  In this article we discuss a possibility to implement a well-known scheme of
proof for contraction mapping theorems in a situation, when convergence,
families of Cauchy sequences, and contractiveness of mappings are defined
axiomatically. We also consider ways to specify families of Cauchy sequences
and contractiveness conditions using distance-like functions with values in
some partially ordered set and establish fixed set and point theorems for
generalized contractions of the \'Ciri\'c and Caristi types.
