A new topological degree theory for densely defined [formula omitted]-perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces

Abstract

Let X be a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space X ∗ . Let T : X ⊃ D ( T ) → 2 X ∗ be maximal monotone, with 0 ∈ D ∘ ( T ) and 0 ∈ T ( 0 ) , and C : X ⊃ D ( C ) → X ∗ . Assume that L ⊂ D ( C ) is a dense linear subspace of X , C is of class ( S + ) L , and 〈 C x , x 〉 ≥ − ψ ( ‖ x ‖ ) , x ∈ D ( C ) , where ψ : R + → R + is nondecreasing. A new topological degree theory is developed for the sum T + C . The current approach utilizes the “approximate” degree d ( T t + C , G , 0 ) , t ↓ 0 , ( T t ≔ ( T − 1 + t J − 1 ) − 1 , G ⊂ X open and bounded) of Kartsatos and Skrypnik for the single-valued mapping T t + C . The subdifferential ∂ φ , for φ belonging to a large class of proper convex lower semicontinuous functions, gives rise to operators T to which this degree theory applies. A theoretical application to an existence problem of nonlinear analysis is included.