# 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 $A:{H}_{1}â†’{H}_{2}$ be a bounded linear operator. Finding a point ${x}^{âˆ—}$ satisfies

${x}^{âˆ—}âˆˆC\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}A{x}^{âˆ—}âˆˆQ.$
(1.1)

This problem, referred to as the split problem, has been studied by some authors. See, e.g., [1â€“8] and [9]. Some algorithms for solving (1.1) have been presented. One is Byrneâ€™s CQ algorithm [1]

${x}_{n+1}={P}_{C}\left({x}_{n}âˆ’\mathrm{Ï„}{A}^{âˆ—}\left(Iâˆ’{P}_{Q}\right)A{x}_{n}\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N},$

where $\mathrm{Ï„}âˆˆ\left(0,\frac{2}{L}\right)$ with L being the largest eigenvalue of the matrix ${A}^{âˆ—}A$, I is the unit matrix or operator, and ${P}_{C}$ and ${P}_{Q}$ denote the orthogonal projections onto C and Q, respectively. Motivated by Byrneâ€™s CQ algorithm, Xu [6] suggested a single step regularized method

${x}_{n+1}={P}_{C}\left(\left(1âˆ’{\mathrm{Î±}}_{n}{\mathrm{Î³}}_{n}\right){x}_{n}âˆ’{\mathrm{Î³}}_{n}{A}^{âˆ—}\left(Iâˆ’{P}_{Q}\right)A{x}_{n}\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N}.$
(1.2)

Very recently, Dang and Gao [5] introduced the following damped projection algorithm

${x}_{n+1}=\left(1âˆ’{\mathrm{Î²}}_{n}\right){x}_{n}+{\mathrm{Î²}}_{n}{P}_{C}\left(\left(1âˆ’{\mathrm{Î±}}_{n}\right)\left({x}_{n}âˆ’\mathrm{Ï„}{A}^{âˆ—}\left(Iâˆ’{P}_{Q}\right)A{x}_{n}\right)\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N}.$

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 ${x}^{âˆ—}$ with the property

${x}^{âˆ—}âˆˆFix\left(U\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}A{x}^{âˆ—}âˆˆFix\left(T\right).$

This problem was first introduced by Censor and Segal [10], who invented an algorithm, which generates a sequence $\left\{{x}_{n}\right\}$ according to the iterative procedure

${x}_{n+1}=U\left({x}_{n}âˆ’\mathrm{Î³}{A}^{âˆ—}\left(Iâˆ’T\right)A{x}_{n}\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N}.$

Recently, Cui, Su and Wang [11] extended the damped projection algorithm to the split common fixed point problems. For some related work, please refer to [12] 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 $S:Qâ†’Q$ and $T:Câ†’C$ 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 $ã€ˆâ‹\dots ,â‹\dots ã€‰$ and the norm $âˆ¥â‹\dots âˆ¥$, respectively. Let C be a nonempty closed convex subset of H.

Definition 2.1 A mapping $T:Câ†’C$ is called nonexpansive if

$âˆ¥Txâˆ’Tyâˆ¥â‰¤âˆ¥xâˆ’yâˆ¥$

for all $x,yâˆˆC$.

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

Definition 2.2 We call ${P}_{C}:Hâ†’C$ the metric projection if for each $xâˆˆH$

$âˆ¥xâˆ’{P}_{C}\left(x\right)âˆ¥=inf\left\{âˆ¥xâˆ’yâˆ¥:yâˆˆC\right\}.$

It is well known that the metric projection ${P}_{C}:Hâ†’C$ is characterized by

$ã€ˆxâˆ’{P}_{C}\left(x\right),yâˆ’{P}_{C}\left(x\right)ã€‰â‰¤0$

for all $xâˆˆH$, $yâˆˆC$. From this, we can deduce that ${P}_{C}$ is firmly-nonexpansive, that is,

${âˆ¥{P}_{C}\left(x\right)âˆ’{P}_{C}\left(y\right)âˆ¥}^{2}â‰¤ã€ˆxâˆ’y,{P}_{C}\left(x\right)âˆ’{P}_{C}\left(y\right)ã€‰$
(2.1)

for all $x,yâˆˆH$. Hence ${P}_{C}$ is also nonexpansive.

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

${âˆ¥tx+\left(1âˆ’t\right)yâˆ¥}^{2}=t{âˆ¥xâˆ¥}^{2}+\left(1âˆ’t\right){âˆ¥yâˆ¥}^{2}âˆ’t\left(1âˆ’t\right){âˆ¥xâˆ’yâˆ¥}^{2}$
(2.2)

for all $x,yâˆˆH$ and $tâˆˆ\left[0,1\right]$, and

${âˆ¥x+yâˆ¥}^{2}={âˆ¥xâˆ¥}^{2}+2ã€ˆx,yã€‰+{âˆ¥yâˆ¥}^{2}$
(2.3)

for all $x,yâˆˆH$. It follows that

${âˆ¥x+yâˆ¥}^{2}â‰¤{âˆ¥xâˆ¥}^{2}+2ã€ˆy,x+yã€‰$
(2.4)

for all $x,yâˆˆH$.

Lemma 2.3 [15]

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

Lemma 2.4 [16]

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

${a}_{n+1}â‰¤\left(1âˆ’{\mathrm{Î³}}_{n}\right){a}_{n}+{\mathrm{Î´}}_{n},\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N},$

where $\left\{{\mathrm{Î³}}_{n}\right\}$ is a sequence in $\left(0,1\right)$, and $\left\{{\mathrm{Î´}}_{n}\right\}$ is a sequence such that

1. (1)

${âˆ‘}_{n=1}^{\mathrm{âˆž}}{\mathrm{Î³}}_{n}=\mathrm{âˆž}$;

2. (2)

${limâ€‰sup}_{nâ†’\mathrm{âˆž}}\frac{{\mathrm{Î´}}_{n}}{{\mathrm{Î³}}_{n}}â‰¤0$ or ${âˆ‘}_{n=1}^{\mathrm{âˆž}}|{\mathrm{Î´}}_{n}|<\mathrm{âˆž}$.

Then ${lim}_{nâ†’\mathrm{âˆž}}{a}_{n}=0$.

## 3 Main results

Let C and Q be two nonempty closed convex subsets of real Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively. Let $A:{H}_{1}â†’{H}_{2}$ be a bounded linear operator with its adjoint ${A}^{âˆ—}$. Let $S:Qâ†’Q$ and $T:Câ†’C$ be two nonexpansive mappings. We use Î“ to denote the set of solutions of (1.3), that is, $\mathrm{Î“}=\left\{{x}^{âˆ—}|{x}^{âˆ—}âˆˆCâˆ©Fix\left(T\right),A{x}^{âˆ—}âˆˆQâˆ©Fix\left(S\right)\right\}$. Now, we present our algorithm.

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

(3.1)

where ${\left\{{\mathrm{Î±}}_{n}\right\}}_{nâˆˆ\mathbb{N}}$ and ${\left\{{\mathrm{Î²}}_{n}\right\}}_{nâˆˆ\mathbb{N}}$ are two real number sequences in $\left(0,1\right)$ and $\mathrm{Î´}âˆˆ\left(0,\frac{1}{{âˆ¥Aâˆ¥}^{2}}\right)$.

Theorem 3.2 Suppose . Assume the sequence ${\left\{{\mathrm{Î±}}_{n}\right\}}_{nâˆˆ\mathbb{N}}$ satisfies three conditions

(C1) ${lim}_{nâ†’\mathrm{âˆž}}{\mathrm{Î±}}_{n}=0$;

(C2) ${âˆ‘}_{n=1}^{\mathrm{âˆž}}{\mathrm{Î±}}_{n}=\mathrm{âˆž}$;

(C3) ${lim}_{nâ†’\mathrm{âˆž}}\frac{{\mathrm{Î±}}_{n+1}}{{\mathrm{Î±}}_{n}}=1$.

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

Proof For the convenience, we write ${z}_{n}={P}_{Q}A{x}_{n}$, ${y}_{n}=\left(1âˆ’{\mathrm{Î±}}_{n}\right)\left({x}_{n}âˆ’\mathrm{Î´}{A}^{âˆ—}\left(Iâˆ’S{P}_{Q}\right)A{x}_{n}\right)$ and ${u}_{n}={P}_{C}\left(\left(1âˆ’{\mathrm{Î±}}_{n}\right)\left({x}_{n}âˆ’\mathrm{Î´}{A}^{âˆ—}\left(Iâˆ’S{P}_{Q}\right)A{x}_{n}\right)\right)$ for all $nâˆˆ\mathbb{N}$. Thus ${u}_{n}={P}_{C}{y}_{n}$ for all $nâˆˆ\mathbb{N}$.

Let ${x}^{âˆ—}={P}_{\mathrm{Î“}}\left(0\right)$. Hence, ${x}^{âˆ—}âˆˆCâˆ©Fix\left(T\right)$ and $A{x}^{âˆ—}âˆˆQâˆ©Fix\left(S\right)$. By the firmly-nonexpansivity of ${P}_{C}$ and ${P}_{Q}$, we can deduce the following conclusions

$âˆ¥{z}_{n}âˆ’A{x}^{âˆ—}âˆ¥=âˆ¥{P}_{Q}A{x}_{n}âˆ’{P}_{Q}A{x}^{âˆ—}âˆ¥â‰¤âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥,$
(3.2)
$âˆ¥{u}_{n}âˆ’{x}^{âˆ—}âˆ¥=âˆ¥{P}_{C}{y}_{n}âˆ’{P}_{C}{x}^{âˆ—}âˆ¥â‰¤âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥,$
(3.3)
${âˆ¥S{z}_{n}âˆ’A{x}^{âˆ—}âˆ¥}^{2}â‰¤{âˆ¥{z}_{n}âˆ’A{x}^{âˆ—}âˆ¥}^{2}â‰¤{âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2},$
(3.4)
$âˆ¥{u}_{n+1}âˆ’{u}_{n}âˆ¥=âˆ¥{P}_{C}{y}_{n+1}âˆ’{P}_{C}{y}_{n}âˆ¥â‰¤âˆ¥{y}_{n+1}âˆ’{y}_{n}âˆ¥$
(3.5)

and

$âˆ¥{z}_{n+1}âˆ’{z}_{n}âˆ¥=âˆ¥{P}_{Q}A{x}_{n+1}âˆ’{P}_{Q}A{x}_{n}âˆ¥â‰¤âˆ¥A{x}_{n+1}âˆ’A{x}_{n}âˆ¥.$
(3.6)

From (3.1) and (3.3), we have

$âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥=âˆ¥T{u}_{n}âˆ’{x}^{âˆ—}âˆ¥â‰¤âˆ¥{u}_{n}âˆ’{x}^{âˆ—}âˆ¥â‰¤âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥.$
(3.7)

Using (2.3), we get

$\begin{array}{rcl}{âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}& =& {âˆ¥\left(1âˆ’{\mathrm{Î±}}_{n}\right)\left({x}_{n}âˆ’{x}^{âˆ—}+\mathrm{Î´}{A}^{âˆ—}\left(S{z}_{n}âˆ’A{x}_{n}\right)\right)âˆ’{\mathrm{Î±}}_{n}{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥\left({x}_{n}âˆ’{x}^{âˆ—}+\mathrm{Î´}{A}^{âˆ—}\left(S{z}_{n}âˆ’A{x}_{n}\right)âˆ¥}^{2}+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}\\ =& \left(1âˆ’{\mathrm{Î±}}_{n}\right)\left[âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥+{\mathrm{Î´}}^{2}{âˆ¥{A}^{âˆ—}\left(S{z}_{n}âˆ’A{x}_{n}\right)âˆ¥}^{2}\\ +2\mathrm{Î´}ã€ˆ{x}_{n}âˆ’{x}^{âˆ—},{A}^{âˆ—}\left(S{z}_{n}âˆ’A{x}_{n}\right)ã€‰\right]+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}.\end{array}$
(3.8)

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

$\begin{array}{c}ã€ˆ{x}_{n}âˆ’{x}^{âˆ—},{A}^{âˆ—}\left(S{z}_{n}âˆ’A{x}_{n}\right)ã€‰\hfill \\ \phantom{\rule{1em}{0ex}}=ã€ˆA\left({x}_{n}âˆ’{x}^{âˆ—}\right),S{z}_{n}âˆ’A{x}_{n}ã€‰\hfill \\ \phantom{\rule{1em}{0ex}}=ã€ˆA{x}_{n}âˆ’A{x}^{âˆ—}+S{z}_{n}âˆ’A{x}_{n}âˆ’\left(S{z}_{n}âˆ’A{x}_{n}\right),S{z}_{n}âˆ’A{x}_{n}ã€‰\hfill \\ \phantom{\rule{1em}{0ex}}=ã€ˆS{z}_{n}âˆ’A{x}^{âˆ—},S{z}_{n}âˆ’A{x}_{n}ã€‰âˆ’{âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}.\hfill \end{array}$
(3.9)

Again using (2.3), we obtain

$ã€ˆS{z}_{n}âˆ’A{x}^{âˆ—},S{z}_{n}âˆ’A{x}_{n}ã€‰=\frac{1}{2}\left({âˆ¥S{z}_{n}âˆ’A{x}^{âˆ—}âˆ¥}^{2}+{âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}âˆ’{âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥}^{2}\right).$
(3.10)

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

$\begin{array}{rcl}ã€ˆ{x}_{n}âˆ’{x}^{âˆ—},{A}^{âˆ—}\left(S{z}_{n}âˆ’A{x}_{n}\right)ã€‰& =& \frac{1}{2}\left({âˆ¥S{z}_{n}âˆ’A{x}^{âˆ—}âˆ¥}^{2}+{âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}âˆ’{âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥}^{2}\right)\\ âˆ’{âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}\\ â‰¤& \frac{1}{2}\left({âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}+{âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}\\ âˆ’{âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥}^{2}\right)âˆ’{âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}\\ =& âˆ’\frac{1}{2}{âˆ¥{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}âˆ’\frac{1}{2}{âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}.\end{array}$
(3.11)

Substituting (3.11) into (3.8), we deduce

$\begin{array}{rcl}{âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}& â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n}\right)\left[{âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}+{\mathrm{Î´}}^{2}{âˆ¥Aâˆ¥}^{2}{âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}\\ +2\mathrm{Î´}\left(âˆ’\frac{1}{2}{âˆ¥{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}âˆ’\frac{1}{2}{âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}\right)\right]+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}\\ =& \left(1âˆ’{\mathrm{Î±}}_{n}\right)\left[{âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}+\left({\mathrm{Î´}}^{2}{âˆ¥Aâˆ¥}^{2}âˆ’\mathrm{Î´}\right){âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}\\ âˆ’\mathrm{Î´}{âˆ¥{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}\right]+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}.\end{array}$
(3.12)

It follows from (3.7) that

$\begin{array}{rcl}{âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥}^{2}& â‰¤& {âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& max\left\{{âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2},{âˆ¥{x}^{âˆ—}âˆ¥}^{2}\right\}.\end{array}$

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

Next, we estimate $âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥$. Set ${v}_{n}={x}_{n}âˆ’\mathrm{Î´}{A}^{âˆ—}\left(Iâˆ’S{P}_{Q}\right)A{x}_{n}$. According to (2.3) and (3.5), we have

$\begin{array}{rcl}{âˆ¥{v}_{n+1}âˆ’{v}_{n}âˆ¥}^{2}& =& {âˆ¥{x}_{n+1}âˆ’{x}_{n}+\mathrm{Î´}\left[{A}^{âˆ—}\left(S{P}_{Q}âˆ’I\right)A{x}_{n+1}âˆ’{A}^{âˆ—}\left(S{P}_{Q}âˆ’I\right)A{x}_{n}\right]âˆ¥}^{2}\\ =& {âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥}^{2}+{\mathrm{Î´}}^{2}{âˆ¥{A}^{âˆ—}\left[\left(S{P}_{Q}âˆ’I\right)A{x}_{n+1}âˆ’\left(S{P}_{Q}âˆ’I\right)A{x}_{n}\right]âˆ¥}^{2}\\ +2\mathrm{Î´}ã€ˆ{x}_{n+1}âˆ’{x}_{n},{A}^{âˆ—}\left[\left(S{P}_{Q}âˆ’I\right)A{x}_{n+1}âˆ’\left(S{P}_{Q}âˆ’I\right)A{x}_{n}\right]ã€‰\\ â‰¤& {âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥}^{2}+{\mathrm{Î´}}^{2}{âˆ¥Aâˆ¥}^{2}{âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)âˆ¥}^{2}\\ +2\mathrm{Î´}ã€ˆA{x}_{n+1}âˆ’A{x}_{n},S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)ã€‰\\ =& {âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥}^{2}+{\mathrm{Î´}}^{2}{âˆ¥Aâˆ¥}^{2}{âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)âˆ¥}^{2}\\ +2\mathrm{Î´}ã€ˆS{z}_{n+1}âˆ’S{z}_{n},S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)ã€‰\\ âˆ’2\mathrm{Î´}{âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)âˆ¥}^{2}\\ =& {âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥}^{2}+{\mathrm{Î´}}^{2}{âˆ¥Aâˆ¥}^{2}{âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)âˆ¥}^{2}\\ +\mathrm{Î´}\left({âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ¥}^{2}+{âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)âˆ¥}^{2}\\ âˆ’{âˆ¥A{x}_{n+1}âˆ’A{x}_{n}âˆ¥}^{2}\right)âˆ’2\mathrm{Î´}{âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)âˆ¥}^{2}\\ =& {âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥}^{2}+\left({\mathrm{Î´}}^{2}{âˆ¥Aâˆ¥}^{2}âˆ’\mathrm{Î´}\right){âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)âˆ¥}^{2}\\ +\mathrm{Î´}\left({âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ¥}^{2}âˆ’{âˆ¥A{x}_{n+1}âˆ’A{x}_{n}âˆ¥}^{2}\right)\\ â‰¤& {âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥}^{2}+\left({\mathrm{Î´}}^{2}{âˆ¥Aâˆ¥}^{2}âˆ’\mathrm{Î´}\right){âˆ¥S{z}_{n+1}âˆ’S{z}_{n}âˆ’\left(A{x}_{n+1}âˆ’A{x}_{n}\right)âˆ¥}^{2}\\ +\mathrm{Î´}\left({âˆ¥{z}_{n+1}âˆ’{z}_{n}âˆ¥}^{2}âˆ’{âˆ¥A{x}_{n+1}âˆ’A{x}_{n}âˆ¥}^{2}\right).\end{array}$
(3.13)

Since $\mathrm{Î´}âˆˆ\left(0,\frac{1}{{âˆ¥Aâˆ¥}^{2}}\right)$, we derive by virtue of (3.6) and (3.13) that

$âˆ¥{v}_{n+1}âˆ’{v}_{n}âˆ¥â‰¤âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥.$
(3.14)

From (3.5) and (3.14), we have

$\begin{array}{rcl}âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥& â‰¤& âˆ¥{y}_{n+1}âˆ’{y}_{n}âˆ¥\\ =& âˆ¥\left(1âˆ’{\mathrm{Î±}}_{n+1}\right){v}_{n+1}âˆ’\left(1âˆ’{\mathrm{Î±}}_{n}\right){v}_{n}âˆ¥\\ =& âˆ¥\left(1âˆ’{\mathrm{Î±}}_{n+1}\right)\left({v}_{n+1}âˆ’{v}_{n}\right)+\left({\mathrm{Î±}}_{n}âˆ’{\mathrm{Î±}}_{n+1}\right){v}_{n}âˆ¥\\ â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n+1}\right)âˆ¥{v}_{n+1}âˆ’{v}_{n}âˆ¥+|{\mathrm{Î±}}_{n+1}âˆ’{\mathrm{Î±}}_{n}|âˆ¥{v}_{n}âˆ¥\\ â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n+1}\right)âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥+|{\mathrm{Î±}}_{n+1}âˆ’{\mathrm{Î±}}_{n}|âˆ¥{v}_{n}âˆ¥.\end{array}$

It follows that

$âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥â‰¤\frac{|{\mathrm{Î±}}_{n+1}âˆ’{\mathrm{Î±}}_{n}|}{{\mathrm{Î±}}_{n+1}}âˆ¥{v}_{n}âˆ¥.$

This, together with condition (C3), implies that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥=0.$
(3.15)

That is,

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’T{u}_{n}âˆ¥=0.$
(3.16)

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

$\begin{array}{rcl}{âˆ¥{u}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}& =& {âˆ¥{P}_{C}{y}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& {âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{P}_{C}{y}_{n}âˆ’{y}_{n}âˆ¥}^{2}\\ =& {âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{u}_{n}âˆ’{y}_{n}âˆ¥}^{2}.\end{array}$
(3.17)

Thus,

$\begin{array}{rcl}{âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥}^{2}& â‰¤& {âˆ¥{u}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& {âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{u}_{n}âˆ’{y}_{n}âˆ¥}^{2}\\ â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{u}_{n}âˆ’{y}_{n}âˆ¥}^{2}.\end{array}$
(3.18)

It follows that

$\begin{array}{rcl}{âˆ¥{u}_{n}âˆ’{y}_{n}âˆ¥}^{2}& â‰¤& {âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥}^{2}+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& \left(âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥+âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥\right)âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}.\end{array}$

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

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{u}_{n}âˆ’{y}_{n}âˆ¥=0.$
(3.19)

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

$\begin{array}{rcl}{âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥}^{2}& â‰¤& {âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)\left({\mathrm{Î´}}^{2}{âˆ¥Aâˆ¥}^{2}âˆ’\mathrm{Î´}\right){âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}\\ âˆ’\left(1âˆ’{\mathrm{Î±}}_{n}\right)\mathrm{Î´}{âˆ¥{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}.\end{array}$

Hence,

$\begin{array}{c}\left(1âˆ’{\mathrm{Î±}}_{n}\right)\left(\mathrm{Î´}âˆ’{\mathrm{Î´}}^{2}{âˆ¥Aâˆ¥}^{2}\right){âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)\mathrm{Î´}{âˆ¥{z}_{n}âˆ’A{x}_{n}âˆ¥}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}â‰¤{âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥}^{2}+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}â‰¤\left(âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥+âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥\right)âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥+{\mathrm{Î±}}_{n}{âˆ¥{x}^{âˆ—}âˆ¥}^{2},\hfill \end{array}$

which implies that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥=\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{z}_{n}âˆ’A{x}_{n}âˆ¥=0.$
(3.20)

So,

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥S{z}_{n}âˆ’{z}_{n}âˆ¥=0.$
(3.21)

Note that

$\begin{array}{rcl}âˆ¥{y}_{n}âˆ’{x}_{n}âˆ¥& =& âˆ¥\mathrm{Î´}{A}^{âˆ—}\left(S{P}_{Q}âˆ’I\right)A{x}_{n}+{\mathrm{Î±}}_{n}{v}_{n}âˆ¥\\ â‰¤& \mathrm{Î´}âˆ¥Aâˆ¥âˆ¥S{z}_{n}âˆ’A{x}_{n}âˆ¥+{\mathrm{Î±}}_{n}âˆ¥{v}_{n}âˆ¥.\end{array}$

It follows from (3.20) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥=0.$
(3.22)

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

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’T{x}_{n}âˆ¥=0.$
(3.23)

Now, we show that

$\underset{nâ†’\mathrm{âˆž}}{limâ€‰sup}ã€ˆ{x}^{âˆ—},{y}_{n}âˆ’{x}^{âˆ—}ã€‰â‰¥0.$

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

$\underset{nâ†’\mathrm{âˆž}}{limâ€‰sup}ã€ˆ{x}^{âˆ—},{y}_{n}âˆ’{x}^{âˆ—}ã€‰=\underset{iâ†’\mathrm{âˆž}}{lim}ã€ˆ{x}^{âˆ—},{y}_{{n}_{i}}âˆ’{x}^{âˆ—}ã€‰.$
(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 ${y}_{{n}_{{i}_{j}}}â‡€z$. For the sake of convenience, we assume (without loss of generality) that ${y}_{{n}_{i}}â‡€z$. Consequently, we derive from the above conclusions that

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

By the demiclosed principle of the nonexpansive mappings S and T (see Lemma 2.3), we deduce that $zâˆˆFix\left(T\right)$ and $AzâˆˆFix\left(S\right)$ (according to (3.23) and (3.21), respectively). Note that ${u}_{{n}_{i}}={P}_{C}{y}_{{n}_{i}}âˆˆC$ and ${z}_{{n}_{i}}={P}_{Q}A{x}_{{n}_{i}}âˆˆQ$. From (3.25), we deduce $zâˆˆC$ and $AzâˆˆQ$. To this end, we deduce that $zâˆˆCâˆ©Fix\left(T\right)$ and $AzâˆˆQâˆ©Fix\left(S\right)$. That is to say, $zâˆˆ\mathrm{Î“}$. Therefore,

$\begin{array}{rcl}\underset{nâ†’\mathrm{âˆž}}{limâ€‰sup}ã€ˆ{x}^{âˆ—},{y}_{n}âˆ’{x}^{âˆ—}ã€‰& =& \underset{iâ†’\mathrm{âˆž}}{lim}ã€ˆ{x}^{âˆ—},{y}_{{n}_{i}}âˆ’{x}^{âˆ—}ã€‰\\ =& \underset{iâ†’\mathrm{âˆž}}{lim}ã€ˆ{x}^{âˆ—},zâˆ’{x}^{âˆ—}ã€‰\\ â‰¥& 0.\end{array}$
(3.26)

Finally, we prove that ${x}_{n}â†’{x}^{âˆ—}$. From (3.1), we have

$\begin{array}{rcl}{âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥}^{2}& â‰¤& {âˆ¥{y}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}\\ =& {âˆ¥\left(1âˆ’{\mathrm{Î±}}_{n}\right)\left({v}_{n}âˆ’{x}^{âˆ—}\right)âˆ’{\mathrm{Î±}}_{n}{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{v}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’2{\mathrm{Î±}}_{n}ã€ˆ{x}^{âˆ—},{y}_{n}âˆ’{x}^{âˆ—}ã€‰\\ â‰¤& \left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’2{\mathrm{Î±}}_{n}ã€ˆ{x}^{âˆ—},{y}_{n}âˆ’{x}^{âˆ—}ã€‰.\end{array}$
(3.27)

Applying Lemma 2.4 and (3.26) to (3.27), we deduce that ${x}_{n}â†’{x}^{âˆ—}$. The proof is completed.â€ƒâ–¡

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## 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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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