Abstract
This paper is concerned with the existence and uniqueness of weak solutions for a class of linear and nonlinear nonlocal homogeneous Dirichlet boundary value problems with a truncated variable-order fractional kernel γ ( x , y ) . By the structural features of the kernel γ ( x , y ) , a new variable-order fractional Banach space [ W 0 α ( x , y ) , p ( Ω ′ ) ] d is introduced as the suitable solution space, and its some qualitative properties are established. Then under such a functional framework, based on the variational methods, we prove the existence results for the linear problem with a nonsymmetric kernel case and for the nonlinear problem with a symmetric kernel case, respectively.
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