On the covering dimension of the set of solutions of some nonlinear equations

Fitzpatrick, P. M.; Massabò, I.; Pejsachowicz, J.

doi:10.1090/S0002-9947-1986-0846606-4

On the covering dimension of the set of solutions of some nonlinear equations
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by P. M. Fitzpatrick, I. Massabò and J. Pejsachowicz PDF

Trans. Amer. Math. Soc. 296 (1986), 777-798 Request permission

Abstract:

We prove an abstract theorem whose sole hypothesis is that the degree of a certain map is nonzero and whose conclusions imply sharp, multidimensional continuation results. Applications are given to nonlinear partial differential equations.

References

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
J. C. Alexander and James A. Yorke, The implicit function theorem and the global methods of cohomology, J. Functional Analysis 21 (1976), no. 3, 330–339. MR 0397772, DOI 10.1016/0022-1236(76)90044-6
Herbert Amann, Antonio Ambrosetti, and Giovanni Mancini, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z. 158 (1978), no. 2, 179–194. MR 481498, DOI 10.1007/BF01320867
Glen E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin-New York, 1967. MR 0214062
Felix E. Browder, On continuity of fixed points under deformations of continuous mappings, Summa Brasil. Math. 4 (1960), 183–191. MR 130683
Felix E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 1–60. MR 0269962
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
F. E. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Functional Analysis 3 (1969), 217–245. MR 0244812, DOI 10.1016/0022-1236(69)90041-x
A. Dold, Lectures on algebraic topology, Die Grundlehren der mathematischen Wissenschaften, Band 200, Springer-Verlag, New York-Berlin, 1972 (German). MR 0415602

Dimension theory

Multiple solutions for nonlinear Fredholm operators

Massimo Furi and Maria Patrizia Pera, On the existence of an unbounded connected set of solutions for nonlinear equations in Banach spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 67 (1979), no. 1-2, 31–38 (1980) (English, with Italian summary). MR 617272
Georg Hetzer, A note on a paper of Howard Shaw concerning a nonlinear elliptic boundary value problem, J. Differential Equations 32 (1979), no. 2, 186–192. MR 534547, DOI 10.1016/0022-0396(79)90058-5
Jean Leray and Jacques-Louis Lions, Quelques résulatats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), 97–107 (French). MR 194733

Topologie et équations fonctionnelles

51

I. Massabò and J. Pejsachowicz, On the connectivity properties of the solution set of parametrized families of compact vector fields, J. Funct. Anal. 59 (1984), no. 2, 151–166. MR 766486, DOI 10.1016/0022-1236(84)90069-7
Roger D. Nussbaum, Degree theory for local condensing maps, J. Math. Anal. Appl. 37 (1972), 741–766. MR 306986, DOI 10.1016/0022-247X(72)90253-3
M. Patrizia Pera, Unbounded components of solutions of nonlinear equations at resonance, with applications to elliptic boundary value problems, Boll. Un. Mat. Ital. B (6) 2 (1983), no. 2, 469–481 (English, with Italian summary). MR 716743
W. V. Petryshyn, On the approximation-solvability of equations involving $A$-proper and psuedo-$A$-proper mappings, Bull. Amer. Math. Soc. 81 (1975), 223–312. MR 388173, DOI 10.1090/S0002-9904-1975-13728-1
Paul H. Rabinowitz, A note on a nonlinear elliptic equation, Indiana Univ. Math. J. 22 (1972/73), 43–49. MR 338554, DOI 10.1512/iumj.1972.22.22006
Howard Shaw, A nonlinear elliptic boundary value problem at resonance, J. Differential Equations 26 (1977), no. 3, 335–346. MR 463687, DOI 10.1016/0022-0396(77)90084-5
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
J. R. L. Webb, Mapping and fixed-point theorems for non-linear operators in Banach-spaces, Proc. London Math. Soc. (3) 20 (1970), 451–468. MR 266008, DOI 10.1112/plms/s3-20.3.451

Remarks on

set contractions

4