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This paper is devoted to the study of evolution problems of the form −dudr(t)∈A(t)u(t)+f(t,u(t)) in a new setting, where, for each t, A(t) : D(A(t)) → 2H is a maximal monotone operator in a Hilbert space H and the mapping t↦A(t) has continuous bounded or Lipschitz variation on [0, T], in the sense of Vladimirov’s pseudo-distance. The measure dr gives an upper bound of that variation. The perturbation f is separately integrable on [0, T] and separately Lipschitz on H. Several versions and new applications are presented. [ABSTRACT FROM AUTHOR]