  Utilizing the stability characterizations of generalized inverses of linear
operator, we investigate the existence of generalized resolvents of linear
pencils in Banach spaces. Some practical criterions for the existence of
generalized resolvents of the linear pencil $\lambda\rightarrow T-\lambda S$
are provided and an explicit expression of the generalized resolvent is given.
As applications, the characterization for the Moore-Penrose inverse of the
linear pencil to be its generalized resolvent and the existence of the
generalized resolvents of linear pencils of finite rank operators, Fredholm
operators and semi-Fredholm operators are also considered. The results obtained
in this paper extend and improve many results in this area.
