A Mathematical Study of A Contact Problem Involving Time-Fractional Derivatives

Abstract

This work studies a nonlinear fractional viscoelastic contact problem arising in continuum mechanics. The model incorporates Caputo fractional derivatives to describe memory effects and hereditary material behavior. The governing equations combine a time fractional constitutive law with contact boundary conditions of normal damped response type. A rigorous variational formulation is established within an appropriate functional framework based on Sobolev spaces and fractional calculus. The existence of a weak solution is proved using the Faedo–Galerkin approximation method, compactness techniques, and fixed-point arguments. The obtained results provide a theoretical basis for future numerical simulations and more advanced fractional contact models.