Differences of solutions of implicit Euler schemes with accretive operators on Banach spaces

TL;DR

This paper derives an upper bound for the difference between solutions of implicit Euler schemes for quasi-accretive operators on Banach spaces, extending previous results and aiding in establishing existence, uniqueness, and regularity of solutions.

Contribution

It generalizes Kobayashi's upper bound to include non-zero forcing terms, enabling analysis of solution differences, existence, and regularity for Euler schemes with accretive operators.

Findings

Established an upper bound for solution differences with non-zero forcing.

Proved existence and uniqueness of Euler solutions as limits of schemes.

Demonstrated regularity properties of Euler solutions.

Abstract

We give an upper bound for the difference of two solutions of Euler schemes approximating the Cauchy problem \[\begin{cases} \dot{u}(t) + Au(t) \ni f(t) \quad (t \in [0, T]), \\ u(0) = u^0, \end{cases}\] where $A \subseteq X \times X$ is a quasi-accretive operator on a Banach space $X$, $T > 0$, $f \in L^1(0, T; X)$ and $u^0 \in X$. This upper bound generalizes a result from Kobayashi, who established an upper bound for the problem with $f = 0$. We show, that the upper bound can be used to establish existence and uniqueness of Euler solutions as limits of solutions of Euler schemes as well as regularity of Euler solutions.