Some generalizations of the first Fredholm theorem to multivalued A-proper mappings with applications to nonlinear elliptic equations

Let X and Y be real normed spaces with an admissible scheme Γ = {E_n, V_n; F_n, W_n} and T: X → 2^Y A-proper with respect to Γ such that dist(y, A(x)) < kc(∥ x ∥) for all y in T(x) with ∥ x ∥ ≥ R for some R > 0 and k > 0, where c: R⁺ → R⁺ is a given function and A: X → 2^Y a suitable possibly not A-proper mapping. Under the assumption that either T or A is odd or that (u, Kx) ≥ 0 for all u in T(x) with ∥ x ∥ ≥ r > 0 and some K: X → Y^*, we obtain (in a constructive way) various generalizations of the first Fredholm theorem. The unique approximation-solvability results for the equation T(x) = f with T such that T(x) - T(y) ε{lunate} A(x - y) for x, y in X or T is Fréchet differentiable are also established. The abstract results for A-proper mappings are then applied to the (constructive) solvability of some boundary value problems for quasilinear elliptic equations. Some of our results include the results of Lasota, Lasota-Opial, Hess, Nečas, Petryshyn, and Babuška.