This paper discusses balls in partial b-metric spaces and cone metric spaces, respectively. Let (X ,pb) be a partial b-metric space in the sense of Mustafa et al. For the family of all pb-open balls in (X ,pb), this paper proves that there are x, y ∈ B ∈ such that B′ B for all B′ ∈ , where B and B′ are with centers x and y, respectively. This result shows that is not a base of any topology on X...

*Correspondence: [email protected] 1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article Abstract We first introduce the concept of b-metric-like space which generalizes the notions of partial metric space, metric-like space and b-metric space. Then we establish the existence and...

‎In this paper‎, ‎we introduce more general contractions called $varphi $-fixed‎ ‎point point for $(F,varphi‎ ,‎alpha )_{s}$ and $(F,varphi‎ ,‎alpha )_{s}$-‎weak contractions‎. ‎We prove the existence and uniqueness of $varphi $-‎fixed point point for $(F,varphi‎ ,‎alpha )_{s}$ and $(F,varphi‎ ,‎alpha‎‎)_{s}$-weak contractions in complete $b$-metric spaces‎. ‎Some examples are‎ ‎supplied to sup...

The $r$ -th generalized Hamming metric and the notation="LaTeX">$b$ -symbol are two different generalizations of metric. former is used on wire-tap channel Type II, latter motivated by limitations reading process in high-density data ...