The implicit Euler scheme for one-sided Lipschitz differential inclusions

An Implicit Function Theorem for One-sided Lipschitz Mappings

“…The practical implementation of the set-valued Crank-Nicolson method is very similar to the implicit Euler method in [3] and we refer to that paper for further details.…”

“…Theorem 5 can be taken as a basis for extending the convergence analysis as h → 0 from the implicit Euler method analyzed in [3] to the set-valued Crank-Nicolson scheme (15). However, expanding the details of such a result is beyond the scope of this paper.…”

“…The following Theorem is a local version of Theorem 2 in [3]. In order to obtain the optimal result, it is necessary to extend the locally defined mapping to the whole space in a suitable way.…”

“…The domains have to be chosen carefully in order to ensure that for all parameters considered, the full sets N( p) are contained in the specified region. Otherwise we encounter various problems related to the continuity properties of the solution mapping N(·) due to the fact that its images need not be convex, see Example 10 in [3]. …”

“…In Section 2, we prove a local version of the solvability theorem presented in [3] by extending a locally defined ROSL mapping to R m in a suitable manner. This solvability theorem can be regarded as a multivalued version of the uniform monotonicity theorem (see [15,17]).…”