Existence analysis of nonlinear q-fractional differential inclusions with p-Laplacian operators and an aerospace vibration-modeling illustration

Abstract

This paper investigates a coupled system of nonlinear q -fractional differential inclusions involving nested p -Laplacian operators and nonlocal boundary conditions. The problem is formulated in a multivalued setting, where the nonlinear inclusion depends on the state vector and its Jackson q -derivative. Using tools from q -fractional calculus, Green-function representations, and multivalued fixed-point theory, we derive sufficient conditions for the existence of at least one solution in the associated product function space. The analysis combines the sequential structure of the Caputo and Riemann–Liouville q -fractional operators with compact convex multifunction values and nonlocal boundary constraints. An aerospace vibration-modeling illustration is also presented to show how a Caputo q -fractional damping term can represent memory-dependent behavior in a base-excited inertial-measurement-unit mounting model. The numerical comparison is reported against rigid mounting and a classical viscous isolator and is interpreted as an illustration of q -fractional memory modeling. The main mathematical contribution of the paper is the existence result for the proposed nonlinear q -fractional differential inclusion system.