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

Abstract

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 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.