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

Abstract

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