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

Keywords:
fixed point theorem, family of Cauchy sequences, Fréchet limit spaceAbstract
In the article, we axiomatically define generalized Lim-spaces (X,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 X2, 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.