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<p>The aim of this paper is to introduce a new class of metric-type spaces called O-metric spaces as a generalization of several metric-type spaces in literature, by constructing a triangle-type inequality that accommodates many binary operations including multiplication and division. Possible metric-type spaces are classified into upward and downward O-metric spaces as O-metrics between identical points are not necessarily minimal. Conditions for passage between upward and downward O-metrics are specified, giving rise to various reverse triangle inequalities. Topologies arising from O-metrics are listed, and properties such as O-convergence, sequential continuity, first countability and T<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1302_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mn>2</mn> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> separation are investigated. It is shown that the topology of an O-metric space can be generated by an upward O-metric on the space hence the focus will be on upward O-metric spaces. With the use of polygon inequalities, a theorem on the existence and uniqueness of fixed points of some contractive-like maps is established in the setting of O-metric spaces, and well known results are obtained as corollaries. Applications to the estimation of distances, polygon inequalities, and optimization of entries in some infinite symmetric matrices are also given.</p>

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A generalized metric-type structure with some applications

  • 期刊论文

  • 发表日期：

    2025年4月18日

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  • Hallowed O. Olaoluwa,
  • Aminat O. Ige,
  • Johnson O. Olaleru,
  • Mujahid Abbas

摘要

The aim of this paper is to introduce a new class of metric-type spaces called O-metric spaces as a generalization of several metric-type spaces in literature, by constructing a triangle-type inequality that accommodates many binary operations including multiplication and division. Possible metric-type spaces are classified into upward and downward O-metric spaces as O-metrics between identical points are not necessarily minimal. Conditions for passage between upward and downward O-metrics are specified, giving rise to various reverse triangle inequalities. Topologies arising from O-metrics are listed, and properties such as O-convergence, sequential continuity, first countability and T \(_2\) 2 separation are investigated. It is shown that the topology of an O-metric space can be generated by an upward O-metric on the space hence the focus will be on upward O-metric spaces. With the use of polygon inequalities, a theorem on the existence and uniqueness of fixed points of some contractive-like maps is established in the setting of O-metric spaces, and well known results are obtained as corollaries. Applications to the estimation of distances, polygon inequalities, and optimization of entries in some infinite symmetric matrices are also given.