Existence of solutions for $[\mathtt{p},\mathtt{q}]$-difference initial value problems: application to the $[\mathtt{p},\mathtt{q}]$-based model of vibrating eardrums

The eardrum is one of the most important organs in the body, and disorders such as infection or injury may affect the proper functioning of the eardrum and lead to hearing problems. In this paper, based on a real-world phenomena, we study some mathematical aspects of an abstract fractional $[\mathtt{p},\mathtt{q}]$-difference equation with initial conditions. Our initial value problem tries to model a vibrating eardrum by using the newly defined fractional Caputo-type $[\mathtt{p},\mathtt{q}]$-derivatives in two nonlinear single-valued and set-valued structures. We obtain a general form of the solutions in the framework of a $[\mathtt{p},\mathtt{q}]$-integral equation, and then we investigate the existence and uniqueness properties with the help of fixed points and the end-points of some special $\mathtt{\beta }$- $\mathtt{\alpha }$-contractions and compact mappings. Finally, we simulate this version of the vibrating eardrum model by giving two numerical examples to validate the established theorems.