Fundamental Properties and Embedding Results in a Novel $(Φ_x, ψ)$-Fractional Musielak Space with an Application to Nonlocal BVP
Ayoub Kasmi, El Houssine Azroul, Mohammed Shimi
TL;DR
The paper develops a new class of fractional Musielak spaces $(\Phi_x,\psi)$-fractional Musielak spaces to model heterogeneous nonlinearities with memory. It builds the functional-analytic foundation by defining $\mathcal{K}_{\Phi_x}^{\alpha,\beta,\psi}$, proving embedding and structural properties, and connecting to known spaces through special cases. Using a variational approach, it proves the existence of a nontrivial weak solution to a nonlinear fractional boundary-value problem under a Mountain Pass framework with an Ambrosetti–Rabinowitz condition. The work provides a unified framework that integrates Musielak–Orlicz theory, variable-growth conditions, and $\psi$-Hilfer fractional calculus to address nonlocal, nonhomogeneous PDEs and paves the way for further analysis of nonlocal boundary-value problems in variable-exponent settings.
Abstract
In this paper, we introduce and study a novel class of generalized $(Φ_x,ψ)$-fractional Musielak spaces $\mathcal{K}_{Φ_x}^{α, β, ψ}$, 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.