The generalized boundary value problem (BVP for short) for $n$-th order ordinary differential equations generates a mapping from a Banach space into another Banach space which mapping in most cases is the sum of a Fredholm mapping of index $0$ and of a completely continuous mapping. The properties of such mappings are very thoroughly studied in the paper, especially generic properties, surjectivity and the bifurcation problem. Applications to the generalized BVP are given and in the case when the corresponding ar homogeneous BVP is positive or self-adjoint some new results have been derived.
V. Šeda. "Fredholm mappings and the generalized boundary value problem." Differential Integral Equations 8 (1) 19 - 40, 1995. https://doi.org/10.57262/die/1369143782