On fixed point and its application to the spread of infectious diseases model in Mvb‐metric space

• Published in: Mathematical methods in the applied sciences Vol. 47; no. 7; pp. 6489 - 6503
• Main Authors: Alam, Khairul Habib, Rohen, Yumnam, Tomar, Anita
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 15.05.2024
• Subjects: F$$ \mathcal{F} $$‐contraction; Infectious diseases; Mathematical models; Metric space; mvb$$ {\mathcal{m}}_{\mathcal{v}}^{\mathcal{b}} $$‐Cauchy; mvb$$ {\mathcal{m}}_{\mathcal{v}}^{\mathcal{b}} $$‐complete; mvb$$ {\mathcal{m}}_{\mathcal{v}}^{\mathcal{b}} $$‐convergent; sequence of iterates
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9933

This work aims to prove new results in an Mvb$$ {\mathcal{M}}_{\mathcal{v}}^{\mathcal{b}} $$‐metric space for a noncontinuous single‐valued self‐map. As a result, we extend, generalize, and unify various fixed‐point conclusions for a single‐valued map and come up with examples to exhibit the theoretical conclusions. Further, we solve a mathematical model of the spread of specific infectious diseases as an application of one of the conclusions. In the sequel, we explain the significance of Mvb$$ {\mathcal{M}}_{\mathcal{v}}^{\mathcal{b}} $$‐metric space because the underlying map is not necessarily continuous even at a fixed point in Mvb$$ {\mathcal{M}}_{\mathcal{v}}^{\mathcal{b}} $$‐metric space thereby adding a new answer to the question concerning continuity at a fixed point posed by Rhoades. Consequently, we may conclude that the results via Mvb$$ {\mathcal{M}}_{\mathcal{v}}^{\mathcal{b}} $$‐metric are very inspiring, and underlying contraction via Mvb$$ {\mathcal{M}}_{\mathcal{v}}^{\mathcal{b}} $$‐metric does not compel the single‐valued self‐map to be continuous even at the fixed point. Our research is greatly inspired by the exciting possibilities of using noncontinuous maps to solve real‐world nonlinear problems.