Existence and uniqueness of weak and capacity solutions to fractional differential equations in fractional Musielak-Orlicz spaces

In this article, we study the existence of capacitary solutions for a nonlinear fractional differential equation problem in fractional Musielak-Orlicz-Sobolev spaces. Using approximation techniques, we establish the existence of weak solutions by introducing a sequence of approximated problems converging, in the sense of capacities, to a solution of the original problem. Additionally, we present a concrete application to illustrate the validity and relevance of the obtained results. This work makes a significant contribution to the analysis of nonlocal and nonlinear problems with memory in this functional framework.