Open Access

Stability and convergence of a new composite implicit iterative sequence in Banach spaces

Fixed Point Theory and Applications20152015:172

https://doi.org/10.1186/s13663-015-0425-z

Received: 3 June 2015

Accepted: 14 September 2015

Published: 22 September 2015

Abstract

The purpose of this paper is to study stability and strong convergence of asymptotically pseudocontractive mappings by using a new composite implicit iteration process in an arbitrary real Banach space. The results in this paper improve and extend the corresponding results in the literature.

Keywords

implicit iteration processstrictly pseudocontractive mappingsasymptotically pseudocontractive mappingsalmost T-stabilityweak T-stability

MSC

47H0947H1047J20

1 Introduction

Throughout this paper we assume that E is an arbitrary real Banach space and \(E^{*}\) denotes the dual space of E. The normalized duality map \(J: E\to2^{E^{*}}\) is defined by
$$ Jx:=\bigl\{ u^{*}\in E^{*}: \bigl\langle x,u^{*} \bigr\rangle =\|x\|^{2}; \bigl\Vert u^{*}\bigr\Vert =\|x\| \bigr\} , $$
where \(\langle\cdot,\cdot\rangle\) denotes the generalized duality pairing between elements of E and \(E^{*}\). If \(E^{*}\) is strictly convex, then J is single-valued.

We first recall some definitions and conclusions.

Definition 1.1

Let \(T: D(T)\subset E\to E\) be a mapping.
  1. (1)
    T is said to be asymptotically nonexpansive (see [1]) if there exists a sequence \(\{k_{n}\}\subset [1,\infty)\), \(\lim_{n\to\infty}k_{n}=1\) such that
    $$ \bigl\Vert T^{n}x-T^{n}y\bigr\Vert \leq k_{n} \|x-y\|, \quad \forall x, y\in D(T), n\geq1; $$
     
  2. (2)
    T is said to be asymptotically pseudocontractive (see [2]) with sequence \(\{k_{n}\}\subset[0, \infty)\), if and only if \(\lim_{n\to\infty }k_{n}=1\), for all \(n\geq1\), \(x, y\in D(T)\) and there exists \(j(x-y)\in J(x-y)\) such that
    $$ \bigl\langle T^{n}x-T^{n}y, j(x-y)\bigr\rangle \leq k_{n}\|x-y\|^{2}; $$
     
  3. (3)
    T is said to be strictly asymptotically pseudocontractive with sequence \(\{k_{n}\}\subset[0, \infty)\), if and only if \(\lim_{n\to \infty}k_{n}=k\in(0, 1)\), for all \(n\geq1\), \(x, y\in D(T)\) and there exists \(j(x-y)\in J(x-y)\) such that
    $$ \bigl\langle T^{n}x-T^{n}y, j(x-y)\bigr\rangle \leq k_{n}\|x-y\|^{2}; $$
     
  4. (4)
    T is said to be uniformly L-Lipschitzian (see [3]) if there exists a constant \(L> 0\) such that
    $$ \bigl\Vert T^{n}x-T^{n}y\bigr\Vert \leq L\|x-y\| \quad \mbox{for all } x, y\in D(T), n\ge1. $$
     

It is easy to see that every asymptotically nonexpansive mapping is uniformly L-Lipschitzian and asymptotically pseudocontractive. In [4], Rhoades constructed an example to show that the class of asymptotically pseudocontractive mappings properly contains the class of asymptotically nonexpansive mappings.

The class of asymptotically pseudocontractive mappings has been studied by several authors (see [2, 47] and the references cited therein) by using the modified Mann iteration process (see [8]) and the modified Ishikawa iteration process (see [9]). Schu [5] proved the following theorem.

Theorem SC

Let H be a Hilbert space, \(K\subset H\) nonempty bounded closed convex, \(L>0\), \(T: K\to K\) completely continuous, uniformly L-Lipschitzian and asymptotically pseudocontractive with sequence \(\{k_{n}\}\subset[1, \infty)\), \(q_{n}=2k_{n}-1\), \(\forall n\ge1\), \(\sum_{n=1}^{\infty}(q_{n}^{2}-1)<\infty\). \(\{\alpha_{n}\}\subset[0, 1]\), \(\epsilon <\alpha_{n}\le b\), \(\forall n\ge1\), and some \(b\in (0, L^{-2}[(1+L^{2})^{1/2}-1])\), \(x_{1}\in K\), for all \(n\ge1\), define
$$ x_{n+1}=(1-\alpha_{n})x_{n}+\alpha _{n}T^{n}x_{n}. $$
(1.1)
Then \(\{x_{n}\}\) converges to some fixed point of T.

The recursion formula (1.1) is a modification of the well-known Mann iteration process (see, e.g., [8]).

Recently, Chang [6] extended Theorem SC to real uniformly smooth Banach spaces and proved the following theorem.

Theorem CH

Let E be a real uniformly smooth Banach space, K a nonempty bounded closed convex subset of E, \(T: K\to K\) be an asymptotically pseudocontractive mapping with sequence \(\{k_{n}\}\subset[1, \infty)\), \(\lim_{n\to\infty}k_{n}=1\), and \(F(T)=\{x\in K: Tx=x\}\neq\emptyset\). Let \(\{\alpha_{n}\}\subset[0, 1]\) satisfy the following conditions: (i) \(\lim_{n\to\infty}\alpha _{n}=0\), (ii) \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\). For an arbitrary \(x_{0}\in K\), let \(\{x_{n}\}\) be iteratively defined by (1.1). If there exists a strictly increasing function \(\phi: [0, \infty)\to[0, \infty)\), \(\phi(0)=0\) such that \(\langle T^{n}x_{n}-x^{*}, j(x_{n}-x^{*}) \rangle \le k_{n}\lVert x_{n}-x^{*}\rVert^{2}-\phi(\lVert x_{n}-x^{*}\rVert)\), \(\forall n\ge 1\), then \(x_{n}\to x^{*}\in F(T)\).

Let K be nonempty closed convex subset of E and \(\{T_{i}\}_{i=1}^{r}\) be a finite family of nonexpansive mappings from K into itself (i.e., \(\lVert T_{i}x-T_{i}y\rVert\le \lVert x-y\rVert\) for \(x, y\in K\) and \(i=1, 2, \ldots, r\)). In 2001, Xu and Ori [10] introduced the following implicit iteration process. For an arbitrary \(x_{0}\in K\) and \(\{\alpha_{n}\}_{n=1}^{\infty}\subset[0, 1)\), the sequence \(\{x_{n}\}_{n=1}^{\infty}\) is generated as follows:
$$ \left \{ \textstyle\begin{array}{l} x_{1}=(1-\alpha_{1})x_{0}+\alpha_{1}T_{1}x_{1}, \\ x_{2}= (1-\alpha_{2})x_{1}+\alpha_{2} T_{2}x_{2}, \\ \vdots \\ x_{r}= (1-\alpha_{r})x_{r-1}+\alpha_{r}T_{r}x_{r}, \\ x_{r+1}= (1-\alpha_{r+1})x_{r}+\alpha_{r+1}T_{1}x_{r+1}, \\ \vdots \end{array}\displaystyle \right . $$
The scheme can be expressed in compact form by
$$ x_{n}=(1-\alpha_{n})x_{n-1}+ \alpha_{n}T_{n (\operatorname{mod} r)}x_{n},\quad n\geq1. $$
(1.2)
Using this iteration, they proved that the sequence \(\{x_{n}\}\) converges weakly to a common fixed point of the finite family of nonexpansive mappings \(\{T_{i}\}_{i=1}^{r}\) in Hilbert spaces under certain conditions. Since then, the construction of fixed points for nonexpansive mappings and strictly pseudocontractive mappings and some other mappings via the implicit iterative algorithm has been extensively investigated by many authors (see, e.g., [1016] and the references cited therein). An implicit process is generally desirable when no explicit scheme is available. Such a process is generally used as a ‘tool’ to establish the convergence of an explicit scheme.
In 2006, Chang et al. [13] introduced another implicit iteration process with error. In the sense of [13], the implicit iteration process with errors for a finite family of asymptotically nonexpansive mappings \(\{T_{i}\}_{i=1}^{r}\) is generated from an arbitrary \(x_{0}\in K\) by
$$ x_{n}=\alpha_{n}x_{n-1}+(1-\alpha _{n})T^{k(n)}_{i(n)}x_{n}+u_{n}, \quad \forall n\ge 1, $$
(1.3)
where \(n=(k(n)-1)r+i(n)\) with \(i(n)\in\{1,2,\ldots, r\}\) and \(k(n)\in \mathbb{N}\) (the positive integer set) and \(k(n)\to\infty\), as \(n\to \infty\). \(\{\alpha_{n}\}_{n=1}^{\infty}\) is a suitable sequence in \([0,1]\) and \(\{u_{n}\}\subset K\) is such that \(\sum_{n=1}^{\infty}\|u_{n}\|<\infty\). They extended the results of [10] from Hilbert spaces to more general uniformly convex Banach spaces and from nonexpansive mappings to asymptotically nonexpansive mappings.

It is clear that even if K is a nonempty convex subset of E and \(\{u_{n}\}\subset K\) is such that \(\sum_{n=1}^{\infty}\|u_{n}\|<\infty\), then the implicit iterative sequence with errors in the sense of [13] need not be well defined, i.e., \(\{x_{n}\}_{n=1}^{\infty}\) may fail to be in K. More precisely, the conditions imposed on the error terms are not compatible with the randomness of the occurrence of errors.

In [16], Thakur proposed another modified composite implicit iteration process for a finite family of asymptotically nonexpansive mappings as follows:
$$ \left \{ \textstyle\begin{array}{l} x_{n}=(1-\alpha_{n})x_{n-1}+\alpha_{n}T^{k(n)}_{i(n)}y_{n}, \\ y_{n}=(1-\beta_{n})x_{n-1}+\beta_{n}T^{k(n)}_{i(n)}x_{n}, \end{array}\displaystyle \right . \quad \{\alpha_{n} \}, \{\beta_{n}\}\subset[0, 1], \forall n\ge1, $$
where \(n=(k-1)N+i\), \(i=i(n)\in\{1,2,\ldots, N\}\), \(k=k(n)\ge1\) is some positive integer and \(k(n)\to\infty\) as \(n\to\infty\).
Inspired and motivated by these facts, we introduce a new modified composite implicit iteration process for an asymptotically pseudocontractive mappings as follows:
$$ \left \{ \textstyle\begin{array}{l} x_{1}\in K, \\ x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n}T^{n}y_{n}, \\ y_{n}=(1-\beta_{n}) x_{n}+\beta_{n}T^{n}x_{n+1}, \end{array}\displaystyle \right . \quad \{ \alpha_{n}\}, \{\beta_{n}\}\subset[0, 1], \forall n\ge1. $$
(1.4)
Unlike iteration methods with errors of [13], our iteration process (1.4) is always well defined, that is, \(\{x_{n}\}\) is always in K if K is convex subset of E.
If \(\{\beta_{n}\}=\{0\}\) for all \(n\ge1\), (1.4) becomes the explicit form as follows:
$$ x_{n+1}=(1-\alpha_{n})x_{n}+ \alpha_{n} T^{n}x_{n}. $$
(1.5)
Equation (1.5) is the modified Mann iterative process (see, e.g., [8]).

Stability results established in metric space, normed linear space, and Banach space settings are available in the literature. There are several authors whose contributions are of colossal value in the study of stability of the fixed point iterative procedures: Imoru and Olatinwo [17], Olatinwo and Postolache [18], Akewe and Okeke [19].

Harder and Hicks [20] mentioned that the study of the stability of iterative schemes is useful for both theoretical and numerical investigations. Consequently, several authors have studied the stability of iterative schemes for various types of nonlinear mappings (see, e.g., [2025] and the references cited therein).

The purpose of this paper is to study the stability and convergence of the composite implicit iterative sequence for an asymptotically pseudocontractive mapping in arbitrary real Banach spaces.

Let K be a nonempty convex subset of E and \(T: K\to K\) be a mapping, \(x_{0}\in K\) and \(\{x_{n}\}\subset K\) defined by
$$ x_{n+1}= f(T, x_{n}), $$
(1.6)
where f is a continuous mapping. Suppose that \(F(T)\neq\emptyset\) and \(x_{n}\to p\in F(T)\). Let \(\{s_{n}\}\) be any bounded sequence in K and \(\{\varepsilon _{n}\}\) a sequence in \([0, \infty)\) defined by
$$ \varepsilon _{n}=\bigl\lVert s_{n+1}-f(T, s_{n})\bigr\rVert , \quad n\ge0. $$
(1.7)
If \(\varepsilon _{n}\to0\) implies that \(s_{n}\to p\), then the iterative sequence \(\{x_{n}\}\) defined by (1.6) is said to be T-stable. If \(\sum_{n=1}^{\infty}\varepsilon _{n} <\infty\) implies that \(s_{n}\to p\), then the sequence \(\{x_{n}\}\) defined by (1.6) is said to be almost T-stable. An example in [25] represents an iterative sequence which is almost T-stable but not T-stable.

2 Preliminaries

In the sequel, \(F(T)=\{x\in K: Tx=x\}\) denotes the set of fixed points of T. We give the stability definition for the sequence \(\{x_{n}\}\) defined by (1.4).

Definition 2.1

Let \(\{x_{n}\}\) be the sequence defined by (1.4) such that \(x_{n}\to p\in F(T)\). Let \(\{s_{n}\}\) be any bounded sequence in K. Define a sequence \(\{\varepsilon _{n}\}\) by
$$ \left \{ \textstyle\begin{array}{l} \varepsilon _{n}=\lVert s_{n+1}-(1-\alpha_{n})s_{n}-\alpha _{n}T^{n}z_{n}\rVert, \\ z_{n}=(1-\beta_{n})s_{n}+\beta_{n}T^{n}s_{n+1}, \end{array}\displaystyle \right . \quad n\ge1. $$
(2.1)
If \(\varepsilon _{n}\to0\) implies that \(s_{n}\to p\), then the sequence \(\{x_{n}\}\) is said to be T-stable. If \(\varepsilon _{n}/\alpha_{n}\to0\) implies that \(s_{n}\to p\), then the sequence \(\{x_{n}\}\) is said to be weakly T-stable. If \(\sum_{n=1}^{\infty}\varepsilon _{n}<\infty\) implies that \(s_{n}\to p\), then the sequence \(\{x_{n}\}\) is said to be almost T-stable.

The following lemmas will be needed in proving our main results.

Lemma 2.2

(Lemma 2 of [26])

Let \(\{a_{n}\}\), \(\{b_{n}\}\), \(\{ c_{n}\}\) be nonnegative real sequences satisfying the following conditions:
$$ a_{n+1}\le (1-t_{n})a_{n}+b_{n}t_{n}+c_{n}, \quad \forall n\ge0, $$
where \(\{t_{n}\}\) is a sequence in \([0, 1]\) such that \(\sum_{n=1}^{\infty}t_{n}=\infty\), \(\lim_{n\to\infty}b_{n}=0\) and \(\sum_{n=1}^{\infty}c_{n}<\infty\). Then \(a_{n}\to0\) (\(n\to\infty\)).

We denote Φ := {\(\phi \mid \phi: [0, \infty)\to[0, \infty)\) be a nondecreasing function such that \(\phi(t)=0\) if and only if \(t=0\)}.

Lemma 2.3

(Lemma 2.1 of [27])

Let \(\{\theta_{n}\}\) be a sequence of nonnegative real numbers, \(\{\lambda_{n}\}\) be a real sequence satisfying
$$ 0\le\lambda_{n}\le1, \quad \sum_{n=0}^{\infty}\lambda _{n}=\infty, $$
and let \(\phi\in\Phi\). If there exists a positive integer \(n_{0}\) such that
$$ \theta_{n+1}^{q}\le \theta_{n}^{q}- \lambda_{n}\phi(\theta_{n+1})+\lambda_{n} \sigma_{n} $$
for some \(q>1\), all \(n\ge n_{0}\) and \(\lim_{n\to\infty}\sigma_{n}=0\), then \(\lim_{n\to\infty}\theta_{n}=0\).

Lemma 2.4

(Lemma 1.1 of [28])

Let E be a real Banach space, \(T: E\to E\) a mapping, and λ any positive real number. Then for any \(x, y\in E\) and \(k>0\),
$$ \lVert x-y\rVert\le \bigl\lVert x-y+\lambda\bigl[(I-T-kI)x-(I-T-kI)y\bigr] \bigr\rVert $$
whenever
$$ \bigl\langle (I-T-kI)x-(I-T-kI)y, j(x-y)\bigr\rangle \ge0,\quad \forall j(x-y)\in J(x-y). $$

Lemma 2.5

Let E be a normed linear space then for all \(x, y\in E\) and for all \(j(x+y)\in J(x+y)\), the following inequality holds:
$$ \lVert x+y\rVert^{2}\le\lVert x\rVert^{2}+2\bigl\langle y, j(x+y)\bigr\rangle . $$

First we give two auxiliary lemmas.

Lemma 2.6

Let K be a nonempty convex subset of an arbitrary real Banach space E and \(T: K\to K \) an asymptotically pseudocontractive mapping with \(\{k_{n}\}\subset[0, \infty)\) and let \(F(T)\neq\emptyset\). If \(\{x_{n}\}\) is the iterative sequence defined by (1.4) and \(p\in F(T)\), then
$$ \lVert x_{n}-p\rVert\le (1+k_{n}\alpha_{n}) \bigl(1-\alpha_{n}+\alpha_{n}^{2}\bigr)\lVert x_{n-1}-p\rVert +\alpha_{n}b_{n}, $$
where
$$ b_{n}=\bigl\lVert T^{n}x_{n+1}-T^{n}y_{n} \bigr\rVert +\alpha_{n} (k_{n}+1)\bigl\lVert x_{n}-T^{n}y_{n}\bigr\rVert . $$
(2.2)

Proof

Since T is an asymptotically pseudocontractive mapping with \(\{k_{n}\}\subset[0, \infty)\), there exists \(j(x_{n+1}-p)\in J(x_{n+1}-p)\) for \(x_{n+1}\in K\) and \(p\in F(T)\) such that
$$ \bigl\langle T^{n}x_{n+1}-p, j(x_{n+1}-p)\bigr\rangle \le k_{n}\lVert x_{n+1}-p\rVert^{2}. $$
Then we have
$$ \bigl\langle \bigl(I-T^{n}+(k_{n}-1)I\bigr)x_{n+1}- \bigl(I-T^{n}+(k_{n}-1)I\bigr)p, j(x_{n+1}-p)\bigr\rangle \ge0. $$
Thus it follows from Lemma 2.4 that
$$ \lVert x_{n+1}-p\rVert\le \biggl\lVert x_{n+1}-p +\frac{\alpha_{n}}{1+\alpha _{n}}\bigl\{ \bigl[I-T^{n}+(k_{n}-1)I \bigr]x_{n+1}-\bigl[I-T^{n}+(k_{n}-1)I\bigr]p\bigr\} \biggr\rVert . $$
(2.3)
It follows from (1.4) that
$$\begin{aligned} x_{n} =& x_{n+1}+\alpha_{n}x_{n}- \alpha_{n}T^{n}y_{n} \\ = & (1+\alpha_{n})x_{n+1}+\alpha_{n} \bigl[I-T^{n}+(k_{n}-1)I\bigr]x_{n+1}+\alpha _{n}\bigl(T^{n}x_{n+1}-T^{n}y_{n} \bigr) \\ &{} -\alpha_{n}k_{n}x_{n}+\alpha_{n}^{2}(k_{n}+1) \bigl(x_{n}-T^{n}y_{n}\bigr). \end{aligned}$$
(2.4)
Since \(p\in F(T)\), we have
$$ p=(1+\alpha_{n})p+\alpha _{n} \bigl[I-T^{n}+(k_{n}-1)I\bigr]p-\alpha_{n} k_{n}p. $$
(2.5)
From (2.4) and (2.5), we have
$$\begin{aligned} \lVert x_{n}-p\rVert \ge& (1+\alpha_{n}) \biggl\Vert x_{n+1}-p+\frac{\alpha_{n}}{1+\alpha _{n}}\bigl[\bigl(I-T^{n}+(k_{n}-1)I \bigr)x_{n+1} \\ &{}-\bigl(I-T^{n}+(k_{n}-1)I\bigr)p\bigr]\biggr\Vert - \alpha_{n}k_{n}\lVert x_{n}-p\rVert \\ &{}-\alpha_{n} \bigl\lVert T^{n}x_{n+1}-T^{n}y_{n} \bigr\rVert -\alpha_{n}^{2}(k_{n}+1) \bigl\lVert x_{n}-T^{n}y_{n} \bigr\rVert . \end{aligned}$$
(2.6)
Therefore from (2.3) we have
$$\begin{aligned} \lVert x_{n}-p\rVert \ge& (1+\alpha_{n})\lVert x_{n+1}-p\rVert-\alpha_{n}k_{n}\lVert x_{n}-p\rVert \\ &{} -\alpha_{n} \bigl\lVert T^{n}x_{n+1}-T^{n}y_{n} \bigr\rVert -\alpha_{n}^{2}(k_{n}+1) \bigl\lVert x_{n}-T^{n}y_{n} \bigr\rVert , \end{aligned}$$
i.e.,
$$\begin{aligned} (1+\alpha_{n})\lVert x_{n+1}-p\rVert \le &(1+ \alpha_{n} k_{n})\lVert x_{n}-p\rVert+ \alpha_{n} \bigl\lVert T^{n}x_{n+1}-T^{n}y_{n} \bigr\rVert \\ &{}+\alpha_{n}^{2}(k_{n}+1) \bigl\lVert x_{n}-T^{n}y_{n} \bigr\rVert . \end{aligned}$$
(2.7)
Since \((1+\alpha_{n})^{-1}\le1\) and \((1+\alpha_{n})^{-1}\le1-\alpha_{n}+\alpha_{n}^{2}\), from (2.7), we have
$$ \lVert x_{n+1}-p\rVert\le(1+\alpha_{n} k_{n}) \bigl(1-\alpha_{n}+\alpha_{n}^{2} \bigr)\lVert x_{n}-p\rVert+\alpha_{n}b_{n}, $$
(2.8)
where \(b_{n}=\lVert T^{n}x_{n+1}-T^{n}y_{n}\rVert+\alpha_{n}(k_{n}+1)\lVert x_{n}-T^{n}y_{n}\rVert\). This completes the proof. □

3 Main results

Theorem 3.1

Let K be a nonempty convex subset of an arbitrary real Banach space E and \(T: K\to K\) be a uniformly Lipschitzian (with a Lipschitzian constant \(L> 0\)) and strictly asymptotically pseudocontractive mapping with sequence \(\{k_{n}\}\subset[0, \infty)\), \(k_{n}\to k\in(0, 1)\), and let \(F(T)\neq\emptyset\). Assume that \(\{x_{n}\}\), \(\{y_{n}\}\) are the sequences defined by (1.4), \(\{\alpha_{n}\}, \{\beta_{n}\} \subset[0, 1]\) satisfy the following conditions:
$$ (\mathrm{i}) \ \alpha_{n}\to0,\qquad \beta_{n}\to0\quad (n \to\infty ); \qquad (\mathrm{ii})\ \sum_{n=1}^{\infty}\alpha_{n}=\infty. $$
If \(\{x_{n}\}\) is bounded in K, then
  1. (1)

    \(\{x_{n}\}\) converges strongly to the unique common fixed point p of T;

     
  2. (2)

    \(\{x_{n}\}\) is both almost T-stable and weakly T-stable.

     

Proof

Assume that \(p_{1}, p_{2}\in F(T)\). Since T is a strictly asymptotically pseudocontractive mapping, there exists \(j(p_{1}-p_{2})\in J(p_{1}-p_{2})\) such that
$$ \lVert p_{1}-p_{2}\rVert^{2}= \bigl\langle T^{n}p_{1}-T^{n}p_{2}, j(p_{1}-p_{2})\bigr\rangle \le k_{n}\lVert p_{1}-p_{2}\rVert^{2}. $$
Letting \(n\to\infty\) we have \(\lVert p_{1}-p_{2}\rVert^{2}\le k\lVert p_{1}-p_{2}\rVert^{2}\), \(k\in (0, 1)\). This implies that \(p_{1}=p_{2}\).
By the strictly asymptotically pseudocontractive property of T, similar to (2.8), we have
$$ \lVert x_{n+1}-p\rVert\le(1+\alpha_{n} k_{n}) \bigl(1-\alpha_{n}+\alpha_{n}^{2} \bigr)\lVert x_{n}-p\rVert+\alpha_{n}b_{n}, $$
(3.1)
where \(b_{n}=\lVert T^{n}x_{n+1}-T^{n}y_{n}\rVert+\alpha_{n}(k_{n}+1)\lVert x_{n}-T^{n}y_{n}\rVert\). Since \(\alpha_{n}\to0\), \(k_{n}\to k\in(0, 1)\), and
$$ (1+\alpha_{n} k_{n}) \bigl(1-\alpha_{n}+ \alpha_{n}^{2}\bigr)\le\bigl(1-\alpha_{n}(1-k_{n}- \alpha _{n})\bigr). $$
Since \(\lim_{n\to\infty}(1-k_{n}-\alpha_{n})=1-k\ge\delta\in(0, 1-k)\), there exists a natural number \(n_{1}\) such that \((1-k_{n}-\alpha_{n})\ge\delta\) for all \(n\ge n_{1}\). Thus we have
$$ (1+\alpha_{n} k_{n}) \bigl(1- \alpha_{n}+\alpha_{n}^{2}\bigr)\le1-\delta \alpha_{n},\quad \forall n\ge n_{1}. $$
(3.2)
Substituting (3.2) into (3.1) we have
$$ \lVert x_{n}-p\rVert\le(1-\delta\alpha_{n}) \lVert x_{n-1}-p\rVert +\alpha_{n}b_{n},\quad \forall n\ge n_{1}. $$
(3.3)
Next we will prove that \(b_{n}\to0\) (\(n\to\infty\)).
Since \(T: K\to K\) is uniformly Lipschitzian with a Lipschitzian constant \(L> 0\), for all \(x, y\in K\), we have
$$ \bigl\lVert T^{n}x_{n+1}-p\bigr\rVert \le L\lVert x_{n+1}-p\rVert, \quad \forall n\ge1. $$
This implies that \(\{T^{n}x_{n+1}\}\) is bounded in K since \(\{x_{n}\}\) is bounded. Similarly, \(\{T^{n}x_{n}\}\) is bounded sequence in K. It follows from (1.4) that
$$ \lVert y_{n}-x_{n}\rVert=\beta_{n} \bigl\lVert x_{n}-T^{n}x_{n+1} \bigr\rVert \le \beta_{n}\bigl(\lVert x_{n}\rVert+ \bigl\lVert T^{n}x_{n+1} \bigr\rVert \bigr)\to0 \quad (n\to \infty). $$
(3.4)
This implies that \(\{y_{n}\}\) is bounded. Thus by the uniform Lipschitzianness of T, we have
$$ \bigl\lVert T^{n}y_{n} \bigr\rVert \le \bigl\lVert T^{n}y_{n}-T^{n}x_{n} \bigr\rVert + \bigl\lVert T^{n}x_{n} \bigr\rVert \le L \lVert y_{n}-x_{n} \rVert+ \bigl\lVert T^{n}x_{n} \bigr\rVert . $$
This implies that \(\{T^{n}y_{n}\}\) is also bounded. From (1.4) and (3.4), we have
$$\begin{aligned} \lVert x_{n+1}-y_{n} \rVert = & \bigl\lVert (x_{n}-y_{n})+\alpha_{n}\bigl(x_{n}-T^{n}y_{n} \bigr) \bigr\rVert \\ \le& \lVert y_{n}-x_{n} \rVert+\alpha_{n}\bigl( \lVert x_{n} \rVert+ \bigl\lVert T^{n}y_{n} \bigr\rVert \bigr)\to0 \quad (n\to\infty). \end{aligned}$$
(3.5)
Therefore we have
$$ \bigl\lVert T^{n}x_{n+1}-T^{n}y_{n} \bigr\rVert \le L \lVert x_{n+1}-y_{n} \rVert\to0\quad (n\to \infty). $$
(3.6)
Observing (3.6), \(\{x_{n}\}\), \(\{T^{n}y_{n}\}\) bounded in K and \(\alpha_{n}\to0\) (\(n\to\infty\)), we know that \(b_{n}\to 0\) (\(n\to\infty\)). By Lemma 2.2 and (3.3), we have \(x_{n}\to p\) (\(n\to\infty\)). The conclusion (1) is proved.
Next we prove the conclusion (2). For any bounded sequence \(\{s_{n}\}\subset K\) defined by (2.1) and \(p\in F(T)\), we have
$$\begin{aligned} \lVert s_{n+1}-p \rVert = & \bigl\lVert s_{n+1}-(1-\alpha_{n})s_{n}-\alpha_{n} T^{n}z_{n}+(1-\alpha _{n})s_{n}+ \alpha_{n}T^{n}z_{n}-p \bigr\rVert \\ \le&\varepsilon _{n}+ \lVert p_{n}-p \rVert, \end{aligned}$$
(3.7)
where
$$ p_{n}=(1-\alpha_{n})s_{n}+\alpha _{n}T^{n}z_{n}, \quad \forall n\ge1. $$
(3.8)
It follows from (3.8) that
$$\begin{aligned} s_{n} = &p_{n}+\alpha_{n}s_{n}- \alpha_{n}T^{n}z_{n} \\ = &(1+\alpha_{n})p_{n}+\alpha_{n} \bigl[I-T^{n}+(k_{n}-1)I\bigr]p_{n}+\alpha _{n}\bigl(T^{n}p_{n}-T^{n}z_{n} \bigr) \\ &{} +\alpha_{n}k_{n}s_{n}+\alpha_{n}^{2}(k_{n}+1) \bigl(s_{n}-T^{n}z_{n}\bigr). \end{aligned}$$
By using a similar method to that given in proving (2.5)-(2.8), (3.2), we can prove that
$$ \lVert p_{n}-p \rVert\le (1-\alpha_{n}\delta) \lVert s_{n}-p \rVert+\alpha_{n}b_{n},\quad \forall n\ge n_{1}, $$
(3.9)
where \(b_{n}= \lVert T^{n}p_{n}-T^{n}z_{n} \rVert+\alpha_{n}(k_{n}+1) \lVert s_{n}-T^{n}z_{n} \rVert\). Since T is uniformly Lipschitzian with Lipschitzian constant \(L> 0\), we have \(\lVert T^{n}s_{n}-p \rVert\le L \lVert s_{n}-p \rVert\). This implies that the sequence \(\{T^{n}s_{n}\}\) is bounded since \(\{s_{n}\}\) is bounded in K. By the same method as in proving (3.4)-(3.6), we can prove that the sequence \(\{T^{n}z_{n}\}\) is bounded and \(\lVert T^{n}p_{n}-T^{n}z_{n} \rVert\to0\) (\(n\to\infty\)). Therefore we have \(b_{n}\to0\) (\(n\to\infty\)). Substituting (3.9) into (3.7) we have
$$ \lVert s_{n+1}-p \rVert\le (1-\alpha_{n}\delta) \lVert s_{n}-p \rVert+\alpha_{n}b_{n}+\varepsilon _{n},\quad \forall n\ge n_{1}. $$
(3.10)
If \(\sum_{n=1}^{\infty}\varepsilon _{n}<\infty\), taking \(a_{n}= \lVert s_{n}-p \rVert\), \(t_{n}=\alpha_{n}\delta\), \(c_{n}=\varepsilon _{n}\) in Lemma 2.2, from (3.10), we have \(s_{n}\to p\) (\(n\to\infty\)), i.e., \(\{x_{n}\}\) is almost T-stable.

If \(\varepsilon _{n}/\alpha_{n}\to0\), taking \(a_{n}= \lVert s_{n}-p \rVert \), \(t_{n}=\alpha_{n}\delta\), \(c_{n}=0\) in Lemma 2.2, from (3.10), we have \(s_{n}\to p\) (\(n\to\infty\)), i.e., \(\{x_{n}\}\) is weakly T-stable. This completes the proof. □

Example 3.2

Let \(E=R=(-\infty, \infty)\) with the usual norm. Take \(K=[0, 1]\) and define \(T: K\to K\) by
$$ Tx= \textstyle\begin{cases} 0& \text{if } x=0, \\ \frac{1}{4}& \text{if }x=1, \\ x-\frac{1}{2^{n+1}} & \text{if } \frac{1}{2^{n+1}}\le x< \frac{1}{2}(\frac{1}{2^{n+1}}+\frac{1}{2^{n}}), \\ \frac {1}{2^{n}}-x& \text{if } \frac{1}{2}(\frac{1}{2^{n+1}}+\frac {1}{2^{n}})\le x< \frac{1}{2^{n}} \end{cases} $$
for all \(n\ge0\). Then \(F(T)=\{0\}\) and T is not continuous at \(x=1\). We can verify that
$$ Tx\le\frac{1}{2}x, \quad x\in K. $$
Thus \(T^{2}\) is continuous in K and \(T^{2}K\subset[0, 2^{-n}]\) for all \(n\ge1\). Then for any \(x\in K\), there exists \(j(x-0)\in J(x-0)\) satisfying
$$ \bigl\langle T^{n}x-T^{n}0, j(x-0) \bigr\rangle =T^{n}x\cdot x\le \frac{1}{2} \lVert x \rVert^{2} $$
for all \(n\ge1\). That is, T is a strictly asymptotically pseudocontractive mapping.

Example 3.3

Let \(E=[0, 1]\). Define \(T: E\to E\) by \(Tx=\frac{x}{5}\), where E has the usual norm. Then \(F(T)=\{0\}\) and T is a strictly asymptotically pseudocontractive mapping with \(k_{n}=\frac{1}{5}\). Consider the following conditions:
$$ \alpha_{n}=\frac{1}{n+1}, \qquad \beta_{n}= \frac {1}{n+2}, \quad \forall n\ge1. $$
Let \(\{x_{n}\}\) be the sequence defined by (1.4). So
$$ \left \{ \textstyle\begin{array}{l} x_{1}=0.25, \\ x_{n+1}=(1-\frac{1}{n+1})x_{n}+\frac{1}{n+1} T^{n}y_{n}, \\ y_{n}=(1-\frac{1}{n+2}) x_{n}+\frac{1}{n+2} T^{n}x_{n+1}, \end{array}\displaystyle \right . \quad \forall n\ge1. $$
We have the results in Table 1.
Table 1

The iteration chart with initial value \(\pmb{x_{1}=0.25}\)

n

\(\boldsymbol{x_{n}}\)

1

0.25

2

0.134228

3

0.090397

4

0.067943

5

0.054373

6

0.045313

7

0.038840

8

0.033985

9

0.030209

10

0.027188

Therefore, the conditions of Theorem 3.1 are fulfilled.

Theorem 3.4

Let K be a nonempty convex subset of an arbitrary real Banach space E, \(T: K\to K\) be a uniformly Lipschitzian (with a Lipschitzian constant \(L> 0\)) and asymptotically pseudocontractive mapping with sequence \(\{k_{n}\}\subset[1, \infty)\), \(\lim_{n\to\infty}k_{n}=1\) and let \(F(T)\neq\emptyset\) and \(p\in F(T)\). Let \(\{x_{n}\}\), \(\{y_{n}\}\) be the sequences defined by (1.4). Assume that \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[0, 1]\) satisfy the following conditions:
$$ (\mathrm{i})\ \alpha_{n}\to0,\qquad \beta_{n}\to0 \quad (n \to \infty);\qquad (\mathrm{ii})\ \sum_{n=1}^{\infty}\alpha_{n}=\infty. $$
If \(\{x_{n}\}\) is bounded in K and there exists a strictly increasing function \(\phi: [0, \infty)\to[0, \infty)\) with \(\phi(0)=0\) such that
$$ \limsup_{n\to\infty} \bigl\{ \bigl\langle T^{n}x_{n+1}-p, j(x_{n+1}-p)\bigr\rangle -k_{n} \lVert x_{n+1}-p \rVert^{2}+\phi\bigl( \lVert x_{n+1}-p \rVert\bigr) \bigr\} \le 0, $$
where \(j(x_{n+1}-p)\in J(x_{n+1}-p)\). Then \(\{x_{n}\}\) converges strongly to the fixed point of T.

Proof

Since \(\{x_{n}\}\) is bounded in K, then \(M=\sup_{n\ge1}\{ \lVert x_{n}-p \rVert\}<\infty\) for \(p\in F(T)\). It follows from Lemma 2.5 and (1.4) that there exists \(j(x_{n+1}-p)\in J(x_{n+1}-p)\) such that
$$\begin{aligned} \lVert x_{n+1}-p \rVert^{2} = & \bigl\lVert (1- \alpha_{n}) (x_{n}-p)+\alpha_{n} \bigl(T^{n}y_{n}-p\bigr) \bigr\rVert ^{2} \\ \le& (1-\alpha_{n})^{2} \lVert x_{n}-p \rVert^{2}+2\alpha_{n}\bigl\langle T^{n}y_{n}-p, j(x_{n+1}-p)\bigr\rangle \\ = & (1-\alpha_{n})^{2} \lVert x_{n}-p \rVert^{2}+2\alpha_{n}\bigl\langle T^{n}x_{n+1}-p, j(x_{n+1}-p)\bigr\rangle \\ &{} +2\alpha_{n}\bigl\langle T^{n}y_{n}-T^{n}x_{n+1}, j(x_{n+1}-p)\bigr\rangle \\ \le& (1-\alpha_{n})^{2} \lVert x_{n}-p \rVert^{2}+2\alpha_{n}\bigl\langle T^{n}x_{n+1}-p, j(x_{n+1}-p)\bigr\rangle \\ &{} +2\alpha_{n}M \bigl\lVert T^{n}y_{n}-T^{n}x_{n+1} \bigr\rVert \\ = & (1-\alpha_{n})^{2} \lVert x_{n}-p \rVert^{2}+2\alpha_{n}d_{n} \\ &{} +2\alpha_{n} \bigl[k_{n} \lVert x_{n+1}-p \rVert^{2}-\phi\bigl( \lVert x_{n+1}-p \rVert\bigr) \bigr]+2M \alpha_{n} e_{n}, \end{aligned}$$
(3.11)
where \(d_{n}=\langle T^{n}x_{n+1}-p, j(x_{n+1}-p)\rangle -k_{n} \lVert x_{n+1}-p \rVert^{2}+\phi( \lVert x_{n+1}-p \rVert)\), \(e_{n}= \lVert T^{n}y_{n}-T^{n}x_{n+1} \rVert\). From (3.6), we have \(e_{n}\to 0\) as \(n\to\infty\). It follows from (3.11) that
$$\begin{aligned} (1-2\alpha_{n} k_{n}) \lVert x_{n+1}-p \rVert^{2} \le& (1-\alpha_{n})^{2} \lVert x_{n}-p \rVert^{2}-2\alpha_{n}\phi\bigl( \lVert x_{n+1}-p \rVert\bigr) \\ &{} +2\alpha_{n} d_{n}+2M\alpha_{n}e_{n}. \end{aligned}$$
Notice that \(\lim_{n\to\infty}(1-2\alpha_{n} k_{n})=1>0\), without loss of generality, we assume that \((1-2\alpha_{n} k_{n})>0\) for all \(n\ge1\). Therefore, we have
$$\begin{aligned} \lVert x_{n+1}-p \rVert^{2} \le& \frac{(1-\alpha_{n})^{2}}{1-2\alpha _{n}k_{n}} \lVert x_{n}-p \rVert^{2} \\ &{} -{2\alpha_{n}\over 1-2\alpha_{n}k_{n}}\phi\bigl( \lVert x_{n+1}-p \rVert \bigr)+ {2\alpha_{n}d_{n}\over 1-2\alpha_{n}k_{n}}+{2M\alpha_{n} e_{n}\over 1-2\alpha_{n}k_{n}} \\ = & \lVert x_{n}-p \rVert^{2}+{\alpha_{n}[2(k_{n}-1)+\alpha_{n}]\over 1-2\alpha_{n} k_{n}} \lVert x_{n}-p \rVert^{2} \\ &{} -{2\alpha_{n}\over 1-2\alpha_{n} k_{n}}\phi\bigl( \lVert x_{n+1}-p \rVert\bigr)+ {2\alpha_{n}d_{n}\over 1-2\alpha _{n}k_{n}}+{2M\alpha_{n} e_{n}\over 1-2\alpha_{n}k_{n}} \\ \le& \lVert x_{n}-p \rVert^{2}-2\alpha_{n}\phi \bigl( \lVert x_{n+1}-p \rVert \bigr)+{\alpha_{n}[2(k_{n}-1)+\alpha_{n}]\over 1-2\alpha_{n}k_{n}}M^{2} \\ &{} +{2M\alpha_{n}e_{n}\over 1-2\alpha_{n}k_{n}}+{2\alpha_{n}d_{n}\over 1-2\alpha_{n}k_{n}} \quad \bigl(\text{since }1-2\alpha_{n}k_{n}\in(0, 1)\bigr) \\ = & \lVert x_{n}-p \rVert^{2}-2\alpha_{n}\phi \bigl( \lVert x_{n+1}-p \rVert \bigr) +{\alpha_{n} \lambda_{n}'\over 1-2\alpha_{n}k_{n}}+{2\alpha_{n}d_{n}\over 1-2\alpha _{n}k_{n}} , \end{aligned}$$
(3.12)
where \(\lambda_{n}'=[2(k_{n}-1)+\alpha_{n}]M^{2}+2e_{n}M\). Taking \(q=2\), \(\theta_{n}= \lVert x_{n}-p \rVert\), \(\lambda_{n}=2\alpha_{n}\), \(\sigma _{n}={\lambda_{n}'+2d_{n} \over 2(1-2\alpha_{n} k_{n})}\) in Lemma 2.3, from (3.12), we have \(x_{n}\to p\) as \(n\to\infty\). This completes the proof. □

If \(\{\beta_{n}\}=\{0\}\) for all \(n\ge1\) in (1.4), it follows from Theorem 3.4 that we have the following result.

Theorem 3.5

Let K be a nonempty convex subset of an arbitrary real Banach space E, \(T: K\to K \) be a uniformly Lipschitzian (with a Lipschitzian constant \(L> 0\)) and asymptotically pseudocontractive mapping with sequence \(\{k_{n}\}\subset[1, \infty)\), \(\lim_{n\to\infty}k_{n}=1\) and let \(F(T)\neq\emptyset\) and \(p\in F(T)\), \(\{x_{n}\}\) be the sequence defined by (1.5). Let \(\{\alpha_{n}\}\subset[0, 1]\) satisfy the following conditions: (i) \(\lim_{n\to\infty}\alpha_{n}=0\), (ii) \(\sum_{n=1}^{\infty}\alpha _{n}=\infty\). If \(\{x_{n}\}\) is bounded in K and there exists a strictly increasing function \(\phi: [0, \infty)\to[0, \infty)\) with \(\phi(0)=0\) such that
$$ \limsup_{n\to\infty} \bigl\{ \bigl\langle T^{n}x_{n+1}-p, j(x_{n+1}-p)\bigr\rangle -k_{n} \lVert x_{n+1}-p \rVert^{2}+\phi\bigl( \lVert x_{n+1}-p \rVert\bigr) \bigr\} \le 0, $$
where \(j(x_{n+1}-p)\in J(x_{n+1}-p)\), \(\forall n\ge1\), then \(x_{n}\to p\) as \(n\to\infty\).

Remark 3.6

Theorem 3.5 extends Theorem CH from real uniformly smooth Banach space to arbitrary real Banach space. The requirement that K be bounded closed imposed in Theorem CH is stronger than the requirement that \(\{x_{n}\}\) be bounded imposed in Theorem 3.5. The condition
$$ \bigl\langle T^{n}x_{n}-p, j(x_{n}-p) \bigr\rangle \le k_{n} \lVert x_{n}-p \rVert^{2}+\phi\bigl( \lVert x_{n}-p \rVert\bigr) $$
in Theorem CH is replaced by
$$\begin{aligned} \limsup_{n\to\infty} \bigl\{ \bigl\langle T^{n}x_{n+1}-p, j(x_{n+1}-p)\bigr\rangle -k_{n} \lVert x_{n+1}-p \rVert^{2}+\phi\bigl( \lVert x_{n+1}-p \rVert\bigr) \bigr\} \le 0 \end{aligned}$$
in Theorem 3.5.

Remark 3.7

We remark that if the error terms are added in (1.4) and are assumed to be bounded, then the results of this paper still hold. On carefully reading Thakur’s work [16], we discovered that there are gaps in the proof of Lemma 2.1 in [16]. In (2.2) and (2.3) of Lemma 2.1, one cannot deduce \(\sum_{n=1}^{\infty}\sigma_{n}<\infty\) from \(\sum_{n=1}^{\infty}\mu_{n}<\infty\), \(\sum_{n=1}^{\infty}d_{n}<\infty\). Thus, his main results would not hold.

Declarations

Acknowledgements

The authors thank the referees for the comments that helped improve the presentation of this article. This work was supported by the Humanity and Social Science Planning Foundation of Ministry of Education of China (Grant No. 14YJAZH095), the National Natural Science Foundation of China (Grant No. 61374081), the High-level Talents Project in Guangdong Province (Grant No. 2014011), and the Natural Science Foundation of Guangdong Province (Grant Nos. S2013010013034, 2015A030313485).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Applied Mathematics, Guangdong University of Technology

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© Yang and Kong 2015