Regularized algorithms for hierarchical fixed-point problems

Let CC be a nonempty closed convex subset of a real Hilbert space HH. Let S:C→CS:C→C be a non-expansive mapping and {Ti}i=1∞:C→C be an infinite family of non-expansive mappings. The purpose of this paper is to find the minimum norm solution of the following general hierarchical fixed point problem Find x̃∈⋂n=1∞Fix(Tn) such that 〈x̃−Sx̃,x̃−x〉≤0,∀x∈⋂n=1∞Fix(Tn). We introduce an explicit regularized algorithm with strong convergence for finding the minimum norm solution of the above hierarchical fixed point problem.