Strong convergence of a proximal point algorithm with bounded error sequence

Given any maximal monotone operator A: D(A) ⊂ H → 2^H in a real Hilbert space H with A^-1(0) ≠ ∅, it is shown that the sequence of proximal iterates x_n+1 = (I+ γ_nA)^-1(λ_nu + (1-λ_n)(x_n+e_n)) converges strongly to the metric projection of u on A^-1(0) for (e_n) bounded, λ_n ∈ (0,1) with λ_n → 1 and γ_n > 0 with γ_n → ∞ as n → ∞. In comparison with our previous paper (Boikanyo and Moroşanu in Optim Lett 4(4):635-641, 2010), 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.