Rotation of Vector Fields: Definition, Basic Properties, and Calculation

Dirichlet Boundary Value Problem for the Second Order Asymptotically Linear System

“…Then [21,22], there is an integer ( , Ω), which is associated with the vector field and called the rotation of the vector field on the boundary Ω. A singular point 0 ∈ R of the vector field is called isolated [21,22], if there is neighbourhood ( 0 ) = {‖ − 0 ‖ < , ∈ R } containing no other singular points. In this case, the rotation ( , ( 0 )) is the same for any sufficiently small radius .…”

Leray-Schauder degree: a half century of extensions and applications

“…The quick search in the Mathematical Reviews disclosing 591 references to papers that make use of it, mentioned by Peter Lax in [42], is surely underestimated, and the real figure should be much larger than one thousand. The reader can consult the monographs [174], [126], [141], [81] and their references to get a first idea of the tremendous bibliography related to the consequences and extensions of [53]. The bibliography of this paper includes a (surely uncomplete) list of some one hundred and twenty monographs dealing with Leray-Schauder theory and its applications, published between 1948 and 1999.…”

“…After some pioneering work of Rothe in 1939 [69], the extension of Leray-Schauder theory to completely continuous perturbations of identity in locally convex spaces was also worked out in details by Nagumo in 1951 [60], but the number of its applications to differential equations has been rather limited. It is still an open problem to know if the extension works for arbitrary topological vector spaces, even if Klee, Kaballo, Kayser, Krauthausen, Riedrich, Hahn, Alex, Kaniok, Van der Bijl, Dobrowolski, Hart, Van der Mill, Pötter, Okoń and others have shown that it can be done in some more general classes than the locally convex ones (see [81] for references).…”

Extension of the Leray–Schauder degree for abstract Hammerstein type mappings

“…An extensive overview may be found in [32] (see also [21,26] The notion of the topological degree is closely connected to 0-epi maps, see [23] and the references therein. For other closely related topics and equivalent concepts, like essential mappings, A-proper mappings, rotation theory of vector fields, degree theory for equivariant vector fields, coincidence degree, index theories on manifolds, winding numbers and discussion on the validity of the Hopf's theorem we refer to [21,26,31,32] and references therein.…”