On the Existence, Uniqueness, and Stability of $\beta $-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 $\beta $ -derivative (Deville et al. 1993), we establish the existence, uniqueness and stability of $\beta $ -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).