The weighted generalized Atangana-Baleanu fractional derivative in banach spaces- definition and applications

In this paper, we introduce the concept of the weighted generalized Atangana-Baleanu fractional derivative. We prove the existence of the stability of solutions of non-local differential equations and non-local differential inclusions, in Banach spaces, with this new fractional derivative in the presence of instantaneous and non-instantaneous impulses. We considered the case in which the lower limit of the fractional derivative was kept at the initial point and where it was changed to the impulsive points. To prove our results, we established the relationship between solutions to each of the four studied problems and those of the corresponding fractional integral equation. There has been no previous study of the weighted generalized Atangana-Baleanu fractional derivative, and so, our findings are new and interesting. The technique we used based on the properties of this new fractional differential operator and suitable fixed point theorems for single valued and set valued functions. Examples are given to illustrate the theoretical results.