# Simultaneous iterative algorithms for the split common fixed-point problem of generalized asymptotically quasi-nonexpansive mappings without prior knowledge of operator norms

- Jing Zhao
^{1}Email author and - Songnian He
^{1}

**2014**:73

https://doi.org/10.1186/1687-1812-2014-73

© Zhao and He; licensee Springer. 2014

**Received: **21 November 2013

**Accepted: **6 March 2014

**Published: **25 March 2014

## Abstract

Let ${H}_{1}$, ${H}_{2}$, ${H}_{3}$ be real Hilbert spaces, let $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms (Trans. Math. Program. Appl. 1:1-11, 2013) with weak convergence for the following split common fixed-point problem:

where $U:{H}_{1}\to {H}_{1}$ and $T:{H}_{2}\to {H}_{2}$ are two firmly quasi-nonexpansive operators with nonempty fixed-point sets $F(U)=\{x\in {H}_{1}:Ux=x\}$ and $F(T)=\{x\in {H}_{2}:Tx=x\}$. Note that by taking ${H}_{2}={H}_{3}$ and $B=I$, we recover the split common fixed-point problem originally introduced in Censor and Segal (J. Convex Anal. 16:587-600, 2009). In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of generalized asymptotically quasi-nonexpansive mappings. To estimate the norm of an operator is a very difficult, if it is not an impossible task. The purpose of this paper is to propose a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information as regards the operator norms.

**MSC:**47H09, 47H10, 47J05, 54H25.

## Keywords

## 1 Introduction and preliminaries

Throughout this paper, we always assume that *H* is a real Hilbert space with the inner product $\u3008\cdot ,\cdot \u3009$ and the norm $\parallel \cdot \parallel $. Let $T:H\to H$ be a mapping. A point $x\in H$ is said to be a fixed point of *T* provided $Tx=x$. In this paper, we use $F(T)$ to denote the fixed point set and use → and ⇀ to denote the strong convergence and weak convergence, respectively. We use ${\omega}_{w}({x}_{k})=\{x:\mathrm{\exists}{x}_{{k}_{j}}\rightharpoonup x\}$ stand for the weak *ω*-limit set of $\{{x}_{k}\}$.

*C*and

*Q*be nonempty closed convex subset of real Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively. The split feasibility problem (SFP) is to find a point

where $A:{H}_{1}\to {H}_{2}$ is a bounded linear operator. The SFP in finite-dimensional Hilbert 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]. Various algorithms have been invented to solve it, *etc*. (see [6–9] and references therein).

*i.e.*, (1.1) has a solution), then (1.1) can be formulated as a fixed point equation by using the fact

That is, ${x}^{\ast}$ solves the SFP (1.1) if and only if ${x}^{\ast}$ solves the fixed point equation (1.2) (see [10] for the details). This implies that we can use fixed point algorithms (see [10–13]) to solve SFP. A popular algorithm that solves the SFP (1.1) is due to Byrne’s CQ algorithm [2], which is found to be a gradient-projection method (GPM) in convex minimization.

where $\gamma \in (0,\frac{2}{\lambda})$, with *λ* being the largest eigenvalue of the matrix ${A}^{t}A$ (${A}^{t}$ stands for matrix transposition). SCFP (1.3) is in itself at the core of the modeling of many inverse problems in various areas of mathematics and physical sciences and has been used to model significant real-world inverse problems in sensor networks, in radiation therapy treatment planning, in resolution enhancement, in wavelet-based denoising, in antenna design, in computerized tomography, in materials science, in watermarking, in data compression, in magnetic resonance imaging, in holography, in color imaging, in optics and neural networks, in graph matching, *etc*. (see [15, 16]).

which allows asymmetric and partial relations between the variables *x* and *y*. The interest is to cover many situations, for instance, in decomposition methods for PDEs, applications in game theory and in intensity-modulated radiation therapy (IMRT). In decision sciences, this allows to consider agents who interplay only via some components of their decision variables (see [18]). In IMRT, this amounts to envisaging a weak coupling between the vector of doses absorbed in all voxels and that of the radiation intensity (see [3]). If ${H}_{2}={H}_{3}$ and $B=I$, then SCFP (1.4) reduces to SCFP (1.3).

for firmly quasi-nonexpansive operators *U* and *T*, where non-decreasing sequence ${\gamma}_{k}\in (\epsilon ,min(\frac{1}{{\lambda}_{A}},\frac{1}{{\lambda}_{B}})-\epsilon )$, ${\lambda}_{A}$, ${\lambda}_{B}$ stand for the spectral radiuses of ${A}^{\ast}A$ and ${B}^{\ast}B$, respectively.

for firmly quasi-nonexpansive operators *U* and *T*, where ${\gamma}_{k}\in (\epsilon ,\frac{2}{{\lambda}_{A}+{\lambda}_{B}}-\epsilon )$, ${\lambda}_{A}$, ${\lambda}_{B}$ stand for the spectral radiuses of ${A}^{\ast}A$ and ${B}^{\ast}B$, respectively.

*A*and

*B*, which is in general not an easy work in practice. To overcome this difficulty, López

*et al.*[20] and Zhao and Yang [21] presented a helpful method for estimating the stepsizes which do not need prior knowledge of the operator norms for solving the split feasibility problems and multiple-set split feasibility problems, respectively. Inspired by them, in this paper, we introduce a new choice of the stepsize sequence $\{{\gamma}_{k}\}$ for the simultaneous iterative algorithm to solve SCFP (1.4) governed by generalized asymptotically quasi-nonexpansive operators as follows:

The advantage of our choice (1.7) of the stepsizes lies in the fact that no prior information as regards the operator norms of *A* and *B* is required, and still convergence is guaranteed.

Now let us recall some definitions, notations and conclusions which will be needed in proving our main results.

**Definition 1.1** A mapping $T:H\to H$ is called demiclosed at the origin if, for any sequence $\{{x}_{n}\}$ which weakly converges to *x*, if the sequence $\{T{x}_{n}\}$ strongly converges to 0, then $Tx=0$.

**Definition 1.2**(1) A mapping $T:H\to H$ is said to be nonexpansive if

*T*is not Lipschitzian. Notice that $F(T)=\{0\}$. For each fixed

*k*, define ${f}_{k}(x)=\parallel {T}^{k}x\parallel -\parallel x\parallel $ for $x\in K$. Set ${\mu}_{k}={sup}_{x\in K}\{{f}_{k}(x),0\}$. Then ${lim}_{k\to \mathrm{\infty}}{\mu}_{k}=0$ and

This shows that *T* is a generalized asymptotically quasi-nonexpansive but it is not asymptotically quasi-nonexpansive and asymptotically nonexpansive because it is not Lipschitzian.

**Definition 1.3**A mapping $T:H\to H$ is said to be uniformly

*L*-Lipschitzian if there exists a constant $L>0$ such that for each $k\ge 1$,

In what follows, we give some preliminary results needed for the convergence analysis of our algorithms.

**Lemma 1.4** ([23])

*Let*

*H*

*be a real Hilbert space*.

*Then for all*$t\in [0,1]$

*and*$x,y\in H$,

**Lemma 1.5** ([24])

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

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

*be sequences of nonnegative real numbers satisfying*

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

## 2 Simultaneous iterative algorithm without prior knowledge of operator norms

In this section we introduce a simultaneous iterative algorithm where the stepsizes don‘t depend on the operator norms $\parallel A\parallel $ and $\parallel B\parallel $ and prove weak convergence of the algorithm to solve SCFP (1.4) governed by generalized asymptotically quasi-nonexpansive operators.

*i.e.*,

**Algorithm 2.1**

*Let*${x}_{0}\in {H}_{1}$, ${y}_{0}\in {H}_{2}$

*be arbitrary and*${\alpha}_{k}\in [0,1]$.

*Assume that the*

*kth iterate*${x}_{k}\in {H}_{1}$, ${y}_{k}\in {H}_{2}$

*has been constructed*;

*then we calculate the*$(k+1)$

*th iterate*$({x}_{k+1},{y}_{k+1})$

*via the formula*:

*The stepsize*${\gamma}_{k}$

*is chosen in such a way that*

*otherwise*, ${\gamma}_{k}=\gamma $ (*γ* *being any nonnegative value*), *where the set of indices* $\mathrm{\Omega}=\{k:A{x}_{k}-B{y}_{k}\ne 0\}$.

**Remark 2.2** Note that in (2.2) the choice of the stepsize ${\gamma}_{k}$ is independent of the norms $\parallel A\parallel $ and $\parallel B\parallel $. The value of *γ* does not influence the considered algorithm, but it was introduced just for the sake of clarity. Furthermore, we will see from Lemma 2.3 that ${\gamma}_{k}$ is well defined.

**Lemma 2.3** *If* Γ *is nonempty*, *then* ${\gamma}_{k}$ *defined by* (2.2) *is well defined*.

*Proof*Taking $(x,y)\in \mathrm{\Gamma}$,

*i.e.*, $x\in F(U)$, $y\in F(T)$ and $Ax=By$. We have

Consequently, for $k\in \mathrm{\Omega}$, that is, $\parallel A{x}_{k}-B{y}_{k}\parallel >0$, we have $\parallel {A}^{\ast}(A{x}_{k}-B{y}_{k})\parallel \ne 0$ or $\parallel {B}^{\ast}(A{x}_{k}-B{y}_{k})\parallel \ne 0$. This leads to the conclusion that ${\gamma}_{k}$ is well defined. □

**Theorem 2.4**

*Assume that*$U-I$, $T-I$

*are demiclosed at origin*,

*and*

*U*,

*T*

*are uniformly L*-

*Lipschitzian*.

*Let the sequence*$\{({x}_{k},{y}_{k})\}$

*be generated by Algorithm*2.1.

*Assume*Γ

*is nonempty and for small enough*$\u03f5>0$,

*where* $k\in \mathrm{\Omega}$. *Then* $\{({x}_{k},{y}_{k})\}$ *weakly converges to a solution* $({x}^{\ast},{y}^{\ast})$ *of* (1.4) *provided that* ${\sum}_{k=1}^{\mathrm{\infty}}({l}_{k}+{\mu}_{k})<\mathrm{\infty}$ *and* $\{{\alpha}_{k}\}\subset (\delta ,1-\delta )$ *for small enough* $\delta >0$. *Moreover*, $\{{x}_{k}\}$ *and* $\{{y}_{k}\}$ *are asymptotically regular and* $\parallel A{x}_{k}-B{y}_{k}\parallel \to 0$.

*Proof*From the condition on ${\gamma}_{k}$, we have

It follows from (2.3) that ${sup}_{k\in \mathrm{\Omega}}{\gamma}_{k}<+\mathrm{\infty}$ and ${\{{\gamma}_{k}\}}_{k\ge 1}$ is bounded.

*i.e.*, $x\in F(U)$; $y\in F(T)$ and $Ax=By$. We have

*U*and

*T*are generalized asymptotically quasi-nonexpansive mappings with $(\{{l}_{k}\},\{{\mu}_{k}\})$, it follows from Lemma 1.4 that

Similarly, ${lim}_{k\to \mathrm{\infty}}\parallel {v}_{k}-{y}_{k}\parallel =0$, $\{{y}_{k}\}$ and $\{{v}_{k}\}$ are asymptotically regular, too.

*U*is uniformly

*L*-Lipschitzian continuous, it follows from (2.13) and (2.16) that

Since *T* is uniformly *L*-Lipschitzian continuous, in the same way as above, we can also prove that $\parallel {v}_{k}-T({v}_{k})\parallel \to 0$ as $k\to \mathrm{\infty}$.

hence $({x}^{\ast},{y}^{\ast})\in \mathrm{\Gamma}$.

By adding the two last equalities, we obtain ${x}^{\ast}=\overline{x}$ and ${y}^{\ast}=\overline{y}$, which implies that the whole sequence $\{({x}_{k},{y}_{k})\}$ weakly converges to a solution of problem (1.4). This completes the proof. □

The following conclusions can be obtained from Theorem 2.4 immediately.

**Theorem 2.5**

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

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

*be two asymptotically quasi*-

*nonexpansive mappings with*$(\{{l}_{k}\})$.

*Assume that*$U-I$, $T-I$

*are demiclosed at origin*,

*and*

*U*,

*T*

*are uniformly L*-

*Lipschitzian*.

*Let the sequence*$\{({x}_{k},{y}_{k})\}$

*be generated by Algorithm*2.1.

*Assume*Γ

*is nonempty and for small enough*$\u03f5>0$,

*where* $k\in \mathrm{\Omega}$. *Then* $\{({x}_{k},{y}_{k})\}$ *weakly converges to a solution* $({x}^{\ast},{y}^{\ast})$ *of* (1.4) *provided that* ${\sum}_{k=1}^{\mathrm{\infty}}{l}_{k}<\mathrm{\infty}$ *and* $\{{\alpha}_{k}\}\subset (\delta ,1-\delta )$ *for small enough* $\delta >0$. *Moreover*, $\{{x}_{k}\}$ *and* $\{{y}_{k}\}$ *are asymptotically regular and* $\parallel A{x}_{k}-B{y}_{k}\parallel \to 0$.

**Theorem 2.6**

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

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

*be two quasi*-

*nonexpansive mappings*.

*Assume that*$U-I$, $T-I$

*are demiclosed at origin*,

*and*

*U*,

*T*

*are uniformly L*-

*Lipschitzian*.

*Let the sequence*$\{({x}_{k},{y}_{k})\}$

*be generated by Algorithm*2.1.

*Assume*Γ

*is nonempty and for small enough*$\u03f5>0$,

*where* $k\in \mathrm{\Omega}$. *Then* $\{({x}_{k},{y}_{k})\}$ *weakly converges to a solution* $({x}^{\ast},{y}^{\ast})$ *of* (1.4) *provided that* $\{{\alpha}_{k}\}\subset (\delta ,1-\delta )$ *for small enough* $\delta >0$. *Moreover*, $\{{x}_{k}\}$ *and* $\{{y}_{k}\}$ *are asymptotically regular and* $\parallel A{x}_{k}-B{y}_{k}\parallel \to 0$.

**Remark 2.7**When $B=I$, Algorithm 2.1 becomes

otherwise ${\gamma}_{k}=\gamma $ (*γ* being any nonnegative value), where the set of indices $\mathrm{\Omega}=\{k:A{x}_{k}-{y}_{k}\ne 0\}$. This solves SCFP (1.3) for generalized asymptotically quasi-nonexpansive operators, asymptotically quasi-nonexpansive operators, quasi-nonexpansive operators, and firmly quasi-nonexpansive operators without prior knowledge of operator norm $\parallel A\parallel $.

## Declarations

### Acknowledgements

The research was supported by the Fundamental Research Funds for the Central Universities (Program No. 3122013k004), it was also supported by the Fundamental Research Funds for the Central Universities (Program No. 3122013C002).

## Authors’ Affiliations

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