Viscosity solutions for fractional hamilton-jacobi equations in banach spaces with smooth norms

This paper investigates the well-posedness of time-fractional Ham-ilton-Jacobi equations in reflexive Banach spaces, extending the theory of Caputo-type viscosity solutions beyond the classical finite-dimensional framework. The transition to infinite dimensions gives rise to serious geometric obstacles, primarily the loss of local compactness, which renders standard doubling-of-variables arguments inapplicable. To overcome this difficulty, we develop a new comparison principle by combining the nonlocal Caputo operator with the Borwein–Preiss Smooth Variational Principle [5] and the Fréchet differentiability of the norm. Specifically, test functions are taken from the class $ \mathcal{A} $ (see Definition 2.7), in which the spatial derivative is understood in the Fréchet sense and the Caputo time-fractional derivative is decomposed into two operators $ J $ and $ K_{(0, t)} $, yielding a framework that is consistent with the structure of the dual space $ X^* $. Furthermore, we establish the existence of solutions via a Perron-type construction and derive stability estimates with respect to both initial data and Hamiltonian perturbations. Our results provide a rigorous theoretical foundation for fractional evolution systems in distributed-parameter settings such as the spaces $ L^p(\Omega) $ and $ \ell^p $.