Strong convergence of a proximal point algorithm with bounded error sequence

Given any maximal monotone operator A : D(A) ⊂ H → 2 in a real Hilbert space H with A−1(0) 6= ∅, it is shown that the sequence of proximal iterates xn+1 = (I + γnA)(λnu+ (1− λn)xn + en) converges strongly to the metric projection of u on A−1(0) for (en) bounded, λn ∈ (0, 1) with λn → 1 and γn > 0 with γn → ∞ as n → ∞. In comparison with our previous paper [Optim. Lett. 4 (2010), 635-641], where the error sequence was supposed to converge to zero, here we consider the classical condition that errors be bounded. In the case when A is the subdifferential of a proper convex lower semicontinuous function φ : H → (−∞,+∞], the algorithm can be used to approximate the minimizer of φ which is nearest to u.