A non-linear Bishop-Phelps-Bollobás type theorem

The main aim of this paper is to prove a Bishop-Phelps-Bollobás 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. Throughout the paper, we consider complex Banach spaces.