A note on the Banach contraction principle in b-metric spaces

https://doi.org/10.5281/zenodo.5728134

• Mohamed Akkouchi Faculty of Sciences-Semlalia, University Cadi Ayyad

Keywords: Banach fixed point principle, b-metric space, stopping rules

Abstract

Let $(X,d; s)$ be a complete $b$-metric space with parameter $s\geq 1$. Let $T$ a contractive map on $X$, that is a selfmap $T$ of $X$ satisfying
$$ d(Tx,Ty) \leq \lambda d(x,y), \, \forall x,y \in X, \eqno(B_{\lambda})$$
with some $\lambda \in [0, 1)$. In 1989, Bakhtin established an alogous to the Banach contraction principle in the context of complete $b$-metric spaces. Precisely, he proved that if $\lambda \in [0, \frac{1}{s})$. Then $T$ has a unique fixed point. The aim of this note is to give a simple proof of the Banach contraction principle in $X$ for all $\lambda \in [0, 1)$. So, in particular, we provide some complements to Bakhtin's result. We establish a fundamental contraction inequality for $T$ and use it to prove convergence of Picard sequences. For such sequences, we give an evaluation of the order of convergence and a posteriori error estimate. We estimate the diameter of the $T$-orbits. As applications, we deduce two stopping rules indicating the sufficient number of iterations of the Picard process which allows a satisfactory approximation for the fixed point of $T$.

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