A Relatively Smooth Bishop-Phelps-Bollobás Theorem In Separable Spaces

In 1970,Bollobas gave a quantized representation of the Bishop-Phelps theorem.A Banach space X is said to have the smooth Bishop-Phelps-Bollobas property(shorted by BPBP),if for every ?>0,and for every x0?SX,x0*?SX*satisfying<x0*,x0>>1-4/?2,there exists a ball relatively Gateaux differentiability point x??SX and x?*?SX*such that<x?*,x?>=1,?x?-x0?<? and ?x?*-x0*?<?.In 2017,a result with ?x?+x0?<? in smooth BPBP theorem has been proved by Sisi Shen.Motivated by it,we further show a smooth Bishop-Phelps-Bollobas theorem of the usual form.Firstly,we prove that every strictly convex and separable Banach space has BPBP of the smooth type by using the Borwein-Preiss smooth variational principle.Then we show that every separable Banach space has the smooth BPBP.Moreover,we prove that every separable strictly convex Asplund space has the BPBP of the strongly smooth type.