On the existence, uniqueness, and stability of ẞ-viscosity solutions to a class of Hamilton-Jacobi equations in Banach spaces

Abstract

This paper is concerned with the qualitative properties of viscosity solutions to a class of Hamilton-Jacobi equations (HJEs) in Banach spaces. Specifically, based on the concept of β-derivative (Deville et al. 1993), we establish the existence, uniqueness and stability of β-viscosity solutions for a class of HJEs in the form u + H(x, u, Du) = 0. The obtained results in this paper extend earlier works in the literature, for example, Crandall and Lions (J. Funct. Anal. 62, 379-398, 1985, J. Funct. Anal., 65, 368-405, 1986) and Deville et al. (1993).