Surjectivity of $\varphi$-accretive operators

Abstract:

Let $X$ and $Y$ be Banach spaces, $\phi :X \to {Y^ * }$ and $P:X \to Y:$; $P$ is said to be strongly $\phi$-accretive if $\langle Px - Py,\;\phi \left ( {x - y} \right )\rangle \geqslant c{|| {x - y} ||^2}$ for some $c > 0$ and each $x$, $y \in X$. These maps constitute a generalization simultaneously of monotone maps (when $Y = {X^ * }$) and accretive maps (when $Y = X$). 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 ${Y^ * }$ 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.