Infinitesimal characterization of a feedback nash equilibrium in a differential game

For mappings defined on metric spaces and with values in Banach spaces, the notions of derivative vectors of first and second order are introduced. These notions are used to establish both necessary and sufficient optimality conditions of first and second order for local ?-minimizers of such mappings, where ? is a strict preorder relation defined on the space of values of the mapping that is minimized. As corollaries of the above results, minimality conditions are also obtained for the case when the mapping is defined on a subset of a normed space.