Variable time-step discretization of degenerate evolution equations in Banach spaces

This work is concerned with an abstract differential inclusion of the form Bu' + partial derivativephi(u) is an element of f, where B: V --> V' is a linear, continuous, symmetric and monotone operator defined over a separable Banach space V, and partial derivativephi is the subdifferential of a proper, convex, l.s.c, positive real function. We consider an approximation of the previous equation by a backward Euler method with variable time-step. Under suitable hypothesis of coercivity we prove that the discrete solution converges uniformly to a strong solution of the equation, in the seminorm induced by B, as the maximum of the time steps goes to 0. We derive computable estimates of the discretization error, which are optimal w.r.t. the order and impose no constrains between consecutive time steps. In addition we prove some regularity and uniqueness results for the solution. Finally we extend some of the previous results to the case in which partial derivativephi is perturbed by a Lipschitz map.