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

• Published in: Karpats'kì matematinì publìkacìï Vol. 15; no. 1; pp. 260 - 269
• Main Authors: Babenko, V.F., Babenko, V.V., Kovalenko, O.V.
• Format: Journal Article
• Language: English
• Published: Vasyl Stefanyk Precarpathian National University 30.06.2023
• Subjects: family of cauchy sequences; fixed point theorem; fréchet limit space
• ISSN: 2075-9827; 2313-0210
• DOI: 10.15330/cmp.15.1.260-269

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.