Fundamental Properties and Embedding Results in a Novel $(\Phi_x, \psi)$-Fractional Musielak Space with an Application to Nonlocal BVP

Abstract

In this paper, we introduce and study a novel class of generalized $(\Phi_x,\psi)$-fractional Musielak spaces $\mathcal{K}_{\Phi_x}^{\alpha, \beta, \psi}$, which extends classical fractional spaces and offers the flexibility to model heterogeneous and nonlinear phenomena with memory and nonlocal effects. A detailed and rigorous analysis of their functional structure is carried out. Several new properties and embedding results are established, highlighting the originality of the proposed framework and its relevance to nonlocal BVPs. To illustrate the significance of this functional setting, we prove the existence of nontrivial solutions to a nonlinear fractional differential problem under an Ambrosetti--Rabinowitz type condition, using the mountain pass theorem. Our results provide new perspectives for the analysis of nonlocal and nonhomogeneous equations in variable-exponent and Musielak-Orlicz settings.