Strongly quasibounded maximal monotone perturbations for the Berkovits–Mustonen topological degree theory

Abstract

Let X be a real reflexive Banach space with dual X ∗ . Let L : X ⊃ D ( L ) → X ∗ be densely defined, linear and maximal monotone. Let T : X ⊃ D ( T ) → 2 X ∗ , with 0 ∈ D ( T ) and 0 ∈ T ( 0 ) , be strongly quasibounded and maximal monotone, and C : X ⊃ D ( C ) → X ∗ bounded, demicontinuous and of type ( S + ) w.r.t. D ( L ) . A new topological degree theory has been developed for the sum L + T + C . This degree theory is an extension of the Berkovits–Mustonen theory (for T = 0 ) and an improvement of the work of Addou and Mermri (for T : X → 2 X ∗ bounded). Unbounded maximal monotone operators with 0 ∈ D ˚ ( T ) are strongly quasibounded and may be used with the new degree theory.