The structure of the solution set of some nonlinear problems
McKenna, P.J. ; Shaw, Howard
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, Tome 64 (1978), p. 239-243 / Harvested from Biblioteca Digitale Italiana di Matematica
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Per equazioni operazionali L⁢u+N⁢u=h, L ed N operatori in uno spazio di Hilbert reale X, L lineare, N non lineare, e sotto moderate ipotesi su L ed N, l'insieme delle soluzioni è, generalmente, una varietà di dimensione uguale all'indice di Fredholm di L. Precisamente, questo accade effettivamente se la proiezione di h su un opportuno sottospazio E di dimensione finita in X non cade su un certo insieme Z di E, di misura zero oppure di prima categoria.

Publié le : 1978-12-01
@article{RLINA_1978_8_65_6_239_0,
     author = {P.J. McKenna and Howard Shaw},
     title = {The structure of the solution set of some nonlinear problems},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
     volume = {64},
     year = {1978},
     pages = {239-243},
     zbl = {0446.47057},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLINA_1978_8_65_6_239_0}
}

McKenna, P.J.; Shaw, Howard. The structure of the solution set of some nonlinear problems. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, Tome 64 (1978) pp. 239-243. http://gdmltest.u-ga.fr/item/RLINA_1978_8_65_6_239_0/

[1] Ambrosetti, A. and Prodi, G. (1972) - On the inversion of some differentiable mappings with singularities between Banach spaces, «Ann. di Mat. Pura Appl.», IV, 93, 231-247. | MR 320844 | Zbl 0288.35020

[2] Berger, M. and Podolak, E. (1975) - On the solutions of a nonlinear Dirichlet problem, «Indiana J. Math.», 24, 837-846. | MR 377274 | Zbl 0329.35026

[3] Brezis, H. and Nirenberg, L. (1978) - Characterisations of the ranges of some nonlinear operators and applications to boundary value problems, «Ann. Scuola Norm. Sup. Pisa», 4, 225-326. | MR 513090 | Zbl 0386.47035

[4] Cesari, L. (1976) - Functional analysis, nonlinear differential equations, and the alternative method, in «Nonlinear Functional Analysis and Differential Equations» (L. Cesari, R. Kannan and J. D. Schuur, Eds.) Dekker, New York, 1-197. | MR 487630

[5] Cesari, L. (1963) - Functional analysis and periodic solutions of nonlinear differential equations. «Contributions to Differential Equations, John Willey», 1, 149-187. | MR 151678

[6] Cesari, L. (1964) - Functional analysis and Galerkin's method, «Michigan Math. Journal», 11, 385-414. | MR 173839 | Zbl 0192.23702

[7] Fucik, S. (1976) - Boundary value problems with jumping nonlinearities, «Cas Prestovani Mat.», 101, 69-87. | MR 447688 | Zbl 0332.34016

[8] Hale, J. K. (1971) - Applications of alternative problems, Lecture notes of the Lefschetz Center for Dynamical Systems, Brown University.

[9] Hess, P. and Ruf, B. - On a superelliptic boundary value problem, to appear. | Zbl 0391.35034

[10] Hess, P. (1974) - On semicoercive problems, «Indiana Univ. Math. J.», 23, 645-654. | MR 341214 | Zbl 0259.47051

[11] Kazdan, J. L. and Warner, F. W. (1975) - Remarks on some quasilinear elliptic equations, «Comm. Pure Appl. Math.», 28, 567-597. | MR 477445 | Zbl 0325.35038

[12] Landesman, E. M. and Lazer, A. C. (1970) - Nonlinear perturbations of elliptic boundary value problems at resonance, «J. Math. Mech.», 19, 609-623. | MR 267269 | Zbl 0193.39203

[13] Leray, J. and Schauder, J. (1934) - Topologie et equations fonctionnelles, «Ann. Sci. École Norm. Sup.», 51, 45-78. | MR 1509338 | Zbl 60.0322.02

[14] Mckenna, P. J. and Shaw Howard, - On the structure of the set of solutions to some nonlinear boundary value problems, «J. Diff. Equations», to appear. | MR 561977 | Zbl 0397.35010

[15] Mckenna, P. J. - On a superlinear elliptic boundary value problem at resonance, «Proc. Amer. Math. Soc.», to appear. | MR 524297

[16] Mckenna, P. J. - On the reduction of a semilinear hyperbolic problem to a Landesman-Lazer type problem, to appear in «Houston Journal of Math.». | MR 523615 | Zbl 0411.35059

[17] Rabinowitz, P. H. (1978) - Some minimax theorems and applications to nonlinear partial differential equations, in «Nonlinear Analysis, a volume in honor of E. H. Rothe», Academic Press, 161-177. | MR 501092

[18] Shaw Howard, (1977) - A nonlinear elliptic boundary value problem at resonance, «J. Diff. Equations», 26, 335-346. | MR 463687 | Zbl 0341.35039

[19] Strauss, W. (1971) - On weak solutions of semilinear hyperbolic equations, «An. Acad. Brasil, Ci.», 42, 645-671. | MR 306715

[20] Williams, S. A. (1970) - A sharp sufficient conditions for solution of a nonlinear elliptic boundary value problem, «J. Diff. Equations», 8, 580-588. | MR 267267 | Zbl 0209.13003

[21] Yosida Kosaku, (1965) - Functional Analysis, Springer Verlag, Berlin. | MR 180824