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

TL;DR

This paper investigates the existence, uniqueness, and stability of $eta$-viscosity solutions to Hamilton-Jacobi equations in Banach spaces, extending previous results using the concept of $eta$-derivative.

Contribution

It introduces new results on $eta$-viscosity solutions for Hamilton-Jacobi equations in Banach spaces, expanding the theoretical framework with the $eta$-derivative approach.

Findings

Established existence of $eta$-viscosity solutions

Proved uniqueness of solutions under certain conditions

Demonstrated stability of solutions with respect to data

Abstract

This paper is concerned with the qualitative properties of viscocity solutions to a class of Hamilton-Jacobi equations (HJEs) in Banach spaces. Specifically, based on the concept of $\beta$-derivative \cite{DGZ93b} 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 ealier works in the literature, for example, \cite{CL85}, \cite{CL86} and \cite{DGZ93b}.