A special quasi-linear mapping and its degree (Q2703137)

This article deals with a modification of the degree of mappings between Banach spaces that was introduced by A. I. Shnirel'man. The author studies some relations between Shnirel'man's classes of Fredholm linear and Fredholm quasilinear mappings and a new class of Fredholm special quasilinear mappings; the latter is defined as the class of uniform limits of Fredholm quasilinear mappings. The author states that the classes of Fredholm quasilinear mappings and Fredholm special quasilinear mappings coincide. Further, he introduces the degree of Fredholm special quasilinear mapping as a limit of degrees of corresponding Fredholm quasilinear mappings and investigates properties of this degree.NEWLINENEWLINENEWLINEThere are numerious vague places in this article. The author writes that his goal is an expansion to the mappings between Banach manifolds of a notion of the degree introduced by A. I. Shnirel'man, however, he considers only mappings between Hilbert spaces and, moreover, proves the coincidence of the class considered by him with the Shnirel'man's class; so, his class of mappings and their degree are not new.