Surjectivity of 𝜑-accretive operators

Let X X and Y Y be Banach spaces, ϕ : X → Y ∗ \phi :X \to {Y^ * } and P : X → Y : P:X \to Y: ; P P is said to be strongly ϕ \phi -accretive if ⟨ P x − P y , ϕ ( x − y ) ⟩ ⩾ c | | x − y | | 2 \langle Px - Py,\;\phi \left ( {x - y} \right )\rangle \geqslant c{|| {x - y} ||^2} for some c > 0 c > 0 and each x x , y ∈ X y \in X . These maps constitute a generalization simultaneously of monotone maps (when Y = X ∗ Y = {X^ * } ) and accretive maps (when Y = X Y = X ). By applying the Caristi-Kirk fixed point theorem, W. O. Ray showed that a localized class of these maps must be surjective under appropriate geometric assumptions on Y ∗ {Y^ * } and continuity assumptions on the duality map. In this paper we show that such geometric assumptions can be removed without affecting the conclusion of Ray.