A GENERALIZATION OF CONTRACTION MAPPING THEOREMS IN A NEWLY DEFINED GENERALIZED METRIC SPACE

Abstract

A newly defined generalized metric space, called \emph{$\varrho_{\mathfrak{g}}$-space} and denoted by $\mathfrak{M}_{\mathfrak{g}}=\left(\Sigma,\varrho_{\mathfrak{g}}\right)$, is introduced axiomatically and various $\varrho_{\mathfrak{g}}$-concepts as \emph{$\varrho_{\mathfrak{g}}$-fundamental sequence}, \emph{$\varrho_{\mathfrak{g}}$-convergence}, \emph{$\varrho_{\mathfrak{g}}$-contraction mapping}, and \emph{$\varrho_{\mathfrak{g}}$-fixed point} are defined in the $\varrho_{\mathfrak{g}}$-space, analogous to those in $\varrho_{\mathfrak{o}}$-spaces. Thereafter, a generalized contraction mapping theorem called \textit{$\varrho_{\mathfrak{g}}$-contraction mapping theorem} is presented in the $\varrho_{\mathfrak{g}}$-space, founding its statement and proof on these $\varrho_{\mathfrak{g}}$-concepts. From the theorem, propositions concerning \textit{$\varrho_{\mathfrak{g}}$-contraction mapping error estimates} and the \textit{implication between $\varrho_{\mathfrak{o}}$, $\varrho_{\mathfrak{g}}$-spaces} are derived. Then, a corollary concerning the implication between $\varrho_{\mathfrak{o}}$, $\varrho_{\mathfrak{g}}$-contraction mappings and the implication between $\varrho_{\mathfrak{o}}$, $\varrho_{\mathfrak{g}}$-fixed points are given. Finally, $\varrho_{\mathfrak{o}}$, $\varrho_{\mathfrak{g}}$-axioms and $\varrho_{\mathfrak{o}}$, $\varrho_{\mathfrak{g}}$-spaces are classified, an illustrative application is presented, highlighting some $\varrho_{\mathfrak{o}}$, $\varrho_{\mathfrak{g}}$-properties, and the work is concluded.

Keywords

• $\mathfrak{g}$-metric
• $\varrho_{\mathfrak{g}}$-space
• $\varrho_{\mathfrak{g}}$-fundamental sequence
• $\varrho_{\mathfrak{g}}$-contraction mapping
• $\varrho_{\mathfrak{g}}$-fixed point
• $\varrho_{\mathfrak{g}}$-Contraction Mapping Theorem

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