Fredholm Mappings and Banach Manifolds

Abstract

Abstract. Two C 1 -mappings, whose domain is a connected compact C 1 -Banach manifold modelled over a Banach space X over K = R or C and whose range is a Banach space Y over K, are introduced. Sufficient conditions are given to assert they share only a value. The proof of the result, which is based upon continuation methods, is constructive. PreliminariesScientific phenomena are locally described by parameters, whose choice is sometimes arbitrary. This implies the importance of the availability of a methodology for the comparison of results of measurements. Locally, a Banach manifold looks like a Banach space. For a local description, different Banach (or coordinate or parameter) spaces are allowed and transformation rules exist for these coordinates.Let X, Y be two Banach spaces. Let u : U ⊂ X → Y be a continuous mapping. One way of solving the equation (1) u(x) = y for any fixed y ∈ Y, is to embed (1) in a continuum of problemswhich is solved when t = 0. When t = 1, problem (2) becomes (1). If it is possible to continue the solution for all t ∈ [0, 1], then (1) is solved. This is the continuation method with respect to a parameter . A continuation method was introduced to solve (1) when u : M → R n , where M is a connected compact C 1 -Banach manifold modelled on R n , and H(·, ·) is a C 1 -mapping [25]. Here M is a Banach manifold modelled on an infinite-dimensional Banach space X over K = R or C and u ranges over an infinite-dimensional Banach space Y over K.