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.

This paper develops a fractional-order chemostat model for biological water treatment using a Caputo fractional derivative with sliding memory (CFDS) to represent history-dependent microbial dynamics. We pose an optimal control problem that minimizes average pollutant concentration through periodic dilution-rate modulation subject to operational constraints. The analysis reduces the dynamics to a one-dimensional fractional differential equation, establishes existence and uniqueness of an optimal periodic solution, and derives the corresponding bang-bang control via the fractional Pontryagin maximum principle combined with a Fourier–Gegenbauer pseudospectral scheme. Sensitivity results show that the fractional order $\alpha$, scaling parameter $\vartheta$, and memory length $L$ significantly influence treatment performance. Numerical simulations demonstrate substantial reductions in substrate levels compared with steady-state operation, underscoring the potential of fractional modeling for improving water treatment efficiency.

This study investigated the stability of certain motions of a heavy, electrically charged gyrostat moving in a central attractive field. The Euler–Poisson equations were derived and then reformulated within the Lie–Poisson framework. We identified and analyzed the system's permanent rotations, presenting a mechanical interpretation of these motions. To assess stability, the Energy–Casimir method is employed to establish sufficient stability criteria. Moreover, the necessary conditions for the stability of the corresponding equilibrium configurations were also presented.