Abstract
The convergence analysis of a variable KM-like method for approximating common fixed points of a possibly infinitely countable family of nonexpansive mappings in a Hilbert space is proposed and proved to be strongly convergent to a common fixed point of a family of nonexpansive mappings. Our variable KM-like technique is applied to solve the split feasibility problem and the multiple-sets split feasibility problem. Especially, the minimum norm solutions of the split feasibility problem and the multiple-sets split feasibility problem are derived. Our results can be viewed as an improvement and refinement of the previously known results. MSC:47H10, 65J20, 65J22, 65J25.
Highlights
Problems of image reconstruction from projections can be represented by a system of linear equations Ax = b. ( . )In practice, the system ( . ) is often inconsistent, and one usually seeks a point which minimizes x ∈ Rn by some predetermined optimization criterion
Takahashi [ ] proved a strong convergence theorem of the following iterative algorithm for countable families of nonexpansive mappings in certain Banach spaces: xn+ = tnfxn + ( – tn)Tnxn, n ∈ N
Let C be a nonempty closed convex subset of a real Hilbert space H, T : C → C a nonexpansive mapping such that F(T) = ∅, and f : C → H be a κ-contraction with κ ∈ [, ) such that PF(T)f (x∗) = x∗ ∈ F(T)
Summary
Takahashi [ ] proved a strong convergence theorem of the following iterative algorithm for countable families of nonexpansive mappings in certain Banach spaces: xn+ = tnfxn + ( – tn)Tnxn, n ∈ N. In Section , we will study the convergence analysis of our variable KM-like algorithm for fixed point problem
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