A relaxed Picard iteration process for set-valued operators of the monotone type

Abstract:

The fixed points $\bar x$ of set-valued operators, $T:X \to {2^X}$, satisfying a condition of the monotonicity type on convex subsets X of a Hilbert space are approximated by a relaxation process, ${x_{n + 1}} = {x_n} + {\omega _n}(T{x_n} - {x_n})$, in which $\tilde T$ is a single-valued branch of T and the relaxation parameter ${\omega _n} \in [0,1]$ is made to depend in a certain way on the prior history of the process. If $\tilde T$ is bounded on bounded subsets of X, then $\left \| {{x_n} - \bar x} \right \|$ converges to 0 like $O({n^{ - 1/2}})$. If $\tilde T$ is also continuous at $\bar x$ and if $\bar x = \tilde T\bar x$, then $\left \| {{x_n} - \bar x} \right \| = o({n^{ - 1/2}})$. If $\tilde T$ satisfies a condition of the Lipschitz type at $\bar x$, then $\left \| {{x_n} - \bar x} \right \| = O({\mu ^{n/2}})$ for some $\mu \in [0,1)$.