In this paper, we establish a new abstract functional space $X^{K,p(·)}(\mathrm{\Omega })$ where K is a uncertain weighted function and p is a variable exponent. Based on the properties of this space, we consider the existence and regularity of weak solutions for non-local fractional differential inclusion with homogeneous Dirichlet boundary conditions. Under a suplinear growth condition we obtain the existence of weak solutions, the compactness and Hölder regularity of the solution set using set-valued analysis and the surjectivity principle of pseudomonotonicity. Furthermore, the existence of extremal solutions and a relaxation result is discussed.

中文翻译：

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具有多值非线性扰动的非局部分数 p(x)-Laplacian 解的特征

在本文中，我们建立了一个新的抽象功能空间 ${十}^{钾,磷(·)}(\mathrm{\Omega })$其中K是不确定的加权函数，p是可变指数。基于该空间的性质，我们考虑了具有齐次狄利克雷边界条件的非局部分数微分包含弱解的存在性和规律性。在上线性增长条件下，我们使用集值分析和伪单调性的满射原理，获得了弱解的存在性、解集的紧致性和 Hölder 正则性。此外，还讨论了极值解和松弛结果的存在。