An iterative process for nonlinear monotonic nonexpansive operators in Hilbert space

Abstract

The following theorem is proved: Suppose H is a complex Hilbert space, and T : H → H T:H \to H is a monotonic, nonexpansive operator on H, and f ∈ H f \in H . Define S : H → H S:H \to H by S u = − T u + f Su = - Tu + f for all u ∈ H u \in H . Suppose 0 ⩽ t n ⩽ 1 0 \leqslant {t_n} \leqslant 1 for all n = 1 , 2 , 3 , … , n = 1,2,3, \ldots , and Σ n = 1 ∞ t n ( 1 − t n ) \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 {V_{n + 1}} = (1 - {t_n}){V_n} + {t_n}S{V_n} converges to the unique solution u = p u = p of the equation u + T u = f u + Tu = f .