1 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 map** \(T:C\to C\) 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 map** \(T:C\to C\) 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 map**s properly includes the class of nonexpansive map**s.

Iterative construction of fixed points of nonlinear map**s 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 map**. 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 [1]; it has been studied extensively in the literature. If T is a nonexpansive map** 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 [2]. Now, it is well known that Mann’s algorithm fails, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces [3]. Iterative methods for nonexpansive map**s have been investigated extensively in the literature; see [2–27] and the references therein. However, iterative methods for strictly pseudo-contractive map**s are far less developed than those for nonexpansive map**s though Browder and Petryshyn [4] initiated their work in 1967. However, strictly pseudo-contractive map**s have more powerful applications than nonexpansive map**s, for example, to solve inverse problems (see Scherzer [21]). Therefore it is interesting to develop the algorithms for finding the fixed points of strictly pseudo-contractive map**s. 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 [28–35]. The first such an attempt was done by Ishikawa [7] 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-map** 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 map**s have been considered by many authors. For the related work, we refer the reader to [22–26, 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.

2 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

([28])

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 map**. 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

([18])

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

([17])

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 \(\lim_{n\rightarrow\infty}a_{n}=0\).

3 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 map** \(f:C\to H\) 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 map** \(P_{C}[\alpha f +(1-k-\alpha)I+kT]\) 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 map** \(P_{C}[\alpha f +(1-k-\alpha )I+kT]\) 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 [28], 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 \(\operatorname{Fix}(T)\).