<inline-formula><math display="inline"><semantics><mi mathvariant="bold-script">F</mi></semantics></math></inline-formula>-Contractions Endowed with Mann’s Iterative Scheme in Convex <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold-script">G</mi><mi mathvariant="bold-italic">b</mi></msub></semantics></math></inline-formula>-Metric Spaces

Abstract

Recently, Ji et al. established certain fixed-point results using Mann’s iterative scheme tailored to Gb-metric spaces. Stimulated by the notion of the F-contraction introduced by Wardoski, the contraction condition of Ji et al. was generalized in this research. Several fixed-point results with Mann’s iterative scheme endowed with F-contractions in Gb-metric spaces were proven. One non-trivial example was elaborated to support the main theorem. Moreover, for application purposes, the existence of the solution to an integral equation is provided by using the axioms of the proven result. The obtained results are generalizations of several existing results in the literature. Furthermore, the results of Ji. et al. are the special case of theorems provided in the present research.

Keywords

• fp )%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> fixed point (<named-content content-type="equation"><inline-formula> <mml:math id="mm702"> <mml:semantics> <mml:mi mathvariant="fraktur">fp</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>)
• M S )%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> metric space (<named-content content-type="equation"><inline-formula> <mml:math id="mm703"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">M</mml:mi> <mml:mi mathvariant="script">S</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>)
• M S )%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <i>b</i>-metric spaces (<i>b</i>-<named-content content-type="equation"><inline-formula> <mml:math id="mm704"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">M</mml:mi> <mml:mi mathvariant="script">S</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>)
• G -metric space ( G - M S )%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula> <mml:math id="mm711"> <mml:semantics> <mml:mi mathvariant="script">G</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-metric space (<named-content content-type="equation"><inline-formula> <mml:math id="mm707"> <mml:semantics> <mml:mi mathvariant="script">G</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-<named-content content-type="equation"><inline-formula> <mml:math id="mm705"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">M</mml:mi> <mml:mi mathvariant="script">S</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>)
• G b -metric space ( G b - M S )%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula> <mml:math id="mm712"> <mml:semantics> <mml:msub> <mml:mi mathvariant="script">G</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:semantics> </mml:math> </inline-formula></named-content>-metric space (<named-content content-type="equation"><inline-formula> <mml:math id="mm708"> <mml:semantics> <mml:msub> <mml:mi mathvariant="script">G</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:semantics> </mml:math> </inline-formula></named-content>-<named-content content-type="equation"><inline-formula> <mml:math id="mm706"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="fraktur">M</mml:mi> <mml:mi mathvariant="script">S</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>)
• cs )%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> Cauchy sequence (<named-content content-type="equation"><inline-formula> <mml:math id="mm709"> <mml:semantics> <mml:mi mathvariant="fraktur">cs</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>)