1 Introduction

Let H be a real Hilbert space, whose inner product and norm are denoted by \(\langle\cdot , \cdot \rangle\) and \(\Vert \cdot \Vert \), respectively. Let K be nonempty, closed, and convex subset of H. We recall that a mapping \(T \colon K\to K\) is called strongly pseudocontractive, if for some \(\beta\in(0,1)\),

$$ \langle Tx - Ty, x - y \rangle \leq\beta{ \Vert x - y\Vert }^{2} , $$
(1.1)

holds, for all \(x,y\in K\), while T is called a pseudocontractive mapping if (1.1) holds for \(\beta=1\). Equivalently T is called pseudocontractive if

$$ \Vert Tx- Ty\Vert ^{2} \leq \Vert x-y\Vert ^{2} + \bigl\Vert (I-T)x - (I-T)y\bigr\Vert ^{2} , \quad \forall x,y \in K . $$
(1.2)

The mapping T is called Lipschitz if there exists a \(L\geq0\) such that \(\Vert Tx-Ty\Vert \leq L\Vert x-y\Vert \), for all \(x,y\in K\). The mapping T is called nonexpansive if \(L=1\) and is called contractive if \(L<1\). Every nonexpansive mapping is pseudocontractive. The converse is not true. For example \(Tx=1-x^{\frac{2}{3}}\), \(0\leq x\leq1\), is a continuous pseudocontractive mapping which is not a nonexpansive mapping.

Let \(T\colon K\to K\) be a mapping and \(\{\alpha_{n} \}\) and \(\{\beta_{n} \}\) be two sequences in \([0,1]\). For arbitrary chosen \(x_{0}\in K\), construct a sequence \(\{x_{n} \}\), where \(x_{n}\) is defined iteratively for each positive integer \(n\geq0\) by

$$\begin{aligned}& x_{n+1}=Tx_{n} , \end{aligned}$$
(1.3)
$$\begin{aligned}& x_{n+1}=(1-\alpha_{n})x_{n}+ \alpha_{n} Tx_{n} , \end{aligned}$$
(1.4)
$$\begin{aligned}& \left. \begin{aligned} &x_{n+1}=(1- \alpha_{n})x_{n}+\alpha_{n} Ty_{n} , \\ &y_{n}=(1-\beta_{n})x_{n} + \beta_{n} Tx_{n} . \end{aligned} \right\} \end{aligned}$$
(1.5)

The sequences \(\{x_{n} \}\) generated by (1.3), (1.4), and (1.5) are called Picard, Mann [1], and Ishikawa [2] iteration sequences, respectively.

In 1955, Krasnoselskii [3] showed that the Picard iteration scheme (1.3) for a nonexpansive mapping T may fail to converge to a fixed point of T even if T has a unique fixed point, but the Mann sequence (1.4) for \(\alpha_{n}=\frac{1}{2} \), \(\forall n\geq0\), converges strongly to the fixed point of T.

In 1974, Ishikawa [2], introduced the iteration scheme (1.5) and proved the following theorem.

Theorem 1.1

If K be a compact convex subset of a Hilbert space H, \(T\colon K\to K\) is a Lipschitzian pseudocontractive mapping and \(x_{0}\) is any point of K, then the sequence \(\{x_{n} \}\) defined iteratively by (1.5) converges strongly to a fixed point of T, where \(\{\alpha_{n} \}\) and \(\{\beta_{n} \}\) satisfy the conditions: \(0\leq\alpha_{n}\leq\beta_{n}<1\), \(\lim_{n\to\infty}\beta_{n}=0\), and \(\sum_{n\geq0}\alpha_{n}\beta_{n}=\infty\).

Since its publication, in 1974, till 2000, it was not known whether or not the Mann iteration process with the simpler iteration (1.4) would converge under the setting of Theorem 1.1 to a fixed point of T if the mapping T is pseudocontractive and continuous (or even Lipschitzian with constant \(L>1\)).

Hicks and Kubicek [4] gave an example of pseudocontractive mapping for which Mann iteration process fails. Borwein and Borwein ([5], Proposition 8) gave an example of a Lipschitzian mapping with a unique fixed point for which the Mann iteration fails to converge. In the example of Borwein and Borwein the mapping was not pseudocontractive, while in the example of Hicks and Kubicek the mapping was not continuous (and hence not Lipschitzian).

The problem for a continuous pseudocontractive mapping still remained open. This question was finally settled by Chidume and Mutangadura [6], by constructing an example of a Lipschitz pseudocontractive mapping with a unique fixed point for which every nontrivial Mann sequence fails to converge.

Example 1.2

Let H be the real Hilbert space \(\mathbb{R}^{2}\) under the usual Euclidean inner product. If \(x=(a,b)\in H\), define \(x^{\bot}\in H\) to be \((b,-a)\). Let \(K:= \{x\in H: \Vert x\Vert \leq 1 \}\) and set

$$K_{1}:= \biggl\{ x\in H : \Vert x\Vert \leq\frac{1}{2} \biggr\} , \qquad K_{2}:= \biggl\{ x\in H : \frac{1}{2}\leq \Vert x \Vert \leq1 \biggr\} . $$

Define \(T\colon K\to K\) as follows:

$$ Tx= \begin{cases} x+ x^{\bot}, &\text{if } x\in K_{1}, \\ \frac{x}{\Vert x\Vert }-x+x^{\bot}, &\text{if } x\in K_{2} . \end{cases} $$
(1.6)

Then:

  1. 1.

    T is Lipschitz and pseudocontractive.

  2. 2.

    The origin is the unique fixed point of T.

  3. 3.

    No Mann sequence converges to the fixed point zero.

  4. 4.

    No Mann sequence converges to any \(x\neq0\).

The Ishikawa iterative method has been studied extensively by various authors. But it was still an open question whether or not this method can be employed to approximate fixed points of Lipschitz pseudocontractive mappings without the compactness assumption on K or T (see, e.g., [7–9]).

In [10], Schu introduced an iteration scheme and proved the following.

Theorem 1.3

Let H be a Hilbert space, K a nonempty, closed, bounded, and convex subset of H, \(w\in K\), \(T\colon K\to K\) pseudocontractive and Lipschitzian with \(L>0\), \(\{\lambda_{n} \}_{n\in\mathbb {N}}\subset (0,1)\) with \(\lim_{n\to\infty}\lambda_{n}=1\), \(\{\alpha _{n} \}_{n\in \mathbb{N}}\subset(0,1)\) with \(\lim_{n\to\infty}\alpha_{n}=0\), such that \(( \{\alpha_{n} \}, \{\mu_{n} \} )\) has certain properties, \(((1-\mu_{n})(1-\lambda_{n})^{-1} )\) is bounded and \(\lim_{n\to\infty} \alpha_{n}^{-1}(1-\mu_{n})=0\), where \(k_{n}:= (1+\alpha_{n}^{2}(1+L^{2}) )^{1/2}\) and \(\mu_{n}:=\lambda _{n}k_{n}^{-1}\) for all \(n\in\mathbb{N}\). For arbitrary \(z_{0}\in K\), define for all \(n\in\mathbb{N}\),

$$ \left. \begin{aligned} &z_{n+1}=(1- \mu_{n+1})w + \mu_{n+1}y_{n} , \\ &y_{n}=(1-\alpha_{n})z_{n} + \alpha_{n} Tz_{n} . \end{aligned} \right\} $$
(1.7)

Then \(\{z_{n} \}\) converges strongly to the unique fixed point of T closest to w.

Theorem 1.3 has some advantages over Theorem 1.1. First, the recursion formula (1.7) is slightly simpler than (1.5). More importantly, no compactness assumption is imposed on K. However, the choices of \(\alpha_{n}\) and \(\mu_{n}\) in Theorem 1.3 are not as simple as those of \(\alpha_{n}\) and \(\beta_{n}\) in Theorem 1.3 (where one can choose \(\alpha_{n}=\beta_{n}=\frac{1}{\sqrt{n+1}}\) for all positive integers n).

Zhou [11] established the hybrid Ishikawa algorithm for Lipschitz pseudocontractive mappings without the compactness assumption. In 2009, Yao et al. [12] introduced the hybrid Mann algorithm for a L-Lipschitz pseudocontractive mapping, which was generalized by Tang et al. [13] to the hybrid Ishikawa iterative process. The schemes given by Zhou [11], Yao et al. [12] and Tang et al. [13] are not easy to compute, since for each \(n\geq1\), it involves the computation of the intersection of \(C_{n}\) and \(Q_{n}\).

Finding a point in the intersection of fixed point sets of a family of nonexpansive mappings is a task that occurs frequently in various areas of mathematical sciences and engineering. For example, the well-known convex feasibility problem reduces to finding a point in the intersection of fixed point sets of a family of nonexpansive mappings; see, e.g., [14, 15].

In 2011, Zegeye et al. [16] extended the result of Tang et al. [13] to the Ishikawa iterative process (not hybrid) for a finite family of Lipschitz pseudocontractive mappings as follows:

$$ \left. \begin{aligned} &y_{n}= (1- \beta_{n})x_{n}+ \beta_{n}T_{n}x_{n}, \\ &x_{n+1}= (1- \alpha_{n})x_{n}+ \alpha_{n}T_{n}y_{n}, \end{aligned} \right\} $$
(1.8)

and they proved a strong convergence theorem under some conditions.

Cheng et al. [17] generalized algorithm (1.8) to a three step iterative process for a countable family of Lipschitz pseudocontractive mappings as follows:

$$ \left. \begin{aligned} &z_{n}= (1- \gamma_{n})x_{n} + \gamma_{n}T_{n}x_{n}, \\ &y_{n}= (1- \beta_{n})x_{n} + \beta_{n}T_{n} z_{n}, \\ &x_{n+1}= (1- \alpha_{n})x_{n} + \alpha_{n}T_{n}y_{n}. \end{aligned} \right\} $$
(1.9)

The iteration (1.9) was initially introduced by Noor [18] for a single mapping.

Recently, Zegeye and Shahzad [19] proved the strong convergence of a three step iteration process for a common fixed point of two Lipschitz pseudocontractive mappings.

In this paper, our purpose is to introduce a new three step iteration process for a countable family of pseudocontractive mappings. The results obtained in this paper improve and extend the results of Zhou [11], Tang et al. [13], Zegeye et al. [16] and Cheng et al. [17] and some other results in this direction.

2 Preliminaries

In the sequel, we also need the following definitions and lemma.

Let H be a real Hilbert space, and \(\phi\colon H \times H \to {\mathbb{R}} \) the function given by

$$ \phi(x, y) :=\Vert x-y\Vert ^{2} = \Vert x \Vert ^{2} - 2 \langle x, y \rangle+ \Vert y \Vert ^{2},\quad \text{for } x, y \in H. $$

It is obvious from the definition of the function ϕ that

$$ {\bigl(\Vert x\Vert - \Vert y\Vert \bigr)}^{2} \leq\phi (x, y) \leq{\bigl(\Vert x\Vert +\Vert y\Vert \bigr)}^{2}, \quad \text{for } x, y \in H. $$

The function ϕ also has the following property:

$$ \phi(y, x)= \phi(y, z)+\phi(z, x)+2 \langle z-y, x-z \rangle, \quad \text{for } x, y, z \in H. $$
(2.1)

A countable family of mappings \(\{T_{n}\}_{n=1}^{\infty}\colon K \to H \) is called uniformly Lipschitz with Lipschitz constant \(L_{n} > 0\), \(n \geq1\), if there exists \(0 < L = \sup_{n\geq1} L_{n}\) such that

$$\Vert T_{n}x - T_{n}y\Vert \leq L \Vert x -y \Vert , \quad \forall x, y \in K, n\geq1. $$

A countable family of mappings \(\{T_{n}\}_{n=1}^{\infty}\colon K \to H \) is called uniformly closed if \(x_{n} \to x^{*}\) and \(\Vert x_{n} - T_{n} x_{n}\Vert \to0\) imply \(x^{*} \in\bigcap_{n=1}^{\infty}F(T_{n})\).

Lemma 2.1

Let H be a real Hilbert space. Then, for all \(x_{i} \in H\) and \(\alpha_{i} \in[0, 1]\) for \(i=1, 2, \ldots, n\) such that \(\alpha_{1}+\alpha_{2}+ \cdots+\alpha_{n}=1\), the following inequality holds:

$$\Vert \alpha_{1}x_{1}+\alpha_{2}x_{2}+ \cdots+\alpha_{n}x_{n}\Vert ^{2} = \sum _{i=1}^{n}\alpha_{i}\Vert x_{i}\Vert ^{2} -\sum_{1\leq i, j \leq n} \alpha_{i}\alpha_{j}\Vert x_{i}-x_{j} \Vert ^{2}. $$

Remark 2.2

We now give recall an example [17] of a countable family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with the interior of the common fixed points nonempty.

Suppose that \(X = {\mathbb{R}}\) and \(K = [- 1, 1]\subset{\mathbb{R}}\). Let \(\{T_{n}\}_{n=1}^{\infty}\colon K \to K \) be given by

$$ T_{n}x = \begin{cases} x,& x \in[-1, 0), \\ (\frac{1}{2^{n}}+ \frac{1}{2} ), & x \in[0, 1]. \end{cases} $$

Then we observe that \(F = \bigcap_{n=1}^{\infty}F(T_{n}) = [-1, 0]\), and hence the interior of the common fixed points is nonempty. Also, it is clear that \(\{T_{n}\}_{n=1}^{\infty}\) is a countable family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with Lipschitz constant \(L = \sup_{n\geq1} L_{n} =2\).

3 Main results

Theorem 3.1

Let K be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(\{T_{n}\}_{n=1}^{\infty}\colon K \to K \) be a countable family of uniformly closed and Lipschitz pseudocontractive mappings with Lipschitzian constants \(L_{n}\). Assume that the interior of \(\mathcal{F} = \bigcap_{n=1}^{\infty}F(T_{n})\) is nonempty. Let \(\{x_{n} \}\) be a sequence generated from an arbitrary \(x_{1} \in K\) by the following algorithm:

$$ \left. \begin{aligned} &z_{n}= \gamma_{n}x_{n}+(1- \gamma_{n})T_{n}x_{n}, \\ &y_{n}= \beta_{n}x_{n}+(1- \beta_{n})T_{n}z_{n}, \\ &x_{n+1}= \alpha_{n}x_{n}+(1- \alpha_{n})[ \theta_{n}x_{n}+\delta_{n}T_{n}y_{n}+ \sigma_{n}T_{n}z_{n}], \end{aligned} \right\} $$
(3.1)

where \(\{\alpha_{n} \}\), \(\{\beta_{n} \}\), \(\{\gamma_{n} \}\), \(\{\theta_{n} \}\), \(\{\delta_{n} \}\), and \(\{\sigma_{n} \}\) are in \((0, 1)\) satisfying the following conditions:

  1. (i)

    \(\theta_{n}+\delta_{n}+\sigma_{n}=1\),

  2. (ii)

    \((1- \alpha_{n})\leq(1-\theta_{n})\leq(1- \beta_{n}) \leq(1- \gamma_{n})\), \(\forall n \geq1\),

  3. (iii)

    \(\liminf_{n \to\infty} (1- \alpha_{n}) = \alpha>0\),

  4. (iv)

    \(\sup_{n\geq1}(1-\gamma_{n}) \leq\gamma\) with \(\gamma^{3}L^{4}+ 2\gamma^{2}L^{3}+\gamma^{2} L^{2}+\gamma L^{2}+2\gamma<1\), where \(L = \sup_{n \geq1} L_{n}\).

Then \(\{x_{n} \}\) converges strongly to \(x^{*} \in\mathcal{F}\).

Proof

Suppose that \(p \in\mathcal{F}\). Using (3.1) and (1.2), we have

$$\begin{aligned}& \Vert z_{n} -p\Vert ^{2} = \gamma_{n} \Vert x_{n} - p\Vert ^{2} + (1- \gamma_{n})\Vert T_{n}x_{n} -p\Vert ^{2} -\gamma_{n}(1- \gamma_{n})\Vert T_{n}x_{n} -x_{n}\Vert ^{2} \\& \hphantom{\Vert z_{n} -p\Vert ^{2}} \leq\gamma_{n}\Vert x_{n} -p\Vert ^{2} + (1- \gamma_{n}) \Vert x_{n} -p\Vert ^{2} +(1-\gamma_{n})\Vert T_{n}x_{n} -x_{n} \Vert ^{2} \\& \hphantom{\Vert z_{n} -p\Vert ^{2}=}{} -\gamma_{n}(1- \gamma_{n})\Vert T_{n}x_{n} -x_{n} \Vert ^{2} \\& \hphantom{\Vert z_{n} -p\Vert ^{2}} = \Vert x_{n} -p\Vert ^{2}+ {(1- \gamma_{n})}^{2}\Vert T_{n}x_{n} -x_{n}\Vert ^{2}, \end{aligned}$$
(3.2)
$$\begin{aligned}& \Vert y_{n} -p\Vert ^{2} = \beta_{n} \Vert x_{n} - p \Vert ^{2} + (1- \beta_{n})\Vert T_{n}z_{n} -p\Vert ^{2} -\beta _{n}(1- \beta_{n})\Vert T_{n}z_{n} -x_{n}\Vert ^{2} \\& \hphantom{\Vert y_{n} -p\Vert ^{2}} \leq \beta_{n}\Vert x_{n} -p\Vert ^{2} +(1- \beta_{n}) \Vert z_{n} -p\Vert ^{2}+(1- \beta_{n})\Vert T_{n}z_{n} -z_{n} \Vert ^{2} \\& \hphantom{\Vert y_{n} -p\Vert ^{2} =}{} -\beta_{n}(1- \beta_{n})\Vert T_{n}z_{n} -x_{n} \Vert ^{2} . \end{aligned}$$
(3.3)

Using Lemma 2.1 and (3.1), we have

$$\begin{aligned}& \Vert x_{n +1} -p\Vert ^{2} \leq \alpha_{n} \Vert x_{n} - p\Vert ^{2} +(1- \alpha_{n}) \bigl[\theta_{n}\Vert x_{n} -p\Vert ^{2} +\delta_{n} \Vert T_{n}y_{n} -p \Vert ^{2} \\& \hphantom{\Vert x_{n +1} -p\Vert ^{2} \leq}{} +\sigma_{n}\Vert T_{n}z_{n} -p \Vert ^{2} -\theta _{n}\delta_{n}\Vert T_{n}y_{n} -x_{n}\Vert ^{2}- \theta_{n}\sigma _{n}\Vert T_{n}z_{n} -x_{n}\Vert ^{2} \bigr] \\& \hphantom{\Vert x_{n +1} -p\Vert ^{2}} \leq\alpha_{n}\Vert x_{n} - p\Vert ^{2}+(1- \alpha_{n}) \theta_{n}\Vert x_{n} -p\Vert ^{2}+(1- \alpha_{n})\bigl[\delta _{n} \Vert y_{n} -p\Vert ^{2} \\& \hphantom{\Vert x_{n +1} -p\Vert ^{2} \leq}{} +\delta_{n} \Vert T_{n}y_{n} -y_{n}\Vert ^{2} \bigr]+(1- \alpha_{n}) \bigl[ \sigma_{n} \Vert z_{n} -p \Vert ^{2}+ \sigma_{n} \Vert T_{n}z_{n} -z_{n} \Vert ^{2} \bigr] \\& \hphantom{\Vert x_{n +1} -p\Vert ^{2} \leq}{} -\theta_{n}\delta_{n}(1- \alpha_{n})\Vert T_{n}y_{n} -x_{n} \Vert ^{2}-\theta_{n}\sigma_{n}(1- \alpha_{n})\Vert T_{n}z_{n} -x_{n} \Vert ^{2}, \end{aligned}$$
(3.4)
$$\begin{aligned}& \Vert T_{n}z_{n} -z_{n}\Vert ^{2} = \gamma_{n} \Vert T_{n}z_{n} - x_{n}\Vert ^{2} + (1- \gamma_{n})\Vert T_{n}z_{n} -T_{n}x_{n} \Vert ^{2} \\& \hphantom{\Vert T_{n}z_{n} -z_{n}\Vert ^{2} =}{} -\gamma_{n}(1- \gamma_{n})\Vert T_{n}x_{n} -x_{n} \Vert ^{2} \\& \hphantom{\Vert T_{n}z_{n} -z_{n}\Vert ^{2}} \leq\gamma_{n}\Vert T_{n}z_{n} - x_{n}\Vert ^{2} + (1- \gamma _{n})^{3}L^{2} \Vert T_{n}x_{n} -x_{n}\Vert ^{2} \\& \hphantom{\Vert T_{n}z_{n} -z_{n}\Vert ^{2} =}{} -\gamma_{n}(1- \gamma_{n})\Vert T_{n}x_{n} -x_{n} \Vert ^{2} \\& \hphantom{\Vert T_{n}z_{n} -z_{n}\Vert ^{2}} = \gamma_{n}\Vert T_{n}z_{n} - x_{n}\Vert ^{2} \\& \hphantom{\Vert T_{n}z_{n} -z_{n}\Vert ^{2} =}{} +(1- \gamma_{n}) \bigl[(1- \gamma_{n})^{2}L^{2}+(1- \gamma_{n}) - 1 \bigr]\Vert T_{n}x_{n} -x_{n}\Vert ^{2}, \end{aligned}$$
(3.5)
$$\begin{aligned}& \Vert T_{n}y_{n} -y_{n}\Vert ^{2} = \beta_{n} \Vert T_{n}y_{n} - x_{n}\Vert ^{2} +(1- \beta_{n})\Vert T_{n}y_{n} -T_{n}z_{n}\Vert ^{2} \\& \hphantom{\Vert T_{n}y_{n} -y_{n}\Vert ^{2} = }{} - \beta_{n}(1- \beta_{n})\Vert T_{n}z_{n} -x_{n} \Vert ^{2} \\& \hphantom{\Vert T_{n}y_{n} -y_{n}\Vert ^{2} } \leq\beta_{n} \Vert T_{n}y_{n} - x_{n}\Vert ^{2} +(1- \beta _{n})L^{2} \Vert y_{n} -z_{n}\Vert ^{2} \\& \hphantom{\Vert T_{n}y_{n} -y_{n}\Vert ^{2} = }{} - \beta_{n}(1- \beta_{n})\Vert T_{n}z_{n} -x_{n} \Vert ^{2} \\& \hphantom{\Vert T_{n}y_{n} -y_{n}\Vert ^{2}} \leq\beta_{n} \Vert T_{n}y_{n} - x_{n}\Vert ^{2} +(1- \beta _{n})L^{2} \Vert y_{n} -z_{n}\Vert ^{2}, \end{aligned}$$
(3.6)
$$\begin{aligned}& \Vert y_{n} -z_{n}\Vert = \bigl\Vert \beta_{n}x_{n}+(1- \beta _{n})T_{n}z_{n} - \gamma_{n}x_{n}-(1- \gamma_{n})T_{n}x_{n} \bigr\Vert \\& \hphantom{\Vert y_{n} -z_{n}\Vert } = \bigl\Vert (1- \gamma_{n})x_{n}-(1- \beta_{n})x_{n}-\bigl\{ (1- \gamma_{n})-(1- \beta_{n})\bigr\} T_{n} x_{n} \\& \hphantom{\Vert y_{n} -z_{n}\Vert =}{} +(1- \beta_{n}) (T_{n}z_{n} - T_{n}x_{n}) \bigr\Vert \\& \hphantom{\Vert y_{n} -z_{n}\Vert } \leq\bigl\{ (1- \gamma_{n})-(1- \beta_{n}) \bigr\} \Vert T_{n}x_{n} -x_{n} \Vert +(1- \beta_{n})L\Vert z_{n} -x_{n}\Vert \\& \hphantom{\Vert y_{n} -z_{n}\Vert } =\bigl\{ (1- \gamma_{n})-(1- \beta_{n})+(1- \beta_{n}) (1- \gamma_{n})L\bigr\} \Vert T_{n}x_{n} -x_{n}\Vert . \end{aligned}$$
(3.7)

Substituting (3.2) and (3.5) in (3.3), we obtain

$$\begin{aligned} \Vert y_{n} -p\Vert ^{2} \leq&\Vert x_{n} -p\Vert ^{2}+ (1- \beta_{n}){(1- \gamma_{n})}^{2}\Vert T_{n}x_{n} -x_{n}\Vert ^{2} +(1- \beta_{n}) \bigl[ \gamma_{n}\Vert T_{n}z_{n} - x_{n} \Vert ^{2} \\ &{} +(1- \gamma_{n}) \bigl[(1- \gamma_{n})^{2}L^{2}+(1- \gamma _{n}) - 1 \bigr]\Vert T_{n}x_{n} -x_{n}\Vert ^{2} \bigr] \\ &{} -\beta_{n}(1- \beta_{n})\Vert T_{n}z_{n} -x_{n} \Vert ^{2} \\ \leq&\Vert x_{n} -p\Vert ^{2}- (1- \beta_{n}) \bigl[(1-\gamma _{n})-(1- \beta_{n}) \bigr]\Vert T_{n}z_{n} - x_{n}\Vert ^{2} \\ &{} +(1- \beta_{n}) (1- \gamma_{n}) \bigl[(1- \gamma _{n})^{2}L^{2}+2(1- \gamma_{n}) - 1 \bigr]\Vert T_{n}x_{n} -x_{n}\Vert ^{2}. \end{aligned}$$
(3.8)

From (3.7) and (3.6), we have

$$\begin{aligned} \Vert T_{n}y_{n} -y_{n}\Vert ^{2} \leq&\beta_{n} \Vert T_{n}y_{n} - x_{n}\Vert ^{2} +(1- \beta_{n})L^{2} \bigl\{ (1- \gamma_{n})-(1- \beta_{n}) \\ &\bigl.\bigl.{} +(1- \beta_{n}) (1- \gamma_{n})L \bigr\} \bigr.^{2} \Vert T_{n}x_{n} -x_{n}\Vert ^{2}. \end{aligned}$$
(3.9)

Substituting (3.2), (3.5), (3.8), and (3.9) in (3.4), we get

$$\begin{aligned}& \Vert x_{n +1} -p\Vert ^{2} \\& \quad \leq \Vert x_{n} -p \Vert ^{2}-(1- \alpha_{n}) \bigl[(1- \gamma_{n})\bigl\{ \delta_{n}(1- \beta _{n})+ \sigma_{n}\bigr\} \bigl\{ 1-2(1- \gamma_{n}) -(1- \gamma_{n})^{2}L^{2}\bigr\} \\& \qquad {} -\delta_{n}(1- \beta_{n})L^{2}\bigl\{ (1- \gamma_{n})-(1- \beta _{n})+(1- \beta_{n}) (1- \gamma_{n})L\bigr\} ^{2} \bigr]\Vert T_{n}x_{n} -x_{n} \Vert ^{2} \\& \qquad {} + (1- \alpha_{n})\sigma_{n} \bigl[(1- \theta_{n})-(1-\gamma _{n}) \bigr]\Vert T_{n}z_{n} -x_{n}\Vert ^{2} \\& \qquad {} +(1- \alpha_{n})\delta_{n} \bigl[(1- \theta_{n})-(1-\beta _{n}) \bigr]\Vert T_{n}y_{n} -x_{n}\Vert ^{2}. \end{aligned}$$

From condition (ii), we have

$$\begin{aligned}& \Vert x_{n +1} -p\Vert ^{2} \\& \quad \leq \Vert x_{n} -p\Vert ^{2}-(1- \alpha _{n}) \bigl[(1-\gamma_{n})\bigl\{ \delta_{n}(1- \beta_{n})+\sigma_{n}\bigr\} \bigl\{ 1-2(1- \gamma_{n}) -(1- \gamma_{n})^{2}L^{2}\bigr\} \\& \qquad {}-\delta_{n}(1- \beta_{n})L^{2}\bigl\{ (1- \gamma_{n})-(1- \beta _{n})+(1- \beta_{n}) (1- \gamma_{n})L\bigr\} ^{2} \bigr]\Vert T_{n}x_{n} -x_{n} \Vert ^{2} \\& \quad \leq \Vert x_{n} -p\Vert ^{2}-(1- \alpha _{n}) \bigl[(1-\gamma_{n}) \bigl\{ \delta_{n}(1- \beta_{n})+\sigma_{n}(1- \beta _{n}) \bigr\} \bigl\{ 1-2(1- \gamma_{n}) \\& \qquad {}-(1- \gamma_{n})^{2}L^{2} \bigr\} - \delta_{n}(1- \beta _{n})L^{2} \bigl\{ (1- \gamma_{n})-(1- \beta_{n}) \\& \qquad\bigl.\bigl.\bigl. {}+(1- \beta_{n}) (1- \gamma_{n})L \bigr\} ^{2} \bigr]\Vert T_{n}x_{n} -x_{n} \Vert \bigr.^{2} \\& \quad \leq \Vert x_{n} -p\Vert ^{2}-(1- \alpha _{n}) \bigl[(1-\gamma_{n}) (1- \beta_{n}) (1- \theta_{n}) \bigl\{ 1-2(1- \gamma _{n}) -(1- \gamma_{n})^{2}L^{2} \bigr\} \\& \qquad {}-\delta_{n}(1- \beta_{n})L^{2} \bigl\{ (1- \gamma_{n})-(1- \beta_{n})+(1- \beta_{n}) (1- \gamma_{n})L \bigr\} ^{2} \bigr]\Vert T_{n}x_{n} -x_{n}\Vert ^{2} \\& \quad \leq \Vert x_{n} -p\Vert ^{2}-(1- \alpha _{n}) \bigl[(1-\gamma_{n}) (1- \beta_{n}) (1- \theta_{n}) \bigl\{ 1-2(1- \gamma _{n}) -(1- \gamma_{n})^{2}L^{2} \bigr\} \\& \qquad {}-(1-\theta_{n}) (1- \beta_{n})L^{2} \bigl\{ (1- \gamma_{n})-(1- \beta_{n})+(1- \beta_{n}) (1- \gamma_{n})L \bigr\} ^{2} \bigr]\Vert T_{n}x_{n} -x_{n}\Vert ^{2} \\& \quad \leq \Vert x_{n} -p\Vert ^{2}-(1- \alpha_{n}) (1- \beta_{n}) (1-\theta_{n}) \bigl[(1- \gamma_{n}) \bigl\{ 1-2(1- \gamma_{n}) -(1- \gamma_{n})^{2}L^{2} \bigr\} \\& \qquad {}-L^{2}\bigl\{ (1- \gamma_{n})-(1- \beta_{n})+(1- \beta_{n}) (1- \gamma_{n})L\bigr\} ^{2} \bigr]\Vert T_{n}x_{n} -x_{n} \Vert ^{2}. \end{aligned}$$
(3.10)

From condition (iv), we have

$$\begin{aligned}& \gamma^{3}L^{4}+ 2\gamma^{2}L^{3}+ \gamma^{2} L^{2}+\gamma L^{2}+2\gamma< 1, \\& (1- \gamma_{n})^{3}L^{4}+ 2(1- \gamma_{n})^{2}L^{3}+(1- \gamma_{n})^{2} L^{2}+(1- \gamma_{n}) L^{2}+2(1- \gamma_{n})<1, \\& (1- \gamma_{n})^{2} L^{2}+2(1- \gamma_{n})-1+(1- \gamma_{n})^{3}L^{4}+ 2(1- \gamma_{n})^{2}L^{3}+(1- \gamma_{n}) L^{2}<0, \\& {\bigl[(1- \gamma_{n})^{2} L^{2}+2(1- \gamma_{n})-1\bigr]}+(1- \gamma_{n})L^{2}\bigl[(1- \gamma _{n})^{2}L^{2}+ 2(1- \gamma_{n})L+ 1\bigr]<0, \\& {\bigl[(1- \gamma_{n})^{2} L^{2}+2(1- \gamma_{n})-1\bigr]}+(1- \gamma_{n})L^{2}\bigl[1+(1- \gamma_{n})L\bigr]^{2}<0, \\& (1- \gamma_{n})\bigl[(1- \gamma_{n})^{2} L^{2}+2(1- \gamma_{n})-1\bigr]+(1- \gamma _{n})^{2}L^{2}\bigl[1+(1- \gamma_{n})L \bigr]^{2}<0, \\& (1-\gamma_{n}) \bigl[(1- \gamma_{n})^{2} L^{2}+2(1- \gamma_{n})-1\bigr]+L^{2}\bigl[(1- \gamma _{n})+(1- \gamma_{n})^{2}L \bigr]^{2}<0, \\& (1-\gamma_{n})\bigl[(1- \gamma_{n})^{2} L^{2}+2(1- \gamma_{n})-1\bigr]+L^{2}\bigl[(1- \gamma _{n})- (1- \beta_{n})+(1- \beta_{n}) (1- \gamma_{n})L\bigr]^{2}<0, \\& (1-\gamma_{n})\bigl[1-2(1- \gamma_{n}) -(1- \gamma_{n})^{2}L^{2}\bigr]-L^{2}\bigl[(1- \gamma_{n})- (1- \beta_{n})+(1- \beta_{n}) (1- \gamma_{n})L\bigr]^{2}>0 . \end{aligned}$$

Then

$$ \Vert x_{n +1} -p\Vert ^{2}\leq \Vert x_{n} -p \Vert ^{2}. $$
(3.11)

It is obvious that if \(\lim_{n\to\infty} \Vert x_{n} -p \Vert \) exists, then \(\{\Vert x_{n} -p\Vert \}\) is bounded. This implies that \(\{x_{n} \}\), \(\{T_{n}x_{n} \}\), \(\{z_{n} \}\), \(\{ T_{n}z_{n} \}\), \(\{y_{n} \}\), and \(\{T_{n}y_{n} \}\) are also bounded.

Furthermore, from (2.1), we have

$$ \phi(p, x_{n})= \phi(p, x_{n+1})+\phi(x_{n+1}, x_{n})+2 \langle x_{n+1}-p, x_{n}-x_{n+1} \rangle. $$

This implies that

$$ \langle x_{n+1}-p, x_{n}-x_{n+1} \rangle+\frac{1}{2}\phi (x_{n+1}, x_{n})= \frac{1}{2} \bigl(\phi(p, x_{n})- \phi(p, x_{n+1}) \bigr). $$
(3.12)

Moreover, since the interior of \(\mathcal{F}\) is nonempty, there exist \(p^{*} \in\mathcal{F}\) and \(r >0\) such that \(p^{*} + rh \in \mathcal{F}\) whenever \(\Vert h\Vert \leq1\). Thus, from the fact that \(\phi(x, y) =\Vert x-y\Vert ^{2}\), and (3.11) and (3.12), we get

$$\begin{aligned} 0& \leq \bigl\langle x_{n+1}- \bigl(p^{*} + rh\bigr), x_{n}-x_{n+1} \bigr\rangle +\frac{1}{2} \phi(x_{n+1}, x_{n}) \\ & =\frac{1}{2} \bigl(\phi \bigl(\bigl(p^{*} + rh\bigr), x_{n} \bigr)- \phi \bigl( \bigl(p^{*} + rh\bigr), x_{n+1} \bigr) \bigr). \end{aligned}$$
(3.13)

Then from (3.12) and (3.13), we obtain

$$\begin{aligned} r \langle h, x_{n}-x_{n+1} \rangle& \leq \bigl\langle x_{n+1}- p^{*} , x_{n}-x_{n+1} \bigr\rangle + \frac{1}{2}\phi(x_{n+1}, x_{n}) \\ & =\frac{1}{2} \bigl(\phi\bigl(p^{*}, x_{n}\bigr)- \phi\bigl( p^{*}, x_{n+1}\bigr) \bigr), \end{aligned}$$

and hence

$$ \langle h, x_{n}-x_{n+1} \rangle\leq\frac{1}{2r} \bigl( \phi\bigl(p^{*}, x_{n}\bigr)- \phi\bigl( p^{*}, x_{n+1}\bigr) \bigr). $$

Since h with \(\Vert h\Vert \leq1\) is arbitrary, we have

$$ \Vert x_{n}-x_{n+1}\Vert \leq\frac{1}{2r} \bigl(\phi \bigl(p^{*}, x_{n}\bigr)- \phi\bigl( p^{*}, x_{n+1}\bigr) \bigr). $$

So, if \(n > m\), then we get

$$\begin{aligned} \Vert x_{m}-x_{n}\Vert & = \Vert x_{m}-x_{m+1}+x_{m+1}- \cdots-x_{n-1}+x_{n-1}-x_{n} \Vert \\ & \leq\sum_{i=m}^{n-1}\Vert x_{i}-x_{i+1}\Vert \\ &\leq\frac{1}{2r}\sum_{i=m}^{n-1} \bigl( \phi\bigl(p^{*}, x_{i}\bigr)- \phi\bigl( p^{*}, x_{i+1}\bigr) \bigr) \\ &\leq\frac{1}{2r} \bigl(\phi\bigl(p^{*}, x_{m}\bigr)- \phi\bigl( p^{*}, x_{n}\bigr) \bigr). \end{aligned}$$

But we know that \(\{\phi( p^{*}, x_{n})\}\) converges. Therefore, we see that \(\{x_{n} \}\) is a Cauchy sequence. Since K is closed subset of H, there exists \(x^{*} \in K\) such that

$$ x_{n} \to x^{*}. $$
(3.14)

Furthermore, (3.10) and conditions (ii), (iii), and (iv), we get

$$\begin{aligned}& \alpha^{4} \bigl[\bigl\{ 1-\gamma^{2}L^{2}-2\gamma \bigr\} - \gamma L^{2}{\{\gamma L+1\} }^{2} \bigr]\sum \Vert T_{n}x_{n} -x_{n}\Vert ^{2} \\& \quad \leq\sum(1-\alpha_{n}) (1-\beta_{n}) (1- \theta_{n}) (1-\gamma_{n})\bigl[\bigl\{ 1-(1- \gamma_{n})^{2}L^{2}-2(1-\gamma_{n})\bigr\} \\& \qquad {}- (1-\gamma_{n}) L^{2}{\bigl\{ (1- \gamma_{n}) L+1\bigr\} }^{2}\bigr]\Vert T_{n}x_{n} -x_{n}\Vert ^{2} \\& \quad =\sum(1-\alpha_{n}) (1-\beta_{n}) (1- \theta_{n})\bigl[(1-\gamma_{n})\bigl\{ 1-(1- \gamma_{n})^{2}L^{2}-2(1-\gamma_{n})\bigr\} \\& \qquad {} - L^{2}{\bigl\{ (1-\gamma_{n})+(1- \gamma_{n})^{2} L\bigr\} }^{2}\bigr]\Vert T_{n}x_{n} -x_{n}\Vert ^{2} \\& \quad \leq\sum(1-\alpha_{n}) (1-\beta_{n}) (1- \theta_{n})\bigl[(1-\gamma_{n})\bigl\{ 1-(1- \gamma_{n})^{2}L^{2}-2(1-\gamma_{n})\bigr\} \\& \qquad {}- L^{2}{\bigl\{ (1-\gamma_{n})-(1- \beta_{n})+(1-\gamma _{n})^{2} L\bigr\} }^{2}\bigr]\Vert T_{n}x_{n} -x_{n} \Vert ^{2} \\& \quad \leq \Vert x_{n} -p\Vert ^{2}- \Vert x_{n +1} -p \Vert ^{2} < \infty, \end{aligned}$$

from which it follows that

$$ \lim_{n\to\infty} \Vert T_{n}x_{n} -x_{n}\Vert =0. $$
(3.15)

Since \(\{T_{n}\}_{n=1}^{\infty}\) is uniformly closed, from (3.14) and (3.15), we obtain \(x^{*} \in\mathcal{F} = \bigcap_{n=1}^{\infty}F(T_{n})\). This completes the proof. □

We get the following result from Theorem 3.1.

Theorem 3.2

Let K and H be as in Theorem  3.1. Let \(T\colon K \to K \) be a uniformly closed and Lipschitz pseudocontractive mapping with Lipschitzian constant L. Assume that the interior of \(F(T)\) is nonempty. Let \(\{x_{n} \}\) be a sequence generated from an arbitrary \(x_{1} \in K\) by the following algorithm:

$$ \left. \begin{aligned} &z_{n}= \gamma_{n}x_{n}+(1- \gamma_{n})Tx_{n}, \\ &y_{n}= \beta_{n}x_{n}+(1- \beta_{n})Tz_{n}, \\ &x_{n+1}= \alpha_{n}x_{n}+(1- \alpha_{n})[ \theta_{n}x_{n}+\delta_{n}Ty_{n}+ \sigma_{n}Tz_{n}], \end{aligned} \right\} $$
(3.16)

where \(\{\alpha_{n} \}\), \(\{\beta_{n} \}\), \(\{\gamma_{n} \}\), \(\{\theta_{n} \}\), \(\{\delta_{n} \}\), and \(\{\sigma_{n} \}\) are in \((0, 1)\) satisfying the conditions (i)-(iv) of Theorem  3.1. Then \(\{x_{n} \}\) converges strongly to \(x^{*} \in \mathcal{F}\).

Remark 3.3

Our results improve and generalize the corresponding results of Zhou [11], Yao et al. [12], Tang et al. [13], Zegeye et al. [16] and Cheng et al. [17] and many others.

4 Numerical examples

In this section, using Example 1.2, we numerically demonstrate the convergence of the algorithm defined in this paper and compare its behavior with the Ishikawa and Noor iterations.

Consider \(T_{n}=T\) for all \(n\in\mathbb{N}\), where T is given by (1.6). Set the control conditions

$$\begin{aligned}& \alpha_{n}=1-\frac{1}{\sqrt{n+1}}, \qquad \beta_{n}= \frac{1}{\sqrt {n+1}} - \frac{1}{(n+1)^{2}},\qquad \gamma_{n}= \frac{1}{\sqrt{2n+9}}, \\& \theta_{n}=1-\frac{1}{\sqrt{n+1}},\qquad \sigma_{n}= \frac {1}{100}(1-\theta_{n}),\qquad \delta_{n}=1- \theta_{n}-\sigma_{n}, \end{aligned}$$

for all \(n\in \mathbb{N}\).

We examine the behavior of the above iterations for the different initial points (see Table 1).

Table 1 Initial point and its norm
Full size table

The details for different cases are as below:

Case I. Using the initial point \(x_{1}=(0.6,0.8)\), a Matlab program leads to the evaluation illustrated in Table 2 and Figure 1.

Figure 1
figure 1

Convergence behavior of first 100 iterates for Case I.

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Table 2 Comparative results, case I
Full size table

Case II. For the initial point \(x_{1}=(0.5,0.7)\), we make the observations as in Table 3 and Figure 2.

Figure 2
figure 2

Convergence behavior of first 100 iterates for Case II.

Full size image
Table 3 Comparative results, case II
Full size table

Case III. Using the initial point \(x_{1}=(0.2,0.7)\), we make the observations as in Table 4 and Figure 3.

Figure 3
figure 3

Convergence behavior of first 100 iterates for Case III.

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Table 4 Comparative results, case III
Full size table

Case IV. Using the initial point \(x_{1}=(0.4,0.3)\), we make the observations as in Table 5 and Figure 4.

Figure 4
figure 4

Comparison of first 100 iterates for Case IV.

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Table 5 Comparative results, case IV
Full size table

Case V. We use the initial point \(x_{1}=(0.2,0.3)\) and obtain Table 6 and Figure 5.

Figure 5
figure 5

Convergence behavior of first 100 iterates for Case V.

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Table 6 Comparative results, case V
Full size table

As discussed in Example 1.2, \(x=(0,0)\) with \(\Vert x\Vert =0\) is the only fixed point of T. From the tables and the graphs it can be seen that the sequence \(\{x_{n} \}\) generated by the new iteration converging toward 0, while for the Ishikawa and Noor iteration \(\Vert x_{n}\Vert \) tends toward 1 in all the above cases.