Let X and Y be Banach spaces,
ϕ : X → Y∗ and P : X → 2Y; P is said to be strongly ϕ-accretive if there exists c > 0
so that (w − v,ϕ(x − y)) ≥ c∥x − y∥2 whenever x,y ∈ X and w ∈ Px, v ∈ Py. Such
mappings constitute a simultaneous generalization of monotone mappings
(when Y = X∗) and accretive mappings (when Y = X). By applying a fixed
point theorem of J. Caristi, it is shown that if P is strongly ϕ-accretive in a
localized sense and if Y can be appropriately renormed, then, under suitable
continuity and range restrictions, P is an open mapping. The results generalize
a number of known theorems and indicate a firm connection between the
theory of ϕ-accretive mappings and the renorming characteristics of the space
Y .
