# On the error estimation and T-stability of the Ishikawa iteration for strongly demicontractive mappings

## Abstract

In this paper, some new formula of error estimations of Ishikawa iteration and some strong convergence theorems of strongly demicontractive mappings are first obtained. And then, convergence speeds and error estimations of the Ishikawa iteration and the Mann iteration are discussed in some examples. Finally, T-stability of the Ishikawa iteration is proved.

## 1 Introduction and preliminaries

Let $$(X, \Vert \cdot \Vert )$$ be a real Hilbert space and C be a closed convex subset of X. Let $$T: C\rightarrow C$$ and $$\operatorname{Fix}(T)$$ denotes the set of fixed points of T, that is, $$\operatorname{Fix}(T)=\{x\in C: Tx=x\}$$. A sequence $$\{x_{n}\}$$ is called the Ishikawa iteration of T if, for an arbitrary $$x_{0}\in C$$,

$$\textstyle\begin{cases} x_{n+1}=(1-\alpha _{n})x_{n}+\alpha _{n}Ty_{n}, \\ y_{n}=(1-\beta _{n})x_{n}+\beta _{n}Tx_{n}, \end{cases}$$
(1.1)

where $$\alpha _{n}, \beta _{n} \in [0,1]$$ and $$n\geq 0$$. We know that if $$\beta _{n}=0$$, then the Ishikawa iteration is called the Mann iteration. Error estimations of the Mann iteration for some contractive and nonexpansive type mappings have been studied in [1,2,3]. In 2015, L. Maruster and St. Maruster  gave the notation of a strongly demicontractive mapping as follows:

The mapping T is called strongly demicontractive if $$\operatorname{Fix}(T)\neq \emptyset$$ and

$$\bigl\Vert Tx-x^{\ast } \bigr\Vert ^{2}\leq a \bigl\Vert x-x^{\ast } \bigr\Vert ^{2}+K \Vert Tx-x \Vert ^{2}$$
(1.2)

for all $$x\in C$$, $$x^{\ast }\in \operatorname{Fix}(T)$$, where $$a\in (0,1)$$ and $$K\geq 0$$. (It is easy to see that if T is a strongly demicontractive mapping, then the fixed point of T is unique.) And then, they considered an error estimation of the Mann iteration and the strong convergence for strongly demicontractive mappings. T-stability of the Mann iteration for a particular case of strongly demicontractive mapping was also proved. Later, [5, 6] provided some other convergence theorems of the Mann iteration for strongly demicontractive mappings. The problem of T-stability for some iterations was discussed in [7, 8]. Recently, comparative studies on some iterations (including the Mann iteration and the Ishikawa iteration) for contractive maps have been reported (see [9, 10]).

By using the idea of [4, 6, 9], we first introduce some new formula of error estimations of Ishikawa iteration and prove some convergence theorems of Ishikawa iteration for strongly demicontractive mappings, and then some examples are given to compare Ishikawa iteration (1.1) with the Mann iteration. Moreover, we will also discuss T-stability of the Ishikawa iteration for strongly demicontractive mappings.

We now show some lemmas to be used in the main result.

### Lemma 1.1

([2, 4])

Let $$\{d_{n}\}$$, $$\{\epsilon _{n}\}$$ be nonnegative sequences of real numbers satisfying

$$d_{n+1}\leq \alpha d_{n}+\epsilon _{n}$$

for all $$n\geq 0$$ and $$0 \leq \alpha <1$$. If $$\lim_{n\rightarrow \infty }\epsilon _{n}=0$$, then $$\lim_{n\rightarrow \infty }d_{n}=0$$.

### Lemma 1.2

()

Let $$\{d_{n}\}$$ be a nonnegative sequence satisfying

$$d_{n+1}\leq \alpha d_{n}+\beta \epsilon _{n},$$
(1.3)

where $$0<\alpha <1$$, $$\beta >0$$ and $$\{\epsilon _{n}\}$$ is a nonnegative sequence that satisfies the condition

$$\frac{\epsilon _{n+1}}{\epsilon _{n}}\geq 2\alpha , \quad \forall n\in \mathbb{N}.$$

Then

$$d_{n+1}\leq d_{0}\alpha ^{n+1}+\beta (2 \alpha \epsilon _{n-1}+\epsilon _{n}).$$

Motivated by the above lemma, Wang  gave the following lemma (which gives a fast convergence condition of $$d_{n}$$).

### Lemma 1.3

()

Let $$\{d_{n}\}$$ be a nonnegative sequence satisfying (1.3), where $$0<\alpha <1$$, $$\beta >0$$ and $$\{\epsilon _{n}\}$$ is a nonnegative sequence that satisfies the condition

$$\frac{\epsilon _{n+1}}{\epsilon _{n}}\leq \frac{\alpha }{2}, \quad \forall n\in \mathbb{N}.$$

Then $$\lim_{n\rightarrow \infty }d_{n}=0$$ and

$$d_{n+1}\leq d_{0}\alpha ^{n+1}+\beta \bigl(\alpha ^{n}\epsilon _{0}+2\alpha ^{n-1}\epsilon _{1}\bigr).$$

## 2 Error estimations of the Ishikawa iteration for strongly demicontractive mappings

In this section, we mainly give two formulas of error estimations of the Ishikawa iteration, and some convergence theorems are also obtained.

### Theorem 2.1

Let T be L-Lipschitzian (that is, there exists $$L>0$$ such that $$\Vert Tx-Ty \Vert \leq L\Vert x-y \Vert$$ for any $$x,y \in C$$) and strongly demicontractive with $$0< a<1$$ and $$K\geq 0$$, $$x^{\ast }\in \operatorname{Fix}(T)$$. Assume that there exist positive numbers $$\theta _{1}$$, $$\theta _{2}$$, $$0< \theta _{1}< \theta _{2}< \min \{1, 1-(1-a)(1-K)\}$$ such that

$$\frac{ \Vert TT_{\alpha _{n},\beta _{n}}x-T_{\alpha _{n},\beta _{n}}x \Vert ^{2}}{ \Vert Tx-x \Vert ^{2}}\geq 2\theta _{2},$$
(2.1)

where $$x\in K$$ and $$T_{\alpha _{n},\beta _{n}}:=(1-\alpha _{n})I+\alpha _{n}T_{\beta _{n}}$$, $$T_{\beta _{n}}:=(1-\beta _{n})I+\beta _{n}T$$. For $$x_{0}\in C$$, let $$\{x_{n}\}$$ be the sequence generated by Ishikawa iteration (1.1) with the control sequences $$\{\alpha _{n}\}$$ and $$\{\beta _{n}\}$$ satisfying

\begin{aligned} & \frac{1-\theta _{2}}{1-a}\leq \alpha _{n} \leq \frac{1-\theta _{1}}{1-a}, \end{aligned}
(2.2)
\begin{aligned} & 0 \leq \beta _{n}\leq q< 1, \quad L^{2}q^{2}+q+a< 1, \end{aligned}
(2.3)

for some $$0 \leq q < 1$$. Then the following error estimation for the sequence $$\{x_{n}\}$$ holds:

$$\bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2}\leq \bigl\Vert x_{0}-x^{\ast } \bigr\Vert ^{2}\theta _{2}^{n+1}+(1+Lq)^{2}M(2 \theta _{2} \epsilon _{n-1}+\epsilon _{n}),$$
(2.4)

where $$M=(\frac{1-\theta _{1}}{1-a})^{2}-(1-K)\frac{1-\theta _{1}}{1-a}$$ and $$\epsilon _{n}= \Vert Tx_{n}-x_{n} \Vert ^{2}$$.

### Proof

From T is a strongly demicontractive mapping in the Hilbert space X, we have

\begin{aligned} \bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2} &= \bigl\Vert (1-\alpha _{n})x_{n}+\alpha _{n}Ty_{n}-x ^{\ast } \bigr\Vert ^{2} \\ &= \bigl\Vert (1-\alpha _{n}) \bigl(x_{n}-x^{\ast } \bigr)+\alpha _{n}\bigl(Ty_{n}-x^{\ast }\bigr) \bigr\Vert ^{2} \\ &=(1-\alpha _{n}) \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\alpha _{n} \bigl\Vert Ty_{n}-x^{ \ast } \bigr\Vert ^{2}-\alpha _{n} (1-\alpha _{n}) \Vert Ty_{n}-x_{n} \Vert ^{2}. \end{aligned}

Since

\begin{aligned} \bigl\Vert Ty_{n}-x^{\ast } \bigr\Vert ^{2} \leq& a \bigl\Vert y_{n}-x^{\ast } \bigr\Vert ^{2}+K \Vert y_{n}-Ty _{n} \Vert ^{2} \\ = &a \bigl\Vert (1-\beta _{n}) \bigl(x_{n}-x^{\ast } \bigr)+\beta _{n}\bigl(Tx_{n}-x^{\ast }\bigr) \bigr\Vert ^{2}+K \bigl\Vert (1-\beta _{n}) (x_{n}-Ty_{n}) \\ & {} +\beta _{n}(Tx_{n}-Ty_{n}) \bigr\Vert ^{2} \\ \leq &a\bigl[(1-\beta _{n}) \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\beta _{n} \bigl\Vert Tx_{n}-x ^{\ast } \bigr\Vert ^{2}-\beta _{n}(1-\beta _{n}) \Vert Tx_{n}-x_{n} \Vert ^{2} \bigr] \\ & {} +K\bigl[(1-\beta _{n}) \Vert x_{n}-Ty_{n} \Vert ^{2}+\beta _{n} \Vert Tx_{n}-Ty_{n} \Vert ^{2}- \beta _{n}(1-\beta _{n}) \Vert Tx_{n}-x_{n} \Vert ^{2}\bigr] \\ \leq &a(1-\beta _{n}) \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+a\beta _{n} \bigl\Vert Tx_{n}-x ^{\ast } \bigr\Vert ^{2}-\bigl[a\beta _{n}(1-\beta _{n}) \\ & {} +K\beta _{n}(1-\beta _{n})\bigr] \Vert Tx_{n}-x_{n} \Vert ^{2}]+K(1-\beta _{n}) \Vert x_{n}-Ty _{n} \Vert ^{2} \\ &{}+K\beta _{n} \Vert Tx_{n}-Ty_{n} \Vert ^{2} \\ \leq &a(1-\beta _{n}) \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+a\beta _{n}\bigl[a \bigl\Vert x_{n}-x ^{\ast } \bigr\Vert ^{2}+K \Vert x_{n}-Tx_{n} \Vert ^{2}\bigr]-\bigl[a\beta _{n}(1-\beta _{n}) \\ & {} +K\beta _{n}(1-\beta _{n})\bigr] \Vert Tx_{n}-x_{n} \Vert ^{2}]+K(1-\beta _{n}) \Vert x_{n}-Ty _{n} \Vert ^{2} \\ &{} +K\beta _{n} \Vert Tx_{n}-Ty_{n} \Vert ^{2} \\ \leq &a(1-\beta _{n}+a\beta _{n}) \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\bigl[aK\beta _{n}-(a+K) \beta _{n}(1-\beta _{n})\bigr] \Vert x_{n}-Tx_{n} \Vert ^{2} \\ & {} +K(1-\beta _{n}) \Vert x_{n}-Ty_{n} \Vert ^{2}+K\beta _{n} \Vert Tx_{n}-Ty_{n} \Vert ^{2}, \end{aligned}

we can get that

\begin{aligned} & \bigl\Vert x_{n+1} -x^{\ast } \bigr\Vert ^{2} \\ &\quad \leq \bigl[1-\alpha _{n}+\alpha _{n}a(1-\beta _{n}+a \beta _{n})\bigr] \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\alpha _{n}\bigl[aK\beta _{n}-(a+K) \beta _{n}(1-\beta _{n})\bigr] \\ &\quad\quad {} \cdot \Vert Tx_{n}-x_{n} \Vert ^{2}+K \alpha _{n} \beta _{n} \Vert Tx_{n}-Ty_{n} \Vert ^{2} +\alpha _{n} \bigl[K(1-\beta _{n})-1+\alpha _{n}\bigr] \Vert x_{n}-Ty_{n} \Vert ^{2}. \quad\hspace{9.5pt} (\ast ) \end{aligned}

Since T is L-Lipschitzian, we have

\begin{aligned} \Vert Tx_{n}-Ty_{n} \Vert \leq L \Vert x_{n}-y_{n} \Vert \leq L\beta _{n} \Vert x_{n}-Tx_{n} \Vert \end{aligned}
(2.5)

and

\begin{aligned} \Vert Ty_{n}-x_{n} \Vert \leq \Vert Ty_{n}-Tx_{n}+Tx_{n}-x_{n} \Vert \leq (1+L\beta _{n}) \Vert x_{n}-Tx_{n} \Vert . \end{aligned}
(2.6)

Using the inequality $$(\ast )$$, (2.5), and (2.6), we obtain

\begin{aligned} \bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2} &\leq \bigl[1-\alpha _{n}+\alpha _{n}a(1-\beta _{n}+a\beta _{n})\bigr] \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\alpha _{n}\beta _{n}\bigl[aK-(a+K) (1- \beta _{n}) \\ & \quad {} +KL^{2}\beta ^{2}_{n}\bigr] \Vert Tx_{n}-x_{n} \Vert ^{2}+\alpha _{n} \bigl[K(1-\beta _{n})-1+ \alpha _{n}\bigr](1+L\beta _{n})^{2} \Vert x_{n}-Tx_{n} \Vert ^{2}. \end{aligned}

Using (2.3), we get

\begin{aligned} \bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2} &\leq \bigl[1-\alpha _{n}+\alpha _{n}a(1-\beta _{n}+a\beta _{n})\bigr] \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\alpha _{n}\bigl[K(1-\beta _{n})-1+ \alpha _{n}\bigr] \\ & \quad {} \cdot (1+L\beta _{n})^{2} \Vert x_{n}-Tx_{n} \Vert ^{2} \\ &\leq \bigl[1-\alpha _{n}(1-a)\bigr] \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+(1+L\beta _{n})^{2}\bigl[ \alpha ^{2}_{n}-(1-K)\alpha _{n}\bigr] \Vert x_{n}-Tx_{n} \Vert ^{2}. \end{aligned}

Similar to the proof of Theorem 1 in , we have

$$\bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2}\leq \theta _{2} \bigl\Vert x_{0}-x^{\ast } \bigr\Vert ^{2}+(1+Lq)^{2}M \epsilon _{n},$$
(2.7)

where $$M=(\frac{1-\theta _{1}}{1-a})^{2}-(1-K)\frac{1-\theta _{1}}{1-a}$$ and $$\epsilon _{n}= \Vert Tx_{n}-x_{n} \Vert ^{2}$$. From Lemma 1.2, it follows that

$$\bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2}\leq \bigl\Vert x_{0}-x^{\ast } \bigr\Vert ^{2}\theta _{2}^{n+1}+(1+Lq)^{2}M(2 \theta _{2} \epsilon _{n-1}+\epsilon _{n}).$$

□

### Corollary 2.1

Suppose that T satisfies all the conditions of Theorem 2.1 and it is also asymptotically T-regular, i.e., $$\Vert Tx_{n}-x_{n} \Vert \rightarrow 0$$. Then the Ishikawa iteration $$\{x_{n}\}$$ with control sequence satisfying (2.2)(2.3) converges strongly to $$x^{\ast }$$.

### Remark 2.1

If $$\beta _{n}=q=0$$, then the Ishikawa iteration changes into the Mann iteration, which was considered in . In this case, $$y_{n}\equiv x _{n}$$, from $$(\ast )$$, we know that

\begin{aligned} \bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2} &\leq \bigl[1-\alpha _{n}(1-a)\bigr] \bigl\Vert x_{n}-x^{\ast } \bigr\Vert ^{2}+\bigl[\alpha ^{2}_{n}-(1-K)\alpha _{n}\bigr] \Vert x_{n}-Tx_{n} \Vert ^{2}. \end{aligned}

Hence, Theorem 1 and Corollary 1 in  can be proved (T does not need to be L-Lipschitzian).

By  and Lemma 1.3, we can also get the following theorem.

### Theorem 2.2

Let T be L-Lipschitzian and strongly demicontractive with $$0< a<1$$ and $$K\geq 0$$, $$x^{\ast }\in \operatorname{Fix}(T)$$. Assume that there exist positive numbers $$\theta _{1}$$, $$\theta _{2}$$, $$0< \theta _{1}< \theta _{2}< \min \{1, 1-(1-a)(1-K)\}$$ such that

$$\frac{ \Vert TT_{\alpha _{n},\beta _{n}}x-T_{\alpha _{n},\beta _{n}}x \Vert ^{2}}{ \Vert Tx-x \Vert ^{2}}\leq \frac{\theta _{2}}{2},$$

where $$x\in K$$ and $$T_{\alpha _{n},\beta _{n}}:=(1-\alpha _{n})I+\alpha _{n}T_{\beta _{n}}$$, $$T_{\beta _{n}}:=(1-\beta _{n})I+\beta _{n}T$$. For $$x_{0}\in C$$, let $$\{x_{n}\}$$ be the sequence generated by Ishikawa iteration (1.1) with the control sequences $$\{\alpha _{n}\}$$ and $$\{\beta _{n}\}$$ satisfying

\begin{aligned} & \frac{1-\theta _{2}}{1-a}\leq \alpha _{n} \leq \frac{1-\theta _{1}}{1-a}, \\ & 0 \leq \beta _{n}\leq q< 1, \quad L^{2}q^{2}+q+a< 1, \end{aligned}

for some $$0 \leq q < 1$$. Then $$\{x_{n}\}$$ converges strongly to the fixed point of T, and the following error estimation for the sequence $$\{x_{n}\}$$ holds:

$$\bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2}\leq \bigl\Vert x_{0}-x^{\ast } \bigr\Vert ^{2}\theta _{2}^{n+1}+(1+Lq)^{2}M\bigl( \theta _{2}^{n}\epsilon _{0}+2\theta _{2}^{n-1}\epsilon _{1}\bigr),$$
(2.8)

where $$M=(\frac{1-\theta _{1}}{1-a})^{2}-(1-K)\frac{1-\theta _{1}}{1-a}$$ and $$\epsilon _{n}= \Vert Tx_{n}-x_{n} \Vert ^{2}$$.

### Proof

From the proof of Theorem 2.1, we have

$$\bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2}\leq \theta _{2} \bigl\Vert x_{0}-x^{\ast } \bigr\Vert ^{2}+(1+Lq)^{2}M \epsilon _{n},$$
(2.9)

where $$M=(\frac{1-\theta _{1}}{1-a})^{2}-(1-K)\frac{1-\theta _{1}}{1-a}$$ and $$\epsilon _{n}= \Vert Tx_{n}-x_{n} \Vert ^{2}$$. From Lemma 1.3, it follows that $$\{x_{n}\}$$ converges strongly to the fixed point of T and

$$\bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2}\leq \bigl\Vert x_{0}-x^{\ast } \bigr\Vert ^{2}\theta _{2}^{n+1}+(1+Lq)^{2}M\bigl( \theta _{2}^{n}\epsilon _{0}+2\theta _{2}^{n-1} \epsilon _{1}\bigr).$$

□

### Remark 2.2

If $$\beta _{n}= 0$$, then the Ishikawa iteration changes into the Mann iteration considered in . In this case, $$q=0$$. From (2.8), we know that

$$\bigl\Vert x_{n+1}-x^{\ast } \bigr\Vert ^{2}\leq \bigl\Vert x_{0}-x^{\ast } \bigr\Vert ^{2}\theta _{2}^{n+1}+M\bigl( \theta _{2}^{n}\epsilon _{0}+2\theta _{2}^{n-1}\epsilon _{1} \bigr).$$
(2.10)

Hence, Theorem 2.1 in  can proved (T does not need to be L-Lipschitzian).

### Remark 2.3

It is worth mentioning that error estimation (2.8) of the Ishikawa iteration depends on L, q, M, $$\theta _{2}$$, $$\epsilon _{0}$$, and $$\epsilon _{1}$$. But if n is large enough, then error estimation (2.8) depends only on $$\theta _{2}$$. Hence, error estimation (2.8) is better than error estimation (2.4) of the Ishikawa iteration.

## 3 The Ishikawa iteration and the Mann iteration for strongly demicontractive mappings in an example

In fact, if T has some particular properties, then the conditions of Theorem 2.2 can be satisfied. Suppose that $$T: \mathbb{R}\rightarrow \mathbb{R}$$ is a differentiable mapping, it is L-Lipschitzian and strongly demicontractive with $$0< a<1$$, $$K\geq 0$$. The sequence $$\{x_{n}\}$$ is generated by the Ishikawa iteration with control sequence satisfying

$$\alpha _{n}\equiv t_{1}, \qquad \beta _{n}\equiv t_{2}, \quad\quad 0 \leq t_{2}< 1, \quad\quad L^{2}t^{2}_{2}+t_{2}+a< 1 \quad \text{and} \quad \frac{1-\theta _{2}}{1-a}\leq t_{1} \leq \frac{1-\theta _{1}}{1-a}.$$

In this case, we have

\begin{aligned} & TT_{t_{1},t_{2}}x-T_{t_{1},t_{2}}x \\ &\quad =T\bigl\{ (1-t_{1})x+t_{1} \bigl[(1-t_{2})x+t _{2}Tx\bigr]\bigr\} -\bigl\{ (1-t_{1})x+t_{1}\bigl[(1-t_{2})x+t_{2}Tx \bigr]\bigr\} \\ &\quad =T\bigl[x+t_{1}t_{2}(Tx-x)\bigr]-Tx+Tx-x-t_{1}t_{2}(Tx-x) \\ &\quad =t_{1}t_{2}T'\xi (Tx-x)+(1-t_{1}t_{2}) (Tx-x) \\ &\quad =\bigl(1-t_{1}t_{2}+t_{1}t_{2}T' \xi \bigr) (Tx-x), \end{aligned}

where $$\xi =x+\eta t_{1}t_{2}(Tx-x)$$, $$0<\eta <1$$. Therefore, if the derivative $$T'$$ of T satisfies

$$\bigl\vert 1-t_{1}t_{2}+t_{1}t_{2}T' \xi \bigr\vert ^{2}\leq \frac{\theta _{2}}{2},$$

then

$$\frac{ \Vert TT_{t_{1},t_{2}}x-T_{t_{1},t_{2}}x \Vert ^{2}}{ \Vert Tx-x \Vert ^{2}}\leq \frac{ \theta _{2}}{2}, \quad \forall x\in C.$$

In order to compare the convergence speeds and error estimations for Ishikawa iteration (1.1) and the Mann iteration, we consider the following example (the same as Example 2.1 in ) which satisfies all the conditions of Theorem 2.2.

### Example 3.1

Let $$C=[0.5,1.5]$$ and define a mapping $$T: C\rightarrow C$$ by

$$Tx=-x+2+(x-1)^{3}, \quad x\in C.$$

This real function is 1-Lipschitzian and strongly demicontractive with $$a=0.15$$, $$K=0.42$$ and $$x^{\ast }=1$$ is the unique fixed point of T. Set $$\theta _{1}=0.2$$, $$\theta _{2}=0.5$$. From

$$\frac{1-\theta _{2}}{1-a}\leq t_{1} \leq \frac{1-\theta _{1}}{1-a}, \quad \quad L^{2}t^{2}_{2}+t_{2}+a< 1,$$

we get $$t_{1}\in [0.5882,0.9412]$$ and $$t_{2}\in [0,0.5488)$$. We choose some $$t_{1}$$ and $$t_{2}$$ satisfying

$$t_{1}t_{2}\in [0.4000,0.6154].$$

Since

$$T'x=3(x-1)^{2}-1\in \biggl[-1,-\frac{1}{4}\biggr], \quad \forall x\in C,$$

we have

$$\bigl\vert 1-t_{1}t_{2}+t_{1}t_{2}T' \xi \bigr\vert ^{2}\leq \biggl(1-\frac{5}{4}t_{1}t_{2} \biggr)^{2} \leq \frac{\theta _{2}}{2}.$$

Hence,

$$\frac{ \Vert TT_{t_{1},t_{2}}x-T_{t_{1},t_{2}}x \Vert ^{2}}{ \Vert Tx-x \Vert ^{2}}\leq \frac{ \theta _{2}}{2}, \quad \forall x\in C.$$

It can be seen that the conditions of Theorem 2.2 are satisfied.

(1) We first compare the convergence speed of Ishikawa iteration (1.1) with that of the Mann iteration. Let $$t=0.6$$ and $$t=0.61$$ (where $$t\in [0.588,0.612]$$ in ) in the Mann iteration. Let $$t_{1}=0.8$$, $$t_{2}=0.5$$ and $$t_{1}=0.9$$, $$t_{2}=0.5$$ in Ishikawa iteration (1.1). Set the stop parameter to $$\Vert x_{n}-x^{\ast } \Vert \leq 10^{-15}$$. Table 1 shows computation results for different initial points. Figure 1 gives 30 times calculating of CPU times for the iterations (see ).

From a mathematical point of view (Table 1), we see that Ishikawa iteration (iv) converges faster than the other three iterations. And convergence speeds of Mann iteration (i) and Mann iteration (ii) are stable with respect to the given initial points. From a computer-calculation point of view (Fig. 1), we also find that Ishikawa iteration (iv) converges faster than the other three iterations.

(2) And then, we want to compare the error estimation of Ishikawa iteration (1.1) with that of the Mann iteration (see (2.8) and (2.10)). Figure 2 shows the error estimations for Mann iteration (i) and Ishikawa iteration (iv). It can be found that the error estimation of Mann iteration (i) is more effective than the error estimation of Ishikawa iteration (iv).

## 4 T-stability of the Ishikawa iteration for strongly demicontractive mappings

Let $$\{x_{n}\}$$ be a sequence given by the iteration procedure

$$x_{n+1}=f(T,x_{n}),$$
(4.1)

where $$T:X\rightarrow X$$ and $$x_{0}\in X$$.

Generally speaking, a fixed point iteration procedure is called stable if small modifications in the initial data that are involved in the computation process produce a small influence on the computed value of the fixed point. Now, we give the specific definition of stability for the iteration procedure as follows.

### Definition 4.1

()

Let $$(X,d)$$ be a metric space and $$T:X\rightarrow X$$ be a mapping, $$x_{0}\in X$$ and the sequence $$\{x_{n}\}$$ produced by (4.1) converges to a fixed point $$x^{\ast }$$ of T. Let $$\{z_{n}\}$$ be an arbitrary sequence in X and set

$$\varepsilon _{n}=d\bigl(z_{n+1},f(T,z_{n})\bigr) \quad \text{for } n=0,1,2,\ldots.$$

We shall say that the fixed point iteration procedure (4.1) is T-stable or stable with respect to T if only if

$$\lim_{n\rightarrow \infty }\epsilon _{n}=0 \quad \Longleftrightarrow \quad \lim_{n\rightarrow \infty }z_{n}=x^{\ast }.$$

Under the same assumptions in , we will prove that the Ishikawa iteration is also T-stable for strongly demicontractive mappings.

### Theorem 4.1

Let T be strongly demicontractive with $$K\in [0,1)$$ and $$x^{\ast }\in \operatorname{Fix}(T)$$. For $$x_{0}\in C$$, let $$\{x_{n}\}$$ be the sequence generated by Ishikawa iteration (1.1) with the control sequence $$\lambda \leq \alpha _{n}<1$$ for some $$\lambda >0$$. Assume that

$$a+\frac{4K}{(1-K)^{2}}< 1.$$
(4.2)

Then $$\{x_{n}\}$$ converges strongly to $$x^{\ast }$$ and Ishikawa iteration (1.1) is stable with respect to T.

### Proof

From the proof of Theorem 2 in , we have

$$\bigl\Vert Tx-x^{\ast } \bigr\Vert \leq M \bigl\Vert x-x^{\ast } \bigr\Vert ,$$

where $$M=\sqrt{a+\frac{4K}{(1-K)^{2}}}<1$$. Suppose that $$\{z_{n}\}$$ is a sequence in C. Define

$$\epsilon _{n}= \bigl\Vert z_{n+1}-(1-\alpha _{n})z_{n}-\alpha _{n}Ts_{n} \bigr\Vert$$

and

$$s_{n}=(1-\beta _{n})z_{n}+\beta _{n}Tz_{n},$$

we have

\begin{aligned} \bigl\Vert z_{n+1}-x^{\ast } \bigr\Vert &\leq \bigl\Vert (1- \alpha _{n})z_{n}-x^{\ast }+\alpha _{n}Ts_{n} \bigr\Vert + \bigl\Vert z_{n+1}-(1-\alpha _{n})z_{n}- \alpha _{n}Ts_{n} \bigr\Vert \\ &\leq (1-\alpha _{n}) \bigl\Vert z_{n}-x^{\ast } \bigr\Vert +\alpha _{n} \bigl\Vert Ts_{n}-x^{ \ast } \bigr\Vert +\epsilon _{n} \\ &\leq \bigl[1-\alpha _{n}+\alpha _{n}(1-\beta _{n})M+\alpha _{n}\beta _{n}M ^{2}\bigr] \bigl\Vert z_{n}-x^{\ast } \bigr\Vert +\epsilon _{n} \\ &\leq \bigl[1-\alpha _{n}+\alpha _{n}(1-\beta _{n})M+\alpha _{n}\beta _{n}M\bigr] \bigl\Vert z_{n}-x^{\ast } \bigr\Vert +\epsilon _{n} \\ &\leq (1-\alpha _{n}+\alpha _{n}M) \bigl\Vert z_{n}-x^{\ast } \bigr\Vert +\epsilon _{n}. \end{aligned}

Now, let $$\alpha =1-\lambda +\lambda M$$, where $$\lambda =\frac{1- \theta _{2}}{1-a}$$. Since $$0< M<1$$, we have $$0<\alpha <1$$ and

$$1-\alpha _{n}+\alpha _{n}M \leq \alpha .$$

Therefore,

$$\bigl\Vert z_{n+1}-x^{\ast } \bigr\Vert \leq \alpha \bigl\Vert z_{n}-x^{\ast } \bigr\Vert +\epsilon _{n}.$$

If $$\lim_{n\rightarrow \infty }\epsilon _{n}=0$$, then it follows by Lemma 1.1 that $$\lim_{n\rightarrow \infty }z_{n}=x^{\ast }$$. Since

$$\bigl\Vert x_{n+1}-(1-\alpha _{n})x_{n}-\alpha _{n}Ty_{n} \bigr\Vert =0,$$

it also results in $$\lim_{n\rightarrow \infty }x_{n}=x^{\ast }$$, i.e., $$\{x_{n}\}$$ converges strongly to $$x^{\ast }$$. Conversely, suppose $$\lim_{n\rightarrow \infty }z_{n}=x^{\ast }$$. It follows from

$$\epsilon _{n} \leq \bigl\Vert z_{n+1}-x^{\ast } \bigr\Vert +(1-\alpha _{n}) \bigl\Vert z_{n+1}-x^{ \ast } \bigr\Vert +\alpha _{n} \bigl\Vert Ts_{n}-x^{\ast } \bigr\Vert$$

that $$\lim_{n\rightarrow \infty }\epsilon _{n}=0$$. Hence, Ishikawa iteration (1.1) is stable with respect to T. □

### Remark 4.1

(1) Theorem 4.1 extends the corresponding results of  (Theorem 2 of ) into the case of the Ishikawa iteration. In order to guarantee that $$0<\alpha <1$$, the condition “$$a+\frac{4K}{(1-K)^{2}}\leq 1$$” (Theorem 2 of ) should be “$$a+\frac{4K}{(1-K)^{2}}< 1$$”.

(2) Theorem 4.1 gives anthor convergence theorem for the strongly demicontractive mapping T with condition (4.2). As we know, condition (4.2) is relatively strong. Indeed, the condition implies that $$0< K<3-\sqrt{8}$$.

(3) In Example 3.1, condition (4.2) does not hold. But we know that

$$\bigl\Vert Tx-x^{\ast } \bigr\Vert \leq \frac{3}{4} \bigl\Vert x-x^{\ast } \bigr\Vert .$$

From the proof of Theorem 4.1, the Mann iteration and the Ishikawa iteration are still T-stable when the control sequence $$\lambda \leq \alpha _{n}<1$$ for some $$\lambda >0$$.

### Example 4.1

Let $$C=[-1,1]$$ and define a mapping $$f: C\rightarrow C$$ by

$$f(x)= \textstyle\begin{cases} 0.6x, & x\in [-1,\frac{1}{2}]; \\ -x+0.4,& x\in (\frac{1}{2},1]. \end{cases}$$

This real function is strongly demicontractive with $$a=0.35$$, $$K=0.1$$, and $$a+\frac{4K}{(1-K)^{2}}\approx 0.84<1$$.

Now, we consider the Mann iterations and the Ishikawa iterations (in Example 3.1) for the above function. Theorem 4.1 shows that the sequences generated by these iterations converge strongly to the fixed point $$x^{\ast }=0$$ and all these iterations are stable with respect to $$f(x)$$. Also set the stop parameter to $$\Vert x_{n}-x^{\ast } \Vert \leq 10^{-15}$$. From Table 2 (Number of iterations) and Fig. 3 (CPU time), we know that Ishikawa iteration (iv) converges faster than the other three iterations.

## 5 Conclusion

In this paper, we have developed some convergence theorems and T-stability of the Ishikawa iteration for strongly demicontractive mappings. By finding fixed points of some strongly demicontractive mappings (see Example 3.1 and Example 4.1), we show that the Ishikawa iteration converges faster than the Mann iteration (see Tables 12, Fig. 1 and Fig. 3), but the Mann iteration is more efficient for the error estimation (see Fig. 2).

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## Acknowledgements

The authors thank the editor and the referees for constructive and pertinent suggestions.

## Funding

This paper was partially supported by the NSF of China (No. 11126290 and No. 61573192), University Science Research Project of Jiangsu Province (No. 13KJB110021), and Scholarship Award for Excellent Doctoral Student granted by the Ministry of Education (No. 1390219098).

## Author information

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### Contributions

Each of the authors contributed to each part of this study equally, all authors read and approved the final manuscript.

### Corresponding author

Correspondence to Chao Wang.

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The authors declare that they have no competing interests. 