Some notes on ``Common fixed point of two $R$-weakly commuting mappings in $b$-metric spaces

Xu, Shaoyuan; Cheng, Suyu; Radenović, Stojan

doi:10.22075/ijnaa.2018.3060.1495

Some notes on ``Common fixed point of two $R$-weakly commuting mappings in $b$-metric spaces"

Document Type : Research Paper

Authors

Shaoyuan Xu ¹

Suyu Cheng ²

Stojan Radenović ³

¹ School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China

² Library, Hanshan Normal University, Chaozhou, 521041, China

³ University of Belgrade, Faculty of Mechanical Engineering, Beograd, Serbia

10.22075/ijnaa.2018.3060.1495

Abstract

Very recently, Kuman et al. [P. Kumam, W. Sintunavarat, S. Sedghi, and N. Shobkolaei. Common Fixed Point of Two $R$-Weakly Commuting Mappings in $b$-Metric Spaces. Journal of Function Spaces, Volume 2015, Article ID 350840, 5 pages] obtained some interesting common fixed point results for two mappings satisfying generalized contractive condition in $b$-metric space without the assumption of the continuity of the $b$-metric, but unfortunately, there exists a gap in the proof of the main result. In this note, we point out and fill such gap by making some remarks and offering a new proof for the result. It should be mentioned that our proofs for some key assertions of the main result are new and somewhat different from the original ones. In addition, we also present a result to check the continuity of the $b$-metrics which is found effective and workable when applied to some examples.

Keywords

$b$-metric spaces

$R$-weakly commuting mappings

the continuity concerning the $b$-metric

common fixed points