Existence Theorems for Equations with Noncoercive Discontinuous Operators

In a Hilbert space, we consider equations with a coercive operator equal to the sum of a linear Fredholm operator of index zero and a compact operator (generally speaking, discontinuous). By using regularization and the theory of topological degree, we establish the existence of solutions that are continuity points of the operator of the equation. We apply general results to the proof of the existence of semiregular solutions of resonance elliptic boundary-value problems with discontinuous nonlinearities.