Conic positive definiteness and sharp minima of fractional orders in vector optimization problems

Motivated by the fact that the usual positive definiteness does not work in an infinite space, we introduce the concept of S-positive definiteness with respect to an ordering cone in a general Banach space and show that the S-positive definiteness plays the same role as the usual positive definiteness in the finite dimensional case. As applications, we study sharp and weak sharp minima of fractional orders in vector optimization.