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.

Automated summary

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, confirming the findings with fixed-point theorems (FPTs) from available literature. The RL FDE of variable order is transformed to alternate RL fractional integral structure, and classical FPTs are used to study existence results and establish Hyers-Ulam stability with the help of standard notions. The approach is broad-based and can be applied to other issues as well.