Fixed sets and fixed points for mappings in generalized $\rm Lim$-spaces of Fréchet

In the article, we axiomatically define generalized $\rm Lim$-spaces $(X,{\rm Lim})$, Cauchy structures, contractive mappings and prove an abstract version of the contraction mapping principle. We also consider ways to specify families of Cauchy sequences and contraction conditions using a base in $X^2$, distance-like or sum-like functions with values in some partially ordered set $Y$. We establish fixed set and fixed point theorems for generalized contractions of the Meir-Keeler and Taylor, Ćirić and Caristi types. The obtained results generalize many known fixed point theorems and are new even in many classical situations.