Topological degree theories for continuous perturbations of resolvent compact maximal monotone operators, existence theorems and applications

Abstract

Let X be a real locally uniformly convex reflexive Banach space. Let T : X ⊇ D(T) → 2X and A : X ⊇ D(A) → 2X be maximal monotone operators such that T is of compact resolvents and A is strongly quasibounded, and C : X ⊇ D(C) → X∗ be a bounded and continuous operator with D(A) ⊆ D(C) or D(C) = U. The set U is a nonempty and open (possibly unbounded) subset of X. New degree mappings are constructed for operators of the type T +A+C. The operator C is neither pseudomonotone type nor defined everywhere. The theory for the case D(C) = U presents a new degree mapping for possibly unbounded U and both of these theories are new even when A is identically zero. New existence theorems are derived. The existence theorems are applied to prove the existence of a solution for a nonlinear variational inequality problem.