An Iterative Process for Nonlinear Monotonic Nonexpansive Operators in Hilbert Space

The following theorem is proved: Suppose H is a complex Hilbert space, and $T:H \to H$ is a monotonic, nonexpansive operator on H, and $f \in H$. Define $S:H \to H$ by $Su = - Tu + f$ for all $u \in H$. Suppose $0 \leqslant {t_n} \leqslant 1$ for all $n = 1,2,3, \ldots ,$ and $\Sigma _{n = 1}^\infty \;{t_n}(1 - {t_n})$ diverges. Then the iterative process ${V_{n + 1}} = (1 - {t_n}){V_n} + {t_n}S{V_n}$ converges to the unique solution $u = p$ of the equation $u + Tu = f$.