In the first part of this paper the authors consider the sequence of abstract Cauchy problems \((1)_n\) \(u'\in - \partial^- f(u)+ {\mathcal G}_n(u)\), \(u(0)= x_n\), \(x_n\in D(f)\), and the limit problem (1) \(u'\in -\partial^- f(u)+ {\mathcal G}(u)\), \(u(0)= \overline x\), \(\overline x\in D(f)\) (where \(\partial^- f\) is the Fréchet subdifferential of a function \(f\) defined on an open subset \(\Omega\) of a real separable Hilbert space \(H\), taking values in \(\mathbb{R}\cup \{+\infty\}\) and \({\mathcal G}_n\), \(\mathcal G\) are multifunction from \(C([0, T], \Omega)\) into the nonempty subsets of \(L^2([0, T], H)\)). An existence theorem for problem (1) in which the existence interval depends neither on \(\mathcal G\) nor on \(\overline x\), but only on \(f\) is established. A sufficient condition for every sequence \((u_n)_n\) of solutions of \((1)_n\) to have a subsequence which converges uniformly to a solution of problem (1) is also proved. Moreover, it is shown the continuous dependence of the solution of problem (1) on the initial point. Applying these theorems in the second part of the paper analogous results are obtained for the multivalued perturbed problem (2) \(x'\in - \partial^- f(x)+ G(t, x)\), \(x(0)= x_0\) (where \(G: [0, T]\times \Omega\to {\mathcal N}(H)\) is a multivalued perturbation).