A note on strong convergence to common fixed points of nonexpansive mappings in Hilbert spaces

TL;DR

This paper explores the strong convergence properties of algorithms for finding common fixed points of nonexpansive mappings in Hilbert spaces, linking various iterative methods and extending existing theorems.

Contribution

It establishes new connections between ${ m T}_C$-class algorithms, CQ Algorithm, and shrinking projection methods, extending convergence results for nonexpansive mappings.

Findings

Strong convergence of algorithms is related to coherent ${ m T}_C$-class sequences.

Examples demonstrate applicability to nonexpansive finite sets and semigroups.

Extensions of existing theorems on fixed point convergence.

Abstract

The aim of this paper is to investigate the links between ${\cal T}_C$-class algorithms, CQ Algorithm and shrinking projection methods. We show that strong convergence of these algorithms are related to coherent ${\cal T}_C$-class sequences of mapping. Some examples dealing with nonexpansive finite set of mappings and nonexpansive semigroups are given. They extend some existing theorems.

Equations

$$\Vert Tx-Ty\Vert \le \Vert x-y\Vert \text{mbox}forallx,y\in C.$$

$$\text{mbox}Fix(T)\stackrel{\text{mbox}def}{=}\left\{x\in C:Tx=x\right\}.$$

$$H_S(x,y)\stackrel{\text{mbox}def}{=}\left\{z\in S|⟨z-y,x-y⟩\le 0\right\}.$$

$$x_{n+1}=Q_C(x_0,x_n,T_n{x}_n)$$

$$\{\begin{array}{l}{y}_n=R_n{x}_n,\\ C_n\stackrel{\text{mbox}def}{=}\left\{z\in C|\Vert y_n-z\Vert \le \Vert x_n-z\Vert \right\},\\ D_n\stackrel{\text{mbox}def}{=}\left\{z\in C|⟨x_n-z,x_0-x_n⟩\ge 0\right\},\\ x_{n+1}=P_{(C_n\cap D_n)}{x}_0.\end{array}$$

$$4⟨z-Tx,x-Tx⟩={\Vert Rx-z\Vert }^2-{\Vert x-z\Vert }^2.$$

$${\mathcal{T}}_C\stackrel{\text{mbox}def}{=}\left\{T:C\mapsto C|\text{mbox}dom(T)=C\text{mbox}and(\forall x\in C)\text{mbox}Fix(T)\subset H(x,Tx)\right\}.$$

$$\displaystyle\left\{\begin{array}[]{l}\sum_{n\geq 0}{\|z_{n+1}-z_{n}\|}^{2}<\infty\\ \sum_{n\geq 0}{\|z_{n}-T_{n}z_{n}\|}^{2}<\infty\end{array}\right.\Rightarrow{\cal M}(z_{n})_{n\geq 0}\subset\bigcap_{n\geq 0}\mathop{\mbox{Fix}}(T_{n})$$

$$\{\begin{array}{l}{C}_{n+1}\stackrel{\text{mbox}def}{=}{C}_n\cap H(x_n,T_n{x}_n)\text{mbox}with{T}_n\stackrel{\text{mbox}def}{=}(R_n+Id)/2,\\ x_{n+1}=P_{C_{n+1}}{x}_0.\end{array}$$

$$F\stackrel{\text{mbox}def}{=}\bigcap _{n\in \mathbb{N}}\text{mbox}Fix(T_n).$$

$${\Vert Tx-Ty\Vert }^2\le ⟨x-y,Tx-Ty⟩\forall (x,y)\in C^2$$

$${\Vert Tx-Ty\Vert }^2\le {\Vert x-y\Vert }^2-{\Vert (T-Id)x-(T-Id)y\Vert }^2\forall (x,y)\in C^2.$$

$$\varnothing \ne \text{mbox}Fix(\mathcal{F})\stackrel{\text{mbox}def}{=}\bigcap _{n\in \mathbb{N}}\text{mbox}Fix(T_n),$$

$${\Gamma }_n^{(j)}x\stackrel{\text{mbox}def}{=}{\alpha }_n^{(j)}x+(1-{\alpha }_n^{(j)})T_n^{(j)}{\Gamma }_n^{(j+1)}x\text{mbox}and{\Gamma }_n^{(N+1)}x\stackrel{\text{mbox}def}{=}x$$

$$T_n^{(j)}x=\frac{1}{t_n}{\int }_0^{t_n}{T}^{(j)}(s)xds,$$

$$\beta (1-\beta ){\Vert x-Tx\Vert }^2\le 2(\Vert x-p\Vert -\Vert T_{\beta }x-p\Vert )\Vert x-p\Vert$$

$${\Vert T_{\beta }x-p\Vert }^2$$

$$\beta (1-\beta ){\Vert Tx-x\Vert }^2$$

$$\beta (1-\beta ){\Vert x-Tx\Vert }^2\le 2\Vert x-S{T}_{\beta }x\Vert \Vert x-p\Vert .$$

$$\Vert x-Sx\Vert \le \Vert x-S{T}_{\beta }x\Vert +\Vert Tx-x\Vert .$$

$$\Vert x-p\Vert$$

$$\Vert x-Sx\Vert$$

$${\alpha }_n^{(j+1)}(1-{\alpha }_n^{(j+1)}){\Vert x_n-T_n^{(j+1)}{\Gamma }_n^{(j+2)}{x}_n\Vert }^2\le 2\Vert x_n-T_n^{(j)}{\Gamma }_n^{(j+1)}{x}_n\Vert \Vert x_n-p\Vert$$

$$\Vert x_n-T_n^{(j)}{x}_n\Vert \le \Vert x_n-T_n^{(j)}{\Gamma }_n^{(j+1)}{x}_n\Vert +\Vert T_n^{(j+1)}{\Gamma }_n^{(j+2)}{x}_n-x_n\Vert$$

$$\Vert T^{(j)}(s)x_n-x_n\Vert \le 2\Vert T_n^{(j)}{x}_n-x_n\Vert +\Vert T(s)(T_n^{(j)}{x}_n)-T_n^{(j)}{x}_n\Vert$$

$$x_{n+1}=x_n=\cdots =x_0⟺\{\begin{array}{ll}{x}_0& \in {\cup }_{k=0}^{n-1}\text{mbox}Fix(T_k)\\ x_0& =x_{n+1}=P_{C\cap {\bigcap }_{k=0}^{n}H(x_k,T_k{x}_k)}\\ & =P_{C\cap H(x_0,T_n{x}_0)}=T_n{x}_0.\end{array}$$

$$⟨x_0-x_n,x_n-x_{n+1}⟩\ge 0.$$

$${\Vert x_{n+1}-x_n\Vert }^2$$

Full text

1 Introduction

Let $C$ be a closed convex subset of a Hilbert space ${\cal H}$. A mapping $T$ of $C$ into itself is called nonexpansive if

[TABLE]

We denote by $\mathop{\mbox{Fix}}(T)$ the set of fixed points of $T$. That is

[TABLE]

There are many iterative methods for approximation of common fixed points of a family of nonexpansive mapping in a Hilbert space. In Section 2 we recall the CQ Algorithm [6, 8] (Algorithm 2) associated to a sequence of mappings $(T_{n})_{n\geq 0}$ of $C$ into itself. The CQ Algorithm when applied to a sequence of mappings of ${\cal H}$ into itself is the same as a Haugazeau method [4] studied in [1, Algorithm 3.1] and applied to ${\cal T}$-class mappings.

We straighforwardly generalize, in Section 2, the ${\cal T}$-class to take into account mappings of $C$ into itself. We denote this new class by the ${\cal T}_{C}$-class. Using this extension, the CQ Algorithm (Algorithm 2) coincides with the Haugazeau method (Algorithm 1) and a strong convergence theorem can be obtained by following results from [1]. Note that the convergence theorem is obtained for ${\cal T}_{C}$-class sequences which are coherent (Definition 5).

In [9] another algorithm called the shrinking projection method is also studied. One of our aims in this article is to prove that, rephrased in the context of ${\cal T}_{C}$-class algorithm, the convergence results of this new algorithm (Algorithm 3) is also related to coherent sequences of ${\cal T}_{C}$-class mappings. We give in Theorem 6 a strong convergence result of Algorithm 3 for ${\cal T}_{C}$-class coherent sequence of mappings. Section 4 is devoted to the proof. The strong convergence of Algorithm 3 is also proved in [9, Theorem 3.3] for sequence of nonexpansive mappings satisfying the NST-condition(I) (Definition 9). It is easy to prove that if $R$ is a nonexpansive mapping of $C$ into itself then $T=(R+Id)/2$ belongs to the ${\cal T}_{C}$-class and that a sequence of nonexpansive mappings satisfying the NST-condition(I) is coherent. Thus Theorem 6 extends [9, Theorem 3.3 and Theorem 3.4].

In Section 3 we show that specific sequences of mappings are coherent. Combined with Theorem 6 it can be considered as an extension to some existing theorems in [6, 9, 7].

2 The ${\cal T}_{C}$-class iterative algorithms, CQ algorithm and the shrinking projection method

We first recall here the ${\cal T}$-class iterative algorithms as defined by H. Bauschke and P. L. Combettes [1].

For $(x,y)\in{\cal H}^{2}$ and $S$ a subset of ${\cal H}$, we define the mapping $H_{S}$ as follows:

[TABLE]

We also define the mapping $H$ by $H\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}H_{{\cal H}}$. Note that $H_{S}(x,x)=S$ and for $x\neq y$, $H(x,y)$ is a closed affine half space. For a nonempty closed convex $C$, we denote by $Q_{C}(x,y,z)$ the projection, when it exists, of $x$ onto $H_{C}(x,y)\cap H_{C}(y,z)$ and $Q$ the projection when $C={\cal H}$, that is $Q\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}Q_{{\cal H}}$. As an intersection of two closed affine half spaces and a closed convex, $H_{C}(x,y)\cap H_{C}(y,z)$ is a possibly empty closed convex.

It is easy to check, from the definition of $H$, that $y$ is the projection of $x$ onto $H(x,y)$ and we therefore have $Q(x,x,y)=P_{H(x,y)}x=y$. Where $P_{C}$ is the metric projection from ${\cal H}$ onto $C$. Moreover, if $y\in C$ then we also have that $y$ is the projection of $x$ onto $H_{C}(x,y)$ which gives $Q_{C}(x,x,y)=y$.

The algorithm studied in [1] is the following

Algorithm 1

Given $x_{0}\in C$ and a sequence $(T_{n})_{n\geq 0}$ of mappings $T_{n}:C\to{\cal H}$, we consider the sequence $(x_{n})_{n\geq 0}$ generated by the following algorithm:

[TABLE]

A very similar algorithm exists under the name of CQ algorithm [6, 8]:

Algorithm 2

Given $x_{0}\in C$, we consider the sequence $(x_{n})_{n\geq 0}$ generated by the following algorithm:

[TABLE]

The link between the two algorithms is described by the following lemma.

Lemma 1

The sequence generated by Algorithm 2 coincides with the sequence given by $x_{n+1}=Q_{C}(x_{0},x_{n},T_{n}x_{n})$ with $T_{n}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}(R_{n}+Id)/2$.

Proof: Following [1], the proof easily follows from the equality

[TABLE]

$\Box$

The convergence of Algorithm 1 and therefore of Algorithm 2 when $C={\cal H}$ is studied in [1]. It relies on two requested properties of the sequence $(T_{n})_{n\geq 0}$. First, the sequence $(T_{n})_{n\geq 0}$ must belong the ${\cal T}$-class which means that for all $n\in{\mathbb{N}}$ we must have $T_{n}\in{\cal T}$ where ${\cal T}$ is defined as follows:

Definition 2

A mapping $T:C\mapsto{\cal H}$ belongs to the ${\cal T}_{C}$-class if it is an element of the set ${\cal T}_{C}$:

[TABLE]

When $C={\cal H}$, we use the notation ${\cal T}={\cal T}_{{\cal H}}$. Second, the sequence $(T_{n})_{n\geq 0}$ must be coherent as defined below.

Definition 3

[1]** A sequence $(T_{n})_{n\geq 0}$ such that $T_{n}\in{\cal T}_{C}$ is coherent if for every bounded sequence $\{z_{n}\}_{n\geq 0}\in C$ the following holds:

[TABLE]

where ${\cal M}(z_{n})_{n\geq 0}$ is the set of weak cluster points of the sequence $(z_{n})_{n\geq 0}$.

Theorem 4

[1, Theorem 4.2]** Suppose that $C={\cal H}$ and the ${\cal T}_{C}$-class sequence $(T_{n})_{n\geq 0}$ is coherent. Then, for an arbitrary orbit of Algorithm 1, exactly one of the following alternatives holds:

1. (a)

$F\neq\emptyset$* and $x_{n}\to_{n}P_{F}x_{0}$;* 2. (b)

$F=\emptyset$* and $x_{n}\to_{n}+\infty$;* 3. (c)

$F=\emptyset$* and the algorithm terminates,*

where the set $F$ is defined by $F\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\bigcap_{n\geq 0}\mathop{\mbox{Fix}}(T_{n})$.

Remark 5

In the previous proof, it is supposed that $C={\cal H}$. If $C$ is a nonempty closed convex subset of ${\cal H}$, Theorem 4 $(a)$ remains valid.

In [9] another iterative algorithm called the shrinking projection method is studied. Using our notation it can be rephrased as follows:

Algorithm 3

Given $x_{0}\in C$ and $C_{0}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}C$, we consider the sequence $(x_{n})_{n\geq 0}$ (when it exists) generated by the following algorithm:

[TABLE]

The previous algorithm is stopped once $C_{n}=\emptyset$. One of the results of this paper is the proof that the convergence of Algorithm 3 is governed by the same rules as for the convergence of Algorithm 1.

Theorem 6

Suppose that the ${\cal T}_{C}$-class sequence $(T_{n})_{n\in{\mathbb{N}}}$ is coherent and let

[TABLE]

Then, if $F\neq\emptyset$ the sequence $(x_{n})_{n\geq 0}$ produced by Algorithm 3 and Algorithm 1 converges to $P_{F}x_{0}$.

Proof: As pointed out in the introduction the case of Algorithm 1 when $C={\cal H}$ is proved in Theorem 4. The extension to the case of a closed nonempty subset $C$ of ${\cal H}$ is straightforward and we will not give an explicit proof. The proof of the case of Algorithm 3 is postponed to Section 4. $\Box$

Remark 7

The first condition for the convergence is the fact that the sequence $(T_{n})_{n\geq 0}$ must belong to the ${\cal T}_{C}$-class. Note that by [1, Proposition 2.3] $T\in{\cal T}$ iff the mapping $2T-Id$ is quasi nonexpansive and $\mathop{\mbox{dom}}(T)={\cal H}$. The equivalence remains true for ${\cal T}_{C}$-class if $\mathop{\mbox{dom}}(T)={\cal H}$ is replaced by $\mathop{\mbox{dom}}(T)=C$.

Thus, if $T_{n}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}(R_{n}+Id)/2$, a necessary and sufficient condition for the sequence $(T_{n})_{n\geq 0}$ to belong to the ${\cal T}_{C}$-class is that the sequence $(R_{n})_{n\geq 0}$ is a sequence of quasi nonexpansive mappings.

Remark 8

Moreover, it is a well known fact [3, Theorem 12.1] that $2T-Id$ is nonexpansive iff $T$ is firmly nonexpansive. Thus, a sufficient condition for the mapping $T$ to belong to the ${\cal T}_{C}$-class is that $T$ is a firmly nonexpansive mapping, i.e:

[TABLE]

or equivalently

[TABLE]

We recall here the definition of the NST-condition (I) [5]. Let $(T_{n})_{n\geq 0}$ and ${\cal F}$ be two families of nonexpansive mappings of $C$ into itself such that

[TABLE]

where $\mathop{\mbox{Fix}}({\cal F})$ is the set of all common fixed points of mappings from the family ${\cal F}$.

Definition 9

The sequence $(T_{n})_{n\geq 0}$ of mappings is said to satisfy the NST-condition (I) with ${\cal F}$ if, for each bounded sequence $(z_{n})_{n\geq 0}\subset C$, we have that $\lim_{n\mapsto\infty}{\|z_{n}-T_{n}z_{n}\|}=0$ implies that $\lim_{n\mapsto\infty}{\|z_{n}-Tz_{n}\|}=0$ for all $T\in{\cal F}$.

Remark 10

Suppose that ${\cal F}$ is a family of nonexpansive mappings. It is easy to see that a sequence $(T_{n})_{n\geq 0}$ of mappings satisfying a NST-condition (I) with ${\cal F}$ is coherent. Indeed, from a demi-closed principle or using [9, Lemma 3.1] if ${\|x_{n}-Tx_{n}\|}\mapsto 0$ for all $T\in{\cal T}$ then ${\cal M}(x_{n})_{n\geq 0}\subset\mathop{\mbox{Fix}}\left(\{T\}_{T\in{\cal T}}\right)$.

3 Coherent sequences of mappings

We consider here Algorithms 1 and 3 for a sequence of mappings $(R_{n})_{n\geq 0}$ built by $N$ level iterations. Our aim is to give conditions under which the sequence $(R_{n})_{n\geq 0}$ or equivalently $(T_{n})_{n\geq 0}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}(R_{n}+Id)/2$ is coherent111By [1, Proposition 4.5] if $(T_{n})_{n\geq 0}\in{\cal T}$ and $T^{\prime}_{n}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}Id+\lambda_{n}(T_{n}-Id)$ with $\lambda_{n}\in[\delta,1]$ and $\delta\in]0,1]$. Then $(T_{n})_{n\geq 0}$ is coherent iff $(T^{\prime}_{n})_{n\geq 0}$ is coherent. and apply Theorem 6 to get convergence results.

Let $N\geq 1$ and $(T^{(j)}_{n})_{n\geq 0}:C\to{\cal H}$ for $1\leq j\leq N$ be a finite set of sequences of nonexpansive mappings. Given also a family of sequences of real parameters $(\alpha^{(j)}_{n})_{n\geq 0}$ for $1\leq j\leq N$, we define new sequences $(\Gamma^{(j)}_{n})_{n\geq 0}:C\to{\cal H}$ by the recursive equations:

[TABLE]

$\mathbf{H}_{\alpha}$: We will assume that the sequences of real parameters $(\alpha^{(j)}_{n})_{n\geq 0}$ satisfy the following condition: for $2\leq j\leq N$ and for all $n\in{\mathbb{N}}$ we have $\alpha^{(j)}_{n}\in(a,b)$ with $0<a<b<1$ and $\alpha^{(1)}_{n}\in[0,b)$.

Using the sequence of mappings $R_{n}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\Gamma_{n}^{(1)}$ in Algorithms 1 and 3 gives $N$ level algorithms. We will consider the following specific examples:

$\mathbf{H}_{1}$

Each sequence $(T^{(j)}_{n})_{n\geq 0}$ is constant, i.e $T^{(j)}_{n}=T^{(j)}$ for $1\leq j\leq N$ and $F\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathop{\mbox{Fix}}\left(\left\{T^{(j)},1\leq j\leq N\right\}\right)$ is nonempty.

$\mathbf{H}_{2}$

The $(T^{(j)}_{n})_{n\geq 0}$ sequences for $1\leq j\leq N$ are given by $T^{(j)}_{n}=T^{(j)}(t_{n})$, where $\left\{T^{(j)}(t):t\geq 0\right\}$ is a finite set of given semigrougs and $(t_{n})_{n\geq 0}$ is a sequence of real numbers such that $\liminf_{n}t_{n}=0$, $\limsup_{n}t_{n}>0$ and $\lim_{n}(t_{n+1}-t_{n})=0$. We assume that $F\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathop{\mbox{Fix}}\left(\left\{T^{(j)}(t),1\leq j\leq N,t\geq 0\right\}\right)$ is nonempty.

$\mathbf{H}_{3}$

The $(T^{(j)}_{n})_{n\geq 0}$ sequences for $1\leq j\leq N$ are given by

[TABLE]

where $\left\{T^{(j)}(t):t\geq 0\right\}$ is a finite set of given semigrougs and $(t_{n})_{n\geq 0}$ is a positive divergent sequence of real numbers. We assume that $F\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathop{\mbox{Fix}}\left(\left\{T^{(j)}(t),1\leq j\leq N,t\geq 0\right\}\right)$ is nonempty.

Theorem 11

Given a finite set of $N$ nonexpansive sequences $(T^{(j)}_{n})_{n\geq 0}$ satisfying $\mathbf{H}_{1}$, $\mathbf{H}_{2}$, or $\mathbf{H}_{3}$. The sequence $(x_{n})_{n\geq 0}$ produced by Algorithm 1 and Algorithm 3 with $R_{n}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\Gamma_{n}^{(1)}$ and $(T_{n})_{n\geq 0}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}(R_{n}+Id)/2$ converges to $P_{F}x_{0}$. The mappings $\Gamma_{n}^{(j)}$ being defined by equation (8) with parameters $\alpha_{n}^{(j)}$ satisfying $\mathbf{H}_{\alpha}$.

Proof: The proof is obtained by showing that the sequence of mappings $(T_{n})_{n\geq 0}$ is coherent in each given case and by applying Theorem 6 to conclude. The coherence is proved in the sequel in Proposition 15 for the case $\mathbf{H}_{1}$, in Proposition 17 for the case $\mathbf{H}_{2}$ and in Proposition 19 for the case $\mathbf{H}_{3}$. $\Box$

We start here by a set of lemmata which are common to all cases.

Lemma 12

Let $T$ be a $F$-quasi nonexpansive mapping and for $\beta\in(0,1)$ the mapping $T_{\beta}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\beta Id+(1-\beta)T$. For $p\in F$ and all $x\in H$ we have :

[TABLE]

Proof: For $p\in F$ and all $x\in H$ we have :

[TABLE]

We thus obtain

[TABLE]

$\Box$

Lemma 13

Let $T$ a $F$-quasi nonexpansive mapping. For $\beta\in(0,1)$ we define the mapping $T_{\beta}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\beta Id+(1-\beta)T$. For $p\in F$, all $x\in H$ and $S$ a $F$-quasi nonexpansive mapping, we have :

[TABLE]

If moreover $S$ is nonexpansive we also have :

[TABLE]

Proof: For $p\in F$ and all $x\in H$ we have :

[TABLE]

We thus have ${\|x-p\|}-{\|T_{\beta}x-p\|}\leq{\|x-ST_{\beta}x\|}$ which combined with Lemma 10 gives equation (11).

Now if $S$ is nonexpansive,

[TABLE]

$\Box$

Lemma 14

*Suppose that $F\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\bigcap_{\{n\in{\mathbb{N}};1\leq j\leq N\}}\mathop{\mbox{Fix}}(T^{(j)}_{n})$ is not empty suppose that for a bounded sequence $(x_{n})_{n\geq 0}$ and a fixed value of $j$ we have

${\|x_{n}-T^{(j)}_{n}\Gamma_{n}^{(j+1)}x_{n}\|}\to 0$. Moreover, suppose that for $2\leq j\leq N$ and all $n\in{\mathbb{N}}$ we have $\alpha^{(j)}_{n}\in(a,b)$ with $0<a<b<1$. Then for all $k\geq j$ we have ${\|x_{n}-T^{(k)}_{n}x_{n}\|}\to 0$.*

Proof: Note first that the sequences $(T^{(j)})_{1\leq j\leq N}$ and $(\Gamma^{(j)})_{1\leq j\leq N+1}$ are composed of nonexpansive mappings. Indeed the composition of nonexpansive mappings is nonexpansive and for $\beta\in(0,1)$ $\beta Id+(1-\beta)S$ is nonexpansive when $S$ is nonexpansive. The sequences are also $F$-quasi nonexpansive since it is straightforward that $F\subset\mathop{\mbox{Fix}}(\Gamma^{(j)}_{n})$ for all $j\in[1,N]$ and $n\in{\mathbb{N}}$ and if $S$ is nonexpansive it is also $\mathop{\mbox{Fix}}(S)$-quasi nonexpansive.

The proof then follows by backward induction on $j$. Assume that the result is true for $j+1$ then we will prove that it is true for $j$. Using the definition of $\Gamma_{n}^{(j+1)}$ and using equation (11) for $p\in F$, $S=T^{(j)}_{n}$, $T=T^{(j+1)}_{n}\Gamma_{n}^{(j+2)}$ and $\beta=\alpha_{n}^{(j+1)}$ (we thus have $T_{\beta}=\Gamma_{n}^{(j+1)}$) we obtain :

[TABLE]

We thus obtain that ${\|x_{n}-T^{(j+1)}_{n}\Gamma_{n}^{(j+2)}x_{n}\|}\to 0$ and by induction hypothesis we obtain ${\|x_{n}-T^{(k)}_{n}x_{n}\|}\to 0$ for $k\geq j+1$. Now using equation (12) with $S\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}T^{(j)}_{n}$, $T\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}T^{(j+1)}_{n}\Gamma_{n}^{(j+2)}$ and $\beta=\alpha_{n}^{(j+1)}$ we get:

[TABLE]

and the result follows for $j$. $\Box$

3.1 The case $\mathbf{H}_{1}$

Proposition 15

*In the case $\mathbf{H}_{1}$, the sequence $(R_{n})_{n\geq 0}$, defined by $R_{n}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\Gamma^{(1)}_{n}$ with parameters satisfying $\mathbf{H}_{\alpha}$, satisfy the NST-condition(I) with

${\cal F}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathop{\mbox{Fix}}{\{T^{(j)}}_{1\leq j\leq N}\}$ and the sequence $T_{n}=(R_{n}+Id)/2$ is a ${\cal T}_{C}$-class and coherent sequence.*

Proof: We have ${\|x_{n}-R_{n}x_{n}\|}={\|x_{n}-T_{n}^{(1)}\Gamma_{n}^{(2)}x_{n}\|}(1-\alpha^{(1)}_{n})$. Thus, if for each bounded sequence $(x_{n})_{n\geq 0}$ ${\|x_{n}-R_{n}x_{n}\|}\mapsto 0$ we also have ${\|x_{n}-T_{n}^{(1)}\Gamma_{n}^{(2)}x_{n}\|}\mapsto 0$ since $(1-\alpha^{(1)}_{n})$ is bounded from zero. Using Lemma 14 we have ${\|x_{n}-T^{(j)}x_{n}\|}\mapsto 0$ for $1\leq j\leq N$ which gives use the NST-condition(I) with ${\cal F}$. Now we consider the sequence $(T_{n})_{n\geq 0}$. The sequence belongs to the ${\cal T}_{C}$-class since $2T_{n}-Id=R_{n}$ is nonexpansive and thus quasi nonexpansive. Now if ${\|x_{n}-T_{n}x_{n}\|}\mapsto 0$ we also have ${\|x_{n}-R_{n}x_{n}\|}\mapsto 0$ and thus using the NST-condition(I) we have ${\|x_{n}-T^{(j)}x_{n}\|}\mapsto 0$ for $1\leq j\leq N$. Since the $T^{(j)}$ are nonexpansive they are also demi-closed [2, Lemma 4] and thus we must have ${\cal M}(x_{n})_{n\geq 0}\subset\mathop{\mbox{Fix}}(\{T^{(j)},1\leq j\leq N\})=\mathop{\mbox{Fix}}(\{T_{n}\}_{n\in{\mathbb{N}}})$. The sequence $(T_{n})_{n\geq 0}$ is thus in the ${\cal T}_{C}$-class and coherent. $\Box$

Remark 16

For $N=1$ we recover [9, Theorem 1.1] and [9, Theorem 4.1].

3.2 The case $\mathbf{H}_{2}$

Let $\left\{T(t):t\geq 0\right\}$ be a family of mappings from a subset $C$ of ${\cal H}$ into itself. We call it a nonexpansive semigroup on $C$ if the following conditions are satisfied:

• (i)

$T(0)x=x$ for all $x\in C$;

• (ii)

$T(s+t)=T(s)T(t)$ for all $s,t\geq 0$;

• (iii)

for each $x\in C$ the mapping $t\mapsto T(t)x$ is continuous;

• (iv)

${\|T(t)x-T(t)y\|}\leq{\|x-y\|}$ for all $x,y\in C$ and $t\geq 0$.

Proposition 17

*In the case $\mathbf{H}_{2}$, the sequence $(R_{n})_{n\geq 0}$, defined by $R_{n}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\Gamma^{(1)}_{n}$ with parameters satisfying $\mathbf{H}_{\alpha}$, satisfy the NST-condition(I) with

${\cal F}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathop{\mbox{Fix}}{\{T^{(j)}(t)}_{1\leq j\leq N,t\geq 0}\}$ and the sequence $T_{n}=(R_{n}+Id)/2$ is a ${\cal T}_{C}$-class and coherent sequence.*

Proof: As in the proof of Proposition 15 we obtain that for each bounded sequence $(x_{n})_{n\geq 0}$ such that ${\|x_{n}-R_{n}x_{n}\|}\mapsto 0$ we also have ${\|x_{n}-T^{(j)}(t_{n})x_{n}\|}\mapsto 0$ for $1\leq j\leq N$. Now it is easy to prove that the weak cluster points of the sequence $(x_{n})_{n\geq 0}$ are in $F$. The proof for each fixed $j$ is the same as in [7, Theorem 2.2, page 6]. We thus obtain the coherence of the sequence $(T_{n})_{n\geq 0}$. $\Box$

Remark 18

For $N=1$ we recover [7, Theorem 2.1] for Algorithm 3 and [7, Theorem 2.2] for Algorithm 1.

3.3 The case $\mathbf{H}_{3}$

Proposition 19

*In the case $\mathbf{H}_{3}$, the sequence $(R_{n})_{n\geq 0}$, defined by $R_{n}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\Gamma^{(1)}_{n}$ with parameters satisfying $\mathbf{H}_{\alpha}$, satisfy the NST-condition(I) with

${\cal F}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathop{\mbox{Fix}}{\{T^{(j)}(t)}_{1\leq j\leq N,t\geq 0}\}$ and the sequence $T_{n}=(R_{n}+Id)/2$ is a ${\cal T}_{C}$-class and coherent sequence.*

Proof: As in the proof of Proposition 15 we obtain that for each bounded sequence $(x_{n})_{n\geq 0}$ such that ${\|x_{n}-R_{n}x_{n}\|}\mapsto 0$ we also have ${\|x_{n}-T^{(j)}(t_{n})x_{n}\|}\mapsto 0$ for $1\leq j\leq N$. Now it is easy to prove that the weak cluster points of the sequence $(x_{n})_{n\geq 0}$ are in $F$. The proof for each fixed $j$ is the same as in [6, Theorem 4.1]. For each fixed $j$, it is a consequence of the inequality [6, Equation (8)]:

[TABLE]

for every $0\leq s<+\infty$ and $n\in{\mathbb{N}}$ with $T^{(j)}_{n}$ and the fact that the right hand side of the above inequality goes to zero as $n$ goes to infinity for a bounded sequence $(x_{n})_{n\geq 0}$ using [6, Lemma 2.1]. We thus obtain the coherence of the sequence $(T_{n})_{n\geq 0}$. $\Box$

Remark 20

For $N=1$ we recover [6, Theorem 4.1] for Algorithm 1 and [9, Theorem 4.4] for Algorithm 3.

4 Proof of Theorem 6

We prove here the strong convergence of Algorithm 3 for a ${\cal T}_{C}$-class sequence of coherent mappings. The proof follows the same steps as the proof of the convergence of Algorithm 1 in [1], we therefore give references to the original propositions.

The proof results from the next proposition and theorem in the following way. Let $(x_{n})_{n\geq 0}$ be an arbitrary orbit of Algorithm 3 and let $F\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathop{\mbox{Fix}}(\{T_{n}\}_{n\in{\mathbb{N}}})$. If $F\neq\emptyset$, then by Proposition 21 $(iv)$ the sequence is defined. By Theorem 22 $(ii)$ the sequence is bounded. Thus $(v)$ is fulfilled and by the coherence property we have ${\cal M}(x_{n})_{n\geq 0}\subset F$. Then, by Theorem 22 $(iv)$, the sequence strongly converges to $P_{F}(x_{0})$.

Proposition 21

[1, Proposition 3.4]** Let $(x_{n})_{n\geq 0}$ be an arbitrary orbit of Algorithm 3. Then:

• $(i)$

If $x_{n+1}$ is defined then ${\|x_{0}-x_{n}\|}\leq{\|x_{0}-x_{n+1}\|}$.

• $(ii)$

If $x_{n}$ is defined then $x_{0}=x_{n}\iff x_{n}=x_{n-1}=\cdots=x_{0}\iff x_{0}\in\bigcup_{k=0}^{n-1}\mathop{\mbox{Fix}}(T_{k})$.

• $(iii)$

If $(x_{n})_{n\geq 0}$ is defined then $({\|x_{0}-x_{n}\|})_{n\in{\mathbb{N}}}$ is increasing.

• $(iv)$

$(x_{n})_{n\geq 0}$* is defined if $F\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathop{\mbox{Fix}}(\{T_{n}\}_{n\in{\mathbb{N}}})\neq\emptyset$.*

Proof: $(i)$: If $x_{n+1}$ is defined we have $x_{n+1}=P_{C_{n+1}}x_{0}$ and thus $x_{n+1}\in C_{n+1}\subset C_{n}$ and since $x_{n}=P_{C_{n}}x_{0}$ we have ${\|x_{0}-x_{n}\|}\leq{\|x_{0}-x_{n+1}\|}$. $(ii)$: The fist equivalence follows from $(i)$. The second one is proved by induction. Note first that $H$ is such that $y=P_{H(x,y)}x$. Now for $y\in C$, we obtain also that $y=P_{C\cap H(x,y)}x$. for $n=1$, we have $x_{1}=P_{C\cap H(x_{0},T_{0}x_{0})}x_{0}=T_{0}x_{0}$ and thus $x_{1}=x_{0}\iff x_{0}\in\mathop{\mbox{Fix}}(T_{0})$. Now assume that the equivalence if fulfilled for $n$. We have

[TABLE]

$(iii)$ follows from $(i)$. $(iv)$: The algorithm is defined if $C_{n}\neq\emptyset$ for all $n\in{\mathbb{N}}$. Thus it is enough to prove that $C\cap\left(\bigcap_{n\in{\mathbb{N}}}H(x_{n},T_{n}x_{n})\right)\neq\emptyset$. By definition of the ${\cal T}_{C}$ class we have $\mathop{\mbox{Fix}}(T_{n})\subset C\cap H(x_{n},T_{n}x_{n})$ and the result follows. $\Box$

Theorem 22

([1, Theorem 3.5]) Let $(x_{n})_{n\geq 0}$ be an arbitrary orbit of Algorithm 3 and let $F\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\bigcap_{n\in{\mathbb{N}}}\mathop{\mbox{Fix}}(T_{n})$. Then

• $(i)$

If $(x_{n})_{n\geq 0}$ is defined then: $(x_{n})_{n\geq 0}$ is bounded $\iff$ $({\|x_{0}-x_{n}\|})_{n\in{\mathbb{N}}}$ converges.

• $(ii)$

If $F\neq\emptyset$, then $(x_{n})_{n\geq 0}$ is bounded and $(\forall n\in{\mathbb{N}})x_{n}\in F\iff x_{n}=P_{F}(x_{0})$.

• $(iii)$

*If $F\neq\emptyset$, then $({\|x_{0}-x_{n}\|})_{n\in{\mathbb{N}}}$ converges and *

$\lim_{n}{\|x_{0}-x_{n}\|}\leq{\|x_{0}-P_{F}x_{0}\|}$.

• $(iv)$

If $F\neq\emptyset$, then: $\lim_{n}x_{n}=P_{F}(x_{0})\iff{\cal M}(x_{n})_{n\in{\mathbb{N}}}\subset F$.

• $(v)$

If $(x_{n})_{n\geq 0}$ is defined and bounded then $\sum_{n\geq 0}{\|x_{n+1}-x_{n}\|}^{2}<+\infty$ and $\sum_{n\geq 0}{\|x_{n}-T_{n}x_{n}\|}^{2}<+\infty$.

Proof: $(i)$ follows from Proposition 21 $(i)$. $(ii)$: If $F\neq\emptyset$ then by Proposition 21 $(iv)$ the sequence is defined. We have $F\subset C\cap\left(\bigcap_{n\in{\mathbb{N}}}H(x_{n},T_{n}x_{n})\right)$ and thus $F\subset C_{n}$. Now, from $P_{F}(x_{0})\in C_{n}$ and $x_{n}=P_{C_{n}}x_{0}$ we obtain ${\|x_{n}-x_{0}\|}\leq{\|x_{0}-P_{F}(x_{0})\|}$ and $(ii)$ follows. $(iii)$ follows from $(i)$, $(ii)$ and the previous inequality. $(iv)$: The forward implication is trivial. For the reverse implication, the proof exactly follows $(iv)$ of [1, Theorem 3.5] since it does not involve $C$. $(v)$: From $x_{n}=P_{C_{n}}x_{0}$ and $x_{n+1}\in C_{n}$ we obtain:

[TABLE]

We thus have :

[TABLE]

Hence $\sum_{n\geq 0}{\|x_{n+1}-x_{n}\|}^{2}\leq\sup_{n\in{\mathbb{N}}}{\|x_{0}-x_{n}\|}^{2}<+\infty$ since $(x_{n})_{n\geq 0}$ is bounded. For all $n\in{\mathbb{N}}$ we have $x_{n+1}\in H(x_{n},T_{n}x_{n})$, which implies,

[TABLE]

and we therefore obtain $\sum_{n\geq 0}{\|x_{n}-T_{n}x_{n}\|}^{2}<+\infty$. $\Box$

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