In this work, we present existence results for some problems which arise in image processing namely image restoration. Our essential tools are Picard's fixed point theorem for a strict contraction and Mountain-pass Theorem for critical point.

The goal of this work is to study the existence of a unique solution and the Ulam-Hyers stability of a coupled system of generalized hybrid pantograph equations with fractional deformable derivatives. Our main tool is Banach's contraction principle. The paper ends with an example to support our results.

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.