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<abstract><p>In this paper, we study a general system of fractional hybrid differential equations with a nonlinear $ \phi_p $-operator, and prove the existence of solution, uniqueness of solution and Hyers-Ulam stability. We use the Caputo fractional derivative in this system so that our system is more general and complex than other nonlinear systems studied before. To establish the results, Green functions are used to transform the considered hybrid boundary problem into a system of fractional integral equations. Then, with the help of the topological degree theorem, we derive some sufficient conditions that ensure the existence and uniqueness of solutions for the proposed system. Finally, an example is presented to show the validity and correctness of the obtained results.</p></abstract>

We investigate the appropriate and sufficient conditions for the existence and uniqueness of a solution for a coupled system of Atangana–Baleanu fractional equations with a p-Laplacian operator. We also study the HU-stability of the solution by using the Atangana–Baleanu–Caputo (ABC) derivative. To achieve these goals, we convert the coupled system of Atangana–Baleanu fractional equations into an integral equation form with the help of Green functions. The existence of the solution is proven by using topological degree theory and Banach’s fixed point theorem, with which we analyze the solution’s continuity, equicontinuity and boundedness. Then, we use Arzela–Ascolli theory to ensure that the solution is completely continuous. Uniqueness is established using the Banach contraction principle. We also investigate several adequate conditions for HU-stability and generalized HU-stability of the solution. An illustrative example is presented to verify our results.

The growth of the world populations number leads to increasing food needs. However, plant diseases can decrease the production and quality of agricultural harvests. Mathematical models are widely used to model and interpret plant diseases, showing viruses’ transmission dynamics and effects. In this paper, we investigate the dynamics of the treatments of plant diseases via the Atangana–Baleanu derivative in the sense of Caputo (ABC). We study the existence and uniqueness of solutions of curative and preventive treatment fractional model for plant disease. By using Lagrange interpolation, we give numerical simulations and investigate the results at various fractional orders under specific parameters. The results show that the increase of the roguing rate for the most infected plant or the decrease of the rate of planting in the infected area will reduce the plant disease transmissions. For balancing the plant production, the decision-makers can plant in other areas in which there are no infected cases.