On Preservation of Common Fixed Points and Coincidences Under a Homotopy of Mapping Families of Ordered Sets

A new approach to coincidence and common fixed points under a homotopy of families of mappings in $b$-metric spaces

“…For formulating the results, we recall some definitions and notions introduced by Fomenko and Podoprikhin [7,9]. Definition 2.1.…”

“…The main objective of the paper is to develop a new approach and obtain sufficient conditions to gurantee the existence of coincidence and common fixed points under order homotopies of families of mappings in preordered s-regular b-metric spaces. We have generalized the results of [7,9] to the case of preordered s-regular b-metric spaces and preordered regular metric spaces. This approach has enabled us to replace the classical approach of considering the underlying space to be complete by a preordered space and b-metric function satisfying the axiom of s-regularity.…”

“…Recently, Fomenko and Podoprikhin [6,10] generalized the results of Arutyunov [1,2] to the case of families of multivalued mappings in partially ordered sets. They [7,10] introduced the notion of order homotopy and proved that common fixed point and coincidence point properties are preserved under homotopies of families of mappings in ordered sets.…”