Some results on parallel iterative algorithms for strictly pseudocontractive mappings

Abstract

In this paper, a parallel iterative algorithm with mixed errors is investigated. Strong and weak convergence theorems of common fixed points of a finite family of strictly pseudocontractive mappings are established in a real Banach space.

AMS Subject Classification:47H05, 47H09, 47J25.

1 Introduction and preliminaries

Throughout this paper, we denote by E and $E^{\ast }$ a real Banach space and a dual space of E, respectively. Let $〈\cdot ,\cdot 〉$ denote the pairing between E and $E^{\ast }$. The normalized duality mapping $J:E\to 2^{E^{\ast }}$ is defined by

$$J(x)=\{f\in E^{\ast }:〈x,f〉={\parallel x\parallel }^2={\parallel f\parallel }^2\},\mathrm{\forall }x\in E.$$

In the sequel, we use j to denote the single-valued normalized duality mapping. Let K be a nonempty subset of E and $T:K\to K$ be a mapping. Recall the following.

T is said to be Lipschitz if there exists a positive constant L such that

$$\parallel Tx-Ty\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in K.$$

T is said to be nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in K.$$

T is said to be strictly pseudocontractive in the terminology of Browder and Petryshyn [1] if there exists $\lambda >0$ such that

$$〈Tx-Ty,j(x-y)〉\le {\parallel x-y\parallel }^2-\lambda {\parallel (x-Tx)-(y-Ty)\parallel }^2,\mathrm{\forall }x,y\in K,$$

(1.1)

for some $j(x-y)\in J(x-y)$. It is clear that the class of strictly pseudocontractive mappings includes the class of nonexpansive mappings as a special case. It is also clear that (1.1) is equivalent to the following:

$$〈(I-T)x-(I_T)y,j(x-y)〉\ge \lambda {\parallel (I-T)x-(I-T)y\parallel }^2,\mathrm{\forall }x,y\in K.$$

(1.2)

We know that strictly pseudocontractive mappings are Lipschitz continuous. Indeed, we find from (1.2) that

$$\parallel Tx-Ty\parallel \le \frac{1+\lambda }{\lambda }\parallel x-y\parallel ,\mathrm{\forall }x,y\in K.$$

T is said to be strongly pseudocontractive if there exists $k\in (0,1)$ such that

$$〈Tx-Ty,j(x-y)〉\le k{\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in K,$$

(1.3)

for some $j(x-y)\in J(x-y)$. T is said to be pseudocontractive if there exists some $j(x-y)\in J(x-y)$ such that

$$〈Tx-Ty,j(x-y)〉\le {\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in K.$$

(1.4)

T is said to be Lipschitz if there exists a positive constant L such that

$$\parallel Tx-Ty\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in K.$$

It is well known that [2] (1.4) is equivalent to the following:

$$\parallel x-y\parallel \le \parallel x-y+s[(I-T)x-(I-T)y]\parallel ,\mathrm{\forall }s>0.$$

(1.5)

We remark that the class of strongly pseudocontractive mappings is independent of the class of strictly pseudocontractive mappings. This can be seen from the following examples.

Example 1.1 [3]

Take $K=(0,\mathrm{\infty })$ and define $T:K\to K$ by

$$Tx=\frac{x^2}{1+x}.$$

Then T is a strictly pseudocontractive mapping but not a strongly pseudocontractive mapping.

Example 1.2 [3]

Take $K=R$ and define $T:K\to K$ by

$$Tx=\{\begin{array}{cc}1,\hfill & x\in (-\mathrm{\infty },-1),\hfill \\ \sqrt{1-{(1+x)}^2},\hfill & x\in [-1,0),\hfill \\ -\sqrt{1-{(x-1)}^2},\hfill & x\in [0,1],\hfill \\ 1,\hfill & x\in (1,\mathrm{\infty }).\hfill \end{array}$$

Example 1.3 [4]

Take $E=R^2$, $B=\{x\in R^2:\parallel x\parallel \le 1\}$, $B_1=\{x\in B:\parallel x\parallel \le \frac{1}{2}\}$, $B_2=\{x\in B:\frac{1}{2}\le \parallel x\parallel \le 1\}$. If $x=(a,b)\in E$, we define $x^{\mathrm{\perp }}$ to be $(b,-a)\in E$. Define $T:B\to B$ by

$$Tx=\{\begin{array}{cc}x+x^{\mathrm{\perp }},\hfill & x\in B_1,\hfill \\ \frac{x}{\parallel x\parallel }-x+x^{\mathrm{\perp }},\hfill & x\in B_2.\hfill \end{array}$$

Then T is a Lipschitz pseudocontractive mapping but not a strictly pseudocontractive mapping.

Let $U=\{x\in E:\parallel x\parallel =1\}$. E is said to be smooth or is said to have a Gâteaux differentiable norm if the limit

$$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

exists for each $x,y\in U$. E is said to have a uniformly Gâteaux differentiable norm if for each $y\in U$, the limit is attained uniformly for all $x\in U$. E is said to be uniformly smooth or is said to have a uniformly Fréchet differentiable norm if the limit is attained uniformly for $x,y\in U$. It is known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single valued and uniformly norm to weak^∗ continuous on each bounded subset of E.

In 2001, Xu and Ori [5], in the framework of Hilbert spaces, introduced the following implicit iteration process for a finite family of nonexpansive mappings $\{T_1,T_2,\dots ,T_N\}$ with $\{{\alpha }_n\}$ a real sequence in $(0,1)$ and an initial point $x_0\in C$:

which can written in the following compact form:

$$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T_n{x}_n,\mathrm{\forall }n\ge 1,$$

(1.6)

where $T_n=T_{n(modN)}$ (here the modN takes values in $\{1,2,\dots ,N\}$).

They obtained the following weak convergence theorem.

Theorem XO Let H be a real Hilbert space, K be a nonempty closed convex subset of H, and $T_i:K\to K$ be a nonexpansive mapping such that $F={\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by (1.6). If $\{{\alpha }_n\}$ is chosen so that ${\alpha }_n\to 0$ as $n\to \mathrm{\infty }$, then $\{x_n\}$ converges weakly to a common fixed point of the family of ${\{T_i\}}_{i=1}^N$.

They further remarked that it is yet unclear what assumptions on the mappings and/or the parameters $\{{\alpha }_n\}$ are sufficient to guarantee the strong convergence of the sequence $\{x_n\}$.

In 2004, Osilike [6] further extended the above results from Hilbert spaces to Banach spaces. To be more precise, he obtain the following results.

Theorem O Let H be a real Hilbert space, K be a nonempty closed convex subset of H, and $T_i:K\to K$ be a strictly pseudocontractive mapping such that $F={\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be defined by (1.6). If $\{{\alpha }_n\}$ is chosen so that ${\alpha }_n\to 0$ as $n\to \mathrm{\infty }$, then $\{x_n\}$ converges weakly to a common fixed point of the family of ${\{T_i\}}_{i=1}^N$.

Subsequently, many authors have investigated the fixed point problem of strictly pseudocontractive mappings based on an implicit or non-implicit iterative algorithm in Banach spaces; see [7–32] and the references therein.

In 2007, Acedo and Xu proposed a parallel iterative algorithm for strictly pseudocontractive mappings in the framework of Hilbert spaces. Weak and strong convergence theorems for common fixed points of a family of strictly pseudocontractive mappings were established; see [26] for more details and the reference therein.

In this paper, motivated by the above results, we consider an implicitly parallel iterative algorithm for a finite family of strictly pseudocontractive mappings. Weak and strong convergence theorems are established in the framework of Banach spaces.

In order to prove our main results, we need the following conceptions and lemmas.

Recall that the space E is said to satisfy Opial’s condition [33] if, for each sequence $\{x_n\}$ in E, the convergence $x_n\to x$ weakly implies that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-x\parallel <\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-y\parallel ,\mathrm{\forall }y\in E(y\ne x).$$

Recall that the mapping $T:K\to K$ is semicompact if any sequence $\{x_n\}$ in K satisfying ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T{x}_n\parallel =0$ has a convergent subsequence.

Lemma 1.1 [34]

Let E be a real Banach space, K be a nonempty closed convex subset of E, and $T:K\to K$ be a continuous strongly pseudocontractive mapping. Then T has a unique fixed point in K.

Lemma 1.2 [11]

Let E be a smooth Banach space and K be a nonempty convex subset of E. Let $r\ge 1$ be some integer. Let $T_i:K\to K$ be a strictly pseudocontractive mapping. Assume that ${\bigcap }_{i=1}^{r}F(T_i)$ is not empty. Assume that ${\{{\mu }_i\}}_{i=1}^r$ is a positive sequence such that ${\sum }_{i=1}^r{\mu }_i=1$. Then ${\bigcap }_{i=1}^{r}F(T_i)=F({\sum }_{i=1}^r{\mu }_i{T}_i)$.

Lemma 1.3 [35]

Let $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be three nonnegative sequences satisfying the following condition:

$$a_{n+1}\le (1+b_n)a_n+c_n,\mathrm{\forall }n\ge n_0,$$

where $n_0$ is some nonnegative integer, ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n<\mathrm{\infty }$, and ${\sum }_{n=1}^{\mathrm{\infty }}{c}_n<\mathrm{\infty }$. Then the limit ${lim}_{n\to \mathrm{\infty }}{a}_n$ exists.

2 Main results

Theorem 2.1 Let E be a smooth and reflexive Banach space which also satisfies Opial’s condition and K be a nonempty closed convex subset of E. Let $N\ge 1$ be some positive integer. Let $T_m:K\to K$, where $m\in \{1,\dots ,N\}$, be a ${\lambda }_i$-strictly pseudocontractive mapping and $\{u_n\}$ be a bounded sequence in K. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated in the following algorithm:

$$x_0\in K,x_n={\alpha }_n{x}_{n-1}+{\beta }_n\sum _{m=1}^N{\delta }_m{T}_m{x}_n+{\gamma }_n{u}_n,n\ge 1,$$

(2.1)

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\delta }_m\}$ are real number sequences in $[0,1]$. Assume that $F:={\bigcap }_{m=1}^{N}F(T_m)\ne \mathrm{\varnothing }$, and the above control sequences satisfy the following restrictions:

• (a) ${\sum }_{m=1}^N{\delta }_m={\alpha }_n+{\beta }_n+{\gamma }_n=1$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n<\mathrm{\infty }$;

• (c) $0<a\le {\alpha }_n\le b<1$, where a and b are constants.

Then $\{x_n\}$ converges weakly to some point in F.

Proof Put $T:={\sum }_{m=1}^N{\delta }_m{T}_m$. We show that T is λ-strictly pseudocontractive mapping, where $\lambda :=min\{{\lambda }_m:1\le m\le N\}$. Notice that

This proves that T is λ-strictly pseudocontractive mapping. Next, we show that the implicit iterative algorithm (2.1) is well defined for the strictly pseudocontractive mappings. Define a mapping

$$P_n(x)={\alpha }_n{x}_{n-1}+{\beta }_n\sum _{m=1}^N{\delta }_m{T}_{m}x+{\gamma }_n{u}_n,\mathrm{\forall }n\ge 1.$$

It follows that

This shows that $P_n$ is strongly pseudocontractive. Since strictly pseudocontractive mappings are Lipschitz continuous, we see that $P_n$ is also continuous. In view of Lemma 1.1, we see that $P_n$ has a unique fixed point. This proves that the implicit iterative algorithm (2.1) is well defined. In view of Lemma 1.2, we see that $F=F({\sum }_{m=1}^N{\delta }_{m}F(T_m))=F(T)$. Fixing $p\in F$, we see that

$$\begin{array}{rcl}{\parallel x_n-p\parallel }^2& =& {\alpha }_n〈x_{n-1}-p,j(x-p)〉+{\beta }_n〈\sum _{m=1}^N{\delta }_m{T}_m{x}_n-p,j(x-y)〉+{\gamma }_n〈u_n-p,j(x-y)〉\\ =& {\alpha }_n〈x_{n-1}-p,j(x-p)〉+{\beta }_n〈T{x}_n-p,j(x-p)〉+{\gamma }_n〈u_n-p,j(x-p)〉\\ \le & {\alpha }_n\parallel x_{n-1}-p\parallel \parallel x-p\parallel +{\beta }_n{\parallel x-p\parallel }^2-{\beta }_n\lambda {\parallel x_n-T{x}_n\parallel }^2+{\gamma }_n\parallel u_n-p\parallel \parallel x-p\parallel .\end{array}$$

It follows that

$$\parallel x_n-p\parallel \le {\alpha }_n\parallel x_{n-1}-p\parallel +{\beta }_n\parallel x_n-p\parallel -{\beta }_n\lambda {\parallel x_n-T{x}_n\parallel }^2+{\gamma }_n\parallel u_n-p\parallel .$$

This implies, from the restriction (c), that

$$\parallel x_n-p\parallel \le \parallel x_{n-1}-p\parallel +{\gamma }_{n}M-{\beta }_n\lambda {\parallel x_n-T{x}_n\parallel }^2,$$

(2.2)

where M is an appropriate constant such that $M\ge {sup}_{n\ge 1}\{\frac{\parallel u_n-p\parallel }{a}\}$. In view of Lemma 1.3, we obtain that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exits. Thanks to (2.2), we find from the restrictions (b) and (c) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T{x}_n\parallel =0.$$

(2.3)

Since the space is reflexive and $\{x_n\}$ is bounded, there exists a subsequence $\{x_{n_i}\}$ of the sequence $\{x_n\}$, which weakly converges to some x∈. In view of Lemma 1.4, we find that $x\in F(T)=F$.

Finally, we show the sequence $\{x_n\}$ weakly converges to x. Suppose the contrary, then there exists some subsequence $\{x_{n_j}\}$ of the sequence $\{x_n\}$ which weakly converges to $x^{\prime }\ne x\in C$. It also follows from Lemma 1.4 that $x^{\prime }\in F$. Since ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exits for any $p\in F$. Put

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x\parallel =d,\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x^{\prime }\parallel =d^{\prime }.$$

Since the space satisfies Opial’s condition, we see that

$$\begin{array}{rl}d& =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x\parallel \\ =\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_i}-x\parallel \\ <\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_i}-x^{\prime }\parallel \\ =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x^{\prime }\parallel \\ =\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-x^{\prime }\parallel \\ <\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-x\parallel \\ =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x\parallel =d.\end{array}$$

This is a contradiction. This shows that $x=x^{\prime }$. This proves that the sequence $\{x_n\}$ converges weakly to $x\in F$. This completes the proof. □

Corollary 2.2 Let E be a smooth and reflexive Banach space which also satisfies Opial’s condition and let K be a nonempty closed convex subset of E. Let $N\ge 1$ be some positive integer. Let $T_m:K\to K$, where $m\in \{1,\dots ,N\}$, be a ${\lambda }_i$-strictly pseudocontractive mapping. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated in the following algorithm:

$$x_0\in K,x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)\sum _{m=1}^N{\delta }_m{T}_m{x}_n,n\ge 1,$$

where $\{{\alpha }_n\}$ and $\{{\delta }_m\}$ are real number sequences in $[0,1]$. Assume that $F:={\bigcap }_{m=1}^{N}F(T_m)\ne \mathrm{\varnothing }$ and the above control sequences satisfy the following restrictions:

• (a) ${\sum }_{m=1}^N{\delta }_m=1$;

• (b) $0<a\le {\alpha }_n\le b<1$, where a and b are constants.

Then $\{x_n\}$ converges weakly to some point in F.

In Hilbert spaces, we find from Theorem 2.1 the following.

Corollary 2.3 Let E be a Hilbert space and K be a nonempty closed convex subset of E. Let $N\ge 1$ be some positive integer. Let $T_m:K\to K$, where $m\in \{1,\dots ,N\}$, be a ${\lambda }_i$-strictly pseudocontractive mapping and $\{u_n\}$ be a bounded sequence in K. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated in the following algorithm:

$$x_0\in K,x_n={\alpha }_n{x}_{n-1}+{\beta }_n\sum _{m=1}^N{\delta }_m{T}_m{x}_n+{\gamma }_n{u}_n,n\ge 1,$$

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\delta }_m\}$ are real number sequences in $[0,1]$. Assume that $F:={\bigcap }_{m=1}^{N}F(T_m)\ne \mathrm{\varnothing }$ and the above control sequences satisfy the following restrictions:

• (a) ${\sum }_{m=1}^N{\delta }_m={\alpha }_n+{\beta }_n+{\gamma }_n=1$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n<\mathrm{\infty }$;

• (c) $0<a\le {\alpha }_n\le b<1$, where a and b are constants.

Then $\{x_n\}$ converges weakly to some point in F.

Next, we give a strong convergence theorem.

Theorem 2.4 Let E be a smooth and reflexive Banach space and K be a nonempty closed convex subset of E. Let $N\ge 1$ be some positive integer. Let $T_m:K\to K$, where $m\in \{1,\dots ,N\}$, be a ${\lambda }_i$-strictly pseudocontractive mapping and $\{u_n\}$ be a bounded sequence in K. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated in the following algorithm:

$$x_0\in K,x_n={\alpha }_n{x}_{n-1}+{\beta }_n\sum _{m=1}^N{\delta }_m{T}_m{x}_n+{\gamma }_n{u}_n,n\ge 1,$$

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\delta }_m\}$ are real number sequences in $[0,1]$. Assume that $F:={\bigcap }_{m=1}^{N}F(T_m)\ne \mathrm{\varnothing }$ and the above control sequences satisfy the following restrictions:

• (a) ${\sum }_{m=1}^N{\delta }_m={\alpha }_n+{\beta }_n+{\gamma }_n=1$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n<\mathrm{\infty }$;

• (c) $0<a\le {\alpha }_n\le b<1$, where a and b are constants.

If ${\sum }_{m=1}^N{\delta }_m{T}_m$ is semicompact, then $\{x_n\}$ converges strongly to some point in F.

Proof Since ${\sum }_{m=1}^N{\delta }_m{T}_m$ is semicompact, we see that there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $x_{n_i}\to x^{\ast }$. Notice that

$$\begin{array}{rcl}\parallel x^{\ast }-\sum _{m=1}^N{\delta }_m{T}_m{x}^{\ast }\parallel & \le & \parallel x^{\ast }-x_{n_i}\parallel +\parallel x_{n_i}-\sum _{m=1}^N{\delta }_m{T}_m{x}_{n_i}\parallel \\ +\parallel \sum _{m=1}^N{\delta }_m{T}_m{x}_{n_i}-\sum _{m=1}^N{\delta }_m{T}_m{x}_{n_i}{x}^{\ast }\parallel .\end{array}$$

Since ${\sum }_{m=1}^N{\delta }_m{T}_m$ is Lipschitz continuous, we see from (2.3) that $x^{\ast }\in F({\sum }_{m=1}^N{\delta }_m{T}_m)=F$. From Theorem 2.1, we know that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exits for any $p\in F$. That is, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x^{\ast }\parallel$ exits. In view of $x_{n_i}\to x^{\ast }$, we find that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x^{\ast }\parallel =0.$$

This completes the proof. □

Corollary 2.5 Let E be a smooth and reflexive Banach space and K be a nonempty closed convex subset of E. Let $N\ge 1$ be some positive integer. Let $T_m:K\to K$, where $m\in \{1,\dots ,N\}$, be a ${\lambda }_i$-strictly pseudocontractive mapping. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated in the following algorithm:

$$x_0\in K,x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)\sum _{m=1}^N{\delta }_m{T}_m{x}_n,n\ge 1,$$

where $\{{\alpha }_n\}$ and $\{{\delta }_m\}$ are real number sequences in $[0,1]$. Assume that $F:={\bigcap }_{m=1}^{N}F(T_m)\ne \mathrm{\varnothing }$ and the above control sequences satisfy the following restrictions:

• (a) ${\sum }_{m=1}^N{\delta }_m=1$;

• (b) $0<a\le {\alpha }_n\le b<1$, where a and b are constants.

If ${\sum }_{m=1}^N{\delta }_m{T}_m$ is semicompact, then $\{x_n\}$ converges strongly to some point in F.

In Hilbert spaces, we find from Theorem 2.1 the following.

Corollary 2.6 Let E be a Hilbert space and K be a nonempty closed convex subset of E. Let $N\ge 1$ be some positive integer. Let $T_m:K\to K$, where $m\in \{1,\dots ,N\}$, be a ${\lambda }_i$-strictly pseudocontractive mapping and $\{u_n\}$ be a bounded sequence in K. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated in the following algorithm:

$$x_0\in K,x_n={\alpha }_n{x}_{n-1}+{\beta }_n\sum _{m=1}^N{\delta }_m{T}_m{x}_n+{\gamma }_n{u}_n,n\ge 1,$$

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\delta }_m\}$ are real number sequences in $[0,1]$. Assume that $F:={\bigcap }_{m=1}^{N}F(T_m)\ne \mathrm{\varnothing }$ and the above control sequences satisfy the following restrictions:

• (a) ${\sum }_{m=1}^N{\delta }_m={\alpha }_n+{\beta }_n+{\gamma }_n=1$;

• (b) ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n<\mathrm{\infty }$;

• (c) $0<a\le {\alpha }_n\le b<1$, where a and b are constants.

If ${\sum }_{m=1}^N{\delta }_m{T}_m$ is semicompact, then $\{x_n\}$ converges strongly to some point in F.