Fundamental Properties and Embedding Results in a Novel (Φ_x, ψ)-Fractional Musielak Space with an Application to Nonlocal BVP

2025/12/17 by Ayoub Kasmi, Kasmi, Ayoub, El Houssine Azroul +3
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fixed Point Theorems Analysis #Nonlinear Differential Equations Analysis #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2512.15972

openalex publication_date 2025/12/17 · openalex created_date 2025/12/21 · openalex updated_date 2026/07/28

Abstract

In this paper, we introduce and study a novel class of generalized (Φ_x,ψ)-fractional Musielak spaces K_Φx^{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.

Citations

• On a novel fractional Caputo-derivative Orlicz space
• [Generalized Telegraph equation with fractional p(x)-Laplacian
• Existence and multiplicity for fractional Dirichlet problem with γ(ξ)-Laplacian equation and Nehari manifold
• On the ψ-Hilfer fractional derivative
• Fractional order Orlicz-Sobolev spaces
• Basic theory of fractional differential equations

Related

• On Fractional Musielak-Sobolev spaces and applications to nonlocal problems
• Ground state solutions for a non-local type problem in fractional Orlicz Sobolev spaces
• On Fractional Anisotropic Musielak-Sobolev Spaces with Applications to Nonlocal Eigenvalue Problems
• Embedding and extension results in Fractional Musielak-Sobolev spaces
• Recent progresses in the theory of nonlinear nonlocal problems