In this paper, we deal with new equivalent conditions for the localized weak Radon-Nikod\'ym property in dual Banach space related to set-valued operators. First, we introduce the geometric definition of the weak Radon-Nikod\'ym property and the weakly fragmented set-valued operator. Next, using the weakly fragmented mapping, we reveal the relation between the weak Radon-Nikod\'ym property and the weakly single-valued operator. Finally, using this relation and the concept of the exposed point, the main theorem is given together with some applications.
"New criteria for the weak Radon-Nikodým property related to set-valued operators." Rocky Mountain J. Math. 45 (5) 1511 - 1526, 2015. https://doi.org/10.1216/RMJ-2015-45-5-1511