The existence and representations of some generalized inverses, including ${A_{T,*}}^{(2)},;{A_{T,*}}^{(1,2)},;{A_{T,*}}^{(2,3)},;{A_{*,S}}^{(2)},;{A_{*,S}}^{(1,2)};and;{A_{*,S}}^{(2,4)}$, are showed. As applications, the perturbation theory for the generalized inverse {A_{T,S}}^{(2)} and the perturbation bound for unique solution of the general restricted system $A_{x}$ = b(dim(AT)=dimT, $b{in}AT$ and $x{in}T$) are studied. Moreover, a characterization and representation of the generalized inverse ${A_{T,*}}^{(2)}$ is obtained.