Abstract
In 2014, Cui and Wang constructed an algorithm for demicontractive operators and proved some weak convergence theorems of their proposed algorithm to show the existence of solutions for the split common fixed point problem without using the operator norm. By Cui and Wang’s motivation, in 2015, Boikanyo constructed also a new algorithm for demicontractive operators and obtained some strong convergence theorems for this problem without using the operator norm. In this paper, we consider a viscosity iterative algorithm in Boikanyo’s algorithm to approximate to a solution of this problem and prove some strong convergence theorems of our proposed algorithm to a solution of this problem. Finally, we apply our main results to some applications, signal processing and others and compare our algorithm with five algorithms such as Cui and Wang’s algorithm, Boikanyo’s algorithm, forward-backward splitting algorithm and the fast iterative shrinkage-thresholding algorithm (FISTA).
Highlights
Assume that C and Q are nonempty closed convex subsets of Hilbert spaces H1 and H2, respectively
In 1994, the split feasibility problem was proposed by Censor and Elfving [1] as follows: Find a point x ∗ ∈ H1 such that x ∗ ∈ C and Ax ∗ ∈ Q
It is interesting to note that, when taking C = H1 and Q = {b}, the split feasibility problem reduces to the linear inverse problem: Find a point x ∗ ∈ H1 such that Ax ∗ = b
Summary
KMUTT-Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory. KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science. Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology. Received: 24 December 2018; Accepted: 20 February 2019; Published: 28 February 2019
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