A non-linear Bishop-Phelps-Bollob\'as type theorem

The main aim of this paper is to prove a Bishop-Phelps-Bollob'as type theorem on the unital uniform algebra A_{w^u}(B_{X^}) consisting of all w^-uniformly continuous functions on the closed unit ball B_{X^} which are holomorphic on the interior of B_{X^}. We show that this result holds for A_{w^u}(B_{X^}) if X^ is uniformly convex or X^* is the uniformly complex convex dual space of an order continuous absolute normed space. The vector-valued case is also studied.