Fredholm alternatives for nonlinear A-proper mappings with applications to nonlinear elliptic boundary value problems

AbstractLet X and Y be real Banach spaces with a projectionally complete scheme Γ = {Xn, Pn; Yn, Qn} and let T: X → Y be an asymptotically linear mapping which is A-proper with respect to Γ and whose asymptotic derivative T∞ ϵ L(X, Y) is also A-proper with respect to Γ. Necessary and sufficient conditions are given in order that the equation T(x) = ƒ be solvable for a given ƒ in Y. Under certain additional conditions it is shown that solutions can be constructed as strong limits of finite dimensional Galerkin type approximates xn ϵ Xn. Theorems 1 and 2 include as special cases the recent results of Kachurovskii, Hess, Nečas, and the author. The abstract results for A-proper mappings are then applied to the (constructive) solvability of boundary value problems for quasilinear elliptic equations of order 2m with asymptotically linear terms of order 2m − 1