Existence of weak solutions for nonlocal Dirichlet problems with variable-order fractional kernels

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 (Formula presented.). By the structural features of the kernel (Formula presented.), a new variable-order fractional Banach space (Formula presented.) 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.