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# An intermixed algorithm for strict pseudo-contractions in Hilbert spaces

## Abstract

An intermixed algorithm for two strict pseudo-contractions in Hilbert spaces have been presented. It is shown that the suggested algorithms converge strongly to the fixed points of two strict pseudo-contractions, independently. As a special case, we can find the common fixed points of two strict pseudo-contractions in Hilbert spaces.

## Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H with its inner product $$\langle\cdot, \cdot\rangle$$ and norm $$\| \cdot\|$$.

### Definition 1.1

A mapping $$T:C\to C$$ is said to be nonexpansive if

$$\|Tx-Ty\|\leq\|x-y\|$$

for all $$x,y\in C$$.

We use $$\operatorname{Fix}(T)$$ to denote the set of fixed points of T.

### Definition 1.2

A mapping $$T:C\to C$$ is said to be strictly pseudo-contractive if there exists a constant $$0\leq\lambda<1$$ such that

$$\|Tx-Ty\|^{2}\leq\|x-y\|^{2}+\lambda\bigl\Vert (I-T)x-(I-T)y\bigr\Vert ^{2},\quad \forall x,y\in C.$$

### Remark 1.3

It is well known that the class of strictly pseudo-contractive mappings properly includes the class of nonexpansive mappings.

Iterative construction of fixed points of nonlinear mappings has a long history and is still an active field in the nonlinear functional analysis. Let C be a nonempty closed convex subset of a real Hilbert space. Let $$T:C\to C$$ be a nonlinear mapping. Let $$\{\alpha _{n}\}$$ be a real number sequence in $$(0,1)$$. For arbitrarily fixed $$x_{0}\in C$$, define a sequence $$\{x_{n}\}$$ in the following manner:

$$x_{n+1}=\alpha_{n}x_{n}+(1- \alpha_{n})Tx_{n},\quad n\ge0.$$
(1.1)

Iteration (1.1) is said to be a Mann iteration ; it has been studied extensively in the literature. If T is a nonexpansive mapping with $$\operatorname{Fix}(T)\ne\emptyset$$ and $$\{\alpha_{n}\}$$ satisfies the condition $$\sum_{n=0}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$$, then the sequence $$\{x_{n}\}$$ generated by Mann’s algorithm converges weakly to a fixed point of T . Now, it is well known that Mann’s algorithm fails, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces . Iterative methods for nonexpansive mappings have been investigated extensively in the literature; see  and the references therein. However, iterative methods for strictly pseudo-contractive mappings are far less developed than those for nonexpansive mappings though Browder and Petryshyn  initiated their work in 1967. However, strictly pseudo-contractive mappings have more powerful applications than nonexpansive mappings, for example, to solve inverse problems (see Scherzer ). Therefore it is interesting to develop the algorithms for finding the fixed points of strictly pseudo-contractive mappings. Now, we know that Mann’s algorithm is not good enough for approximating fixed points of (even if Lipschitz continuous) pseudo-contractions. Thus, we have to find other type of iterative algorithms; see . The first such an attempt was done by Ishikawa  who introduced the following Ishikawa algorithm:

\begin{aligned} &y_{n}=(1-\beta_{n})x_{n}+ \beta_{n}Tx_{n}, \\ &x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}Ty_{n}, \end{aligned} \quad n\ge0,

where $$\{\alpha_{n}\}$$ and $$\{\beta_{n}\}$$ are sequences in the interval $$[0,1]$$, T is a (nonlinear) self-mapping of C, and the initial guess $$x_{0}\in C$$ is selected arbitrarily. (Ishikawa’s algorithm can be viewed as a double-step (or two-level) Mann’s algorithm.) Ishikawa proved that his algorithm converges in norm to a fixed point of a Lipschitz pseudo-contraction T if $$\{\alpha_{n}\}$$ and $$\{\beta_{n}\}$$ satisfy certain conditions and if T is compact.

On the other hand, iterative methods for approximating the common fixed points of a finite (or an infinite) family of nonlinear mappings have been considered by many authors. For the related work, we refer the reader to [2226, 32, 33]. Above discussion suggests the following question.

### Question 1.4

Could we construct an iterative algorithm such that it converges strongly to the fixed points of a finite family of strict pseudo-contractions?

It is our purpose in this paper to construct redundant intermixed algorithms for two strict pseudo-contractions. It is shown that the suggested algorithms converge strongly to the fixed points of two strict pseudo-contractions, independently. As a special case, we can find the common fixed points of two strict pseudo-contractions in Hilbert spaces.

## Preliminaries

Let C be a nonempty closed convex subset of H. The (nearest point or metric) projection from H onto C is defined as follows: for each point $$x\in H$$, $$P_{C}x$$ is the unique point in C with the property:

$$\|x-P_{C}x\|\leq\|x-y\|, \quad y\in C.$$

Note that $$P_{C}$$ is characterized by the inequality:

$$P_{C}x\in C,\quad \langle x-P_{C}x, y-P_{C}x \rangle\leq0,\quad y\in C.$$

Consequently, $$P_{C}$$ is nonexpansive.

In order to prove our main results, we need the following well-known lemmas.

### Lemma 2.1

()

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $$T:C\to C$$ be a λ-strictly pseudo-contractive mapping. Then $$I-T$$ is demi-closed at 0, i.e., if $$x_{n} \rightharpoonup x\in C$$ and $$x_{n}-Tx_{n}\to0$$, then $$x=Tx$$.

### Lemma 2.2

()

Let $$\{x_{n}\}$$ and $$\{y_{n}\}$$ be bounded sequences in a Banach space E and $$\{\beta_{n}\}$$ be a sequence in $$[0,1]$$ with $$0<\liminf_{n\rightarrow\infty}\beta_{n}\leq \limsup_{n\rightarrow \infty}\beta_{n}<1$$. Suppose that $$x_{n+1}=(1-\beta_{n})x_{n}+\beta_{n}z_{n}$$ for all $$n\geq0$$ and $$\limsup_{n\rightarrow \infty}(\|z_{n+1}-z_{n}\|-\|x_{n+1}-x_{n}\|)\leq0$$. Then $$\lim_{n\rightarrow\infty}\|z_{n}-x_{n}\|=0$$.

### Lemma 2.3

()

Assume $$\{ a_{n}\}$$ is a sequence of nonnegative real numbers such that $$a_{n+1}\leq (1-\gamma_{n})a_{n}+\gamma_{n}\delta_{n}$$, $$n\geq0$$ where $$\{\gamma_{n}\}$$ is a sequence in $$(0,1)$$ and $$\{\delta_{n}\}$$ is a sequence in R such that

1. (i)

$$\sum_{n=0}^{\infty}\gamma_{n}=\infty$$;

2. (ii)

$$\limsup_{n\rightarrow\infty}\delta_{n}\leq0$$ or $$\sum_{n=0}^{\infty}|\delta_{n}\gamma_{n}|<\infty$$.

Then $$\lim_{n\rightarrow\infty}a_{n}=0$$.

## Main results

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $$T:C\to C$$ be a λ-strict pseudo-contraction. Let $$f:C\to H$$ be a $$\rho_{1}$$-contraction and $$g:C\to H$$ be a $$\rho _{2}$$-contraction. (A mapping $$f:C\to H$$ is said to be contractive if $$\| f(x)-f(y)\|\le\rho\|x-y\|$$ for some $$\rho\in[0,1)$$ and for all $$x, y\in C$$.) Let $$k\in(0,1-\lambda)$$ be a constant.

Now we propose the following redundant intermixed algorithm for two strict pseudo-contractions S and T.

### Algorithm 3.1

For arbitrarily given $$x_{0}\in C$$, $$y_{0}\in C$$, let the sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ be generated iteratively by

$$\left \{ \textstyle\begin{array}{l} x_{n+1} =(1-\beta_{n})x_{n}+\beta_{n}P_{C}[\alpha_{n}f(y_{n})+(1-k-\alpha _{n})x_{n}+kTx_{n}], \quad n\geq0, \\ y_{n+1} =(1-\beta_{n})y_{n}+\beta_{n}P_{C}[\alpha_{n}g(x_{n})+(1-k-\alpha _{n})y_{n}+kSy_{n}], \quad n\geq0, \end{array}\displaystyle \right .$$
(3.1)

where $$\{\alpha_{n}\}$$ and $$\{\beta_{n}\}$$ are two real number sequences in $$(0,1)$$.

### Remark 3.2

Note that the definition of the sequence $$\{x_{n}\}$$ is involved in the sequence $$\{y_{n}\}$$ and the definition of the sequence $$\{y_{n}\}$$ is also involved in the sequence $$\{x_{n}\}$$. So, this algorithm is said to be the redundant intermixed algorithm. We can use this algorithm to find the fixed points of S and T, independently.

### Theorem 3.3

Suppose that $$\operatorname{Fix}(S)\ne\emptyset$$ and $$\operatorname{Fix}(T)\neq \emptyset$$. Assume the following conditions are satisfied:

1. (C1)

$$\lim_{n\to\infty}\alpha_{n}=0$$ and $$\sum_{n=0}^{\infty}\alpha_{n}=\infty$$;

2. (C2)

$$\beta_{n}\in[\xi_{1}, \xi_{2}]\subset(0,1)$$ for all $$n\ge0$$.

Then the sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ generated by (3.1) converge strongly to the fixed points $$P_{\operatorname{Fix}(T)} f(y^{*})$$ and $$P_{\operatorname{Fix}(S)} g(x^{*})$$ of T and S, respectively, where $$x^{*}\in \operatorname{Fix}(T)$$ and $$y^{*}\in \operatorname{Fix}(S)$$.

### Proof

First, we give the following propositions.

### Proposition 3.4

The sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ are bounded.

In order to prove this proposition, we need the following result.

### Proposition 3.5

The mapping $$P_{C}[\alpha f +(1-k-\alpha)I+kT]$$ is contractive for small enough α.

### Proof

Let $$x,y\in C$$. Then we have

\begin{aligned}& \bigl\Vert P_{C}\bigl[\alpha f(x) +(1-k-\alpha)x+kTx \bigr]-P_{C}\bigl[\alpha f(y) +(1-k-\alpha )y+kTy\bigr]\bigr\Vert ^{2} \\& \quad \leq \bigl\Vert \alpha\bigl(f(x)-f(y)\bigr)+(1-k-\alpha) (x-y)+k(Tx-Ty) \bigr\Vert ^{2} \\& \quad = \biggl\Vert \alpha\bigl(f(x)-f(y)\bigr)+(1-\alpha) \biggl[ \frac{1-k-\alpha}{1-\alpha }(x-y)+\frac{k}{1-\alpha}(Tx-Ty) \biggr]\biggr\Vert ^{2} \\& \quad \le \alpha\bigl\Vert f(x)-f(y)\bigr\Vert ^{2}+(1-\alpha) \biggl\Vert \frac{1-k-\alpha}{1-\alpha }(x-y)+\frac{k}{1-\alpha}(Tx-Ty)\biggr\Vert ^{2} \\& \quad \le \alpha\rho_{1}\Vert x-y\Vert ^{2}+ \frac{(1-k-\alpha)^{2}}{1-\alpha} \Vert x-y\Vert ^{2}+\frac{k^{2}}{1-\alpha} \Vert Tx-Ty \Vert ^{2} \\& \qquad {} +\frac{2(1-k-\alpha)k}{1-\alpha}\langle Tx-Ty, x-y\rangle \\& \quad \leq \alpha\rho_{1}\Vert x-y\Vert ^{2}+ \frac{(1-k-\alpha)^{2}}{1-\alpha} \Vert x-y\Vert ^{2}+\frac{k^{2}}{1-\alpha}\bigl[\Vert x-y\Vert ^{2}+\lambda\bigl\Vert (I-T)x-(I-T)y\bigr\Vert ^{2}\bigr] \\& \qquad {} +\frac{2(1-k-\alpha)k}{1-\alpha} \biggl[\Vert x-y\Vert ^{2}- \frac{1-\lambda}{2}\bigl\Vert (I-T)x-(I-T)y\bigr\Vert ^{2} \biggr] \\& \quad = \alpha\rho_{1}\Vert x-y\Vert ^{2}+ \frac{1}{1-\alpha}\bigl[\lambda k^{2}-(1-\lambda ) (1-k-\alpha)k\bigr] \bigl\Vert (I-T)x-(I-T)y\bigr\Vert ^{2} \\& \qquad {} +(1-\alpha)\Vert x-y\Vert ^{2} \\& \quad = \frac{k}{1-\alpha}\bigl[k-(1-\alpha) (1-\lambda)\bigr]\bigl\Vert (I-T)x-(I-T)y\bigr\Vert ^{2}+\bigl[1-(1-\rho_{1})\alpha \bigr]\Vert x-y\Vert ^{2}. \end{aligned}

Thus, we get

\begin{aligned}& \bigl\Vert P_{C}\bigl[\alpha f(x) +(1-k-\alpha)x+kTx \bigr]-P_{C}\bigl[\alpha f(y) +(1-k-\alpha )y+kTy\bigr]\bigr\Vert \\& \quad \leq \biggl[1-\frac{(1-\rho_{1})\alpha}{2} \biggr]\|x-y\| \end{aligned}

for all $$x,y\in C$$ as $$k\leq(1-\alpha)(1-\lambda)$$ (that is, $$\alpha \le1-\frac{k}{1-\lambda}$$). □

Next, we prove Proposition 3.4.

### Proof

Since $$\operatorname{Fix}(S)\ne\emptyset$$ and $$\operatorname{Fix}(T)\neq \emptyset$$, we can choose $$x^{*}\in \operatorname{Fix}(T)$$ and $$y^{*}\in \operatorname{Fix}(S)$$. From (3.1), we have

\begin{aligned} \bigl\Vert x_{n+1}-x^{*}\bigr\Vert =&\bigl\Vert (1- \beta_{n})x_{n}+\beta_{n}P_{C}\bigl[ \alpha_{n}f(y_{n})+(1-k-\alpha _{n})x_{n}+kTx_{n} \bigr]-x^{*}\bigr\Vert \\ \leq& \beta_{n}\bigl\Vert P_{C}\bigl[ \alpha_{n}f(y_{n})+(1-k-\alpha_{n})x_{n}+kTx_{n} \bigr]-x^{*}\bigr\Vert \\ &{}+(1-\beta_{n})\bigl\Vert x_{n}-x^{*}\bigr\Vert \\ \le&\beta_{n}\alpha_{n}\bigl\Vert f(y_{n})-x^{*} \bigr\Vert +\beta_{n}\bigl\Vert (1-k-\alpha _{n}) \bigl(x_{n}-x^{*}\bigr)+k\bigl(Tx_{n}-Tx^{*}\bigr)\bigr\Vert \\ &{}+(1-\beta_{n})\bigl\Vert x_{n}-x^{*}\bigr\Vert \\ \le&\beta_{n}\alpha_{n}\bigl\Vert f(y_{n})-f \bigl(y^{*}\bigr)\bigr\Vert +\beta_{n}\alpha_{n}\bigl\Vert f \bigl(y^{*}\bigr)-x^{*}\bigr\Vert +(1-\beta_{n})\bigl\Vert x_{n}-x^{*}\bigr\Vert \\ &{}+\beta_{n}(1-\alpha_{n})\bigl\Vert x_{n}-x^{*} \bigr\Vert \\ \le&\rho_{1}\beta_{n}\alpha_{n}\bigl\Vert y_{n}-y^{*}\bigr\Vert +\beta_{n}\alpha_{n}\bigl\Vert f\bigl(y^{*}\bigr)-x^{*}\bigr\Vert +(1-\alpha_{n} \beta_{n})\bigl\Vert x_{n}-x^{*}\bigr\Vert \\ \le&\rho\beta_{n}\alpha_{n}\bigl\Vert y_{n}-y^{*} \bigr\Vert +\beta_{n}\alpha_{n}\bigl\Vert f\bigl(y^{*} \bigr)-x^{*}\bigr\Vert +(1-\alpha_{n}\beta_{n})\bigl\Vert x_{n}-x^{*}\bigr\Vert , \end{aligned}

where $$\rho=\max\{\rho_{1},\rho_{2}\}$$. Similarly, we have

\begin{aligned} \bigl\Vert y_{n+1}-y^{*}\bigr\Vert \le&\rho_{2} \beta_{n}\alpha_{n}\bigl\Vert x_{n}-x^{*}\bigr\Vert +\beta_{n}\alpha_{n}\bigl\Vert g\bigl(x^{*}\bigr)-y^{*} \bigr\Vert +(1-\alpha_{n}\beta_{n})\bigl\Vert y_{n}-y^{*}\bigr\Vert \\ \le&\rho\beta_{n}\alpha_{n}\bigl\Vert x_{n}-x^{*} \bigr\Vert +\beta_{n}\alpha_{n}\bigl\Vert g\bigl(x^{*} \bigr)-y^{*}\bigr\Vert +(1-\alpha_{n}\beta_{n})\bigl\Vert y_{n}-y^{*}\bigr\Vert . \end{aligned}

Hence, we obtain

\begin{aligned}& \bigl\Vert x_{n+1}-x^{*}\bigr\Vert +\bigl\Vert y_{n+1}-y^{*} \bigr\Vert \\& \quad \le \bigl[1-(1-\rho)\alpha_{n}\beta_{n}\bigr]\bigl( \bigl\Vert x_{n}-x^{*}\bigr\Vert +\bigl\Vert y_{n}-y^{*} \bigr\Vert \bigr) +\alpha_{n}\beta_{n}\bigl(\bigl\Vert f \bigl(y^{*}\bigr)-x^{*}\bigr\Vert +\bigl\Vert g\bigl(x^{*}\bigr)-y^{*}\bigr\Vert \bigr) \\& \quad \le \max \biggl\{ \bigl\Vert x_{n}-x^{*}\bigr\Vert +\bigl\Vert y_{n}-y^{*}\bigr\Vert ,\frac{\Vert f(y^{*})-x^{*}\Vert +\Vert g(x^{*})-y^{*}\Vert }{1-\rho} \biggr\} . \end{aligned}

By induction, we have

\begin{aligned}& \bigl\Vert x_{n}-x^{*}\bigr\Vert +\bigl\Vert y_{n}-y^{*} \bigr\Vert \\& \quad \le \max \biggl\{ \bigl\Vert x_{0}-x^{*}\bigr\Vert +\bigl\Vert y_{0}-y^{*}\bigr\Vert ,\frac{\Vert f(y^{*})-x^{*}\Vert +\Vert g(x^{*})-y^{*}\Vert }{1-\alpha} \biggr\} . \end{aligned}

So, $$\{x_{n}\}$$ and $$\{y_{n}\}$$ are bounded. □

### Proposition 3.6

$$\|x_{n}-Tx_{n}\|\to0$$ and $$\|y_{n}-Sy_{n}\|\to0$$.

### Proof

We first estimate $$\|x_{n+1}-x_{n}\|$$. Set $$u_{n}=P_{C}[\alpha _{n}f(y_{n})+(1-k-\alpha_{n})x_{n}+kTx_{n}]$$, $$n\ge0$$. It follows that

\begin{aligned} \Vert u_{n+1}-u_{n}\Vert \le&\bigl\Vert \alpha_{n+1}f(y_{n+1})+(1-k-\alpha _{n+1})x_{n+1}+kTx_{n+1} \\ &{}-\alpha_{n}f(y_{n})-(1-k-\alpha_{n})x_{n}+kTx_{n} \bigr\Vert \\ \le&\bigl\Vert (1-k-\alpha_{n+1}) (x_{n+1}-x_{n})+k(Tx_{n+1}-Tx_{n}) \bigr\Vert \\ &{}+\alpha_{n+1}\bigl(\bigl\Vert f(y_{n+1})\bigr\Vert + \Vert x_{n}\Vert \bigr)+\alpha_{n}\bigl(\bigl\Vert f(y_{n})\bigr\Vert +\Vert x_{n}\Vert \bigr) \\ \le&(1-\alpha_{n+1})\Vert x_{n+1}-x_{n}\Vert + \alpha_{n+1}\bigl(\bigl\Vert f(y_{n+1})\bigr\Vert +\Vert x_{n}\Vert \bigr) \\ &{}+\alpha_{n}\bigl(\bigl\Vert f(y_{n})\bigr\Vert + \Vert x_{n}\Vert \bigr). \end{aligned}

Since $$\alpha_{n}\to0$$, we deduce that

$$\limsup_{n\to\infty}\bigl(\Vert u_{n+1}-u_{n} \Vert -\Vert x_{n+1}-x_{n}\Vert \bigr)\le0.$$

From Lemma 2.2, we get

$$\lim_{n\to\infty}\|u_{n}-x_{n}\|=0 \quad \mbox{and} \quad \lim_{n\to\infty}\| x_{n+1}-x_{n} \|=0 .$$

From (3.1), we derive

\begin{aligned} \Vert x_{n+1}-Tx_{n}\Vert \le&(1-\beta_{n}) \Vert x_{n}-Tx_{n}\Vert +\beta_{n} \alpha_{n}\bigl\Vert f(y_{n})-Tx_{n}\bigr\Vert \\ &{}+\beta_{n}(1-k-\alpha_{n})\Vert x_{n}-Tx_{n} \Vert \\ =&\bigl[1-(k+\alpha_{n})\beta_{n}\bigr]\Vert x_{n}-Tx_{n}\Vert +\beta_{n}\alpha_{n} \bigl\Vert f(y_{n})-Tx_{n}\bigr\Vert . \end{aligned}

Thus,

\begin{aligned} \Vert x_{n}-Tx_{n}\Vert \le&\Vert x_{n}-x_{n+1}\Vert +\Vert x_{n+1}-Tx_{n} \Vert \\ \le&\bigl[1-(k+\alpha_{n})\beta_{n}\bigr]\Vert x_{n}-Tx_{n}\Vert +\beta_{n}\alpha_{n} \bigl\Vert f(y_{n})-Tx_{n}\bigr\Vert \\ &{}+\Vert x_{n}-x_{n+1}\Vert . \end{aligned}

It follows that

\begin{aligned} \Vert x_{n}-Tx_{n}\Vert \le&\frac{1}{(k+\alpha_{n})\beta_{n}}\bigl( \Vert x_{n}-x_{n+1}\Vert +\beta _{n} \alpha_{n}\bigl\Vert f(y_{n})-Tx_{n}\bigr\Vert \bigr) \\ \to& 0. \end{aligned}

Similarly, we can obtain

$$\lim_{n\to\infty}\|y_{n}-Sy_{n}\|=0.$$

□

By Proposition 3.5, we know that the mapping $$P_{C}[\alpha f +(1-k-\alpha )I+kT]$$ is contractive for small enough α. Thus, the equation $$x=P_{C}[tf(x) +(1-k-t)x+kTx]$$ has a unique fixed point, denoted by $$x_{t}$$, that is,

$$x_{t}=P_{C}\bigl[tf(x_{t}) +(1-k-t)x_{t}+kTx_{t}\bigr]$$
(3.2)

for small enough t. In order to prove Theorem 3.3, we need the following lemma.

### Lemma 3.7

Suppose $$\operatorname{Fix}(T)\neq\emptyset$$. Then, as $$t\to0$$, the net $$\{x_{t}\}$$ defined by (3.2) converges strongly to a fixed point of T.

### Proof

Let $$x^{*}\in \operatorname{Fix}(T)$$. From (3.2), we have

\begin{aligned} \bigl\Vert x_{t}-x^{*}\bigr\Vert =&\bigl\Vert P_{C} \bigl[tf(x_{t})+(1-k-t)x_{t}+kTx_{t}\bigr]-x^{*}\bigr\Vert \\ \leq& t\bigl\Vert f(x_{t})-x^{*}\bigr\Vert +\bigl\Vert (1-k-t) \bigl(x_{t}-x^{*}\bigr)+k\bigl(Tx_{t}-x^{*}\bigr)\bigr\Vert \\ \leq& t\rho_{1}\bigl\Vert x_{t}-x^{*}\bigr\Vert +t\bigl\Vert f\bigl(x^{*}\bigr)-x^{*}\bigr\Vert +(1-t)\bigl\Vert x_{t}-x^{*} \bigr\Vert , \end{aligned}

hence,

$$\bigl\Vert x_{t}-x^{*}\bigr\Vert \le\frac{1}{1-\rho_{1}}\bigl\Vert f \bigl(x^{*}\bigr)-x^{*}\bigr\Vert .$$

Thus, $$\{x_{t}\}$$ is bounded. Again, from (3.2), we get

$$\Vert x_{t}-Tx_{t}\Vert \leq t\bigl\Vert f(x_{t})-Tx_{t}\bigr\Vert +(1-k-t)\Vert x_{t}-Tx_{t}\Vert .$$

It follows that

$$\Vert x_{t}-Tx_{t}\Vert \leq\frac{t}{k+t}\bigl\Vert f(x_{t})-Tx_{t}\bigr\Vert \to0.$$

Let $$\{t_{n}\}\subset(0,1)$$. Assume that $$t_{n}\to0$$ as $$n\to\infty$$. Put $$x_{n}:=x_{t_{n}}$$. We have $$\lim_{n\to\infty}\| x_{n}-Tx_{n}\|=0$$. Set $$y_{t}=tf(x_{t})+(1-k-t)x_{t}+kTx_{t}$$, for all t. Then we have $$x_{t}=P_{C}y_{t}$$, and for any $$x^{*}\in \operatorname{Fix}(T)$$,

\begin{aligned} x_{t}-x^{*} =&x_{t}-y_{t}+y_{t}-x^{*} \\ =&x_{t}-y_{t}+t\bigl(f(x_{t})-x^{*}\bigr)+(1-k-t) \bigl(x_{t}-x^{*}\bigr)+k\bigl(Tx_{t}-x^{*}\bigr). \end{aligned}

From the property of the metric projection, we deduce

$$\bigl\langle x_{t}-y_{t},x_{t}-x^{*}\bigr\rangle \leq0.$$

So,

\begin{aligned} \bigl\Vert x_{t}-x^{*}\bigr\Vert ^{2} =&\bigl\langle x_{t}-y_{t}, x_{t}-x^{*}\bigr\rangle +\bigl\langle (1-k-t) \bigl(x_{t}-x^{*}\bigr)+k\bigl(Tx_{t}-x^{*} \bigr),x_{t}-x^{*}\bigr\rangle \\ &{}+t\bigl\langle f(x_{t})-x^{*}, x_{t}-x^{*}\bigr\rangle \\ \leq& \bigl\Vert (1-k-t) \bigl(x_{t}-x^{*}\bigr)+k \bigl(Tx_{t}-x^{*}\bigr)\bigr\Vert \bigl\Vert x_{t}-x^{*}\bigr\Vert \\ &{}+t\bigl\langle f(x_{t})-f\bigl(x^{*}\bigr), x_{t}-x^{*}\bigr\rangle +t\bigl\langle f\bigl(x^{*}\bigr)-x^{*}, x_{t}-x^{*}\bigr\rangle \\ \leq& \bigl[1-(1-\rho_{1})t\bigr]\bigl\Vert x_{t}-x^{*} \bigr\Vert ^{2}+t\bigl\langle f\bigl(x^{*}\bigr)-x^{*}, x_{t}-x^{*}\bigr\rangle . \end{aligned}

Hence,

$$\bigl\Vert x_{t}-x^{*}\bigr\Vert ^{2}\le\frac{1}{(1-\rho_{1})} \bigl\langle f\bigl(x^{*}\bigr)-x^{*}, x_{t}-x^{*}\bigr\rangle , \quad \forall x^{*} \in \operatorname{Fix}(T).$$

By similar arguments to , we find that the net $$\{x_{t}\}$$ converges strongly to $$x^{*}\in \operatorname{Fix}(T)$$. This completes the proof. □

### Remark 3.8

From Lemma 3.7, we know that the net $$\{x_{t}\}$$ defined by $$x_{t}=P_{C}[tu +(1-k-t)x_{t}+kTx_{t}]$$ where $$u\in H$$, converges to $$P_{\operatorname{Fix}(T)} u$$. Let $$x^{*}\in \operatorname{Fix}(T)$$ and $$y^{*}\in \operatorname{Fix}(S)$$. If we take $$u=f(y^{*})$$, then the net $$\{x_{t}\}$$ defined by $$x_{t}=P_{C}[tf(y^{*}) +(1-k-t)x_{t}+kTx_{t}]$$, converges to $$P_{\operatorname{Fix}(T)} f(y^{*})$$.

Finally, we prove that $$x_{n}\to P_{\operatorname{Fix}(T)} f(y^{*})$$ and $$y_{n}\to P_{\operatorname{Fix}(S)}g(x^{*})$$, where $$x^{*}\in \operatorname{Fix}(T)$$ and $$y^{*}\in \operatorname{Fix}(S)$$. We note the following fact. If the sequence $$\{w_{n}\}$$ is bounded and $$\| w_{n}-Tw_{n}\|\to0$$, we easily deduce that

$$\limsup_{n\to\infty}\bigl\langle f\bigl(P_{\operatorname{Fix}(S)} g\bigl(x^{*} \bigr)\bigr)-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr), w_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\rangle \le0.$$

Set $$v_{n}=P_{C}[\alpha_{n}g(x_{n})+(1-k-\alpha_{n})y_{n}+kSy_{n}]$$ for all $$n\ge0$$. Thus, we deduce that the sequences $$\{u_{n}\}$$ and $$\{v_{n}\}$$ satisfy: (1) $$\{u_{n}\}$$ and $$\{ v_{n}\}$$ are bounded; (2) $$\|u_{n}-Tu_{n}\|\to0$$ and $$\|v_{n}-Sv_{n}\|\to0$$. Therefore,

$$\limsup_{n\to\infty}\bigl\langle f\bigl(P_{\operatorname{Fix}(S)} g\bigl(x^{*} \bigr)\bigr)-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr), u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\rangle \le0$$

and

$$\limsup_{n\to\infty}\bigl\langle g\bigl(P_{\operatorname{Fix}(T)} f\bigl(y^{*} \bigr)\bigr)-P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr), v_{n}-P_{\operatorname{Fix}(S)} g \bigl(x^{*}\bigr)\bigr\rangle \le0.$$

Next, we estimate $$\|u_{n}-P_{\operatorname{Fix}(T)} f(y^{*})\|$$. Set $$\tilde{u}_{n}=\alpha _{n}f(y_{n})+(1-k-\alpha_{n})x_{n}+kTx_{n}$$ and $$\tilde{v}_{n}=\alpha _{n}g(x_{n})+(1-k-\alpha_{n})y_{n}+kSy_{n}$$ for all n. We have

\begin{aligned} \begin{aligned} &\bigl\Vert u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2} \\ &\quad = \bigl\Vert P_{C}[\tilde{u}_{n}]-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\Vert ^{2} \\ &\quad \le \bigl\langle \tilde{u}_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr), u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\rangle \\ &\quad = \bigl\langle \alpha_{n} f(y_{n})+(1-k- \alpha_{n})x_{n}+kTx_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*} \bigr), u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\rangle \\ &\quad \le \alpha_{n}\bigl\langle f(y_{n})-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr), u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\rangle \\ &\qquad {} +(1-\alpha_{n})\bigl\Vert x_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert \bigl\Vert u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\Vert \\ &\quad \le \frac{1-\alpha_{n}}{2}\bigl\Vert x_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\Vert ^{2}+\frac{1}{2}\bigl\Vert u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2} \\ &\qquad {} +\alpha_{n}\bigl\langle f(y_{n})-f \bigl(P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr)\bigr), u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\rangle \\ &\qquad {} +\alpha_{n}\bigl\langle f\bigl(P_{\operatorname{Fix}(S)} g\bigl(x^{*} \bigr)\bigr)-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr), u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\rangle \\ &\quad \le \frac{1-\alpha_{n}}{2}\bigl\Vert x_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\Vert ^{2}+\frac{1}{2}\bigl\Vert u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2} \\ &\qquad {} +\alpha_{n}\rho\bigl\Vert y_{n}-P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr)\bigr\Vert \bigl\Vert u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\Vert \\ &\qquad {} +\alpha_{n}\bigl\langle f\bigl(P_{\operatorname{Fix}(S)} g\bigl(x^{*} \bigr)\bigr)-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr), u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\rangle \\ &\quad \le \frac{1-\alpha_{n}}{2}\bigl\Vert x_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\Vert ^{2}+\frac{1}{2}\bigl\Vert u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2} \\ &\qquad {} +\frac{\alpha_{n}\rho}{2}\bigl(\bigl\Vert y_{n}-P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr)\bigr\Vert ^{2}+\bigl\Vert u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2}\bigr) \\ &\qquad {} +\alpha_{n}\bigl\langle f\bigl(P_{\operatorname{Fix}(S)} g\bigl(x^{*} \bigr)\bigr)-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr), u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\rangle . \end{aligned} \end{aligned}

It follows that

\begin{aligned}& \bigl\Vert u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2} \\& \quad \le \frac{1-\alpha_{n}}{1-\alpha_{n}\rho}\bigl\Vert x_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\Vert ^{2}+\frac {\alpha_{n}\rho}{1-\alpha_{n}\rho}\bigl\Vert y_{n}-P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr)\bigr\Vert ^{2} \\& \qquad {} +\frac{2\alpha_{n}}{1-\alpha_{n}\rho}\bigl\langle f\bigl(P_{\operatorname{Fix}(S)} g\bigl(x^{*} \bigr)\bigr)-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr), u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\rangle . \end{aligned}

Thus,

\begin{aligned}& \bigl\Vert x_{n+1}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2} \\& \quad \le (1-\beta_{n})\bigl\Vert x_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\Vert ^{2}+\beta_{n}\bigl\Vert u_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2} \\& \quad \le \biggl(1-\frac{1-\rho}{1-\alpha_{n}\rho}\alpha_{n}\beta_{n} \biggr)\bigl\Vert x_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2}+\frac{\alpha_{n}\beta_{n}\rho}{1-\alpha_{n}\rho}\bigl\Vert y_{n}-P_{\operatorname{Fix}(S)} g \bigl(x^{*}\bigr)\bigr\Vert ^{2} \\& \qquad {} +\frac{2\alpha_{n}\beta_{n}}{1-\alpha_{n}\rho}\bigl\langle f\bigl(P_{\operatorname{Fix}(S)} g\bigl(x^{*} \bigr)\bigr)-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr), u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\rangle . \end{aligned}

Similarly, we also have

\begin{aligned}& \bigl\Vert y_{n+1}-P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr)\bigr\Vert ^{2} \\& \quad \le \biggl(1-\frac{1-\rho}{1-\alpha_{n}\rho}\alpha_{n}\beta_{n} \biggr)\bigl\Vert y_{n}-P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr)\bigr\Vert ^{2}+\frac{\alpha_{n}\beta_{n}\rho}{1-\alpha_{n}\rho}\bigl\Vert x_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\Vert ^{2} \\& \qquad {} +\frac{2\alpha_{n}\beta_{n}}{1-\alpha_{n}\rho}\bigl\langle g\bigl(P_{\operatorname{Fix}(T)} f\bigl(y^{*} \bigr)\bigr)-P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr), v_{n}-P_{\operatorname{Fix}(S)} g \bigl(x^{*}\bigr)\bigr\rangle . \end{aligned}

Therefore,

\begin{aligned}& \bigl\Vert x_{n+1}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2}+\bigl\Vert y_{n+1}-P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr) \bigr\Vert ^{2} \\& \quad \le \biggl(1-\frac{1-2\rho}{1-\alpha_{n}\rho}\alpha_{n}\beta_{n} \biggr) \bigl(\bigl\Vert x_{n}-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr)\bigr\Vert ^{2}+\bigl\Vert y_{n}-P_{\operatorname{Fix}(S)} g\bigl(x^{*} \bigr)\bigr\Vert ^{2}\bigr) \\& \qquad {} +\frac{2\alpha_{n}\beta_{n}}{1-\alpha_{n}\rho}\bigl\langle f\bigl(P_{\operatorname{Fix}(S)} g\bigl(x^{*} \bigr)\bigr)-P_{\operatorname{Fix}(T)} f\bigl(y^{*}\bigr), u_{n}-P_{\operatorname{Fix}(T)} f \bigl(y^{*}\bigr)\bigr\rangle \\& \qquad {} +\frac{2\alpha_{n}\beta_{n}}{1-\alpha_{n}\rho}\bigl\langle g\bigl(P_{\operatorname{Fix}(T)} f\bigl(y^{*} \bigr)\bigr)-P_{\operatorname{Fix}(S)} g\bigl(x^{*}\bigr), v_{n}-P_{\operatorname{Fix}(S)} g \bigl(x^{*}\bigr)\bigr\rangle . \end{aligned}

We can check that all assumptions of Lemma 2.3 are satisfied. Therefore, $$x_{n}\to P_{\operatorname{Fix}(T)} f(y^{*})$$ and $$y_{n}\to P_{\operatorname{Fix}(S)} g(x^{*})$$. This completes the proof. □

### Algorithm 3.9

For arbitrarily given $$x_{0}\in C$$, let the sequence $$\{x_{n}\}$$ be generated iteratively by

$$x_{n+1}=(1-\beta_{n})x_{n}+ \beta_{n}P_{C}\bigl[(1-k-\alpha_{n})x_{n}+kTx_{n} \bigr],\quad n\geq0,$$
(3.3)

where $$\{\alpha_{n}\}$$ and $$\{\beta_{n}\}$$ are two real number sequences in $$(0,1)$$.

### Theorem 3.10

Suppose $$\operatorname{Fix}(T)\neq \emptyset$$. Assume the following conditions are satisfied:

1. (C1)

$$\lim_{n\to\infty}\alpha_{n}=0$$ and $$\sum_{n=0}^{\infty}\alpha_{n}=\infty$$;

2. (C2)

$$\beta_{n}\in[\xi_{1}, \xi_{2}]\subset(0,1)$$ for all $$n\ge0$$.

Then the sequence $$\{x_{n}\}$$ generated by (3.3) converge strongly to the fixed points $$P_{\operatorname{Fix}(T)}(0)$$, which is the minimum norm element in $$\operatorname{Fix}(T)$$.

## References

1. 1.

Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510 (1953)

2. 2.

Reich, S: Weak convergence theorems for non-expansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274-276 (1979)

3. 3.

Genel, A, Lindenstrauss, J: An example concerning fixed points. Isr. J. Math. 22, 81-86 (1975)

4. 4.

Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings. J. Math. Anal. Appl. 20, 197-228 (1967)

5. 5.

Browder, FE: Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces. Arch. Ration. Mech. Anal. 24, 82-90 (1967)

6. 6.

Halpern, B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 73, 957-961 (1967)

7. 7.

Ishikawa, S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147-150 (1974)

8. 8.

Lions, PL: Approximation de points fixes de contractions. C. R. Acad. Sci., Sér. A-B Paris 284, 1357-1359 (1977)

9. 9.

Opial, Z: Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Am. Math. Soc. 73, 595-597 (1967)

10. 10.

Wittmann, R: Approximation of fixed points of non-expansive mappings. Arch. Math. 58, 486-491 (1992)

11. 11.

Moudafi, A: Viscosity approximation methods for fixed-point problems. J. Math. Anal. Appl. 241, 46-55 (2000)

12. 12.

Shioji, N, Takahashi, W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 125, 3641-3645 (1997)

13. 13.

Suzuki, T: A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 135, 99-106 (2007)

14. 14.

Reich, S, Zaslavski, AJ: Convergence of Krasnoselskii-Mann iterations of nonexpansive operators. Math. Comput. Model. 32, 1423-1431 (2000)

15. 15.

Xu, HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279-291 (2004)

16. 16.

Geobel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)

17. 17.

Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240-256 (2002)

18. 18.

Suzuki, T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 103-123 (2005)

19. 19.

Mainge, PE: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325, 469-479 (2007)

20. 20.

Chidume, CE, Chidume, CO: Iterative approximation of fixed points of nonexpansive mappings. J. Math. Anal. Appl. 318, 288-295 (2006)

21. 21.

Scherzer, O: Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl. 194, 911-933 (1991)

22. 22.

Atsushiba, S, Takahashi, W: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian J. Math. 41, 435-453 (1999)

23. 23.

Bauschke, HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 202, 150-159 (1996)

24. 24.

Ceng, LC, Cubiotti, P, Yao, JC: Strong convergence theorems for finitely many nonexpansive mappings and applications. Nonlinear Anal. 67, 1464-1473 (2007)

25. 25.

Chang, SS: Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 323, 1402-1416 (2006)

26. 26.

Jung, JS: Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 302, 509-520 (2005)

27. 27.

Yao, Y, Shahzad, N, Liou, YC: Modified semi-implicit midpoint rule for nonexpansive mappings. Fixed Point Theory Appl. 2015, 166 (2015)

28. 28.

Marino, G, Xu, HK: Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336-349 (2007)

29. 29.

Chidume, CE, Mutangadura, SA: An example on the Mann iteration method for Lipschitz pseudo-contractions. Proc. Am. Math. Soc. 129, 2359-2363 (2001)

30. 30.

Yao, Y, Liou, YC, Marino, G: A hybrid algorithm for pseudo-contractive mappings. Nonlinear Anal. 71, 997-5002 (2009)

31. 31.

Zhou, H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear Anal. 70, 4039-4046 (2009)

32. 32.

Guo, W, Choi, M, Cho, YJ: Convergence theorems for continuous pseudocontractive mappings in Banach spaces. J. Inequal. Appl. 2014, 384 (2014)

33. 33.

Hussain, N, Ćirić, LB, Cho, YJ, Rafiq, A: On Mann-type iteration method for a family of hemicontractive mappings in Hilbert spaces. J. Inequal. Appl. 2013, 41 (2013)

34. 34.

Yao, Y, Liou, YC, Yao, JC: Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction. Fixed Point Theory Appl. 2015, 127 (2015)

35. 35.

Yao, Y, Postolache, M, Liou, YC, Yao, Z: Construction algorithms for a class of monotone variational inequalities. Optim. Lett. (2015). doi:10.1007/s11590-015-0954-8

## Acknowledgements

The authors are grateful to the three reviewers for their valuable comments and suggestions. Zhangsong Yao was supported by the Scientific Research Project of Nanjing Xiaozhuang University (2015NXY46).

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Correspondence to Shin Min Kang.

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