Degree theory for \(C^1\) Fredholm mappings of index 0
This paper provides an integer-valued degree theory for \(C^{1}\) Fredholm mappings of index zero between Banach spaces. The authors also discuss the extension to Banach manifolds. The authors' degree theory is based on the ideas introduced in the \(C^{2}\) case by \textit{P. M. Fitzpatrick, J. Pejsachowicz} and \textit{P. J. Rabier} [J. Reine Angew. Math. 427, 1-33 (1992; Zbl 0745.47049); J. Funct. Anal. 124, 1-39 (1994; Zbl 0802.47056)]. This degree theory addresses orientation of the mapping rather than of the spaces. In brief the sign associated to a regular point arises as follows: choose a regular point as base point; connect it to the point in question by a path; consider the associated path of derivatives; create a path of invertible almost-inverses of these derivatives; and take the product of the Leray-Schauder degrees of the compositions at the endpoints. The degree discussed here is not homotopy-invariant, but the authors provide a way to track its behavior under homotopy. The improvement from \(C^{2}\) to \(C^{1}\) depends on addressing homotopies by using an approximation theorem for \(C^{1}\) Fredholm mappings of arbitrary index in place of the Sard-Smale theorem. This approximation theorem is proven in \textit{J. Pejsachowicz} and \textit{P. J. Rabier} [J. Anal. Math. 76, 265-288 (1998; Zbl 0924.58001)]. Degree theory for Fredholm mappings is important for the study of nonlinear partial differential equations. The authors comment that the improvement from \(C^{2}\) to \(C^{1}\) brings additional meaningful problems within the scope of the theory and considerably reduces the difficulty of checking that other problems fall within the scope of the theory.
scientific article; zbMATH DE number 1274833
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