The implicit Euler scheme for one-sided Lipschitz differential inclusions (Q1956539)

The object of the present paper is to study the set-valued implicit Euler numerical procedure for the differential inclusion of \(\mathbb{R}^{m}:\) \[ \dot{x}\in F(t,x(t)) \text{ a.e. in} \quad [0,T], \quad x(0)=x_{0} \] where \(F\) is continuous, has convex and compact values and satisfies the so called relaxed one-sided Lipschitz condition (ROSL). Error estimates are given for: \[ y\in \mathbb{R}^{m}:y\in x+hF(t+h,x) \] under ROSL condition. After proving the solvability of the implicit scheme: \[ y\in \mathbb{R}^{m}:y\in x+hF(t+h,y) \] the authors give error estimates and a discrete stability theorem. Finally the paper gives a fully discretized implicit Euler scheme and shows how the solution of the above implicit inclusion can be approximated. The model example is chosen from the theoretical biology. It describes a simple biochemical process, where an organic substrate molecule is changed into a product,by means of an enzyme.