Hyers-Ulam and Hyers-Ulam-Rassias Stability for a Class of Fractional Evolution Differential Equations with Neutral Time Delay

In this paper, we demonstrate that neutral fractional evolution equations with finite delay possess a stable mild solution. Our model incorporates a mixed fractional derivative that combines the Riemann-Liouville and Caputo fractional derivatives with orders 0<alpha<1 and 1<beta<2. We identify the infinitesimal generator of the cosine family and analyze the stability of the mild solution using both Hyers-Ulam-Rassias and Hyers-Ulam stability methodologies, ensuring robust and reliable results for fractional dynamic systems with delay. In order to guarantee that the features of invariance under transformations, such as rotations or reflections, result in the presence of fixed points that remain unchanging and represent the consistency and balance of the underlying system, fixed-point theorems employ the symmetry idea. Lastly, the results obtained are applied to a fractional order nonlinear wave equation with finite delay with respect to time.