Some results of new definition of N_F^α-Fourier transform and their applications

Abstract

Based on a new definition of fractional derivative, called N_F^α-derivative, we introduce a new definition and present some results on N_α^F-periodic functions and N_F^α-Fourier transform related to this type of functions. We also use these results to study the N_F^α-partial differential equation N_F^αx(t) = A x(t) + I_F^αx(t) + f(t), where A is a linear closed operator on a Banach space X.