In the present work, simulation function is applied to establish fixed point results of $\mathcal{F}$-contraction in the setting of Bi-polar metric space. Our results are extensions or generalizations of results proved in the literature. The derived results are substantiated with suitable examples and an application to find the solution to the integral equation.

The Department of Economic and Social Affairs of the United Nations has released seventeen goals for sustainable development and SDG No. 3 is “Good Health and Well-being”, which mainly emphasizes the strategies to be adopted for maintaining a healthy life. The Monkeypox Virus disease was first reported in 1970. Since then, various health initiatives have been taken, including by the WHO. In the present work, we attempt a fractional model of Monkeypox virus disease, which we feel is crucial for a better understanding of this disease. We use the recently introduced $𝒜ℬ𝒞$ fractional derivative to closely examine the Monkeypox virus disease model. The evaluation of this model determines the existence of two equilibrium states. These two stable points exist within the model and include a disease-free equilibrium and endemic equilibrium. The disease-free equilibrium has undergone proof to demonstrate its stability properties. The system remains stable locally and globally whenever the effective reproduction number remains below one. The effective reproduction number becoming greater than unity makes the endemic equilibrium more stable both globally and locally than unity. To comprehensively study the model’s solutions, we employ the Picard-Lindelof approach to investigate their existence and uniqueness. We investigate the Ulam-Hyers and UlamHyers Rassias stability of the fractional order nonlinear framework for the Monkeypox virus disease model. Furthermore, the approximate solutions of the $𝒜ℬ𝒞$ fractional order Monkeypox virus disease model are obtained with the help of a numerical technique combining the Lagrange polynomial interpolation and fundamental theorem of fractional calculus with the $𝒜ℬ𝒞$ fractional derivative.

In this present paper, we introduce graphical Branciari $\mathrm{\aleph }$-metric space and prove the fixed point theorem for $\Omega$- $\mathcal{Q}$-contraction on complete graphical Branciari $\mathrm{\aleph }$ -metric spaces. Our result has been supplemented with suitable, non trivial examples. We have applied the derived fixed point result to solve non-linear Fredholm integral equations and fractional differential equation.