Existence of Solutions for a Multi-term Fractional Boundary Value Problem with a Separated Fractional Boundary Condition

Multi-term fractional differential equations offer a more detailed view of processes with different levels of memory and relaxation. Hence, they provide enhanced accuracy in modelling real-world systems in viscoelasticity, finance, control theory, and many other fields. However, the consideration of both the perturbation and the convection terms in this type of differential equations complicates the study of this type of problems. Accordingly, they are usually addressed in the literature using numerical methods, such as the finite difference method. In this work, we investigate a fractional multi-term differential equation where the leading derivative is the Riemann-Liouville- Caputo fractional derivative, combined with separated fractional boundary conditions. In effect, recent applications suggest that this new derivative is be more suitable in certain situations. Moreover, fractional boundary conditions offer a powerful extension of classical ones, but they complicate the solution’s form. Nevertheless, employing fractional order derivatives in boundary conditions is more advantageous, as the latter corresponds to a broader situation, permitting more flexibility in the choice of the derivative, resulting in better outcomes when modelling real-world models in contrast to the classical integer order models. To solve the problem at hand we convert it into a Volterra equation. Then by means of corresponding Green’s functions, we apply fixed point theory techniques to obtain existence results under different conditions imposed on the nonlinearity. Mainly, the Banach contraction principle, the Krasnoselskii fixed point theorem, and the Leray-Schauder nonlinear alternative are utilised to prove the existence of solutions.