Structure and dimension of global branches of solutions to multiparameter nonlinear equations
Authors:
J. Ize, I. Massabò, J. Pejsachowicz and A. Vignoli
Journal:
Trans. Amer. Math. Soc. 291 (1985), 383-435
MSC:
Primary 58E07; Secondary 47H15
DOI:
https://doi.org/10.1090/S0002-9947-1985-0800246-0
MathSciNet review:
800246
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Abstract | References | Similar Articles | Additional Information
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.
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/0022-1236%2879%2990027-2
- 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 https://doi.org/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 https://doi.org/10.1016/0022-1236%2869%2990041-x Yu. G. Borisovich, V. G. Zvyagin and Yu. I. Sapronov, Nonlinear Fredholm maps and the Leray-Schauder theory, Russian Math. Surveys 32 (1977), 1-54.
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1090/S0002-9947-1985-0784013-2
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, 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 https://doi.org/10.1016/0022-0396%2872%2990028-9
- 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 https://doi.org/10.1016/0022-1236%2884%2990069-7
- Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 312341, DOI https://doi.org/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 https://doi.org/10.1016/0022-1236%2871%2990030-9
- 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
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Keywords:
Global branches of solutions,
multiparameter continuation problems,
multiparameter bifurcation problems,
covering dimension,
essential maps,
cantor manifolds
Article copyright:
© Copyright 1985
American Mathematical Society
