 1978. · Zbl 0368.46014
[5] Walter Rudin, Principles of mathematical analysis, Second edition, McGraw-Hill Book Co., New York, 1964. · Zbl 0052.05301
[6] Schauder, J., Über den Zusammenhang zwischen der Eindeutigkeit und Lösbarkeit partieller Differentialgleichungen zweiter Ordnung von Elliptischen Typus, Math. Ann. 106 (1932), 661–721. · Zbl 0004.35001
[7] Sine, Robert C., Fixed Points and Nonexpansive Mappings, Amer. Math. Soc. (Contemporary Mathematics Vol. 18), Providence, R.I., 1983. · Zbl 0599.47085
[8] Smart, D. R., Fixed Point Theorems, Cambridge Univ. Press, Cambridge, 1980. · Zbl 0427.47036
[9] Angus E. Taylor and W. Robert Mann, Advanced calculus, 3rd ed., John Wiley & Sons, Inc., New York, 1983. · Zbl 0584.26001
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