Fixed sets and fixed points for mappings in generalized \(\mathrm{Lim}\)-spaces of Fréchet. (English) Zbl 1517.54013
Summary: In the article, we axiomatically define generalized \(\mathrm{Lim}\)-spaces \((X,\mathrm{Lim})\), 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.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E15 | Uniform structures and generalizations |
Keywords:
fixed set theorem; family of Cauchy sequences; Fréchet limit space; weak orbital continuityReferences:
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