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

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.