# Split feasibility problems for total quasi-asymptotically nonexpansive mappings

- Xiong Rui Wang
^{1}, - Shih-sen Chang
^{2}Email author, - Lin Wang
^{2}and - Yun-he Zhao
^{2}

**2012**:151

https://doi.org/10.1186/1687-1812-2012-151

© Wang et al.; licensee Springer 2012

**Received: **23 April 2012

**Accepted: **30 August 2012

**Published: **18 September 2012

## Abstract

The purpose of this paper is to propose an algorithm for solving the *split feasibility problems* for *total quasi-asymptotically nonexpansive mappings* in infinite-dimensional Hilbert spaces. The results presented in the paper not only improve and extend some recent results of Moudafi [Nonlinear Anal. 74:4083-4087, 2011; Inverse Problem 26:055007, 2010], but also improve and extend some recent results of Xu [Inverse Problems 26:105018, 2010; 22:2021-2034, 2006], Censor and Segal [J. Convex Anal. 16:587-600, 2009], Censor *et al.* [Inverse Problems 21:2071-2084, 2005], Masad and Reich [J. Nonlinear Convex Anal. 8:367-371, 2007], Censor *et al.* [J. Math. Anal. Appl. 327:1244-1256, 2007], Yang [Inverse Problem 20:1261-1266, 2004] and others.

**MSC:**47J05, 47H09, 49J25.

## Keywords

## 1 Introduction

Throughout this paper, we always assume that ${H}_{1}$, ${H}_{2}$ are real Hilbert spaces, ‘→’, ‘⇀’ denote strong and weak convergence, respectively, and $F(T)$ is a fixed point set of a mapping *T*.

The *split feasibility problem* (SFP) in finite-dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph and radiation therapy treatment planning [3–5]. The *split feasibility problem* in an infinite-dimensional real Hilbert space can be found in [2, 4, 6–10].

*split feasibility problem*for

*total quasi-asymptotically nonexpansive mappings*in the framework of infinite-dimensional real Hilbert spaces:

*i.e.*,

## 2 Preliminaries

We first recall some definitions, notations and conclusions which will be needed in proving our main results.

Let *E* be a Banach space. A mapping $T:E\to E$ is said to be *demi-closed at origin* if for any sequence $\{{x}_{n}\}\subset E$ with ${x}_{n}\rightharpoonup {x}^{\ast}$ and $\parallel (I-T){x}_{n}\parallel \to 0$, ${x}^{\ast}=T{x}^{\ast}$.

*E*is said to have

*the Opial property*, if for any sequence $\{{x}_{n}\}$ with ${x}_{n}\rightharpoonup {x}^{\ast}$,

**Remark 2.1** It is well known that each Hilbert space possesses the Opial property.

**Definition 2.2**Let

*H*be a real Hilbert space.

- (1)A mapping $G:H\to H$ is said to be a $(\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )$-
*total quasi-asymptotically nonexpansive mapping*if $F(G)\ne \mathrm{\varnothing}$; and there exist nonnegative real sequences $\{{\nu}_{n}\}$, $\{{\mu}_{n}\}$ with ${\nu}_{n}\to 0$ and ${\mu}_{n}\to 0$ and a strictly increasing continuous function $\zeta :{\mathcal{R}}^{+}\to {\mathcal{R}}^{+}$ with $\zeta (0)=0$ such that for each $n\ge 1$,${\parallel p-{G}^{n}x\parallel}^{2}\le {\parallel p-x\parallel}^{2}+{\nu}_{n}\zeta (\parallel p-x\parallel )+{\mu}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F(G),x\in H.$(2.1)

Now, we give an example of total quasi-asymptotically nonexpansive mapping.

*C*be a unit ball in a real Hilbert space ${l}^{2}$, and let $T:C\to C$ be a mapping defined by

where $\{{a}_{i}\}$ is a sequence in (0, 1) such that ${\prod}_{i=2}^{\mathrm{\infty}}{a}_{i}=\frac{1}{2}$.

- (i)
$\parallel Tx-Ty\parallel \le 2\parallel x-y\parallel $, $\mathrm{\forall}x,y\in C$;

- (ii)
$\parallel {T}^{n}x-{T}^{n}y\parallel \le 2{\prod}_{j=2}^{n}{a}_{j}\parallel x-y\parallel $, $\mathrm{\forall}x,y\in C$, $\mathrm{\forall}n\ge 2$.

*T*defined as above is a total quasi-asymptotically nonexpansive mapping.

- (2)A mapping $G:H\to H$ is said to be $(\{{k}_{n}\})$-
*quasi-asymptotically nonexpansive*if $F(G)\ne \mathrm{\varnothing}$; and there exists a sequence $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ with ${k}_{n}\to 1$ such that for all $n\ge 1$,${\parallel p-{G}^{n}x\parallel}^{2}\le {k}_{n}{\parallel p-x\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F(G),x\in H.$(2.4) - (3)A mapping $G:H\to H$ is said to be
*quasi-nonexpansive*if $F(G)\ne \mathrm{\varnothing}$ such that$\parallel p-Gx\parallel \le \parallel p-x\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F(G),x\in H.$(2.5)

**Remark 2.3** It is easy to see that every quasi-nonexpansive mapping is a $(\{1\})$-quasi-asymptotically nonexpansive mapping and every $\{{k}_{n}\}$-quasi-asymptotically nonexpansive mapping is a $(\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )$-total quasi-asymptotically nonexpansive mapping with $\{{\nu}_{n}={k}_{n}-1\}$, $\{{\mu}_{n}=0\}$ and $\zeta (t)={t}^{2}$, $t\ge 0$.

**Definition 2.4**(1) A mapping $G:H\to H$ is said to be uniformly

*L*-Lipschitzian if there exists a constant $L>0$ such that

- (2)
A mapping $G:H\to H$ is said to be

*semi-compact*if for any bounded sequence $\{{x}_{n}\}\subset H$ with ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-G{x}_{n}\parallel =0$, there exists a subsequence $\{{x}_{{n}_{i}}\}\subset \{{x}_{n}\}$ such that ${x}_{{n}_{i}}$ converges strongly to some point ${x}^{\ast}\in H$.

**Proposition 2.5**

*Let*$G:H\to H$

*be a*$(\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )$-

*total quasi*-

*asymptotically nonexpansive mapping*.

*Then for each*$q\in F(G)$

*and for each*$x\in H$,

*the following inequalities are equivalent*:

*for each*$n\ge 1$

*Proof*(I) (2.1) ⇔ (2.6) In fact, since

Simplifying it, inequality (2.6) is obtained.

- (II)(2.6) ⇔ (2.7) In fact, since$\begin{array}{rcl}\u3008x-{G}^{n}x,x-q\u3009& =& \u3008x-{G}^{n}x,x-{G}^{n}x+{G}^{n}x-q\u3009\\ =& {\parallel x-{G}^{n}x\parallel}^{2}+\u3008x-{G}^{n}x,{G}^{n}x-q\u3009\end{array}$

Simplifying it, the inequality (2.7) is obtained.

Conversely, from (2.7) the inequality (2.6) can be obtained immediately.

This completes the proof of Proposition 2.5. □

**Lemma 2.6** [11]

*Let*$\{{a}_{n}\}$, $\{{b}_{n}\}$

*and*$\{{\delta}_{n}\}$

*be sequences of nonnegative real numbers satisfying*

*If* ${\sum}_{i=1}^{\mathrm{\infty}}{\delta}_{n}<\mathrm{\infty}$ *and* ${\sum}_{i=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}$, *then the limit* ${lim}_{n\to \mathrm{\infty}}{a}_{n}$ *exists*.

## 3 Split feasibility problem

- 1.
${H}_{1}$ and ${H}_{2}$ are two real Hilbert spaces, $A:{H}_{1}\to {H}_{2}$ is a bounded linear operator;

- 2.
$S:{H}_{1}\to {H}_{1}$ and $T:{H}_{2}\to {H}_{2}$ are two uniformly

*L*-Lipschitzian and ($\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta $)-total quasi-asymptotically nonexpansive mappings satisfying the following conditions: - (i)
*T*and*S*both are demi-closed at origin; - (ii)
${\sum}_{n=1}^{\mathrm{\infty}}({\mu}_{n}+{\nu}_{n})<\mathrm{\infty}$;

- (iii)
there exist positive constants

*M*and ${M}^{\ast}$ such that $\zeta (t)\le \zeta (M)+{M}^{\ast}{t}^{2}$, $\mathrm{\forall}t\ge 0$.

We are now in a position to give the following result.

**Theorem 3.1**

*Let*${H}_{1}$, ${H}_{2}$,

*A*,

*S*,

*T*,

*L*, $\{{\mu}_{n}\}$, $\{{\nu}_{n}\}$,

*ζ*

*be the same as above*.

*Let*$\{{x}_{n}\}$

*be the sequence generated by*:

*where*$\{{\alpha}_{n}\}$

*is a sequence in*$[0,1]$,

*and*$\gamma >0$

*is a constant satisfying the following conditions*:

- (iv)
$0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$;

*and*$\gamma \in (0,\frac{1}{{\parallel A\parallel}^{2}})$, - (I)
*If*$\mathrm{\Gamma}\ne \mathrm{\varnothing}$ (*where*Γ*is the set of solutions to*((*SFP*)-(1.1)),*then*$\{{x}_{n}\}$*converges weakly to a point*${x}^{\ast}\in \mathrm{\Gamma}$. - (II)
*In addition*,*if**S**is also semi*-*compact*,*then*$\{{x}_{n}\}$*and*$\{{u}_{n}\}$*both converge strongly to*${x}^{\ast}\in \mathrm{\Gamma}$.

*The proof of conclusion (I)*(1) First, we prove that for each $p\in \mathrm{\Gamma}$, the following limits exist:

- (2)Next, we prove that$\underset{n\to \mathrm{\infty}}{lim}\parallel {x}_{n+1}-{x}_{n}\parallel =0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty}}{lim}\parallel {u}_{n+1}-{u}_{n}\parallel =0.$(3.16)

- (3)Next, we prove that$\parallel {u}_{n}-S{u}_{n}\parallel \to 0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\parallel A{x}_{n}-TA{x}_{n}\parallel \to 0\phantom{\rule{1em}{0ex}}(\text{as}n\to \mathrm{\infty}).$(3.19)

*S*is uniformly

*L*-Lipschitzian continuous, it follows from (3.16) and (3.20) that

*T*is uniformly

*L*-Lipschitzian continuous, by the same way as above, from (3.16) and (3.21), we can also prove that

- (4)
Finally, we prove that ${x}_{n}\rightharpoonup {x}^{\ast}$ and ${u}_{n}\rightharpoonup {x}^{\ast}$, which is a solution of (SFP)-(1.1).

By the assumption that *S* is demi-closed at zero, we get that ${x}^{\ast}\in F(S)$.

*A*is a linear bounded operator, we get $A{x}_{{n}_{i}}\rightharpoonup A{x}^{\ast}$. In view of (3.19), we have

Since *T* is demi-closed at zero, we have $A{x}^{\ast}\in F(T)$. Summing up the above argument, it is clear that ${x}^{\ast}\in \mathrm{\Gamma}$, *i.e.*, ${x}^{\ast}$ is a solution to the (SFP)-(1.1).

Now, we prove that ${x}_{n}\rightharpoonup {x}^{\ast}$ and ${u}_{n}\rightharpoonup {x}^{\ast}$.

□

*The proof of conclusion (II)* By the assumption that *S* is semi-compact, it follows from (3.23) that there exists a subsequence of $\{{u}_{{n}_{i}}\}$ (without loss of generality, we still denote it by $\{{u}_{{n}_{i}}\}$) such that ${u}_{{n}_{i}}\to {u}^{\ast}\in H$ (some point in *H*). Since ${u}_{{n}_{i}}\rightharpoonup {x}^{\ast}$. This implies that ${x}^{\ast}={u}^{\ast}$, and so ${u}_{{n}_{i}}\to {x}^{\ast}\in \mathrm{\Gamma}$. By virtue of (3.2), we know that ${lim}_{n\to \mathrm{\infty}}\parallel {u}_{n}-{x}^{\ast}\parallel =0$ and ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{x}^{\ast}\parallel =0$, *i.e.*, $\{{u}_{n}\}$ and $\{{x}_{n}\}$ both converge strongly to ${x}^{\ast}\in \mathrm{\Gamma}$.

This completes the proof of Theorem 3.1. □

**Theorem 3.2**

*Let*${H}_{1}$, ${H}_{2}$

*and*

*A*

*be the same as in Theorem*3.1.

*Let*$S:{H}_{1}\to {H}_{1}$

*and*$T:{H}_{2}\to {H}_{2}$

*be two*$(\{{k}_{n}\})$-

*quasi*-

*asymptotically nonexpansive mappings with*$\{{k}_{n}\}\subset [1,\mathrm{\infty})$, ${k}_{n}\to 1$

*satisfying the following conditions*:

- (i)
*T**and**S**both are demi*-*closed at origin*; - (ii)
${\sum}_{n=1}^{\mathrm{\infty}}({k}_{n}-1)<\mathrm{\infty}$.

*Let*$\{{x}_{n}\}$

*be the sequence generated by*

*where* $\{{\alpha}_{n}\}$ *is a sequence in* $[0,1]$ *and* $\gamma >0$ *is a constant satisfying the condition* (*iv*) *in Theorem * 3.1. *Then the conclusions in Theorem * 3.1 *still hold*.

*Proof* By assumptions, $S:{H}_{1}\to {H}_{1}$ and $T:{H}_{2}\to {H}_{2}$ both are $(\{{k}_{n}\})$-quasi-asymptotically nonexpansive mappings with $\{{k}_{n}\}\subset [1,\mathrm{\infty})$, ${k}_{n}\to 1$; by Remark 2.3, *S* and *T* both are uniformly *L*-Lipschitzian (where $L={sup}_{n\ge 1}{k}_{n}$) and ($\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta $)-total quasi-asymptotically nonexpansive mapping with $\{{\nu}_{n}={k}_{n}-1\}$, $\{{\mu}_{n}=0\}$ and $\zeta (t)={t}^{2}$, $t\ge 0$. Therefore, all conditions in Theorem 3.1 are satisfied. The conclusions of Theorem 3.2 can be obtained from Theorem 3.1 immediately. □

**Theorem 3.3**

*Let*${H}_{1}$, ${H}_{2}$

*and*

*A*

*be the same as in Theorem*3.1.

*Let*$S:{H}_{1}\to {H}_{1}$

*and*$T:{H}_{2}\to {H}_{2}$

*be two quasi*-

*nonexpansive mappings and demi*-

*closed at origin*.

*Let*$\{{x}_{n}\}$

*be the sequence generated by*

*where* $\{{\alpha}_{n}\}$ *is a sequence in* $[0,1]$ *and* $\gamma >0$ *is a constant satisfying the condition* (*iv*) *in Theorem * 3.1. *Then the conclusions in Theorem * 3.1 *still hold*.

*Proof* By the assumptions, $S:{H}_{1}\to {H}_{1}$ and $T:{H}_{2}\to {H}_{2}$ are quasi-nonexpansive mappings. By Remark 2.3, *S* and *T* both are uniformly *L*-Lipschitzian (where $L=1$) and ($\{1\}$)- quasi-asymptotically nonexpansive mappings. Therefore, all conditions in Theorem 3.2 are satisfied. The conclusions of Theorem 3.3 can be obtained from Theorem 3.2 immediately. □

**Remark 3.4** Theorems 3.1, 3.2 and 3.3 not only improve and extend the corresponding results of Moudafi [12, 13], but also improve and extend the corresponding results of Censor *et al.* [4, 5], Yang [7], Xu [14], Censor and Segal [15], Masad and Reich [16] and others.

## Declarations

### Acknowledgements

This work was supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ010) and the Natural Science Foundation of Yunnan Province (Grant No.2011FB074).

## Authors’ Affiliations

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