ABSTRACT
If D is an open and bounded subset of X with D ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137735/46b2ebee-a99f-4a7d-b17e-b7dad9c42178/content/eq1198.tif"/> and ∂D its closure and boundary in X, T: D ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137735/46b2ebee-a99f-4a7d-b17e-b7dad9c42178/content/eq1199.tif"/> → Y a given mapping that satisfies suitable conditions, and f ∉ T(dD), then the topological degree of T on D over f, deg(T,D,f), is in principle an algebraic count of the number of solutions x ∈ D of the equation Tx = f. For this count to be useful, it must have several crucial properties: additivity on the domain D, invariance under suitable homotopies on T, existence of a solution x in D of Tx = f if deg(T,D,f) ≠ 0, and so on.
