Abstract
In this manuscript, we coined pentagonal controlled fuzzy metric spaces and fuzzy controlled hexagonal metric space as generalizations of fuzzy triple controlled metric spaces and fuzzy extended hexagonal b-metric spaces. We use a control function in fuzzy controlled hexagonal metric space and introduce five noncomparable control functions in pentagonal controlled fuzzy metric spaces. In the scenario of pentagonal controlled fuzzy metric spaces, we prove the Banach fixed point theorem, which generalizes the Banach fixed point theorem for the aforementioned spaces. An example is offered to support our main point. We also presented an application to dynamic market equilibrium.
Highlights
Fuzzy notions are used to describe the degrees of possession of a certain property
/α i−1 for all α with the continuous t-norms (CTNs) ∗ such that α1 ∗ α2 = α1α2: ðX, M,∗Þ is fuzzy controlled hexagonal metric spaces (FCHMSs) with control functions Qðz, dÞ = 1 + z + d: Definition 6
The definitions of FCHMS and pentagonal controlled fuzzy metric spaces (PCFMSs) are presented as well as proofs of fixed point findings
Summary
Fuzzy notions are used to describe the degrees of possession of a certain property. The ability of fuzzy set (FS) theory to address circumstances that fixed point theory finds problematic originates from its attractiveness in tackling control problems. Kamran et al [9] presented the approach of extended metric space and proved several fixed point results for contraction mappings. Saleem et al [11] coined the notion of fuzzy double controlled metric spaces and proved several fixed point results. Badshah-e-Rome and Sarwar [12] presented the approach of extended fuzzy rectangular b-metric spaces and proved fixed point results for contraction mappings via α-admissibility. Zubair et al [14] presented fuzzy extended hexagonal b-metric spaces (FEHBMSs) and proved several fixed point results. In this manuscript, we generalized the ideas of FTCMSs and FEHBMSs and present the approaches of pentagonal controlled fuzzy metric spaces (PCFMSs) and fuzzy controlled hexagonal metric spaces (FCHMSs). An application to dynamic market equilibrium is given to validate the main result
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