A Strong Convergence Algorithm for the Two-Operator Split Common Fixed Point Problem in Hilbert Spaces

Chung-Chien Hong,Young-Ye Huang

doi:10.1155/2014/350479

Abstract

The two-operator split common fixed point problem (two-operator SCFP) with firmly nonexpansive mappings is investigated in this paper. This problem covers the problems of split feasibility, convex feasibility, and equilibrium and can especially be used to model significant image recovery problems such as the intensity-modulated radiation therapy, computed tomography, and the sensor network. An iterative scheme is presented to approximate the minimum norm solution of the two-operator SCFP problem. The performance of the presented algorithm is compared with that of the last algorithm for the two-operator SCFP and the advantage of the presented algorithm is shown through the numerical result.

Highlights

Throughout this paper, H denotes a real Hilbert space with inner product ⟨⋅, ⋅⟩ and its induced norm ‖ ⋅ ‖, I the identity mapping on H, N the set of all natural numbers, R the set of all real numbers, and PΩ the metric projection onto set Ω. x is the upper bound of sequence {xn}, while x is the lower bound
Inspired by the work of [25, 26], this paper presents another algorithm to find the minimum norm solution of two-operator SCFP
(iii) directed if ⟨Tx − x, Tx − q⟩ ≤ 0, for x ∈ H, q ∈ Fix (T) . (13). It is well-known that the fixed point set Fix(S) of a nonexpansive mapping S is closed and convex; compare [27]

Summary

Introduction

Throughout this paper, H denotes a real Hilbert space with inner product ⟨⋅, ⋅⟩ and its induced norm ‖ ⋅ ‖, I the identity mapping on H, N the set of all natural numbers, R the set of all real numbers, and PΩ the metric projection onto set Ω. x is the upper bound of sequence {xn}, while x is the lower bound. The sequence {xn} generated by the CQ algorithm converges weakly to a solution of SFP (5); compare [14,15,16]. Moudafi [25] named SCFP (6) with p = 1 the two-operator SCFP and gave an algorithm which generates a sequence weakly converging to the solution of the two-operator SCFP. Inspired by the work of [25, 26], this paper presents another algorithm to find the minimum norm solution of two-operator SCFP. We note that the two-operator SCFP contains the SFP and the zero point problem of maximal monotone operators. Putting A = I, the above two-operator SCFP is reduced to the common zero point problem of two maximal monotone operators M and N: Find x∗ ∈ H so that x∗ ∈ M−10 ∩ N−10.

Preliminaries

Main Theorems

Numerical Results