Forward-backward approximation of nonlinear semigroups in finite and infinite horizon (Q2032126)

The authors consider the problem \[ \begin{aligned} -&\dot{u}(t)\in\left( A+B\right) u(t) \text{ for a.e. }t>0,\\ &u(0)=u_{0}\in D(A), \end{aligned} \] in a class of Banach spaces, where \(A\) is \(m\)-accretive and \(B\) is coercive. First, the approximation of solutions is investigated. Solutions are approximated by trajectories constructed by interpolation of sequences generated using forward-backward iteration and these are shown to converge uniformly on a finite time interval, proving existence and uniqueness of solutions. Second, asymptotic equivalence results are given that connect the behaviour of forward-backward iterations as the number of iterations goes to infinity with the behaviour of the solution as time goes to infinity, for step sizes that are sufficiently small. These results are based on a certain inequality which the authors trace back to \textit{E. Hille} [Fysiogr. Sällsk. Lund Förh. 21, No. 14, 130--142 (1951; Zbl 0044.32902)].