A substitute for the Sard-Smale theorem in the \(C^1\) case
The purpose of this paper is to find a Fredholm perturbation of a Fredholm mapping \(F\) of index \(\tau \in\mathbb{Z},\) which is small in the \(C^1\) sense, has 0 as a regular value and can be connected to \(F\) by a \(C^1\) Fredholm homotopy with compact zero set if \(F\) is proper. Let us observe that the Fredholm mapping \(F\) is only \(C^1,\) yet of arbitrary index. The Sard-Smale theorem yields the existence of arbitrary small perturbations of a Fredholm mapping of index \(\tau \in\mathbb{Z}\) and class \(C^{{\max(\tau,0)}+1}\) between separable Banach spaces that have 0 as a regular value. In this theorem, the smoothness hypotheses can be weakened if \(\tau \geq 0,\) but the authors prove in this paper that the approximation results remain valid irrespective of the index \(\tau\) when \(F\) only belongs to \(C^1\) and verifies properness-like conditions. The hypotheses of separable spaces is not needed either. The authors also prove that in Banach spaces the hypotheses about \(C^1\) norm can be dropped. In order to state these results, the authors point out that all the arguments go over to the setting of Banach manifolds modeled on spaces equipped with a norm of \(C^1\) class away from the origin.
scientific article; zbMATH DE number 1274832