The existence and uniqueness of solution for a system of nonlinear generalized fractional differential equations of different orders on the bounded interval \([0,T]\) subject to a nonlocal boundary condition containing Riemann-Stieltjes and generalized fractional derivatives integral operators are considered in this work. Two fixed point theorems are used by the authors. The Leray-Schauder alternative is used to prove existence of solution while the Banach contraction mapping theorem is used to prove the uniqueness of the solution for the boundary value problem.
The authors define two Banach spaces \(X\) and \(Y\) and an operator \(\mathcal{F}: X \times Y \to X \times Y\) as \(\mathcal{F}(u,v)(t)(t)=(\mathcal{F}_1(u,v)(t),\mathcal{F}_2(u,v)(t))\).
Two theorems are used by the authors to prove the main results. Theorem 1 considers the uniqueness of solution while Theorem 2 considers the existence of solution. In Theorem 1, the authors use two functions of bounded variations \(H_1, H_2:[0,T] \times \mathbb{R} \to \mathbb{R}\), two continuous functions \(f, g \in C([0,T] \times \mathbb{R}^3,\mathbb{R})\) and assume the existence of two positive constants \(L_f\) and \(L_g\) whose values are expected to be less then \(\frac{1}{2}\) so that \(L_f + L_g <1\). The operator \(\mathcal{F}\), is shown to be a contraction operator. Hence, by the contraction mapping theorem, \(\mathcal{F}\) has a unique fixed point. This implies a unique solution of the system of nonlinear generalized fractional differential equations exists on \([0,T]\).
In Theorem 2, the authors address the issue of existence of solutions using the Leray-Schauder alternative. The conditions for existence of solution are stated and proved. The proof is done in three steps. Firstly, the operator \(\mathcal{F}\) is shown to be uniformly bounded, then completely continuous and finally, shown to be bounded. The conclusion of existence of at least one fixed point which corresponds to at least one solution follows from the Leray-Schauder alternative.
The study concludes with two examples to demonstrate the results that have been obtained in the study.