Existence and Uniqueness Theorems for a Variable-Order Fractional Differential Equation with Delay

This study establishes the existence and stability of solutions for a general class of Riemann-Liouville (RL) fractional differential equations (FDEs) with a variable order and finite delay. Our findings are confirmed by the fixed-point theorems (FPTs) from the available literature. We transform the RL FDE of variable order to alternate RL fractional integral structure, then with the use of classical FPTs, the existence results are studied and the Hyers-Ulam stability is established by the help of standard notions. The approach is more broad-based and the same methodology can be used for a number of additional issues.