Algorithm Reasearch Of Solving Split Common Fixed-Point Problem For Firmly Quasi-Nonexpansive Mappings

The split common fixed-point problem is gradual developed on the basis of the convex feasibility problem, the split feasibility problem and the common fixed-piont problem. For decades, it is widely used in the inverse problem of senor networks, radiotherapy planning,wavelet denoising, practical problems such as computer tomography technonogy. The core of this article is mainly to solve split common fixed-point problem of firmly quai-nonexpansive mappings in Hilbert spaces, which is divided into three parts.The first section: very recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms for the split common fixed-point problem concerned two bounded linear operators. However, to employ Moudafi’s algorithms, one needs to know a prior norm(or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, it is not an impossible task. It is the purpose of this paper to introduce a viscosity iterative algorithm with a way of selecting the step-sizes such that the implementation of the algorithm does not need any prior information about the operator norms. We prove the strong convergence of the proposed algorithms for split common fixed-point problem governed by the firmly quasi-nonexpansive operators.The second section: we introduce a iterative algorithm for finding common solution of variational inequality for Lipschitzian and strongly monotone operator and the split common fixed-point problem for firmly quasi-nonexpansive operators. We prove the strong convergence of the proposed algorithm which does not need any prior information about the bounded operator norms.The third section: Yang and He proposed a general regularization method for finding common solution of variational inequality and fixed point of nonexpansive operator. Inspired by them, we introduce a iterative algorithm for finding common solution of variational inequality for Lipschitzian strongly pseudo-contractive operators and the split common fixed-point problem for firmly nonexpansive operators. We prove the strong convergence of the proposed algorithm which does not need any prior information about the bounded operator norms.