The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

Abstract

Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and $T:D\to D$ be a generalized Lipschitz Φ-hemi-contractive mapping with $q\in F(T)\ne \mathrm{\varnothing }$. Let $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{d_n\}$ be four real sequences in $[0,1]$ and satisfy the conditions (i) $a_n,b_n,d_n\to 0$ as $n\to \mathrm{\infty }$ and $c_n=o(a_n)$; (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$. For some $x_0\in D$, let $\{u_n\}$, $\{v_n\}$ be any bounded sequences in D, and $\{x_n\}$ be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φ-quasi-accretive mapping.

MSC:47H10.

1 Introduction and preliminary

Let E be a real Banach space and $E^{\ast }$ be its dual space. The normalized duality mapping $J:E\to 2^{E^{\ast }}$ is defined by

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

where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing. It is well known that

• (i) If E is a smooth Banach space, then the mapping J is single-valued;

• (ii) $J(\alpha x)=\alpha J(x)$ for all $x\in E$ and $\alpha \in \mathrm{\Re }$;

• (iii) If E is a uniformly smooth Banach space, then the mapping J is uniformly continuous on any bounded subset of E. We denote the single-valued normalized duality mapping by j.

Definition 1.1 ([1])

Let D be a nonempty closed convex subset of E, $T:D\to D$ be a mapping.

• (1) T is called strongly pseudocontractive if there is a constant $k\in (0,1)$ such that for all $x,y\in D$,

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

• (2) T is called ϕ-strongly pseudocontractive if for all $x,y\in D$, there exist $j(x-y)\in J(x-y)$ and a strictly increasing continuous function $\varphi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ with $\varphi (0)=0$ such that

  $$〈Tx-Ty,j(x-y)〉\le {\parallel x-y\parallel }^2-\varphi (\parallel x-y\parallel )\parallel x-y\parallel ;$$

• (3) T is called Φ-pseudocontractive if for all $x,y\in D$, there exist $j(x-y)\in J(x-y)$ and a strictly increasing continuous function $\mathrm{\Phi }:[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ with $\mathrm{\Phi }(0)=0$ such that

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

It is obvious that Φ-pseudocontractive mappings not only include ϕ-strongly pseudocontractive mappings, but also strongly pseudocontractive mappings.

Definition 1.2 ([1])

Let $T:D\to D$ be a mapping and $F(T)=\{x\in D:Tx=x\}\ne \mathrm{\varnothing }$.

• (1) T is called ϕ-strongly-hemi-pseudocontractive if for all $x\in D$, $q\in F(T)$, there exist $j(x-q)\in J(x-q)$ and a strictly increasing continuous function $\varphi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ with $\varphi (0)=0$ such that

  $$〈Tx-Tq,j(x-q)〉\le {\parallel x-q\parallel }^2-\varphi (\parallel x-q\parallel )\parallel x-q\parallel .$$

• (2) T is called Φ-hemi-pseudocontractive if for all $x\in D$, $q\in F(T)$, there exist $j(x-q)\in J(x-q)$ and the strictly increasing continuous function $\mathrm{\Phi }:[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ with $\mathrm{\Phi }(0)=0$ such that

  $$〈Tx-Tq,j(x-q)〉\le {\parallel x-q\parallel }^2-\mathrm{\Phi }(\parallel x-q\parallel ).$$

Closely related to the class of pseudocontractive-type mappings are those of accretive type.

Definition 1.3 ([1])

Let $N(T)=\{x\in E:Tx=0\}\ne \mathrm{\varnothing }$. The mapping $T:E\to E$ is called strongly quasi-accretive if for all $x\in E$, $x^{\ast }\in N(T)$, there exist $j(x-x^{\ast })\in J(x-x^{\ast })$ and a constant $k\in (0,1)$ such that $〈Tx-T{x}^{\ast },j(x-x^{\ast })〉\ge k{\parallel x-x^{\ast }\parallel }^2$; T is called ϕ-strongly quasi-accretive if for all $x\in E$, $x^{\ast }\in N(T)$, there exist $j(x-x^{\ast })\in J(x-x^{\ast })$ and a strictly increasing continuous function $\varphi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ with $\varphi (0)=0$ such that $〈Tx-T{x}^{\ast },j(x-x^{\ast })〉\ge \varphi (\parallel x-x^{\ast }\parallel )\parallel x-x^{\ast }\parallel$; T is called Φ-quasi-accretive if for all $x\in E$, $x^{\ast }\in N(T)$, there exist $j(x-x^{\ast })\in J(x-x^{\ast })$ and a strictly increasing continuous function $\mathrm{\Phi }:[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ with $\mathrm{\Phi }(0)=0$ such that $〈Tx-T{x}^{\ast },j(x-x^{\ast })〉\ge \mathrm{\Phi }(\parallel x-x^{\ast }\parallel )$.

Definition 1.4 ([2])

For arbitrary given $x_0\in D$, the Ishikawa iterative process with errors ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ is defined by

$$\{\begin{array}{c}{y}_n=(1-b_n-d_n)x_n+b_{n}T{x}_n+d_n{v}_n,n\ge 0,\hfill \\ x_{n+1}=(1-a_n-c_n)x_n+a_{n}T{y}_n+c_n{u}_n,n\ge 0,\hfill \end{array}$$

(1.1)

where $\{u_n\}$, $\{v_n\}$ are any bounded sequences in D; $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{d_n\}$ are four real sequences in $[0,1]$ and satisfy $a_n+c_n\le 1$, $b_n+d_n\le 1$, for all $n\ge 0$. If $b_n=d_n=0$, then the sequence $\{x_n\}$ defined by

$$x_{n+1}=(1-a_n-c_n)x_n+a_{n}T{x}_n+c_n{u}_n,n\ge 0$$

(1.2)

is called the Mann iterative process with errors.

Definition 1.5 ([3, 4])

A mapping $T:D\to D$ is called generalized Lipschitz if there exists a constant $L>0$ such that $\parallel Tx-Ty\parallel \le L(1+\parallel x-y\parallel )$, $\mathrm{\forall }x,y\in D$.

The aim of this paper is to prove the convergent results of the above Ishikawa and Mann iterations with errors for generalized Lipschitz Φ-hemi-contractive mappings in uniformly smooth real Banach spaces. For this, we need the following lemmas.

Lemma 1.6 ([5])

Let E be a uniformly smooth real Banach space, and let $J:E\to 2^{E^{\ast }}$ be a normalized duality mapping. Then

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,J(x+y)〉$$

(1.3)

for all $x,y\in E$.

Lemma 1.7 ([6])

Let ${\{{\rho }_n\}}_{n=0}^{\mathrm{\infty }}$ be a nonnegative sequence which satisfies the following inequality:

$${\rho }_{n+1}\le (1-{\lambda }_n){\rho }_n+{\sigma }_n,n\ge 0,$$

(1.4)

where ${\lambda }_n\in [0,1]$ with ${\sum }_{n=0}^{\mathrm{\infty }}{\lambda }_n=\mathrm{\infty }$, ${\sigma }_n=o({\lambda }_n)$. Then ${\rho }_n\to 0$ as $n\to \mathrm{\infty }$.

2 Main results

Theorem 2.1 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and $T:D\to D$ be a generalized Lipschitz Φ-hemi-contractive mapping with $q\in F(T)\ne \mathrm{\varnothing }$. Let $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{d_n\}$ be four real sequences in $[0,1]$ and satisfy the conditions (i) $a_n,b_n,d_n\to 0$ as $n\to \mathrm{\infty }$ and $c_n=o(a_n)$; (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$. For some $x_0\in D$, let $\{u_n\}$, $\{v_n\}$ be any bounded sequences in D, and $\{x_n\}$ be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the unique fixed point q of T.

Proof Since $T:D\to D$ is a generalized Lipschitz Φ-hemi-contractive mapping, there exists a strictly increasing continuous function $\mathrm{\Phi }:[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ with $\mathrm{\Phi }(0)=0$ such that

$$〈Tx-Tq,J(x-q)〉\le {\parallel x-q\parallel }^2-\mathrm{\Phi }(\parallel x-q\parallel ),$$

i.e.,

$$-〈x-Tx,J(x-q)〉\le -\mathrm{\Phi }(\parallel x-q\parallel ),$$

and

$$\parallel Tx-Ty\parallel \le L(1+\parallel x-y\parallel ),$$

for any $x,y\in D$ and $q\in F(T)$.

Step 1. There exists $x_0\in D$ and $x_0\ne T{x}_0$ such that $r_0=\parallel x_0-T{x}_0\parallel \cdot \parallel x_0-q\parallel \in R(\mathrm{\Phi })$ (range of Φ). Indeed, if $\mathrm{\Phi }(r)\to +\mathrm{\infty }$ as $r\to +\mathrm{\infty }$, then $r_0\in R(\mathrm{\Phi })$; if $sup\{\mathrm{\Phi }(r):r\in [0,+\mathrm{\infty })\}=r_1<+\mathrm{\infty }$ with $r_0<r_1$, then for $q\in D$, there exists a sequence $\{w_n\}$ in D such that $w_n\to q$ as $n\to \mathrm{\infty }$ with $w_n\ne q$. Furthermore, we obtain that $\{w_n-T{w}_n\}$ is bounded. Hence, there exists a natural number $n_0$ such that $\parallel w_n-T{w}_n\parallel \cdot \parallel w_n-q\parallel <\frac{r_1}{2}$ for $n\ge n_0$, then we redefine $x_0=w_{n_0}$ and $\parallel x_0-T{x}_0\parallel \cdot \parallel x_0-q\parallel \in R(\mathrm{\Phi })$.

Step 2. For any $n\ge 0$, $\{x_n\}$ is bounded. Set $R={\mathrm{\Phi }}^{-1}(r_0)$, then from Definition 1.2(2), we obtain that $\parallel x_0-q\parallel \le R$. Denote $B_1=\{x\in D:\parallel x-q\parallel \le R\}$, $B_2=\{x\in D:\parallel x-q\parallel \le 2R\}$. Since T is generalized Lipschitz, so T is bounded. We may define $M={sup}_{x\in B_2}\{\parallel Tx-q\parallel +1\}+{sup}_n\{\parallel u_n-q\parallel \}+{sup}_n\{\parallel v_n-q\parallel \}$. Next, we want to prove that $x_n\in B_1$. If $n=0$, then $x_0\in B_1$. Now, assume that it holds for some n, i.e., $x_n\in B_1$. We prove that $x_{n+1}\in B_1$. Suppose it is not the case, then $\parallel x_{n+1}-q\parallel >R$. Since J is uniformly continuous on a bounded subset of E, then for ${ϵ}_0=\frac{\mathrm{\Phi }(\frac{R}{4})}{24L(1+2R)}$, there exists $\delta >0$ such that $\parallel Jx-Jy\parallel <ϵ$ when $\parallel x-y\parallel <\delta$, $\mathrm{\forall }x,y\in B_2$. Now, denote

$$\begin{array}{rcl}{\tau }_0& =& min\{1,\frac{R}{2[L(1+2R)+2R+M]},\frac{R}{4[L(1+R)+2R+M]},\\ \frac{\delta }{2[L(1+2R)+2R+M]},\frac{\mathrm{\Phi }(\frac{R}{4})}{24R^2},\frac{\mathrm{\Phi }(\frac{R}{4})}{24L(1+2R)},\frac{\mathrm{\Phi }(\frac{R}{4})}{48MR}\}.\end{array}$$

Owing to $a_n,b_n,c_n,d_n\to 0$ as $n\to \mathrm{\infty }$, without loss of generality, assume that $0\le a_n,b_n,c_n,d_n\le {\tau }_0$ for any $n\ge 0$. Since $c_n=o(a_n)$, denote $c_n<a_n{\tau }_0$. So, we have

(2.1)

(2.2)

(2.3)

(2.4)

and

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

Therefore,

$$\begin{array}{c}\parallel J(x_n-q)-J(y_n-q)\parallel <{ϵ}_0;\hfill \\ \parallel J(x_{n+1}-q)-J(x_n-q)\parallel <{ϵ}_0.\hfill \end{array}$$

Using Lemma 1.6 and the above formulas, we obtain

(2.10)

and

(2.11)

Substitute (2.11) into (2.10)

(2.12)

this is a contradiction. Thus, $x_{n+1}\in B_1$, i.e., $\{x_n\}$ is a bounded sequence. So, $\{y_n\}$, $\{T{y}_n\}$, $\{T{x}_n\}$ are all bounded sequences.

Step 3. We want to prove $\parallel x_n-q\parallel \to 0$ as $n\to \mathrm{\infty }$. Set $M_1=max\{{sup}_n\parallel x_n-q\parallel ,{sup}_n\parallel y_n-q\parallel ,{sup}_n\parallel T{x}_n-q\parallel ,{sup}_n\parallel T{y}_n-q\parallel ,{sup}_n\parallel u_n-q\parallel ,{sup}_n\parallel v_n-q\parallel \}$.

By (2.10), (2.11), we have

(2.13)

and

(2.14)

where $A_n=\parallel J(x_{n+1}-q)-J(x_n-q)\parallel$, $B_n=\parallel J(x_n-q)-J(y_n-q)\parallel$ and $A_n,B_n\to 0$ as $n\to \mathrm{\infty }$.

Taking (2.14) into (2.13),

(2.15)

where $C_n=\frac{a_n}{2}{M}_1^2+M_1{A}_n+3M_1{B}_n+2b_n{M}_1^2+2d_n{M}_1^2+\frac{c_n{M}_1^2}{a_n}\to 0$ as $n\to \mathrm{\infty }$.

Set ${inf}_{n\ge 0}\frac{\mathrm{\Phi }(\parallel y_n-q\parallel )}{1+{\parallel x_{n+1}-q\parallel }^2}=\lambda$, then $\lambda =0$. If it is not the case, we assume that $\lambda >0$. Let $0<\gamma <min\{1,\lambda \}$, then $\frac{\mathrm{\Phi }(\parallel y_n-q\parallel )}{1+{\parallel x_{n+1}-q\parallel }^2}\ge \gamma$, i.e., $\mathrm{\Phi }(\parallel y_n-q\parallel )\ge \gamma +\gamma {\parallel x_{n+1}-q\parallel }^2\ge \gamma {\parallel x_{n+1}-q\parallel }^2$. Thus, from (2.15) it follows that

(2.16)

This implies that

(2.17)

Let ${\rho }_n={\parallel x_n-q\parallel }^2$, ${\lambda }_n=\frac{2a_n\gamma }{1+2a_n\gamma }$, ${\sigma }_n=\frac{2a_n{C}_n}{1+2a_n\gamma }$. Then we get that

$${\rho }_{n+1}\le (1-{\lambda }_n){\rho }_n+{\sigma }_n.$$

Applying Lemma 1.7, we get that ${\rho }_n\to 0$ as $n\to \mathrm{\infty }$. This is a contradiction and so $\lambda =0$. Therefore, there exists an infinite subsequence such that $\frac{\mathrm{\Phi }(\parallel y_{n_i}-q\parallel )}{1+{\parallel x_{n_i+1}-q\parallel }^2}\to 0$ as $i\to \mathrm{\infty }$. Since $0\le \frac{\mathrm{\Phi }(\parallel y_{n_i}-q\parallel )}{1+M_1^2}\le \frac{\mathrm{\Phi }(\parallel y_{n_i}-q\parallel )}{1+{\parallel x_{n_i+1}-q\parallel }^2}$, then $\mathrm{\Phi }(\parallel y_{n_i}-q\parallel )\to 0$ as $i\to \mathrm{\infty }$. In view of the strictly increasing continuity of Φ, we have $\parallel y_{n_i}-q\parallel \to 0$ as $i\to \mathrm{\infty }$. Hence, $\parallel x_{n_i}-q\parallel \to 0$ as $i\to \mathrm{\infty }$. Next, we want to prove $\parallel x_n-q\parallel \to 0$ as $n\to \mathrm{\infty }$. Let $\mathrm{\forall }\epsilon \in (0,1)$, there exists $n_{i_0}$ such that $\parallel x_{n_i}-q\parallel <ϵ$, $a_n,c_n<\frac{ϵ}{8M_1}$, $b_n,d_n<\frac{ϵ}{16M_1}$, $C_n<\frac{\mathrm{\Phi }(ϵ)}{2}$, for any $n_i,n\ge n_{i_0}$. First, we want to prove $\parallel x_{n_i+1}-q\parallel <ϵ$. Suppose it is not the case, then $\parallel x_{n_i+1}-q\parallel \ge ϵ$. Using (1.1), we may get the following estimates:

(2.18)

(2.19)

Since Φ is strictly increasing, then (2.19) leads to $\mathrm{\Phi }(\parallel y_{n_i}-q\parallel )\ge \mathrm{\Phi }(\frac{ϵ}{4})$. From (2.15), we have

(2.20)

which is a contradiction. Hence, $\parallel x_{n_i+1}-q\parallel <ϵ$. Suppose that $\parallel x_{n_i+m}-q\parallel <ϵ$ holds. Repeating the above course, we can easily prove that $\parallel x_{n_i+m+1}-q\parallel <ϵ$ holds. Therefore, for any m, we obtain that $\parallel x_{n_i+m}-q\parallel <ϵ$, which means $\parallel x_n-q\parallel \to 0$ as $n\to \mathrm{\infty }$. This completes the proof. □

Theorem 2.2 Let E be an arbitrary uniformly smooth real Banach space, and $T:E\to E$ be a generalized Lipschitz Φ-quasi-accretive mapping with $N(T)\ne \mathrm{\varnothing }$. Let $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{d_n\}$ be four real sequences in $[0,1]$ and satisfy the conditions (i) $a_n,b_n,d_n\to 0$ as $n\to \mathrm{\infty }$ and $c_n=o(a_n)$; (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$. For some $x_0\in D$, let $\{u_n\}$, $\{v_n\}$ be any bounded sequences in E, and $\{x_n\}$ be an Ishikawa iterative sequence with errors defined by

$$\{\begin{array}{c}{y}_n=(1-b_n-d_n)x_n+b_{n}S{x}_n+d_n{v}_n,n\ge 0,\hfill \\ x_{n+1}=(1-a_n-c_n)x_n+a_{n}S{y}_n+c_n{u}_n,n\ge 0,\hfill \end{array}$$

(2.21)

where $S:E\to E$ is defined by $Sx=x-Tx$ for any $x\in E$. Then $\{x_n\}$ converges strongly to the unique solution of the equation $Tx=0$ (or the unique fixed point of S).

Proof Since T is a generalized Lipschitz and Φ-quasi-accretive mapping, it follows that

$$\parallel Tx-Ty\parallel \le L(1+\parallel x-y\parallel ),$$

i.e.,

$$\begin{array}{c}\parallel Sx-Sy\parallel \le L_1(1+\parallel x-y\parallel ),L_1=1+L;\hfill \\ 〈Tx-Tq,J(x-q)〉\ge \mathrm{\Phi }(\parallel x-q\parallel ),\hfill \end{array}$$

i.e.,

$$〈Sx-Sq,J(x-q)〉\le {\parallel x-q\parallel }^2-\mathrm{\Phi }(\parallel x-q\parallel ),$$

for all $x,y\in E$, $q\in N(T)$. The rest of the proof is the same as that of Theorem 2.1. □

Corollary 2.3 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and $T:D\to D$ be a generalized Lipschitz Φ-hemi-contractive mapping with $q\in F(T)\ne \mathrm{\varnothing }$. Let $\{a_n\}$, $\{c_n\}$ be two real sequences in $[0,1]$ and satisfy the conditions (i) $a_n\to 0$ as $n\to \mathrm{\infty }$ and $c_n=o(a_n)$; (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$. For some $x_0\in D$, let $\{u_n\}$ be any bounded sequence in D, and $\{x_n\}$ be the Mann iterative sequence with errors defined by (1.2). Then (1.2) converges strongly to the unique fixed point q of T.

Corollary 2.4 Let E be an arbitrary uniformly smooth real Banach space, and $T:E\to E$ be a generalized Lipschitz Φ-quasi-accretive mapping with $N(T)\ne \mathrm{\varnothing }$. Let $\{a_n\}$, $\{d_n\}$ be two real sequences in $[0,1]$ and satisfy the conditions (i) $a_n\to 0$ as $n\to \mathrm{\infty }$ and $c_n=o(a_n)$; (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n=\mathrm{\infty }$. For some $x_0\in D$, let $\{u_n\}$ be any bounded sequence in E, and $\{x_n\}$ be the Mann iterative sequence with errors defined by

$$x_{n+1}=(1-a_n-c_n)x_n+a_{n}S{x}_n+c_n{u}_n,n\ge 0,$$

(2.22)

where $S:E\to E$ is defined by $Sx=x-Tx$ for any $x\in E$. Then $\{x_n\}$ converges strongly to the unique solution of the equation $Tx=0$ (or the unique fixed point of S).

Remark 2.5 It is mentioned that in 2006, Chidume and Chidume [1] proved the approximative theorem for zeros of generalized Lipschitz generalized Φ-quasi-accretive operators. This result provided significant improvements of some recent important results. Their result is as follows.

Theorem CC ([[1], Theorem 3.1])

Let E be a uniformly smooth real Banach space and $A:E\to E$ be a mapping with $N(A)\ne \mathrm{\varnothing }$. Suppose A is a generalized Lipschitz Φ-quasi-accretive mapping. Let $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ be real sequences in $[0,1]$ satisfying the following conditions: (i) $a_n+b_n+c_n=1$; (ii) ${\sum }_{n=0}^{\mathrm{\infty }}(b_n+c_n)=\mathrm{\infty }$; (iii) ${\sum }_{n=0}^{\mathrm{\infty }}{c}_n<\mathrm{\infty }$; (iv) ${lim}_{n\to \mathrm{\infty }}{b}_n=0$. Let $\{x_n\}$ be generated iteratively from arbitrary $x_0\in E$ by

$$x_{n+1}=a_n{x}_n+b_{n}S{x}_n+c_n{u}_n,n\ge 0,$$

(2.23)

where $S:E\to E$ is defined by $Sx:=f+x-Ax$, $\mathrm{\forall }x\in E$ and $\{u_n\}$ is an arbitrary bounded sequence in E. Then, there exists ${\gamma }_0\in \mathrm{\Re }$ such that if $b_n+c_n\le {\gamma }_0$, $\mathrm{\forall }n\ge 0$, the sequence $\{x_n\}$ converges strongly to the unique solution of the equation $Au=0$.

However, there exists a gap in the proof process of above Theorem CC. Here, $c_n=min\{\frac{ϵ}{4\beta },\frac{1}{2\sigma }\mathrm{\Phi }(\frac{ϵ}{2}){\alpha }_n\}$ (${\alpha }_n=b_n+c_n$) does not hold in line 14 of Claim 2 on page 248, i.e., $c_n\le \frac{1}{2\sigma }\mathrm{\Phi }(\frac{ϵ}{2}){\alpha }_n$ is a wrong case. For instance, set the iteration parameters: $a_n=1-b_n-c_n$, where $\{b_n\}:b_1=\frac{1}{4}$, $b_n=\frac{1}{n}$, $n\ge 2$; $\{c_n\}:\frac{1}{4},\frac{1}{2^2},\frac{1}{3^2},\frac{1}{4},\frac{1}{5^2},\dots ,\frac{1}{8^2},\frac{1}{9},\frac{1}{{10}^2},\dots ,\frac{1}{{15}^2},\frac{1}{16},\frac{1}{{17}^2},\dots ,\frac{1}{{24}^2},\frac{1}{25},\frac{1}{{26}^2},\dots ,\frac{1}{{35}^2},\frac{1}{36},\frac{1}{{37}^2},\dots$ . Then ${\sum }_{n=0}^{\mathrm{\infty }}{c}_n<+\mathrm{\infty }$, but $c_n\ne o(b_n+c_n)$. Therefore, the proof of above Theorem CC is not reasonable. Up to now, we do not know the validity of Theorem CC. This will be an open question left for the readers!