Abstract
In this paper, the most basic idea is to apply the viscosity approximation method to study the split feasibility problem (SFP), we will be in the infinite-dimensional Hilbert space to study the problem . We defined $x_{0}\in C$ as arbitrary and $x_{n+1}=(1-\alpha_{n})P_{C}(I-\lambda_{n}A^{*}(I-P_{Q})A)x_{n}+\alpha_{n}f(x_{n})$, for $n\geq0,$ where $\{\alpha_{n}\}\subset(0,1)$. Under the proper control conditions of some parameters, we show that the sequence $\{x_{n}\}$ converges strongly to a solution of SFP. The results in this paper extend and further improve the relevant conclusions in Deepho (Deepho, J. \& Kumam, P., 2015).
Highlights
In recent years, a large number of scholars have done a lot of meaningful research on the split feasibility problem (SFP), because the problem in signal processing and linear constrained optimization problems such as the feasible solution plays an important role
The most basic idea is to apply the viscosity approximation method to study the split feasibility problem (SFP), we will be in the infinite-dimensional Hilbert space to study the problem
A large number of scholars have done a lot of meaningful research on the split feasibility problem (SFP), because the problem in signal processing and linear constrained optimization problems such as the feasible solution plays an important role
Summary
A large number of scholars have done a lot of meaningful research on the split feasibility problem (SFP), because the problem in signal processing and linear constrained optimization problems such as the feasible solution plays an important role In order to find the solution of the problem SFP (1), many authors have proposed a variety of algorithms, it is worth noting that Byrne (Byrne, C., 2002) proposed the so-called CQ algorithm, the algorithm is this: take an initial point x0 ∈ H1 arbitrarily, and define the iterative step as xn+1 = PC(xn − λA∗(I − PQAxn), n ≥ 0,. & Kumam, P., 2015) proposed the following algorithm: xn+1 = (1 − αn)PC(I − λA∗(I − PQ)A)xn + αn f (xn), n ≥ 1,. Where {αn} ∈ (0, 1), 0 < λ < 2/∥A∥2, f : C → C is a contraction on C, and they proved that when the parameter {αn} satisfied certain conditions , the algorithm (3) is strong converges to a solution of SFP(1). We will show that the sequence {xn}n≥0 defined by (4) strongly converges to a solution of SFP(1)
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