A Viscosity Approximation Method for the Split Feasibility Problems in Hilbert Space

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 x0 ∈ C as arbitrary and xn+1 = (1 − αn)PC(I − λnA(I − PQ)A)xn + αn f (xn), for n ≥ 0, where {αn} ⊂ (0, 1). Under the proper control conditions of some parameters, we show that the sequence {xn} 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).


Introduction
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 (Censor, Y., et al, 2006;Byrne, C., 2002;Byrne, C., 2004.;Yang, Q., 2004.;Qu, B. & Xiu, N., 2005.;Xu, H. K., 2006.;Xu, H. K., 2010).In 1994,the SFP was first introduced by Censor and Elfving (Censor, Y. & Elfving, T., 1994), which is to find a point x * satisfying the property: where C and Q be nonempty closed convex subsets of the real Hilbert spaces H 1 and H 2 , A : H 1 → H 2 be a bounded linear operator.
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 x 0 ∈ H 1 arbitrarily, and define the iterative step as Where 0 < λ < 2/ρ(A * A) and P C denotes the projector onto C and ρ(A * A) is the spectral radius of the self-adjoint operator A * A, I denotes the identity operator.Then the sequence {x n } n≥0 generated by (2) converges strongly to a solution of SFP whenever H 1 is finite-dimensional and whenever there exists a solution to SFP(1).By Byrne's CQ algorithm and Xu's viscosity approximatiom method (Xu, H. K., 2004), In 2015, Deepho and Kumam (Deepho, J. & Kumam, P., 2015) proposed the following algorithm: where {α n } ∈ (0, 1), 0 < λ < 2/∥A∥ 2 , f : C → C is a contraction on C, and they proved that when the parameter {α n } satisfied certain conditions ,then the algorithm (3) is strong converges to a solution of SFP(1).In this paper, we study the following more general algorithm which generates a sequence according to the recursive formula: And we will show that the sequence {x n } n≥0 defined by (4) strongly converges to a solution of SFP(1).

Preliminaries
Throughout this paper, we always assumes that H 1 and H 2 are two real Hilbert spaces with inner product ⟨•⟩ and norm ∥ • ∥, we use Ω to denote the solution set of SFP(1), that is The notation: ⇀ denotes weak convergence and → denotes strong convergence.Below we first list the definitions and theorems to be used in this paper.
Definition 2.1.Assume H is a real Hilbert space.Let T : H → H be the nonlinear operators, In this case ,we also say that T is α− averaged.Thus firmly nonexpansive mappings (in particular ,the projections ) is 1 2 − averaged mappings.Definition 2.2.An operator T : H → H is called oriented operator if Fix(T ) Φ, and In fact, we know that the oriented operator also contains firmly nonexpansive operator.The following is a useful characterization of projections.Proposition 2.1 Given x ∈ H and z ∈ C. Then z = P C x if and only if We collect some basic properties of averaged mappings and inverse strongly monotone operators in the following lemma.

NSTL Condition
Let C be a nonempty closed convex subset of a real Hilbert space H. Motivated by Nakajo, Shimoji and Takahashi (Takahashi, W., 2009), we give the following definition: Let T n be families of nonexpansive mappings of C into itself such that ∩ ∞ n=1 F(T n ) Φ ,where F(T n ) is the set of all fixed points of T n .Then T n is said to be satisfy NSTL-condition if for each bounded sequence {z n } ⊂ C,

Main Results
Theorem 3.1 Suppose the SFP(1) is consistent and 0 < λ ′ < λ n < λ ′′ < 2 ∥A∥ 2 .Let C be a nonempty closed convex subset of a real Hilbert space H 1 .Let f : C → C be a contraction with constant ρ ∈ (0, 1).Take an initial guess x 0 ∈ H 1 arbitrarily, and we define the sequence {x n } by where Then the sequence {x n } generated by algorithm(7) converges strongly to x ∈ Ω, where x = P Ω f ( x).
Proof.The proof of the process will be divided into four steps.First we show that the sequence {x n } is bounded.For our convenience, we take T n = P C (I −λ n A * (I − P Q )A).We assume ∩ ∞ n=1 F(T n ) Φ and T n satisfy NSTL condition.By lemma 2.2, we know that ∩ ∞ n=1 F(T n ) is a solution of SFP(1).Now, we note that the condition 0 < λ ′ < λ n < λ ′′ < 2 ∥A∥ 2 implies that the operator P C (I − λ n A * (I − P Q )A) is averaged.Since I − P Q is firmly nonexpansive mappings and so is 1 2 -averaged, which is 1 -ism.Also observe that A * (I λ n ∥A∥ 2 averaged and P C is 1 2 -averaged, by lemma 2.1, we may obtain that P C (I − λ n A * (I − P Q )A) is µ n -averaged, where ∈ (0, 1) for some nonexpansive mappings S. Note that T n is also nonexpansive mappings, then for p ∈ So {x n } is bounded, we also have that {T n x n } and { f (x n )} are bounded.Next, we claim that Indeed, by the definition of the (7),that is Next, we will show that lim sup Indeed take a subquence We may assume that x n k ⇀ x.It follows from Lemma 2.3 and Finally, we will show that x n → x in norm.It follows from Lemma 2.4, we obtain Thus, we have where Lemma 2.1(Qu, B. & Xiu, N., 2005; Xu, H. K., 2011)  Let T : H → H be a given mapping.
By the condition (C1), we have lim n→∞ ∥x n+1 − T n x n ∥ = 0.This together with the NSTL condition, Thus,(8) is clearly established.