Ulam-type stability for Caputo-type fractional delay differential equations

Abstract This study focuses on differential equations incorporating generalized fractional derivatives of the Caputo type. The concept of Ulam-type stability (US) is analyzed in the context of both initial value problems and boundary value problems (BVPs) for the fractional differential equations under investigation. Particular attention is given to addressing certain misconceptions that arise when applying US to BVPs. To mitigate these issues, we propose incorporating a parameter into the boundary conditions as a potential solution. The dependency of the solution on this parameter is established, and a method is outlined for selecting the parameter appropriately. This approach ensures that the solution of the fractional equation is strongly influenced by the arbitrarily chosen solution of the associated inequality. The theoretical findings are further clarified through illustrative examples.