Basic results for fractional anisotropic spaces and applications

In this paper, we introduce a new space that generalizes the \(\phi \)-Hilfer space with the \(\xi (\cdot )\)-Laplacian operator, denoted \((\phi ,{\xi }(\cdot ))\)-HFDS. We refer to this new space as the \(\phi \)-fractional space with anisotropic \(\overrightarrow{\xi }(\cdot )\)-Laplacian operator, abbreviated as \((\phi ,\overrightarrow{\xi }(\cdot ))\)-HFDAS. We prove that \((\phi ,\overrightarrow{\xi }(\cdot ))\)-HFDAS is a separable, and reflexive Banach space. Furthermore, we extend some well-known properties and embedding results of the \((\phi ,\xi (\cdot ))\)-HFDS space to \((\phi ,\overrightarrow{\xi }(\cdot ))\)-HFDAS. Moreover, we illustrate an application of \((\phi ,\overrightarrow{\xi }(\cdot ))\)-HFDAS by solving a differential equation via variational methods.