Abstract
We develop a degree theory forC 1 Fredholm mappings of index 0 between Banach spaces and Banach manifolds. As in earlier work devoted to theC 2 case, our approach is based upon the concept of parity of a curve of linear Fredholm operators of index 0. This avoids considerations about Fredholm structures involved in other approaches and leads to a theory as complete as that of Leray-Schauder in a much broader setting. In particular, the well-known possible sign change under homotopy is fully elucidated. The technical difficulty arising withC 1 versusC 2 Fredholm mappings of index 0 is notorious: with onlyC 1 smoothness, the Sard-Smale theorem is no longer available to handle crucial issues involving homotopy. In this work, this difficulty is overcome by using a new approximation theorem forC 1 Fredholm mappings of arbitrary index instead of the Sard—Smale theorem when dealing with homotopies.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Benevieri and M. Furi,A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree theory, Annales des Sciences Mathématiques du Québec (to appear).
R. Bonic and J. Frampton,Smooth functions on Banach manifolds, J. Math. Mech.15 (1966), 877–898.
Y. G. Borisovich, V. G. Zvyagin and Y. I. Sapronov,Nonlinear Fredholm maps and the LeraySchauder theory, Russian Math. Surveys324 (1977), 1–54.
F. E. Browder,Nonlinear operators and nonlinear equations of evolution in Banach spaces, inNonlinear Functional Analysis (F. E. Browder, ed.), Proc. Symp. Pure Math.18 (Part 2) (1976).
F. E. Browder and R. D. Nussbaum,The topological degree for noncompact nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. (N.S.)74 (1968), 671–676.
R. Brussee,The canonical class and the C∞ properties of KÄhler surfaces, preprint.
R. Caccioppoli,Sulle corrispondenze funzionali inverse diramate, teoria generale e applicazioni ad alcune equazioni funzionali nonlineari e alproblema di Plateau, I, II, Rend. Acad. Naz. Lincei(6)24 (1936), 258–263, 416–421.
K. D. Elworthy and A. J. Tromba,Differential structures and Fredholm maps on Banach manifolds, inGlobal Analysis (S. S. Chern and S. Smale, eds.), Proc. Symp. Pure Math.15 (1970), 45–94.
K. D. Elworthy and A. J. Tromba,Degree theory on Banach manifolds, inNonlinear Functional Analysis (F. E. Browder, ed.), Proc. Symp. Pure Math., Part118 (1970), 86–94.
P. M. Fitzpatrick and J. Pejsachowicz,The fundamental group of linear Fredholm operators and the global analysis of semilinear equations, Contemp. Math.72 (1988), 47–87.
P. M. Fitzpatrick and J. Pejsachowicz,Parity and generalized multiplicity, Trans. Amer. Math. Soc.326 (1991), 281–305.
P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier,The degree of proper C 2 Fredholm mappings I, J. Reine Angew. Math.427 (1992), 1–33.
P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier,Orientability of Fredholm families and topological degree for orientable Fredholm mappings, J. Funct. Anal.124 (1994), 1–39.
C. A. S. Isnard,The topological degree on Banach manifolds, Global Analysis and Its Applications, Vol. 2, International Atomic Energy Agency, Vienna, 1974.
H. Jeanjean, M. Lucia and C. A. Stuart,Branches of solutions to semilinear elliptic equations on ℝN, Math. Z. (to appear).
M. A. Krasnoselskii and P. P. Zabreiko,Geometrical methods of nonlinear analysis, Grundlehren Math. Wiss.263, Springer-Verlag, New York, 1984.
N. H. Kuiper,The homotopy type of the unitary group of Hilbert space, Topology3 (1965), 19–30.
N. Moulis,Approximation de fonctions différentiables sur certains espaces de Banach, Ann. Inst. Fourier (Grenoble)21 (1971), 293–345.
L. Nirenberg,Variational and topological methods in nonlinear problems. Bull. Amer. Math. Soc. (N.S.)4 (1981), 267–301.
R. D. Nussbaum,The fixed point index and fixed point theorems, inTopological Methods for Ordinary Differential Equations (M.G. Furi and P. Zecca, eds.), Lecture Notes in Math.1537, Springer-Verlag, New York, 1993.
R. S. Palais,Lusternik—Schnirelman theory on Banach manifolds, Topology5 (1966), 115–132.
J. Pejsachowicz and P. J. Rabier,A substitute for the Sard—Smale theorem in the C1case, J. Analyse Math., this volume.
F. Quinn and A. Sard,Hausdorff conullity of critical images of Fredholm maps, Amer. J. Math.94 (1972), 1101–1110.
PH. Rabinowitz,Some global results for nonlinear eigenvalue problems, J. Funct. Anal.7 (1971), 487–513.
Y. I. Sapronov,On the degree theory for nonlinear Fredholm maps, Voronezh Univ. Trudy Nauchn. Issled. Inst. Mat. No.11 (1973), 93–101.
S. Smale,An infinite dimensional generalization of Sard’s theorem, Amer. J. Math.87 (1965), 861–866.
C. Troestler and M. Willem,Nontrivial solutions of a semilinear Schrödinger equation, Comm. Partial Differential Equations21 (1996), 1431–1449.
A. J. Tromba,The Euler characteristic of vector fields on Banach manifolds and a globalization of Leray-Schauder degree, Adv. Math.28 (1978), 148–173.
V. G. Zvyagin,A study of the topological properties of nonlinear operators, Doctoral dissertation, Voronezh (1974).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pejsachowicz, J., Rabier, P.J. Degree theory forC 1 Fredholm mappings of index 0. J. Anal. Math. 76, 289–319 (1998). https://doi.org/10.1007/BF02786939
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02786939