A Minimizing Sequence Proof of the Banach Fixed Point Theorem
Keywords:
Fixed point theorems, Contraction mappings, Complete metric spaces, b-metric spaces, Direct methods, Distance minimization, Minimizing sequences, Picard iteration, Infimum, Incomplete metric spaces
Abstract
We present a novel proof of the Banach contraction mapping theorem based on minimizing sequences. By analyzing the set $\mathcal{A} = \{d(x, T(x)) : x \in X\}$ of point-to-image distances, we construct a sequence that converges to the unique fixed point. A key technical contribution is Lemma~\ref{lemma1}, which establishes an optimal inequality between contraction coefficients and the b-metric constant. We demonstrate applications to b-metric spaces and discuss extensions to incomplete metric spaces. Examples show the effectiveness of this approach, which provides a geometric alternative to classical methods.
Published
2025-11-02
How to Cite
Bataihah, A. (2025). A Minimizing Sequence Proof of the Banach Fixed Point Theorem. Statistics, Optimization & Information Computing. https://doi.org/10.19139/soic-2310-5070-2991
Issue
Section
Research Articles
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