Abstract
If we leave out of consideration some works by H. Poincaré, E. Picard, and E. Le Roy, relating to equations of particular type, then the date of initiation of the modern theory of nonlinear elliptic equations of the second order can be fixed at 1900. In that year, in fact, at the International Congress of Paris, D. Hilbert stated his conjecture that every solution of an analytic elliptic equation is analytic.
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References
Birkhof, G.D., and O.D. Kellogg: Invariant points in function space. Trans. Amer. Math. Soc. 23 (1922) 96–115.
Levy, P.: Sur les fonctions de lignes implicites. Bull. Soc. Math. de France 48 (1920) 13–27.
Hildebrandt, T. H., and L. M. Graves, Implicit functions and their differentials in general analysis. Trans. Amer. Math. Soc. 29 (1927) 127–153.
According to the more general result obtained by J. Schauder [4], theorem 41, VII holds under the sole hypothesis that K is bounded and convex. On this question see also R. Caccioppoli [1, 2] and for a first generalization Tichonoff: Ein Fixpunktsatz. Math. Ann. 111 (1935) 767–776. Other noteworthy extensions have now been obtained in recent years by F. E. Browder. For bibliographic indications relating to this we refer to Browder’s more recent works on the argument: A further generalization of the Schauder fixed point theorem, Duke Math. J. 32 (1965) 575-578. Fixed point theorems for non linear semicontractive mappings in Banach spaces, Arch. Rat. Mech. Anal. 21 (1966) 259-269.
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© 1970 Springer-Verlag Berlin Heidelberg
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Miranda, C. (1970). Nonlinear equations. In: Partial Differential Equations of Elliptic Type. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 2 . Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-35147-5_6
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DOI: https://doi.org/10.1007/978-3-662-35147-5_6
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