Strong convergence of a proximal point algorithm with bounded error sequence

Given any maximal monotone operator $A:D(A)\subset <!--\; ?\; -->H\to <!--\; ?\; -->2^H$ in a real Hilbert space H with $A^{-<!--\; -\; -->1}(0)\ne <!--\; ?\; -->\mathrm{\varnothing <!--\; Ø\; -->}$ , it is shown that the sequence of proximal iterates $x_{n+1}=(I+{\mathrm{\gamma <!--\; ?\; -->}}_{n}A{)}^{-<!--\; -\; -->1}({\mathrm{\lambda <!--\; ?\; -->}}_{n}u+(1-<!--\; -\; -->{\mathrm{\lambda <!--\; ?\; -->}}_n)(x_n+e_n))$ converges strongly to the metric projection of u on A ⁻¹(0) for (e n ) bounded, ${\mathrm{\lambda <!--\; ?\; -->}}_n\in <!--\; ?\; -->(0,1)$ with ${\mathrm{\lambda <!--\; ?\; -->}}_n\to <!--\; ?\; -->1$ and γ n > 0 with ${\mathrm{\gamma <!--\; ?\; -->}}_n\to <!--\; ?\; -->\mathrm{\infty <!--\; 8\; -->}$ as $n\to <!--\; ?\; -->\mathrm{\infty <!--\; 8\; -->}$ . 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 $\mathrm{\phi <!--\; f\; -->}:H\to <!--\; ?\; -->(-<!--\; -\; -->\mathrm{\infty <!--\; 8\; -->},+\mathrm{\infty <!--\; 8\; -->}]$ , the algorithm can be used to approximate the minimizer of φ which is nearest to u.