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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Surjectivity of $ \varphi $-accretive operators

Authors: Jong An Park and Sehie Park
Journal: Proc. Amer. Math. Soc. 90 (1984), 289-292
MSC: Primary 47H15
MathSciNet review: 727252
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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{\vert\vert {x - y} \vert\vert^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.

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Keywords: Strongly $ \phi $-accretive, locally strongly $ \phi $-accretive, strongly upper semicontinuous
Article copyright: © Copyright 1984 American Mathematical Society