  We give a class of bounded closed sets $C$ in a Banach space satisfying a
generalized and stronger form of the Bishop-Phelps property studied by Bourgain
in \cite{Bj} for dentable sets. A version of the {\it
``Bishop-Phelps-Bollob\'as"} theorem will be also given. The density and the
residuality of bounded linear operators attaining their maximum on $C$ (known
in the literature) will be replaced, for this class of sets, by being the
complement of a $\sigma$-porous set. The result of the paper is applicable for
both linear operators and non-linear mappings. When we apply our result to
subsets (from this class) whose closed convex hull is the closed unit ball, we
obtain a new class of Banach spaces involving property $(A)$ introduced by
Lindenstrauss. We also establish that this class of Banach spaces is stable
under $\ell_1$-sum when the spaces have a same ``modulus". Applications to norm
attaining bounded multilinear mappings and Lipschitz mappings will also be
given.
