On the existence, uniqueness, and stability of β-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).