# A damped algorithm for the split feasibility and fixed point problems

## Abstract

The purpose of this paper is to study the split feasibility problem and the fixed point problem. We suggest a damped algorithm. Convergence theorem is proven.

MSC:47J25, 47H09, 65J15, 90C25.

## 1 Introduction

Let C and Q be two closed convex subsets of two Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively, and let $A:{H}_{1}\to {H}_{2}$ be a bounded linear operator. Finding a point ${x}^{\ast }$ satisfies

${x}^{\ast }\in C\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}A{x}^{\ast }\in Q.$
(1.1)

This problem, referred to as the split problem, has been studied by some authors. See, e.g.,  and . Some algorithms for solving (1.1) have been presented. One is Byrne’s CQ algorithm 

${x}_{n+1}={P}_{C}\left({x}_{n}-\tau {A}^{\ast }\left(I-{P}_{Q}\right)A{x}_{n}\right),\phantom{\rule{1em}{0ex}}n\in \mathbb{N},$

where $\tau \in \left(0,\frac{2}{L}\right)$ with L being the largest eigenvalue of the matrix ${A}^{\ast }A$, I is the unit matrix or operator, and ${P}_{C}$ and ${P}_{Q}$ denote the orthogonal projections onto C and Q, respectively. Motivated by Byrne’s CQ algorithm, Xu  suggested a single step regularized method

${x}_{n+1}={P}_{C}\left(\left(1-{\alpha }_{n}{\gamma }_{n}\right){x}_{n}-{\gamma }_{n}{A}^{\ast }\left(I-{P}_{Q}\right)A{x}_{n}\right),\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.$
(1.2)

Very recently, Dang and Gao  introduced the following damped projection algorithm

${x}_{n+1}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}{P}_{C}\left(\left(1-{\alpha }_{n}\right)\left({x}_{n}-\tau {A}^{\ast }\left(I-{P}_{Q}\right)A{x}_{n}\right)\right),\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.$

If every closed convex subset of a Hilbert space is the fixed point set of its associating projection, then the split feasibility problem becomes a special case of the split common fixed point problem of finding a point ${x}^{\ast }$ with the property

${x}^{\ast }\in Fix\left(U\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}A{x}^{\ast }\in Fix\left(T\right).$

This problem was first introduced by Censor and Segal , who invented an algorithm, which generates a sequence $\left\{{x}_{n}\right\}$ according to the iterative procedure

${x}_{n+1}=U\left({x}_{n}-\gamma {A}^{\ast }\left(I-T\right)A{x}_{n}\right),\phantom{\rule{1em}{0ex}}n\in \mathbb{N}.$

Recently, Cui, Su and Wang  extended the damped projection algorithm to the split common fixed point problems. For some related work, please refer to  and [13, 14].

Motivated by these results, the purpose of this paper is to study the following split feasibility problem and fixed point problem

(1.3)

where $S:Q\to Q$ and $T:C\to C$ are two nonexpansive mappings. We suggest a damped algorithm for solving (1.3). Convergence theorem is proven.

## 2 Preliminaries

Let H be a real Hilbert space with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$, respectively. Let C be a nonempty closed convex subset of H.

Definition 2.1 A mapping $T:C\to C$ is called nonexpansive if

$\parallel Tx-Ty\parallel \le \parallel x-y\parallel$

for all $x,y\in C$.

We will use $Fix\left(T\right)$ to denote the set of fixed points of T, that is, $Fix\left(T\right)=\left\{x\in C:x=Tx\right\}$.

Definition 2.2 We call ${P}_{C}:H\to C$ the metric projection if for each $x\in H$

$\parallel x-{P}_{C}\left(x\right)\parallel =inf\left\{\parallel x-y\parallel :y\in C\right\}.$

It is well known that the metric projection ${P}_{C}:H\to C$ is characterized by

$〈x-{P}_{C}\left(x\right),y-{P}_{C}\left(x\right)〉\le 0$

for all $x\in H$, $y\in C$. From this, we can deduce that ${P}_{C}$ is firmly-nonexpansive, that is,

${\parallel {P}_{C}\left(x\right)-{P}_{C}\left(y\right)\parallel }^{2}\le 〈x-y,{P}_{C}\left(x\right)-{P}_{C}\left(y\right)〉$
(2.1)

for all $x,y\in H$. Hence ${P}_{C}$ is also nonexpansive.

It is well known that in a real Hilbert space H, the following two equalities hold

${\parallel tx+\left(1-t\right)y\parallel }^{2}=t{\parallel x\parallel }^{2}+\left(1-t\right){\parallel y\parallel }^{2}-t\left(1-t\right){\parallel x-y\parallel }^{2}$
(2.2)

for all $x,y\in H$ and $t\in \left[0,1\right]$, and

${\parallel x+y\parallel }^{2}={\parallel x\parallel }^{2}+2〈x,y〉+{\parallel y\parallel }^{2}$
(2.3)

for all $x,y\in H$. It follows that

${\parallel x+y\parallel }^{2}\le {\parallel x\parallel }^{2}+2〈y,x+y〉$
(2.4)

for all $x,y\in H$.

Lemma 2.3 

Let C be a closed convex subset of a real Hilbert space H, and let $S:C\to C$ be a nonexpansive mapping. Then, the mapping $I-S$ is demiclosed. That is, if $\left\{{x}_{n}\right\}$ is a sequence in C such that ${x}_{n}\to {x}^{\ast }$ weakly and $\left(I-S\right){x}_{n}\to y$ strongly, then $\left(I-S\right){x}^{\ast }=y$.

Lemma 2.4 

Assume that $\left\{{a}_{n}\right\}$ is a sequence of nonnegative real numbers such that

${a}_{n+1}\le \left(1-{\gamma }_{n}\right){a}_{n}+{\delta }_{n},\phantom{\rule{1em}{0ex}}n\in \mathbb{N},$

where $\left\{{\gamma }_{n}\right\}$ is a sequence in $\left(0,1\right)$, and $\left\{{\delta }_{n}\right\}$ is a sequence such that

1. (1)

${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_{n}=\mathrm{\infty }$;

2. (2)

${lim sup}_{n\to \mathrm{\infty }}\frac{{\delta }_{n}}{{\gamma }_{n}}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n}|<\mathrm{\infty }$.

Then ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$.

## 3 Main results

Let C and Q be two nonempty closed convex subsets of real Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively. Let $A:{H}_{1}\to {H}_{2}$ be a bounded linear operator with its adjoint ${A}^{\ast }$. Let $S:Q\to Q$ and $T:C\to C$ be two nonexpansive mappings. We use Γ to denote the set of solutions of (1.3), that is, $\mathrm{\Gamma }=\left\{{x}^{\ast }|{x}^{\ast }\in C\cap Fix\left(T\right),A{x}^{\ast }\in Q\cap Fix\left(S\right)\right\}$. Now, we present our algorithm.

Algorithm 3.1 For ${x}_{0}\in {H}_{1}$ arbitrarily, let $\left\{{x}_{n}\right\}$ be a sequence defined by

(3.1)

where ${\left\{{\alpha }_{n}\right\}}_{n\in \mathbb{N}}$ and ${\left\{{\beta }_{n}\right\}}_{n\in \mathbb{N}}$ are two real number sequences in $\left(0,1\right)$ and $\delta \in \left(0,\frac{1}{{\parallel A\parallel }^{2}}\right)$.

Theorem 3.2 Suppose $\mathrm{\Gamma }\ne \mathrm{\varnothing }$. Assume the sequence ${\left\{{\alpha }_{n}\right\}}_{n\in \mathbb{N}}$ satisfies three conditions

(C1) ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$;

(C2) ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$;

(C3) ${lim}_{n\to \mathrm{\infty }}\frac{{\alpha }_{n+1}}{{\alpha }_{n}}=1$.

Then the sequence $\left\{{x}_{n}\right\}$, generated by algorithm (3.1), converges strongly to ${x}^{\ast }={P}_{\mathrm{\Gamma }}\left(0\right)$.

Proof For the convenience, we write ${z}_{n}={P}_{Q}A{x}_{n}$, ${y}_{n}=\left(1-{\alpha }_{n}\right)\left({x}_{n}-\delta {A}^{\ast }\left(I-S{P}_{Q}\right)A{x}_{n}\right)$ and ${u}_{n}={P}_{C}\left(\left(1-{\alpha }_{n}\right)\left({x}_{n}-\delta {A}^{\ast }\left(I-S{P}_{Q}\right)A{x}_{n}\right)\right)$ for all $n\in \mathbb{N}$. Thus ${u}_{n}={P}_{C}{y}_{n}$ for all $n\in \mathbb{N}$.

Let ${x}^{\ast }={P}_{\mathrm{\Gamma }}\left(0\right)$. Hence, ${x}^{\ast }\in C\cap Fix\left(T\right)$ and $A{x}^{\ast }\in Q\cap Fix\left(S\right)$. By the firmly-nonexpansivity of ${P}_{C}$ and ${P}_{Q}$, we can deduce the following conclusions

$\parallel {z}_{n}-A{x}^{\ast }\parallel =\parallel {P}_{Q}A{x}_{n}-{P}_{Q}A{x}^{\ast }\parallel \le \parallel A{x}_{n}-A{x}^{\ast }\parallel ,$
(3.2)
$\parallel {u}_{n}-{x}^{\ast }\parallel =\parallel {P}_{C}{y}_{n}-{P}_{C}{x}^{\ast }\parallel \le \parallel {y}_{n}-{x}^{\ast }\parallel ,$
(3.3)
${\parallel S{z}_{n}-A{x}^{\ast }\parallel }^{2}\le {\parallel {z}_{n}-A{x}^{\ast }\parallel }^{2}\le {\parallel A{x}_{n}-A{x}^{\ast }\parallel }^{2}-{\parallel {z}_{n}-A{x}_{n}\parallel }^{2},$
(3.4)
$\parallel {u}_{n+1}-{u}_{n}\parallel =\parallel {P}_{C}{y}_{n+1}-{P}_{C}{y}_{n}\parallel \le \parallel {y}_{n+1}-{y}_{n}\parallel$
(3.5)

and

$\parallel {z}_{n+1}-{z}_{n}\parallel =\parallel {P}_{Q}A{x}_{n+1}-{P}_{Q}A{x}_{n}\parallel \le \parallel A{x}_{n+1}-A{x}_{n}\parallel .$
(3.6)

From (3.1) and (3.3), we have

$\parallel {x}_{n+1}-{x}^{\ast }\parallel =\parallel T{u}_{n}-{x}^{\ast }\parallel \le \parallel {u}_{n}-{x}^{\ast }\parallel \le \parallel {y}_{n}-{x}^{\ast }\parallel .$
(3.7)

Using (2.3), we get

$\begin{array}{rcl}{\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}& =& {\parallel \left(1-{\alpha }_{n}\right)\left({x}_{n}-{x}^{\ast }+\delta {A}^{\ast }\left(S{z}_{n}-A{x}_{n}\right)\right)-{\alpha }_{n}{x}^{\ast }\parallel }^{2}\\ \le & \left(1-{\alpha }_{n}\right){\parallel \left({x}_{n}-{x}^{\ast }+\delta {A}^{\ast }\left(S{z}_{n}-A{x}_{n}\right)\parallel }^{2}+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}\\ =& \left(1-{\alpha }_{n}\right)\left[\parallel {x}_{n}-{x}^{\ast }\parallel +{\delta }^{2}{\parallel {A}^{\ast }\left(S{z}_{n}-A{x}_{n}\right)\parallel }^{2}\\ +2\delta 〈{x}_{n}-{x}^{\ast },{A}^{\ast }\left(S{z}_{n}-A{x}_{n}\right)〉\right]+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}.\end{array}$
(3.8)

Since A is a linear operator with its adjoint ${A}^{\ast }$, we have

$\begin{array}{c}〈{x}_{n}-{x}^{\ast },{A}^{\ast }\left(S{z}_{n}-A{x}_{n}\right)〉\hfill \\ \phantom{\rule{1em}{0ex}}=〈A\left({x}_{n}-{x}^{\ast }\right),S{z}_{n}-A{x}_{n}〉\hfill \\ \phantom{\rule{1em}{0ex}}=〈A{x}_{n}-A{x}^{\ast }+S{z}_{n}-A{x}_{n}-\left(S{z}_{n}-A{x}_{n}\right),S{z}_{n}-A{x}_{n}〉\hfill \\ \phantom{\rule{1em}{0ex}}=〈S{z}_{n}-A{x}^{\ast },S{z}_{n}-A{x}_{n}〉-{\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}.\hfill \end{array}$
(3.9)

Again using (2.3), we obtain

$〈S{z}_{n}-A{x}^{\ast },S{z}_{n}-A{x}_{n}〉=\frac{1}{2}\left({\parallel S{z}_{n}-A{x}^{\ast }\parallel }^{2}+{\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}-{\parallel A{x}_{n}-A{x}^{\ast }\parallel }^{2}\right).$
(3.10)

By (3.4), (3.9) and (3.10), we get

$\begin{array}{rcl}〈{x}_{n}-{x}^{\ast },{A}^{\ast }\left(S{z}_{n}-A{x}_{n}\right)〉& =& \frac{1}{2}\left({\parallel S{z}_{n}-A{x}^{\ast }\parallel }^{2}+{\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}-{\parallel A{x}_{n}-A{x}^{\ast }\parallel }^{2}\right)\\ -{\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}\\ \le & \frac{1}{2}\left({\parallel A{x}_{n}-A{x}^{\ast }\parallel }^{2}-{\parallel {z}_{n}-A{x}_{n}\parallel }^{2}+{\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}\\ -{\parallel A{x}_{n}-A{x}^{\ast }\parallel }^{2}\right)-{\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}\\ =& -\frac{1}{2}{\parallel {z}_{n}-A{x}_{n}\parallel }^{2}-\frac{1}{2}{\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}.\end{array}$
(3.11)

Substituting (3.11) into (3.8), we deduce

$\begin{array}{rcl}{\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}& \le & \left(1-{\alpha }_{n}\right)\left[{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\delta }^{2}{\parallel A\parallel }^{2}{\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}\\ +2\delta \left(-\frac{1}{2}{\parallel {z}_{n}-A{x}_{n}\parallel }^{2}-\frac{1}{2}{\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}\right)\right]+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}\\ =& \left(1-{\alpha }_{n}\right)\left[{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+\left({\delta }^{2}{\parallel A\parallel }^{2}-\delta \right){\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}\\ -\delta {\parallel {z}_{n}-A{x}_{n}\parallel }^{2}\right]+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}\\ \le & \left(1-{\alpha }_{n}\right){\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}.\end{array}$
(3.12)

It follows from (3.7) that

$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& \le & {\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}\\ \le & \left(1-{\alpha }_{n}\right){\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}\\ \le & max\left\{{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2},{\parallel {x}^{\ast }\parallel }^{2}\right\}.\end{array}$

The boundedness of the sequence $\left\{{x}_{n}\right\}$ yields.

Next, we estimate $\parallel {x}_{n+1}-{x}_{n}\parallel$. Set ${v}_{n}={x}_{n}-\delta {A}^{\ast }\left(I-S{P}_{Q}\right)A{x}_{n}$. According to (2.3) and (3.5), we have

$\begin{array}{rcl}{\parallel {v}_{n+1}-{v}_{n}\parallel }^{2}& =& {\parallel {x}_{n+1}-{x}_{n}+\delta \left[{A}^{\ast }\left(S{P}_{Q}-I\right)A{x}_{n+1}-{A}^{\ast }\left(S{P}_{Q}-I\right)A{x}_{n}\right]\parallel }^{2}\\ =& {\parallel {x}_{n+1}-{x}_{n}\parallel }^{2}+{\delta }^{2}{\parallel {A}^{\ast }\left[\left(S{P}_{Q}-I\right)A{x}_{n+1}-\left(S{P}_{Q}-I\right)A{x}_{n}\right]\parallel }^{2}\\ +2\delta 〈{x}_{n+1}-{x}_{n},{A}^{\ast }\left[\left(S{P}_{Q}-I\right)A{x}_{n+1}-\left(S{P}_{Q}-I\right)A{x}_{n}\right]〉\\ \le & {\parallel {x}_{n+1}-{x}_{n}\parallel }^{2}+{\delta }^{2}{\parallel A\parallel }^{2}{\parallel S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)\parallel }^{2}\\ +2\delta 〈A{x}_{n+1}-A{x}_{n},S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)〉\\ =& {\parallel {x}_{n+1}-{x}_{n}\parallel }^{2}+{\delta }^{2}{\parallel A\parallel }^{2}{\parallel S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)\parallel }^{2}\\ +2\delta 〈S{z}_{n+1}-S{z}_{n},S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)〉\\ -2\delta {\parallel S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)\parallel }^{2}\\ =& {\parallel {x}_{n+1}-{x}_{n}\parallel }^{2}+{\delta }^{2}{\parallel A\parallel }^{2}{\parallel S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)\parallel }^{2}\\ +\delta \left({\parallel S{z}_{n+1}-S{z}_{n}\parallel }^{2}+{\parallel S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)\parallel }^{2}\\ -{\parallel A{x}_{n+1}-A{x}_{n}\parallel }^{2}\right)-2\delta {\parallel S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)\parallel }^{2}\\ =& {\parallel {x}_{n+1}-{x}_{n}\parallel }^{2}+\left({\delta }^{2}{\parallel A\parallel }^{2}-\delta \right){\parallel S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)\parallel }^{2}\\ +\delta \left({\parallel S{z}_{n+1}-S{z}_{n}\parallel }^{2}-{\parallel A{x}_{n+1}-A{x}_{n}\parallel }^{2}\right)\\ \le & {\parallel {x}_{n+1}-{x}_{n}\parallel }^{2}+\left({\delta }^{2}{\parallel A\parallel }^{2}-\delta \right){\parallel S{z}_{n+1}-S{z}_{n}-\left(A{x}_{n+1}-A{x}_{n}\right)\parallel }^{2}\\ +\delta \left({\parallel {z}_{n+1}-{z}_{n}\parallel }^{2}-{\parallel A{x}_{n+1}-A{x}_{n}\parallel }^{2}\right).\end{array}$
(3.13)

Since $\delta \in \left(0,\frac{1}{{\parallel A\parallel }^{2}}\right)$, we derive by virtue of (3.6) and (3.13) that

$\parallel {v}_{n+1}-{v}_{n}\parallel \le \parallel {x}_{n+1}-{x}_{n}\parallel .$
(3.14)

From (3.5) and (3.14), we have

$\begin{array}{rcl}\parallel {x}_{n+1}-{x}_{n}\parallel & \le & \parallel {y}_{n+1}-{y}_{n}\parallel \\ =& \parallel \left(1-{\alpha }_{n+1}\right){v}_{n+1}-\left(1-{\alpha }_{n}\right){v}_{n}\parallel \\ =& \parallel \left(1-{\alpha }_{n+1}\right)\left({v}_{n+1}-{v}_{n}\right)+\left({\alpha }_{n}-{\alpha }_{n+1}\right){v}_{n}\parallel \\ \le & \left(1-{\alpha }_{n+1}\right)\parallel {v}_{n+1}-{v}_{n}\parallel +|{\alpha }_{n+1}-{\alpha }_{n}|\parallel {v}_{n}\parallel \\ \le & \left(1-{\alpha }_{n+1}\right)\parallel {x}_{n+1}-{x}_{n}\parallel +|{\alpha }_{n+1}-{\alpha }_{n}|\parallel {v}_{n}\parallel .\end{array}$

It follows that

$\parallel {x}_{n+1}-{x}_{n}\parallel \le \frac{|{\alpha }_{n+1}-{\alpha }_{n}|}{{\alpha }_{n+1}}\parallel {v}_{n}\parallel .$

This, together with condition (C3), implies that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n+1}-{x}_{n}\parallel =0.$
(3.15)

That is,

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-T{u}_{n}\parallel =0.$
(3.16)

Using the firmly-nonexpansiveness of ${P}_{C}$, we have

$\begin{array}{rcl}{\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}& =& {\parallel {P}_{C}{y}_{n}-{x}^{\ast }\parallel }^{2}\\ \le & {\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {P}_{C}{y}_{n}-{y}_{n}\parallel }^{2}\\ =& {\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {u}_{n}-{y}_{n}\parallel }^{2}.\end{array}$
(3.17)

Thus,

$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& \le & {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}\\ \le & {\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {u}_{n}-{y}_{n}\parallel }^{2}\\ \le & \left(1-{\alpha }_{n}\right){\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}-{\parallel {u}_{n}-{y}_{n}\parallel }^{2}.\end{array}$
(3.18)

It follows that

$\begin{array}{rcl}{\parallel {u}_{n}-{y}_{n}\parallel }^{2}& \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}\\ \le & \left(\parallel {x}_{n}-{x}^{\ast }\parallel +\parallel {x}_{n+1}-{x}^{\ast }\parallel \right)\parallel {x}_{n+1}-{x}_{n}\parallel +{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}.\end{array}$

This, together with (3.15) and (C1), implies that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {u}_{n}-{y}_{n}\parallel =0.$
(3.19)

Returning to (3.18) and using (3.12), we have

$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& \le & {\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}\\ \le & \left(1-{\alpha }_{n}\right){\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+\left(1-{\alpha }_{n}\right)\left({\delta }^{2}{\parallel A\parallel }^{2}-\delta \right){\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}\\ -\left(1-{\alpha }_{n}\right)\delta {\parallel {z}_{n}-A{x}_{n}\parallel }^{2}+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}.\end{array}$

Hence,

$\begin{array}{c}\left(1-{\alpha }_{n}\right)\left(\delta -{\delta }^{2}{\parallel A\parallel }^{2}\right){\parallel S{z}_{n}-A{x}_{n}\parallel }^{2}+\left(1-{\alpha }_{n}\right)\delta {\parallel {z}_{n}-A{x}_{n}\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}+{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \left(\parallel {x}_{n}-{x}^{\ast }\parallel +\parallel {x}_{n+1}-{x}^{\ast }\parallel \right)\parallel {x}_{n+1}-{x}_{n}\parallel +{\alpha }_{n}{\parallel {x}^{\ast }\parallel }^{2},\hfill \end{array}$

which implies that

$\underset{n\to \mathrm{\infty }}{lim}\parallel S{z}_{n}-A{x}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {z}_{n}-A{x}_{n}\parallel =0.$
(3.20)

So,

$\underset{n\to \mathrm{\infty }}{lim}\parallel S{z}_{n}-{z}_{n}\parallel =0.$
(3.21)

Note that

$\begin{array}{rcl}\parallel {y}_{n}-{x}_{n}\parallel & =& \parallel \delta {A}^{\ast }\left(S{P}_{Q}-I\right)A{x}_{n}+{\alpha }_{n}{v}_{n}\parallel \\ \le & \delta \parallel A\parallel \parallel S{z}_{n}-A{x}_{n}\parallel +{\alpha }_{n}\parallel {v}_{n}\parallel .\end{array}$

It follows from (3.20) that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{y}_{n}\parallel =0.$
(3.22)

From (3.16), (3.19) and (3.22), we get

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-T{x}_{n}\parallel =0.$
(3.23)

Now, we show that

$\underset{n\to \mathrm{\infty }}{lim sup}〈{x}^{\ast },{y}_{n}-{x}^{\ast }〉\ge 0.$

Choose a subsequence $\left\{{y}_{{n}_{i}}\right\}$ of $\left\{{y}_{n}\right\}$ such that

$\underset{n\to \mathrm{\infty }}{lim sup}〈{x}^{\ast },{y}_{n}-{x}^{\ast }〉=\underset{i\to \mathrm{\infty }}{lim}〈{x}^{\ast },{y}_{{n}_{i}}-{x}^{\ast }〉.$
(3.24)

Since the sequence $\left\{{y}_{{n}_{i}}\right\}$ is bounded, we can choose a subsequence $\left\{{y}_{{n}_{{i}_{j}}}\right\}$ of $\left\{{y}_{{n}_{i}}\right\}$ such that ${y}_{{n}_{{i}_{j}}}⇀z$. For the sake of convenience, we assume (without loss of generality) that ${y}_{{n}_{i}}⇀z$. Consequently, we derive from the above conclusions that

${x}_{{n}_{i}}⇀z,\phantom{\rule{2em}{0ex}}{u}_{{n}_{i}}⇀z,\phantom{\rule{2em}{0ex}}A{x}_{{n}_{i}}⇀Az\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{z}_{{n}_{i}}⇀Az.$
(3.25)

By the demiclosed principle of the nonexpansive mappings S and T (see Lemma 2.3), we deduce that $z\in Fix\left(T\right)$ and $Az\in Fix\left(S\right)$ (according to (3.23) and (3.21), respectively). Note that ${u}_{{n}_{i}}={P}_{C}{y}_{{n}_{i}}\in C$ and ${z}_{{n}_{i}}={P}_{Q}A{x}_{{n}_{i}}\in Q$. From (3.25), we deduce $z\in C$ and $Az\in Q$. To this end, we deduce that $z\in C\cap Fix\left(T\right)$ and $Az\in Q\cap Fix\left(S\right)$. That is to say, $z\in \mathrm{\Gamma }$. Therefore,

$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim sup}〈{x}^{\ast },{y}_{n}-{x}^{\ast }〉& =& \underset{i\to \mathrm{\infty }}{lim}〈{x}^{\ast },{y}_{{n}_{i}}-{x}^{\ast }〉\\ =& \underset{i\to \mathrm{\infty }}{lim}〈{x}^{\ast },z-{x}^{\ast }〉\\ \ge & 0.\end{array}$
(3.26)

Finally, we prove that ${x}_{n}\to {x}^{\ast }$. From (3.1), we have

$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& \le & {\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}\\ =& {\parallel \left(1-{\alpha }_{n}\right)\left({v}_{n}-{x}^{\ast }\right)-{\alpha }_{n}{x}^{\ast }\parallel }^{2}\\ \le & \left(1-{\alpha }_{n}\right){\parallel {v}_{n}-{x}^{\ast }\parallel }^{2}-2{\alpha }_{n}〈{x}^{\ast },{y}_{n}-{x}^{\ast }〉\\ \le & \left(1-{\alpha }_{n}\right){\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-2{\alpha }_{n}〈{x}^{\ast },{y}_{n}-{x}^{\ast }〉.\end{array}$
(3.27)

Applying Lemma 2.4 and (3.26) to (3.27), we deduce that ${x}_{n}\to {x}^{\ast }$. The proof is completed. □

## References

1. Byrne C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 2002, 18: 441–453. 10.1088/0266-5611/18/2/310

2. Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692

3. Ceng LC, Ansari QH, Yao JC: An extragradient method for split feasibility and fixed point problems. Comput. Math. Appl. 2012, 64: 633–642. 10.1016/j.camwa.2011.12.074

4. Wang F, Xu HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2010., 2010: Article ID 102085

5. Dang Y, Gao Y: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 2011., 27: Article ID 015007

6. Xu HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010., 26: Article ID 105018

7. Yao Y, Wu J, Liou YC: Regularized methods for the split feasibility problem. Abstr. Appl. Anal. 2012., 2012: Article ID 140679

8. Yao Y, Kim TH, Chebbi S, Xu HK: A modified extragradient method for the split feasibility and fixed point problems. J. Nonlinear Convex Anal. 2012, 13: 383–396.

9. Yao Y, Postolache M, Liou YC: Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory Appl. 2013., 2013: Article ID 201

10. Censor Y, Segal A: The split common fixed point problem for directed operators. J. Convex Anal. 2009, 16: 587–600.

11. Cui H, Su M, Wang F: Damped projection method for split common fixed point problems. J. Inequal. Appl. 2013., 2013: Article ID 123 10.1186/1029-242X-2013-123

12. Moudafi A: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 2011, 74: 4083–4087. 10.1016/j.na.2011.03.041

13. He ZH: The split equilibrium problems and its convergence algorithms. J. Inequal. Appl. 2012., 2012: Article ID 162

14. He ZH, Du WS: On hybrid split problem and its nonlinear algorithms. Fixed Point Theory Appl. 2013., 2013: Article ID 47

15. Geobel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.

16. Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332

## Acknowledgements

Cun-lin Li was supported in part by NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3. Yonghong Yao was supported in part by NSFC 11071279, NSFC 71161001-G0105 and LQ13A010007.

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Correspondence to Yonghong Yao.

### Competing interests

The authors declare that they have no competing interests.

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Li, Cl., Liou, YC. & Yao, Y. A damped algorithm for the split feasibility and fixed point problems. J Inequal Appl 2013, 379 (2013). https://doi.org/10.1186/1029-242X-2013-379

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• DOI: https://doi.org/10.1186/1029-242X-2013-379

### Keywords

• split feasibility problem
• fixed point problem
• nonexpansive mapping
• damped algorithm 