Parity and generalized multiplicity
HTML articles powered by AMS MathViewer
- by P. M. Fitzpatrick and Jacobo Pejsachowicz PDF
- Trans. Amer. Math. Soc. 326 (1991), 281-305 Request permission
Abstract:
Assuming that $X$ and $Y$ are Banach spaces and $\alpha :[a,b] \to \mathcal {L}(X,Y)$ is a path of linear Fredholm operators with invertible endpoints, in $[{\text {F}} - \text {P}1]$ we defined a homotopy invariant of $\alpha ,\sigma (\alpha ,I) \in {{\mathbf {Z}}_2}$, the parity of $\alpha$ on $I$. The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove (a) that, generically, the parity is a $\bmod 2$ count of the number of transversal intersections of $\alpha (I)$ with the set of singular operators, (b) that if ${\lambda _0}$ is an isolated singular point of $\alpha$, then the local parity \[ \sigma (\alpha ,{\lambda _0}) \equiv \lim \limits _{\varepsilon \to 0} \sigma (\alpha ,[{\lambda _0} - \varepsilon ,{\lambda _0} + \varepsilon ])\] remains invariant under Lyapunov-Schmidt reduction, and (c) that $\sigma (\alpha ,{\lambda _0}) = {(- 1)^{{M_G}({\lambda _0})}}$, where ${M_G}({\lambda _0})$ is any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data.References
- J. C. Alexander and P. M. Fitzpatrick, Galerkin approximations in several parameter bifurcation problems, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 3, 489–500. MR 556928, DOI 10.1017/S0305004100056929
- Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
- V. M. Eni, The multiplicity of an eigenvalue of an operator pencil, Mat. Issled. 4 (1969), no. vyp. 2 (12), 32–41 (Russian). MR 0281042
- J. Esquinas, Optimal multiplicity in local bifurcation theory. II. General case, J. Differential Equations 75 (1988), no. 2, 206–215. MR 961153, DOI 10.1016/0022-0396(88)90136-2
- J. Esquinas and J. López-Gómez, Optimal multiplicity in local bifurcation theory. I. Generalized generic eigenvalues, J. Differential Equations 71 (1988), no. 1, 72–92. MR 922199, DOI 10.1016/0022-0396(88)90039-3
- P. M. Fitzpatrick, Homotopy, linearization, and bifurcation, Nonlinear Anal. 12 (1988), no. 2, 171–184. MR 926211, DOI 10.1016/0362-546X(88)90033-8
- P. M. Fitzpatrick and Jacobo Pejsachowicz, An extension of the Leray-Schauder degree for fully nonlinear elliptic problems, Nonlinear functional analysis and its applications, Part 1 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 425–438. MR 843576
- P. M. Fitzpatrick and Jacobo Pejsachowicz, The fundamental group of the space of linear Fredholm operators and the global analysis of semilinear equations, Fixed point theory and its applications (Berkeley, CA, 1986) Contemp. Math., vol. 72, Amer. Math. Soc., Providence, RI, 1988, pp. 47–87. MR 956479, DOI 10.1090/conm/072/956479
- P. M. Fitzpatrick and Jacobo Pejsachowicz, A local bifurcation theorem for $C^1$-Fredholm maps, Proc. Amer. Math. Soc. 109 (1990), no. 4, 995–1002. MR 1009988, DOI 10.1090/S0002-9939-1990-1009988-5 —, Nonorientability of the index bundle and several parameter bifurcation, J. Funct. Anal. (in press). —, The Leray-Schauder theory and fully nonlinear elliptic boundary value problems, Mem. Amer. Math. Soc. (to appear).
- 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
- Avner Friedman and Marvin Shinbrot, Nonlinear eigenvalue problems, Acta Math. 121 (1968), 77–125. MR 250096, DOI 10.1007/BF02391910
- I. C. Gohberg and M. G. Kreĭn, The basic propositions on defect numbers, root numbers and indices of linear operators, Amer. Math. Soc. Transl. (2) 13 (1960), 185–264. MR 0113146
- M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
- 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, Necessary and sufficient conditions for multiparameter bifurcation, Rocky Mountain J. Math. 18 (1988), no. 2, 305–337. Nonlinear Partial Differential Equations Conference (Salt Lake City, UT, 1986). MR 951940, DOI 10.1216/RMJ-1988-18-2-305 T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., Vol. 132, Springer-Verlag, New York, 1980.
- Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 107819, DOI 10.1007/BF02790238
- Hansjörg Kielhöfer, Multiple eigenvalue bifurcation for Fredholm operators, J. Reine Angew. Math. 358 (1985), 104–124. MR 797678, DOI 10.1515/crll.1985.358.104
- Ulrich Koschorke, Infinite dimensional $K$-theory and characteristic classes of Fredholm bundle maps, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 95–133. MR 0279838
- M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by A. H. Armstrong; translation edited by J. Burlak. MR 0159197
- M. A. Krasnosel′skiĭ and P. P. Zabreĭko, Geometrical methods of nonlinear analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 263, Springer-Verlag, Berlin, 1984. Translated from the Russian by Christian C. Fenske. MR 736839, DOI 10.1007/978-3-642-69409-7
- David C. Lay, Spectral analysis using ascent, descent, nullity and defect, Math. Ann. 184 (1969/70), 197–214. MR 259644, DOI 10.1007/BF01351564
- N. G. Lloyd, Degree theory, Cambridge Tracts in Mathematics, No. 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978. MR 0493564
- B. Laloux and J. Mawhin, Multiplicity, Leray-Schauder formula, and bifurcation, J. Differential Equations 24 (1977), no. 3, 309–322. MR 442775, DOI 10.1016/0022-0396(77)90001-8
- R. J. Magnus, A generalization of multiplicity and the problem of bifurcation, Proc. London Math. Soc. (3) 32 (1976), no. 2, 251–278. MR 402561, DOI 10.1112/plms/s3-32.2.251
- A. S. Markus and E. I. Sigal, The multiplicity of the characteristic number of an analytic operator function, Mat. Issled. 5 (1970), no. 3(17), 129–147 (Russian). MR 0312306
- J. Pejsachowicz, $K$-theoretic methods in bifurcation theory, Fixed point theory and its applications (Berkeley, CA, 1986) Contemp. Math., vol. 72, Amer. Math. Soc., Providence, RI, 1988, pp. 193–206. MR 956492, DOI 10.1090/conm/072/956492
- Jacobo Pejsachowicz and Alfonso Vignoli, On the topological coincidence degree for perturbations of Fredholm operators, Boll. Un. Mat. Ital. B (5) 17 (1980), no. 3, 1457–1466 (English, with Italian summary). MR 770860
- W. V. Petryshyn, Bifurcation and asymptotic bifurcation for equations involving $A$-proper mappings with applications to differential equations, J. Differential Equations 28 (1978), no. 1, 124–154. MR 477926, DOI 10.1016/0022-0396(78)90082-7
- Patrick J. Rabier, Generalized Jordan chains and two bifurcation theorems of Krasnosel′skiĭ, Nonlinear Anal. 13 (1989), no. 8, 903–934. MR 1009078, DOI 10.1016/0362-546X(89)90021-7 —, Generalized Jordan chains and bifurcation with one-dimensional null-space, preprint.
- 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
- Paul Sarreither, Transformationseigenschaften endlicher Ketten und allgemeine Verzweigungsaussagen, Math. Scand. 35 (1974), 115–128 (German). MR 361859, DOI 10.7146/math.scand.a-11539
- C. A. Stuart, Some bifurcation theory for $k$-set contractions, Proc. London Math. Soc. (3) 27 (1973), 531–550. MR 333856, DOI 10.1112/plms/s3-27.3.531
- Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
- J. F. Toland, A Leray-Schauder degree calculation leading to nonstandard global bifurcation results, Bull. London Math. Soc. 15 (1983), no. 2, 149–154. MR 689248, DOI 10.1112/blms/15.2.149
- J. R. L. Webb and S. C. Welsh, $A$-proper maps and bifurcation theory, Ordinary and partial differential equations (Dundee, 1984) Lecture Notes in Math., vol. 1151, Springer, Berlin, 1985, pp. 342–349. MR 826304, DOI 10.1007/BFb0074743
- Stewart C. Welsh, Global results concerning bifurcation for Fredholm maps of index zero with a transversality condition, Nonlinear Anal. 12 (1988), no. 11, 1137–1148. MR 969494, DOI 10.1016/0362-546X(88)90048-X
- David Westreich, Bifurcation at eigenvalues of odd multiplicity, Proc. Amer. Math. Soc. 41 (1973), 609–614. MR 328707, DOI 10.1090/S0002-9939-1973-0328707-9
- M. G. Zaĭdenberg, S. G. Kreĭn, P. A. Kučment, and A. A. Pankov, Banach bundles and linear operators, Uspehi Mat. Nauk 30 (1975), no. 5(185), 101–157 (Russian). MR 0415661
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 326 (1991), 281-305
- MSC: Primary 58E07; Secondary 47H15, 58C99
- DOI: https://doi.org/10.1090/S0002-9947-1991-1030507-7
- MathSciNet review: 1030507