Abstract
Abstract The convergence of sequences and non-unique fixed points are established in ℳ-orbitally complete cone metric spaces over the strongly minihedral cone, and scalar weighted cone assuming the cone to be strongly minihedral. Appropriate examples and applications validate the established theory. Further, we provide one more answer to the question of the existence of the contractive condition in Cone metric spaces so that the fixed point is at the point of discontinuity of a map. Also, we provide a negative answer to a natural question of whether the contractive conditions in the obtained results can be replaced by its metric versions.
Highlights
K-metric and K-normed spaces were familiarized ([1], [5], [13]) using an ordered Banach space as the range of a metric, in place of the set of real numbers
Bogdan Rzepecki ([11]), familiarized a generalized metric dE : U × U → S, where, S is a normal cone in a Banach space E with a partial order
The established results and illustrative examples show that the cone metric space is a real generalization of a metric space
Summary
K-metric and K-normed spaces were familiarized ([1], [5], [13]) using an ordered Banach space as the range of a metric, in place of the set of real numbers. Key words and phrases: Cone metric space, strongly minihedral, normal cone, nonunique fixed point. Cone metric spaces and defined convergence and Cauchy sequences in terms of the interior points of the cone under consideration. In this manuscript, the convergence of sequences and non-unique fixed points are established in an M-orbitally complete strongly minihedral cone metric space and scalar weighted cone metric space. The established results and illustrative examples show that the cone metric space is a real generalization of a metric space. We provide novel answers in cone metric spaces to the open question posed by Rhoades ([10]) regarding the existence of a contractive map having the discontinuity at a fixed point. It is worth mentioning here that Khamsi ([7]) claimed that the majority of the cone fixed point results are identical to the classical results and extensions of known fixed point theorems to cone metric spaces are superfluous
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
