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# New iterative schemes for a finite family of nonself uniformly quasi-Lipschitzian mappings in Banach spaces

Fixed Point Theory and Applications20122012:72

https://doi.org/10.1186/1687-1812-2012-72

• Received: 23 August 2011
• Accepted: 2 May 2012
• Published:

## Abstract

In this article, we introduce the concept of nonself uniformly quasi-Lipschitzian mapping and consider a new iterative scheme with errors to converge to a common fixed point for a finite family of nonself uniformly quasi-Lipschitzian mappings in Banach spaces. The results of this article improve and extend many known results.

## Keywords

• nonself uniformly quasi-Lipschitzian mapping
• new iterative scheme with errors
• common fixed point
• Banach spaces

## 1 Introduction and preliminaries

Throughout the article, we assume that X is a real Banach space, C is a nonempty subset of X, and Fix(T) is the set of fixed points of mapping T, i.e., Fix(T) = {x C : Tx = x}.

Definition 1.1. Let T : CC be a mapping.
1. (1)
T is said to be asymptotically nonexpansive if there exists a sequence {k n } [1, ) with $\underset{n\to \infty }{lim}{k}_{n}=1$ such that
$||{T}^{n}x-{T}^{n}y||\phantom{\rule{2.77695pt}{0ex}}\le {k}_{n}||x-y||$

for all x, y C and n ≥ 1.
1. (2)
T is said to be uniformly Lipschitzian if there exists a constant L > 0 such that
$||{T}^{n}x-{T}^{n}y||\phantom{\rule{2.77695pt}{0ex}}\le L||x-y||$

for all x, y C and n ≥ 1.
1. (3)
T is called asymptotically quasi-nonexpansive if there exists a sequence {k n } [1, ) with $\underset{n\to \infty }{lim}{k}_{n}=1$ such that
$||{T}^{n}x-p||\phantom{\rule{2.77695pt}{0ex}}\le {k}_{n}||x-p||$

for all x C, p Fix(T) and n ≥ 1.

Remark 1.1. (i) The concept of asymptotically nonexpansive mapping was initially introduced by Geobel and Kirk . Meanwhile, they proved that if C is a nonempty closed, convex, and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping has a fixed point.
1. (ii)

It is easy to see that if T is an asymptotically nonexpansive mapping, then T is a uniformly Lipschitzian mapping (taking L = supn ≥ 1k n ), and if Fix(T) ≠ Ø, then every asymptotically nonexpansive mapping T is an asymptotically quasi-nonexpansive mapping.

Definition 1.2. Let X be a real Banach space and C be a nonempty subset of X.
1. (1)

A mapping P : XC is said to be retraction if P2 = P.

2. (2)

If there exists a nonexpansive retraction P : XC such that Px = x for all x C, then the set C is said to be nonexpansive retract of X.

Next, we introduce some concepts for nonself mappings.

Definition 1.3. Let X be a real Banach space, C be a nonempty subset of X, and P : XC the nonexpansive retraction of X onto C. Let T : CX be a nonself mapping.
1. (1)
T is said to be nonself asymptotically nonexpansive if there exists a sequence {k n } [1, ) with $\underset{n\to \infty }{lim}{k}_{n}=1$ such that
$||T{\left(PT\right)}^{n-1}x-T{\left(PT\right)}^{n-1}y||\phantom{\rule{2.77695pt}{0ex}}\le {k}_{n}||x-y||$

for all x, y C and n ≥ 1.
1. (2)
T is said to be nonself uniformly Lipschitzian if there exists a constant L > 0 such that
$||T{\left(PT\right)}^{n-1}x-T{\left(PT\right)}^{n-1}y||\phantom{\rule{2.77695pt}{0ex}}\le L||x-y||$

for all x, y C and n ≥ 1.
1. (3)
T is said to be nonself asymptotically quasi-nonexpansive if there exists a sequence {k n } [1, ∞) with $\underset{n\to \infty }{lim}{k}_{n}=1$ such that
$||T{\left(PT\right)}^{n-1}x-p||\phantom{\rule{2.77695pt}{0ex}}\le {k}_{n}||x-p||$

for all x C, p Fix(T) and n ≥ 1.
1. (4)
T is said to be nonself uniformly quasi-Lipschitzian if there exists a constant L > 0 such that
$||T{\left(PT\right)}^{n-1}x-p||\phantom{\rule{2.77695pt}{0ex}}\le L||x-p||$

for all x C, p Fix(T) and n ≥ 1.

Remark 1.2. (i) The concept of nonself asymptotically nonexpansive mapping was introduced by Chidume et al.  which is a generalization of asymptotically nonexpansive self-mapping.
1. (ii)

If T is a nonself asymptotically nonexpansive mapping, then it must be nonself uni-formly Lipschitzian, but the converse does not hold .

2. (iii)

If T is a nonself uniformly Lipschitzian mapping or a nonself asymptotically quasi-nonexpansive mapping, then it must be a nonself uniformly quasi-Lipschitzian mapping.

Fixed points iterative technique for (self or nonself) asymptotically nonexpansive mappings in Banach spaces, including Mann type iteration, Ishikawa type iteration, and three-step type iteration, have been studied by many authors (see, e.g., ). Recently, Khan et al.  introduced an iterative scheme (which generalizes Mann iteration, Ishikawa iteration, and three-step iteration) for a finite family of asymptotically quasi-nonexpansive self-mappings {T i : i I}: CC, where I = {1, 2, . . ., k} and C be a convex set. For any initial point x1 C:
$\left\{\begin{array}{c}{x}_{n+1}=\left(1-{\alpha }_{kn}\right){x}_{n}+{\alpha }_{kn}{T}_{k}^{n}{y}_{\left(k-1\right)n},\hfill \\ {y}_{\left(k-1\right)n}=\left(1-{\alpha }_{\left(k-1\right)n}\right){x}_{n}+{\alpha }_{\left(k-1\right)n}{T}_{\left(k-1\right)}^{n}{y}_{\left(k-2\right)n},\hfill \\ {y}_{\left(k-2\right)n}=\left(1-{\alpha }_{\left(k-2\right)n}\right){x}_{n}+{\alpha }_{\left(k-2\right)n}{T}_{\left(k-2\right)}^{n}{y}_{\left(k-3\right)n},\hfill \\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}⋮\hfill \\ {y}_{1n}=\left(1-{\alpha }_{1n}\right){x}_{n}+{\alpha }_{1n}{T}_{1}^{n}{y}_{0n},\hfill \end{array}\right\$
(1.1)

where y0n= x n and {α in } are real sequences in [0, 1] for all n ≥ 1. They proved the convergence to a common fixed point for a finite family of asymptotically quasi-nonexpansive self-mappings in Banach spaces by using the iterative (1.1).

Inspired and motivated by the above research, we introduce a new iterative process as follows:

Let C be a nonempty convex subset of a real Banach space X and P : XC the nonexpansive retraction of X onto C. Assume T i : CX, i I be a finite family of nonself uniformly quasi-Lipschitzian mappings. For any x1 C, the sequence {x n } is defined by
$\left\{\begin{array}{c}{x}_{n+1}={y}_{kn}=P\left({\alpha }_{kn}{x}_{n}+{\beta }_{kn}{T}_{k}{\left(P{T}_{k}\right)}^{n-1}{y}_{\left(k-1\right)n}+{\gamma }_{kn}{u}_{kn}\right),\hfill \\ {y}_{\left(k-1\right)n}=P\left({\alpha }_{\left(k-1\right)n}{x}_{n}+{\beta }_{\left(k-1\right)n}{T}_{\left(k-1\right)}{\left(P{T}_{\left(k-1\right)}\right)}^{n-1}{y}_{\left(k-2\right)n}+{\gamma }_{\left(k-1\right)n}{u}_{\left(k-1\right)n}\right),\hfill \\ {y}_{\left(k-2\right)n}=P\left({\alpha }_{\left(k-2\right)n}{x}_{n}+{\beta }_{\left(k-2\right)n}{T}_{\left(k-2\right)}{\left(P{T}_{\left(k-2\right)}\right)}^{n-1}{y}_{\left(k-3\right)n}+{\gamma }_{\left(k-2\right)n}{u}_{\left(k-2\right)n}\right),\hfill \\ \phantom{\rule{2.77695pt}{0ex}}⋮\hfill \\ {y}_{1n}=P\left({\alpha }_{1n}{x}_{n}+{\beta }_{1n}{T}_{1}{\left(P{T}_{1}\right)}^{n-1}{y}_{0n}+{\gamma }_{1n}{u}_{1n}\right),\hfill \end{array}\right\$
(1.2)

where y0n= x n for all n ≥ 1, {α in }, {β in }, {γ in } are real sequences in [0, 1] with α in + β in + γ in = 1 and {u in } is a bounded sequence in C, for i I.

Remark 1.3. The iterative sequence (1.2) is a natural generalization of the well-known iteration:
1. (i)

If {T i : i I} is asymptotically quasi-nonexpansive self-mappings and γ in = 0 for i I and n ≥ 1, then the iterative sequence (1.2) reduces to (1.1).

2. (ii)

If k = 2 and T1, T2 are nonself asymptotically nonexpansive mappings, γ1n= γ2n= 0, then the iterative sequence (1.2) reduces to the Ishikawa type iteration in Wang .

3. (iii)

If k = 1 and T1 is a nonself asymptotically nonexpansive mapping, γ1n= 0, then the iterative sequence (1.2) reduce to the Mann type iteration in Chidume et al. .

In this article, we are concerned with the convergence to a common fixed point for a finite family of nonself uniformly quasi-Lipschitzian mappings in Banach spaces by using the iterative sequence (1.2). As one will see, our results extend and generalize the corresponding results in  as follows: (i) the condition ${\sum }_{n=1}^{\infty }\left({k}_{in}-1\right)<\infty$ is dropped; (ii) the condition ${\sum }_{n=1}^{\infty }{\gamma }_{in}<\infty ,i\in I$ is replaced with ${\sum }_{n=1}^{\infty }\left({\beta }_{kn}+{\gamma }_{kn}\right)<\infty$; (iii) a more general mapping is considered.

We need the following lemma for proving our main results.

Lemma 1.1. () Let {a n }, {b n }, and {c n } be sequences of nonnegative real numbers satisfying the following conditions:
${a}_{n+1}\le \left(1+{b}_{n}\right){a}_{n}+{c}_{n},\forall n\ge 1.$

where ${\sum }_{n=1}^{\infty }{b}_{n}<\infty$ and ${\sum }_{n=1}^{\infty }{c}_{n}<\infty$, then $\underset{n\to \infty }{lim}{a}_{n}$ exists.

## 2 Main results

In this section, we shall prove the strong convergence of the iteration sequence (1.2) to a common fixed point for a finite family of nonself uniformly quasi-Lipschitzian mappings T i (i I) in real Banach spaces. We first prove the following lemma.

Lemma 2.1. Let C be a nonempty convex subset of a real normed linear space X. Let {T i : i I}: CX be a finite family of nonself uniformly quasi-Lipschitzian mappings with L i > 0, i.e.,
$\parallel {T}_{i}{\left(P{T}_{i}\right)}^{n-1}x-{p}_{i}\parallel \phantom{\rule{2.77695pt}{0ex}}\le {L}_{i}\parallel x-{p}_{i}\parallel$
for all x C and p i Fix(T i ), i I. Define the sequence {x n } as in (1.2) with ${\sum }_{n=1}^{\infty }\left({\beta }_{kn}+{\gamma }_{kn}\right)<\infty$. If $F={\bigcap }_{i=1}^{k}Fix\left({T}_{i}\right)\ne \varnothing$, then
1. (i)
there exist two constants M0, M1 > 0 such that
$||{x}_{n+1}-p||\phantom{\rule{2.77695pt}{0ex}}\le \left[1+{\theta }_{n}{M}_{0}\right]||{x}_{n}-p||+\phantom{\rule{2.77695pt}{0ex}}{\theta }_{n}{M}_{1}$
(2.1)

where θ n = β kn + γ kn for all n ≥ 1, p F.
1. (ii)
there exists a constant M2 > 0, such that
$||{x}_{n+m}-p||\phantom{\rule{2.77695pt}{0ex}}\le {M}_{2}||{x}_{n}-p||+{M}_{1}{M}_{2}\sum _{j=n}^{n+m-1}{\theta }_{j}$
(2.2)

for all n, m ≥ 1, p F.

Proof. (i)We take p F. Since {u in } is a bounded sequence in C for all i I, there exists M > 0 such that
$M=\underset{1\le i\le k}{max}\left\{||{u}_{in}-p||\right\}.$
Let L = max1 ≤ i ≤ k{L i } > 0. Using (1.2), we have
$\begin{array}{cc}\hfill ||{y}_{1n}-p||& =||P\left({\alpha }_{1n}{x}_{n}+{\beta }_{1n}{T}_{1}{\left(P{T}_{1}\right)}^{n-1}{x}_{n}+{\gamma }_{1n}{u}_{1n}\right)-p||\hfill \\ \le \phantom{\rule{0.3em}{0ex}}||{\alpha }_{1n}{x}_{n}+{\beta }_{1n}{T}_{1}{\left(P{T}_{1}\right)}^{n-1}{x}_{n}+{\gamma }_{1n}{u}_{1n}-\left({\alpha }_{1n}+{\beta }_{1n}+{\gamma }_{1n}\right)p||\hfill \\ \le {\alpha }_{1n}||{x}_{n}-p||+{\beta }_{1n}||{T}_{1}{\left(P{T}_{1}\right)}^{n-1}{x}_{n}-p||+{\gamma }_{1n}||{u}_{1n}-p||\hfill \\ \le {\alpha }_{1n}||{x}_{n}-p||+{\beta }_{1n}L||{x}_{n}-p||+{\gamma }_{1n}M\hfill \\ \le \left(1+L\right)||{x}_{n}-p||+M.\hfill \end{array}$
Assume that
$||{y}_{in}-p||\phantom{\rule{2.77695pt}{0ex}}\le {\left(1+L\right)}^{i}||{x}_{n}-p||+\sum _{j=0}^{i-1}{L}^{j}M$
holds for some 1 ≤ ik - 1. Then
$\begin{array}{cc}\hfill ||{y}_{\left(i+1\right)n}-p||& =||P\left({\alpha }_{\left(i+1\right)n}{x}_{n}+{\beta }_{\left(i+1\right)n}{T}_{\left(i+1\right)}{\left(P{T}_{\left(i+1\right)}\right)}^{n-1}{y}_{in}+{\gamma }_{\left(i+1\right)n}{u}_{\left(i+1\right)n}\right)-p||\hfill \\ \le {\alpha }_{\left(i+1\right)n}||{x}_{n}-p||+{\beta }_{\left(i+1\right)n}||{T}_{\left(i+1\right)}{\left(P{T}_{\left(i+1\right)}\right)}^{n-1}{y}_{in}-p||+{\gamma }_{\left(i+1\right)n}||{u}_{\left(i+1\right)n}-p||\hfill \\ \le {\alpha }_{\left(i+1\right)n}||{x}_{n}-p||+{\beta }_{\left(i+1\right)n}L||{y}_{in}-p||+{\gamma }_{\left(i+1\right)n}||{u}_{\left(i+1\right)n}-p||\hfill \\ \le {\alpha }_{\left(i+1\right)n}||{x}_{n}-p||+{\beta }_{\left(i+1\right)n}L\left[{\left(1+L\right)}^{i}||{x}_{n}-p||+\sum _{j=0}^{i-1}{L}^{j}M\right]+{\gamma }_{\left(i+1\right)n}||{u}_{\left(i+1\right)n}-p||\hfill \\ \le \left[{\alpha }_{\left(i+1\right)n}+{\beta }_{\left(i+1\right)n}L{\left(1+L\right)}^{i}\right]||{x}_{n}-p||+{\beta }_{\left(i+1\right)n}L\sum _{j=0}^{i-1}{L}^{j}M+{\gamma }_{\left(i+1\right)n}M\hfill \\ \le \left[1+L{\left(1+L\right)}^{i}\right]||{x}_{n}-p||+\sum _{j=1}^{i}{L}^{j}M+M\hfill \\ \le {\left(1+L\right)}^{i+1}||{x}_{n}-p||+\sum _{j=0}^{i}{L}^{j}M\hfill \end{array}$
Therefore, by induction, we get for all i I
$||{y}_{in}-p||\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\le {\left(1+L\right)}^{i}||{x}_{n}-p||+\sum _{j=0}^{i-1}{L}^{j}M.$
Now, from (1.2), it implies that
$\begin{array}{cc}\hfill ||{x}_{n+1}-p||& =||P\left({\alpha }_{kn}{x}_{n}+{\beta }_{kn}{T}_{k}{\left(P{T}_{k}\right)}^{n-1}{y}_{\left(k-1\right)n}+{\gamma }_{kn}{u}_{kn}\right)-p||\hfill \\ \le {\alpha }_{kn}||{x}_{n}-p||+{\beta }_{kn}L||{y}_{\left(k-1\right)n}-p||+{\gamma }_{kn}||{u}_{kn}-p||\hfill \\ \le {\alpha }_{kn}||{x}_{n}-p||+{\beta }_{kn}L\left[{\left(1+L\right)}^{k-1}||{x}_{n}-p||+\sum _{j=0}^{k-2}{L}^{j}M\right]+{\gamma }_{kn}M\hfill \\ \le \left[{\alpha }_{kn}+{\beta }_{kn}L{\left(1+L\right)}^{k-1}\right]||{x}_{n}-p||+{\beta }_{kn}L\sum _{j=0}^{k-2}{L}^{j}M+{\gamma }_{kn}M\hfill \\ \le \left[1+{\theta }_{n}{\left(1+L\right)}^{k}\right]||{x}_{n}-p||+{\theta }_{n}\left[L\sum _{j=0}^{k-2}{L}^{j}M+M\right]\hfill \\ \le \left[1+{\theta }_{n}{M}_{0}\right]||{x}_{n}-p||+{\theta }_{n}{M}_{1},\hfill \end{array}$
where θ n = β kn +γ kn , and M0 = (1 + L) k , and ${M}_{1}={\sum }_{j=0}^{k-1}{L}^{j}M$.
1. (ii)
It is well known that 1 + x ≤ e x for all x ≥ 0. Estimate (2.1) yields
$\begin{array}{cc}\hfill ||{x}_{n+m}-p||& \le \left(1+{\theta }_{n+m-1}{M}_{0}\right)||{x}_{n+m-1}-p||+{\theta }_{n+m-1}{M}_{1}\hfill \\ \le {e}^{{\theta }_{n+m-1}{M}_{0}}\left[\left(1+{\theta }_{n+m-2}\right){M}_{0}||{x}_{n+m-2}-p||+{\theta }_{n+m-2}{M}_{1}\right]+{\theta }_{n+m-1}{M}_{1}\hfill \\ \le {e}^{\left({\theta }_{n+m-1}+{\theta }_{n+m-2}\right){M}_{0}}||{x}_{n+m-2}-p||+{e}^{{\theta }_{n+m-1}{M}_{0}}{M}_{1}\left({\theta }_{n+m-1}+{\theta }_{n+m-2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\cdots \cdots \hfill \\ \le {e}^{{M}_{0}{\mathrm{\Sigma }}_{j=1}^{\infty }{\theta }_{j}}||{x}_{n}-p||+{e}^{{M}_{0}{\mathrm{\Sigma }}_{j=1}^{\infty }{\theta }_{j}}\cdot {M}_{1}\cdot \sum _{j=n}^{n+m-1}{\theta }_{j}\hfill \\ \le {M}_{2}||{x}_{n}-p||+{M}_{1}{M}_{2}\sum _{j=n}^{n+m-1}{\theta }_{j},\hfill \end{array}$

where ${M}_{2}={e}^{{M}_{0}{\mathrm{\Sigma }}_{j=1}^{\infty }{\theta }_{j}}$.

Theorem 2.1. Let C be a nonempty closed convex subset of a real Banach space X. Let {T i : i I}: CX be a finite family of nonself uniformly quasi-Lipschitzian mappings with L i > 0, i.e.,
$||{T}_{i}{\left(P{T}_{i}\right)}^{n-1}x-{p}_{i}||\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le {L}_{i}||x-{p}_{i}||,$

for all x C and p i F (T i ), i I. Define the sequence {x n } as in (1.2) with ${\sum }_{n=1}^{\infty }\left({\beta }_{kn}+{\gamma }_{kn}\right)<\infty$. Suppose that $F={\bigcap }_{i=1}^{k}Fix\left({T}_{i}\right)\ne \varnothing$ and closed. Then {x n } converges to a common fixed point of {T i : i I} if and only if $\underset{n\to \infty }{lim inf}d\left({x}_{n},F\right)=0$, where d(x, F) = inf{||x - p||: p F}.

Proof. The necessity is obvious. Next, we will prove the sufficiency. It follows from Lemma 2.1 that {x n } is bounded. From Lemma 2.1 (i), we have
$\begin{array}{cc}\hfill ||{y}_{1n}-p||& \le {\alpha }_{1n}||{x}_{n}-p||+{\beta }_{1n}L||{x}_{n}-p||+{\gamma }_{1n}||{u}_{1n}-p||\hfill \\ \le {\alpha }_{1n}||{x}_{n}-p||+{\beta }_{1n}L||{x}_{n}-p||+{\gamma }_{1n}\left[||{u}_{1n}-{x}_{n}||+||{x}_{n}-p||\right]\hfill \\ \le \left(1+L\right)||{x}_{n}-p||+{M}^{\prime },\hfill \end{array}$
where
${M}^{\prime }=\underset{1\le i\le k}{max}\left\{||{u}_{in}-{x}_{n}||\right\}.$
The same to the proof of Lemma 2.1, we get that
$||{x}_{n+1}-p||\phantom{\rule{2.77695pt}{0ex}}\le \left[1+{\theta }_{n}{M}_{0}\right]||{x}_{n}-p||+{\theta }_{n}{M}_{1}^{\prime },$
where θ n = β kn + γ kn , and M0 = (1 +L) k , and ${M}_{1}^{\prime }={\sum }_{j=0}^{k-1}{L}^{j}{M}^{\prime }$. Taking infimum over all p in F, we obtain
$d\left({x}_{n+1},F\right)\le \left(1+{\theta }_{n}{M}_{0}\right)d\left({x}_{n},F\right)+{\theta }_{n}{M}_{1}^{\prime }.$

Note that ${M}_{1}^{\prime }$ does not depend on p.

Since ${\sum }_{n=1}^{\infty }{\theta }_{n}={\sum }_{n=1}^{\infty }\left({\beta }_{kn}+{\gamma }_{kn}\right)<\infty$ and Lemma 1.1, we get that limn → ∞d(x n , F) exists. Furthermore, from $\underset{n\to \infty }{lim inf}d\left({x}_{n},F\right)=0$, we obtain that ${lim}_{n\to \infty }d\left({x}_{n},F\right)=0$.

We claim that {x n } is a Cauchy sequence. Indeed, for any ε > 0, there exists a constant N0 such that for all nN0, we have
$d\left({x}_{n},F\right)\le \frac{\epsilon }{4{M}_{2}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}\sum _{i={N}_{0}}^{\infty }{\theta }_{i}\le \frac{\epsilon }{4{M}_{1}{M}_{2}}.$
In particular, there exists a p1 F and a constant N1 > N0, such that
$||{x}_{{N}_{1}}-{p}_{1}||\phantom{\rule{2.77695pt}{0ex}}\le \frac{\epsilon }{4{M}_{2}}.$
It follows from (2.2) that when n > N1, we have
$\begin{array}{cc}\hfill ||{x}_{n+m}-{x}_{n}||& \le ||{x}_{n+m}-{p}_{1}||+||{x}_{n}-{p}_{1}||\phantom{\rule{2.77695pt}{0ex}}\hfill \\ \le 2{M}_{2}||{x}_{{N}_{1}}-{p}_{1}||+{M}_{1}{M}_{2}\left(\sum _{j={N}_{1}}^{n+m-1}{\theta }_{j}+\sum _{j={N}_{1}}^{n-1}{\theta }_{j}\right)\hfill \\ \le 2{M}_{2}\frac{\epsilon }{4{M}_{2}}+{M}_{1}{M}_{2}\left(\frac{\epsilon }{4{M}_{1}{M}_{2}}+\frac{\epsilon }{4{M}_{1}{M}_{2}}\right)=\epsilon .\hfill \end{array}$

Hence, {x n } is a Cauchy sequence in closed convex subset of real Banach spaces. Clearly, {x n } converges to a point of C.

Suppose that $\underset{n\to \infty }{lim}{x}_{n}=p\in C$. We notice that
$|d\left(p,F\right)-d\left({x}_{n},F\right)|\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}||{x}_{n}-p||,$

for all n ≥ 1. Since $\underset{n\to \infty }{lim}{x}_{n}=p$ and ${lim}_{n\to \infty }d\left({x}_{n},F\right)=0$, we conclude that d(p, F) = 0. Therefore, p F. □

Corollary 2.1. Let C be a nonempty closed convex subset of a real Banach space X. Let {T i : i I}: CX be a finite family of nonself uniformly Lipschitzian mappings with L i > 0. Define the sequence {x n } as in (1.2) with ${\sum }_{n=1}^{\infty }\left({\beta }_{kn}+{\gamma }_{kn}\right)<\infty$. Suppose that $F={\bigcap }_{i=1}^{k}\mathsf{\text{Fix}}\left({T}_{i}\right)\ne \varnothing$ and closed. Then{x n } converges to a common fixed point of {T i : i I} if and only if $\underset{n\to \infty }{lim inf}d\left({x}_{n},F\right)=0$, where d(x, F) = inf{||x - p||: p F}.

Corollary 2.2. Let C be a nonempty closed convex subset of a real Banach space X. Let {T i : i I}: CX be a finite family of nonself asymptotically nonexpansive mappings (or nonself asymptotically quasi-nonexpansive mappings) with {k in }. Define the sequence {x n } as in (1.2) with ${\sum }_{n=1}^{\infty }\left({\beta }_{kn}+{\gamma }_{kn}\right)<\infty$. Suppose that $F={\bigcap }_{i=1}^{k}\mathsf{\text{Fix}}\left({T}_{i}\right)\ne \varnothing$ and closed. Then {x n } converges to a common fixed point of {T i : i I} if and only if $\underset{n\to \infty }{lim inf}d\left({x}_{n},F\right)=0$, where d(x, F) = inf{||x - p|| : p F}.

Proof. Since for all i I, {k in } [1, ) and $\underset{n\to \infty }{lim}{k}_{in}=1$, there exists L i > 0 such that ${L}_{i}=\underset{n\ge 1}{sup}\left\{{k}_{in}\right\}<\infty$. Consequently, {T i : i I} is a finite family of nonself uniformly quasi-Lipschitzian mappings with L i > 0. From Theorem 2.1, we get the desired result. □

Remark 2.1. (i) When {T i : i I} is a finite family of asymptotically nonexpansive self-mappings or asymptotically quasi-nonexpansive self-mappings with {k in }, Corollary 2.2 also holds.
1. (ii)

In Corollary 2.2, we remove the condition: "${\sum }_{n=1}^{\infty }\left({k}_{in}-1\right)<\infty$", which is required in many other article (see, e.g., [2, 49]).

2. (iii)

When considering iterative schemes with errors, many authors need the conditions: "${\sum }_{n=1}^{\infty }{\gamma }_{in}<\infty ,i\in I$", see for example . But in Corollary 2.2, we only need the condition: "${\sum }_{n=1}^{\infty }{\theta }_{n}<\infty$", where θ n = β kn + γ kn .

Theorem 2.2. Let C be a nonempty closed convex subset of a real Banach space X. Let {T i : i I}: CX be a finite family of nonself uniformly quasi-Lipschitzian mappings with L i > 0. Define the sequence {x n } as in (1.2) with ${\sum }_{n=1}^{\infty }\left({\beta }_{kn}+{\gamma }_{kn}\right)<\infty$. Suppose that $F={\bigcap }_{i=1}^{k}\mathsf{\text{Fix}}\left({T}_{i}\right)\ne \varnothing$, and closed. If for any given 1 ≤ lk,
1. (i)

$\underset{n\to \infty }{lim}||{x}_{n}-{T}_{l}{x}_{n}||\phantom{\rule{0.3em}{0ex}}=\mathsf{\text{0}}$;

2. (ii)

there exists a constant α > 0 such that || x n - T l x n || ≥ αd(x n , F) for all n ≥ 1.

Then {x n } converges to a common fixed point of {T i : i I}.

Proof. From the conditions (i) and (ii), it implies that ${lim}_{n\to \infty }d\left({x}_{n},F\right)=0$. Therefore, from the proof of Theorem 2.1, {x n } converges to a common fixed point of {T i : i I}. □

Remark 2.2. A mapping T : C → X is said to be semi-compact, if for any sequence {x n } C such that $\underset{n\to \infty }{lim}||{x}_{n}-T{x}_{n}||\phantom{\rule{2.77695pt}{0ex}}=0$, there exists a subsequence $\left\{{x}_{{n}_{j}}\right\}$ of {x n } such that $\left\{{x}_{{n}_{j}}\right\}$ converges strongly to x* C.

Theorem 2.3. Let C be a nonempty closed convex subset of a real Banach space X. Let {T i : i I}: C → X be a finite family of nonself uniformly quasi-Lipschitzian mappings with L i > 0. Define the sequence {x n } as in (1.2) with ${\sum }_{n=1}^{\infty }\left({\beta }_{kn}+{\gamma }_{kn}\right)<\infty$. Suppose that $F={\bigcap }_{i=1}^{k}F\left({T}_{i}\right)\ne \varnothing$ and closed. If
1. (i)

$\underset{n\to \infty }{lim}||{x}_{n}-{T}_{i}{x}_{n}||\phantom{\rule{2.77695pt}{0ex}}=0$ for all 1 ≤ ik;

2. (ii)

for some 1 ≤ lk, T l is semi-compact.

Then {x n } converges to a common fixed point of {T i : i I}.

Proof. Since T l is semi-compact and $\underset{n\to \infty }{lim}||{x}_{n}-{T}_{l}{x}_{n}||\phantom{\rule{0.3em}{0ex}}=\mathsf{\text{0}}$, there exist a subsequence $\left\{{x}_{{n}_{j}}\right\}\subset \left\{{x}_{n}\right\}$ such that ${x}_{{n}_{j}}\to x*\in C$. Consequently, we have
$||{x}^{*}-{T}_{i}{x}^{*}||=\underset{{n}_{j}\to \infty }{lim}||{x}_{{n}_{j}}-{T}_{i}{x}_{{n}_{j}}||\phantom{\rule{0.3em}{0ex}}=\underset{{n}_{j}\to \infty }{lim}||{x}_{{n}_{j}}-{T}_{l}{x}_{{n}_{j}}||\phantom{\rule{0.3em}{0ex}}=0.$
This implies that x* F. From Theorem 2.1, it follows that
$||{x}_{n+1}-{x}^{*}||\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le \left[1+{\theta }_{n}{M}_{0}\right]||{x}_{n}-{x}^{*}||+{\theta }_{n}{M}_{1}^{\prime }.$
Since ${\sum }_{n=1}^{\infty }{\theta }_{n}<\infty$, it implies from Lemma 1.1 that there exist a constant b ≥ 0 such that
$\underset{n\to \infty }{lim}||{x}_{n}-{x}^{*}||\phantom{\rule{0.3em}{0ex}}=b.$

From ${x}_{{n}_{j}}\to {x}^{*}$, we know that b = 0, i.e., x n x*. Thus, {x n } converge to a common fixed point of {T i : i I}.

## Declarations

### Acknowledgements

The authors thank the editor and the referees for constructive and pertinent suggestions. CW was partially supported by the NSF of China (Tianyuan fund for Mathematics, NO.11126290) and Foundation of Nanjing University of Information Science and Technology. JL was partially supported by MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Project No. 12YJC630091).

## Authors’ Affiliations

(1)
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, People's Republic of China
(2)
School of Computer and Information Engineering, Zhejiang Gongshang University, Hangzhou, 310018, People's Republic of China
(3)
Department of Mathematics, Ningbo University, Zhejiang, 315211, People's Republic of China
(4)
Department of Mathematics, Tongji University, Shanghai, 200092, People's Republic of China

## References

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