Stability, Existence and Uniqueness of Boundary Value Problems for a Coupled System of Fractional Differential Equations

The current thesis investigates four different nonlinear systems of fractional differential equations and deals with the existence, uniqueness, and stability of their solutions. The first studied problem is a coupled system of fractional differential equations with four-point integral boundary conditions. Existence and uniqueness of solutions are established by applying the contraction mapping principle and Leray–Schauder’s alternative theorem. Finding and results are demonstrated and supported with numerical examples. The second studied case is a boundary value problem for a coupled system of nonlinear fractional differential equations, where the existence and uniqueness of solutions is proven by using the Banach’s fixed point theorem and Schauder’s alternative. Furthermore, the Hyers-Ulam stability of solutions is discussed, sufficient stability conditions are drawn, and supporting numerical results are presented. In the third problem, a coupled system of Caputo type sequential fractional differential equations with integral boundary conditions is studied. Similarly, existence and uniqueness of solutions are discussed and established by employing contraction mapping principle and Leray–Schauder’s alternative theorem, and Hyers-Ulam stability of the boundary value problem is investigated. The last problem is a nonlinear Caputo type sequential fractional differential equation with non-separated non-local integral fractional boundary conditions. Existence, uniqueness, and Hyers-Ulam stability of solutions are discussed and established, and theoretical findings are presented and supported by numerical examples. Keywords: fractional differential equation, sequential, Caputo, integral boundary conditions, stability, Hyers-Ulam stability, existence and uniqueness of solutions.