Structure and dimension of global branches of solutions to multiparameter nonlinear equations

Ize, J.; Massabò, I.; Pejsachowicz, J.; Vignoli, A.

doi:10.1090/S0002-9947-1985-0800246-0

Structure and dimension of global branches of solutions to multiparameter nonlinear equations
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by J. Ize, I. Massabò, J. Pejsachowicz and A. Vignoli PDF

Trans. Amer. Math. Soc. 291 (1985), 383-435 Request permission

Abstract:

This paper is concerned with the topological dimension of global branches of solutions appearing in different problems of Nonlinear Analysis, in particular multiparameter (including infinite dimensional) continuation and bifurcation problems. By considering an extension of the notion of essential maps defined on sets and using elementary point set topology, we are able to unify and extend, in a selfcontained fashion, most of the recent results on such problems. Our theory applies whenever any generalized degree theory with the boundary dependence property may be used, but with no need of algebraic structures. Our applications to continuation and bifurcation follow from the nontriviality of a local invariant, in the stable homotopy group of a sphere, and give information on the local dimension and behavior of the sets of solutions, of bifurcation points and of continuation points.

References

J. C. Alexander and Stuart S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Rational Mech. Anal. 76 (1981), no. 4, 339–354. MR 628173, DOI 10.1007/BF00249970
J. C. Alexander and Stuart S. Antman, Global behavior of solutions of nonlinear equations depending on infinite-dimensional parameters, Indiana Univ. Math. J. 32 (1983), no. 1, 39–62. MR 684754, DOI 10.1512/iumj.1983.32.32004
J. C. Alexander and P. M. Fitzpatrick, The homotopy of certain spaces of nonlinear operators, and its relation to global bifurcation of the fixed points of parametrized condensing operators, J. Functional Analysis 34 (1979), no. 1, 87–106. MR 551112, DOI 10.1016/0022-1236(79)90027-2
J. C. Alexander, I. Massabò, and J. Pejsachowicz, On the connectivity properties of the solution set of infinitely-parametrized families of vector fields, Boll. Un. Mat. Ital. A (6) 1 (1982), no. 2, 309–312 (English, with Italian summary). MR 663297
Ju. G. Borisovič, Topology and nonlinear functional analysis, Uspekhi Mat. Nauk 34 (1979), no. 6(210), 14–22 (Russian). MR 562811
Felix E. Browder, Degree of mapping for nonlinear mappings of monotone type, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 6, i, 1771–1773. MR 699437, DOI 10.1073/pnas.80.6.1771
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

Nonlinear Fredholm maps and the Leray-Schauder theory

32

James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
P. M. Fitzpatrick, I. Massabò, and J. Pejsachowicz, On the covering dimension of the set of solutions of some nonlinear equations, Trans. Amer. Math. Soc. 296 (1986), no. 2, 777–798. MR 846606, DOI 10.1090/S0002-9947-1986-0846606-4
P. M. Fitzpatrick, I. Massabò, and J. Pejsachowicz, Complementing maps, continuation and global bifurcation, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 79–81. MR 699319, DOI 10.1090/S0273-0979-1983-15161-3
M. Furi, M. Martelli, and A. Vignoli, Contributions to the spectral theory for nonlinear operators in Banach spaces, Ann. Mat. Pura Appl. (4) 118 (1978), 229–294 (English, with Italian summary). MR 533609, DOI 10.1007/BF02415132
M. Furi, M. Martelli, and A. Vignoli, On the solvability of nonlinear operator equations in normed spaces, Ann. Mat. Pura Appl. (4) 124 (1980), 321–343 (English, with Italian summary). MR 591562, DOI 10.1007/BF01795399
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
A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces. I, Rozprawy Mat. 30 (1962), 93. MR 149253
Kazimierz Gęba and Andrzej Granas, Infinite dimensional cohomology theories, J. Math. Pures Appl. (9) 52 (1973), 145–270. MR 380865
J. Grispolakis and E. D. Tymchatyn, On confluent mappings and essential mappings—a survey, Rocky Mountain J. Math. 11 (1981), no. 1, 131–153. MR 612135, DOI 10.1216/RMJ-1981-11-1-131
Jorge Ize, Bifurcation theory for Fredholm operators, Mem. Amer. Math. Soc. 7 (1976), no. 174, viii+128. MR 425696, DOI 10.1090/memo/0174
Jorge Ize, Introduction to bifurcation theory, Differential equations (S ao Paulo, 1981) Lecture Notes in Math., vol. 957, Springer, Berlin-New York, 1982, pp. 145–202. MR 679145
Jorge Ize, Obstruction theory and multiparameter Hopf bifurcation, Trans. Amer. Math. Soc. 289 (1985), no. 2, 757–792. MR 784013, DOI 10.1090/S0002-9947-1985-0784013-2
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
Jean Leray and Jules Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78 (French). MR 1509338
J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636. MR 328703, DOI 10.1016/0022-0396(72)90028-9
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, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 312341, DOI 10.1007/BF02414948
A. R. Pears, Dimension theory of general spaces, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975. MR 0394604
Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9
B. N. Sadovskiĭ, Limit-compact and condensing operators, Uspehi Mat. Nauk 27 (1972), no. 1(163), 81–146 (Russian). MR 0428132
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Gordon Thomas Whyburn, Topological analysis, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N. J., 1958. MR 0099642