Krasnoselski-Mann iteration for hierarchical fixed-point problems

This paper deals with a method for approximating a solution of the following fixed-point problem: find (x) over tilde is an element of H; (x) over tilde = (proj(Fix(T)) o P)(x) over tilde, where H is a Hilbert space, P and T are two nonexpansive mappings on a closed convex subset D and proj(Fix(T)) denotes the metric projection on the set of fixed points of T. This amounts to saying that (x) over tilde is the fixed point of T which satisfies a variational inequality depending on a given criterion P, namely: find (x) over tilde is an element of H; 0 is an element of (I - P) (x) over tilde + N-Fix(T) (x) over tilde, where N-Fix(T) denotes the normal cone to the set of fixed points of T. Convergence results for the proposed method are proved. It should be noted that the proposed method can be regarded as a generalized version of Krasnoselski - Mann's iteration for solving a broader class of problems than the original KM algorithm, namely hierarchical fixed-point problems. This class is very interesting because it covers monotone variational inequality on fixed-point sets, minimization problems over equilibrium constraints, hierarchical minimization problems,.... The special aspect of the algorithm together with convergence results makes it an original and theoretically interesting scheme. On the other hand, the framework is general enough and permits us to treat in a unified way several iterative schemes, recovering, developing and improving some known related convergence results in this field.