Method of semidiscretization in time to nonlinear retarded differential equations with nonlocal history conditions (Q1777659)

The authors investigate nonlinear retarded differential equations in a real Hilbert space \(H\) of the form \[ u'(t)+A(u(t))=f(t,u(t),u(r_1(t)),\ldots,u(r_m(t))),\quad t\in(0,T], \] \[ h(u_{[-\tau,0]})=\phi_0, \] where \(\phi_0\in C([-\tau,0],H)\) and \(A\) is a single valued maximal monotone operator. The authors show under suitable Lipschitz conditions on \(f\) and surjectivity type conditions on \(h\) the existence of a maximal solution. Further, they estimate the rate of convergence for certain time-discretization schemes.