arXiv subjects

Mohammed Shimi

Publications and source records attributed to Mohammed Shimi.

6 recordsLinked to original sources

arXiv2025

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

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.

math.AP↗

arXiv2025

Embedding Results and Fractional $p(x,.)$-Laplacian Problem in Unbounded Domains

In this paper, we prove a new continuous embedding theorem for fractional Sobolev spaces with variable exponents into variable exponent Lebesgue spaces on unbounded domains. As an application, we study a class of nonlocal elliptic problems driven by the fractional $p(x, \cdot)$-Laplacian operator. Using variational methods combined with the established embedding result, we prove the existence of nontrivial weak solutions under suitable growth and regularity conditions on the nonlinearity.\\ A significant analytical challenge addressed in this work arises from both the nonlocal behavior of the fractional $p(x, \cdot)$-Laplacian and the lack of compactness induced by the unboundedness of the domain.

math.AP↗

arXiv2024

Anisotropic fractional Sobolev spaces with variable exponent and application to nonlocal problems

The main goal of this paper is to introduce a new fractional anisotropic Sobolev space with variable exponent where the basic qualitative properties (completeness, separability, reflexivity, ...) are established, including the continuous and compact embedding results. Moreover, some functional proprieties of anisotropic fractional $\vec{p}(.,.)$-Laplacian operator are proved. As an application, we use the mountain pass theorem and Ekeland's variational principle to ensure the existence of a weak solution for a nonlocal anisotropic problem with variable exponent.

math.AP↗

arXiv2020

Embedding and extension results in Fractional Musielak-Sobolev spaces

In this paper, we are concerned with some qualitative properties of the new fractional Musielak-Sobolev spaces $W^sL_{\varPhi_{x,y}}$ such that the generalized Poincaré type inequality and some continuous and compact embedding theorems of these spaces. Moreover, we prove that any function in $W^sL_{\varPhi_{x,y}}(Ω)$ may be extended to a function in $W^sL_{\varPhi_{x,y}}(\R^N)$, with $Ω\subset \R^N$ is a bounded domain of class $C^{0,1}$. In addition, we establish a result relates to the complemented subspace in $W^s{L_{\varPhi_{x,y}}}\left( \R^N\right)$. As an application, using the mountain pass theorem and some variational methods, we investigate the existence of a nontrivial weak solution for a class of nonlocal fractional type problems with Dirichlet boundary data.

math.AP↗

arXiv2019

General fractional Sobolev Space with variable exponent and applications to nonlocal problems

In this paper, we extend the fractional Sobolev spaces with variable exponents $W^{s,p(x,y)}$ to include the general fractional case $W^{K,p(x,y)}$, where $p$ is a variable exponent, $s\in (0,1)$ and $K$ is a suitable kernel. We are concerned with some qualitative properties of the space $W^{K,p(x,y)}$ (completeness, reflexivity, separability, and density). Moreover, we prove a continuous and compact embedding theorem of these spaces into variable exponent Lebesgue spaces. As applications, we discuss the existence of a nontrivial solution for a nonlocal $p(x,.)$-Kirchhoff type problem. Further, we establish the existence and uniqueness of a solution for a variational problem involving the integro-differential operator of elliptic type $\mathcal{L}^{p(x,.)}_K$.

math.AP↗

arXiv2019

Existence of solutions for a nonlocal Kirchhoff type problem in Fractional Orlicz-Sobolev spaces

In this paper, we investigate the existence of weak solution for a Kirchhoff type problem driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions {\small$$ (D_{K,A}) \hspace*{0.5cm} \left\{ \begin{array}{clclc} M\left( \displaystyle \int_{\R^{2N}}A\left( [u(x)-u(y)] K(x,y)\right) dxdy\right) \mathcal{L}^K_A u & = & f(x,u) & \text{ in }& Ω, \hspace*{7cm} u & = & 0 \hspace*{0.2cm} \hspace*{0.2cm} & \text{ in } & \R^N\setminus Ω. \label{eq1} \end{array} \right. $$ } Where $\mathcal{L}^K_A$ is a nonlocal operator with singular kernel $K$ and $A$ is an $N$-function, $Ω$ is an open bounded subset in $\R^N$ with Lipschitz boundary $\partial Ω$.

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