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

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.

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 : C → C be a mapping.

• (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

  $$\left|\right|T^{n}x-T^{n}y\left|\right|\le k_n\left|\right|x-y\left|\right|$$

for all x, y ∈ C and n ≥ 1.

• (2) T is said to be uniformly Lipschitzian if there exists a constant L > 0 such that

  $$\left|\right|T^{n}x-T^{n}y\left|\right|\le L\left|\right|x-y\left|\right|$$

for all x, y ∈ C and n ≥ 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

  $$\left|\right|T^{n}x-p\left|\right|\le k_n\left|\right|x-p\left|\right|$$

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 [1]. 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.

• (ii) It is easy to see that if T is an asymptotically nonexpansive mapping, then T is a uniformly Lipschitzian mapping (taking L = sup_{n ≥ 1}k _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) A mapping P : X → C is said to be retraction if P² = P.

• (2) If there exists a nonexpansive retraction P : X → C 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 : X → C the nonexpansive retraction of X onto C. Let T : C → X be a nonself mapping.

• (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

  $$\left|\right|T{\left(PT\right)}^{n-1}x-T{\left(PT\right)}^{n-1}y\left|\right|\le k_n\left|\right|x-y\left|\right|$$

for all x, y ∈ C and n ≥ 1.

• (2) T is said to be nonself uniformly Lipschitzian if there exists a constant L > 0 such that

  $$\left|\right|T{\left(PT\right)}^{n-1}x-T{\left(PT\right)}^{n-1}y\left|\right|\le L\left|\right|x-y\left|\right|$$

for all x, y ∈ C and n ≥ 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

  $$\left|\right|T{\left(PT\right)}^{n-1}x-p\left|\right|\le k_n\left|\right|x-p\left|\right|$$

for all x ∈ C, p ∈ Fix(T) and n ≥ 1.

• (4) T is said to be nonself uniformly quasi-Lipschitzian if there exists a constant L > 0 such that

  $$\left|\right|T{\left(PT\right)}^{n-1}x-p\left|\right|\le L\left|\right|x-p\left|\right|$$

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. [2] which is a generalization of asymptotically nonexpansive self-mapping.

• (ii) If T is a nonself asymptotically nonexpansive mapping, then it must be nonself uni-formly Lipschitzian, but the converse does not hold [3].

• (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., [2–8]). Recently, Khan et al. [9] 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}: C → C, where I = {1, 2, . . ., k} and C be a convex set. For any initial point x₁ ∈ C:

$$\left\{\begin{array}{c}{x}_{n+1}=\left(1-{\alpha }_{kn}\right)x_n+{\alpha }_{kn}{T}_k^n{y}_{\left(k-1\right)n},\hfill \\ y_{\left(k-1\right)n}=\left(1-{\alpha }_{\left(k-1\right)n}\right)x_n+{\alpha }_{\left(k-1\right)n}{T}_{\left(k-1\right)}^n{y}_{\left(k-2\right)n},\hfill \\ y_{\left(k-2\right)n}=\left(1-{\alpha }_{\left(k-2\right)n}\right)x_n+{\alpha }_{\left(k-2\right)n}{T}_{\left(k-2\right)}^n{y}_{\left(k-3\right)n},\hfill \\ ⋮\hfill \\ y_{1n}=\left(1-{\alpha }_{1n}\right)x_n+{\alpha }_{1n}{T}_1^n{y}_{0n},\hfill \end{array}\right.$$

(1.1)

where y_0n= 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 : X → C the nonexpansive retraction of X onto C. Assume T _i : C → X, i ∈ I be a finite family of nonself uniformly quasi-Lipschitzian mappings. For any x₁ ∈ C, the sequence {x _n } is defined by

$$\left\{\begin{array}{c}{x}_{n+1}=y_{kn}=P\left({\alpha }_{kn}{x}_n+{\beta }_{kn}{T}_k{\left(P{T}_k\right)}^{n-1}{y}_{\left(k-1\right)n}+{\gamma }_{kn}{u}_{kn}\right),\hfill \\ y_{\left(k-1\right)n}=P\left({\alpha }_{\left(k-1\right)n}{x}_n+{\beta }_{\left(k-1\right)n}{T}_{\left(k-1\right)}{\left(P{T}_{\left(k-1\right)}\right)}^{n-1}{y}_{\left(k-2\right)n}+{\gamma }_{\left(k-1\right)n}{u}_{\left(k-1\right)n}\right),\hfill \\ y_{\left(k-2\right)n}=P\left({\alpha }_{\left(k-2\right)n}{x}_n+{\beta }_{\left(k-2\right)n}{T}_{\left(k-2\right)}{\left(P{T}_{\left(k-2\right)}\right)}^{n-1}{y}_{\left(k-3\right)n}+{\gamma }_{\left(k-2\right)n}{u}_{\left(k-2\right)n}\right),\hfill \\ ⋮\hfill \\ y_{1n}=P\left({\alpha }_{1n}{x}_n+{\beta }_{1n}{T}_1{\left(P{T}_1\right)}^{n-1}{y}_{0n}+{\gamma }_{1n}{u}_{1n}\right),\hfill \end{array}\right.$$

(1.2)

where y_0n= 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:

• (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).

• (ii) If k = 2 and T₁, T₂ are nonself asymptotically nonexpansive mappings, γ_1n= γ_2n= 0, then the iterative sequence (1.2) reduces to the Ishikawa type iteration in Wang [8].

• (iii) If k = 1 and T₁ is a nonself asymptotically nonexpansive mapping, γ_1n= 0, then the iterative sequence (1.2) reduce to the Mann type iteration in Chidume et al. [2].

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 [2–10] 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. ([5]) 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}: C → X be a finite family of nonself uniformly quasi-Lipschitzian mappings with L _i > 0, i.e.,

$$\parallel T_i{\left(P{T}_i\right)}^{n-1}x-p_i\parallel \le L_i\parallel x-p_i\parallel$$

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

• (i) there exist two constants M₀, M₁ > 0 such that

  $$\left|\right|x_{n+1}-p\left|\right|\le \left[1+{\theta }_n{M}_0\right]\left|\right|x_n-p\left|\right|+{\theta }_n{M}_1$$

  (2.1)

where θ _n = β _kn + γ _kn for all n ≥ 1, p ∈ F.

• (ii) there exists a constant M₂ > 0, such that

  $$\left|\right|x_{n+m}-p\left|\right|\le M_2\left|\right|x_n-p\left|\right|+M_1{M}_2\sum _{j=n}^{n+m-1}{\theta }_j$$

  (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

$$M=\underset{1\le i\le k}{max}\left\{\left|\right|u_{in}-p\left|\right|\right\}.$$

Let L = max_{1 ≤ i ≤ k}{L _i } > 0. Using (1.2), we have

$$\begin{array}{cc}\hfill \left|\right|y_{1n}-p\left|\right|& =\left|\right|P\left({\alpha }_{1n}{x}_n+{\beta }_{1n}{T}_1{\left(P{T}_1\right)}^{n-1}{x}_n+{\gamma }_{1n}{u}_{1n}\right)-p\left|\right|\hfill \\ \le \left|\right|{\alpha }_{1n}{x}_n+{\beta }_{1n}{T}_1{\left(P{T}_1\right)}^{n-1}{x}_n+{\gamma }_{1n}{u}_{1n}-\left({\alpha }_{1n}+{\beta }_{1n}+{\gamma }_{1n}\right)p\left|\right|\hfill \\ \le {\alpha }_{1n}\left|\right|x_n-p\left|\right|+{\beta }_{1n}\left|\right|T_1{\left(P{T}_1\right)}^{n-1}{x}_n-p\left|\right|+{\gamma }_{1n}\left|\right|u_{1n}-p\left|\right|\hfill \\ \le {\alpha }_{1n}\left|\right|x_n-p\left|\right|+{\beta }_{1n}L\left|\right|x_n-p\left|\right|+{\gamma }_{1n}M\hfill \\ \le \left(1+L\right)\left|\right|x_n-p\left|\right|+M.\hfill \end{array}$$

Assume that

$$||y_{in}-p||\le {\left(1+L\right)}^i||x_n-p||+\sum _{j=0}^{i-1}{L}^{j}M$$

holds for some 1 ≤ i ≤ k - 1. Then

$$\begin{array}{cc}\hfill \left|\right|y_{\left(i+1\right)n}-p\left|\right|& =\left|\right|P\left({\alpha }_{\left(i+1\right)n}{x}_n+{\beta }_{\left(i+1\right)n}{T}_{\left(i+1\right)}{\left(P{T}_{\left(i+1\right)}\right)}^{n-1}{y}_{in}+{\gamma }_{\left(i+1\right)n}{u}_{\left(i+1\right)n}\right)-p\left|\right|\hfill \\ \le {\alpha }_{\left(i+1\right)n}\left|\right|x_n-p\left|\right|+{\beta }_{\left(i+1\right)n}\left|\right|T_{\left(i+1\right)}{\left(P{T}_{\left(i+1\right)}\right)}^{n-1}{y}_{in}-p\left|\right|+{\gamma }_{\left(i+1\right)n}\left|\right|u_{\left(i+1\right)n}-p\left|\right|\hfill \\ \le {\alpha }_{\left(i+1\right)n}\left|\right|x_n-p\left|\right|+{\beta }_{\left(i+1\right)n}L\left|\right|y_{in}-p\left|\right|+{\gamma }_{\left(i+1\right)n}\left|\right|u_{\left(i+1\right)n}-p\left|\right|\hfill \\ \le {\alpha }_{\left(i+1\right)n}\left|\right|x_n-p\left|\right|+{\beta }_{\left(i+1\right)n}L\left[{\left(1+L\right)}^i||x_n-p||+\sum _{j=0}^{i-1}{L}^{j}M\right]+{\gamma }_{\left(i+1\right)n}\left|\right|u_{\left(i+1\right)n}-p\left|\right|\hfill \\ \le \left[{\alpha }_{\left(i+1\right)n}+{\beta }_{\left(i+1\right)n}L{\left(1+L\right)}^i\right]\left|\right|x_n-p\left|\right|+{\beta }_{\left(i+1\right)n}L\sum _{j=0}^{i-1}{L}^{j}M+{\gamma }_{\left(i+1\right)n}M\hfill \\ \le \left[1+L{\left(1+L\right)}^i\right]\left|\right|x_n-p\left|\right|+\sum _{j=1}^i{L}^{j}M+M\hfill \\ \le {\left(1+L\right)}^{i+1}\left|\right|x_n-p\left|\right|+\sum _{j=0}^i{L}^{j}M\hfill \end{array}$$

Therefore, by induction, we get for all i ∈ I

$$\left|\right|y_{in}-p\left|\right|\le {\left(1+L\right)}^i\left|\right|x_n-p\left|\right|+\sum _{j=0}^{i-1}{L}^{j}M.$$

Now, from (1.2), it implies that

$$\begin{array}{cc}\hfill \left|\right|x_{n+1}-p\left|\right|& =\left|\right|P\left({\alpha }_{kn}{x}_n+{\beta }_{kn}{T}_k{\left(P{T}_k\right)}^{n-1}{y}_{\left(k-1\right)n}+{\gamma }_{kn}{u}_{kn}\right)-p\left|\right|\hfill \\ \le {\alpha }_{kn}\left|\right|x_n-p\left|\right|+{\beta }_{kn}L\left|\right|y_{\left(k-1\right)n}-p\left|\right|+{\gamma }_{kn}\left|\right|u_{kn}-p\left|\right|\hfill \\ \le {\alpha }_{kn}||x_n-p||+{\beta }_{kn}L\left[{\left(1+L\right)}^{k-1}||x_n-p||+\sum _{j=0}^{k-2}{L}^{j}M\right]+{\gamma }_{kn}M\hfill \\ \le \left[{\alpha }_{kn}+{\beta }_{kn}L{\left(1+L\right)}^{k-1}\right]\left|\right|x_n-p\left|\right|+{\beta }_{kn}L\sum _{j=0}^{k-2}{L}^{j}M+{\gamma }_{kn}M\hfill \\ \le \left[1+{\theta }_n{\left(1+L\right)}^k\right]\left|\right|x_n-p\left|\right|+{\theta }_n\left[L\sum _{j=0}^{k-2}{L}^{j}M+M\right]\hfill \\ \le \left[1+{\theta }_n{M}_0\right]\left|\right|x_n-p\left|\right|+{\theta }_n{M}_1,\hfill \end{array}$$

where θ _n = β _kn +γ _kn , and M₀ = (1 + L)^k, and $M_1={\sum }_{j=0}^{k-1}{L}^{j}M$.

• (ii) It is well known that 1 + x ≤ e^x for all x ≥ 0. Estimate (2.1) yields

  $$\begin{array}{cc}\hfill \left|\right|x_{n+m}-p\left|\right|& \le \left(1+{\theta }_{n+m-1}{M}_0\right)\left|\right|x_{n+m-1}-p\left|\right|+{\theta }_{n+m-1}{M}_1\hfill \\ \le e^{{\theta }_{n+m-1}{M}_0}\left[\left(1+{\theta }_{n+m-2}\right)M_0\left|\right|x_{n+m-2}-p\left|\right|+{\theta }_{n+m-2}{M}_1\right]+{\theta }_{n+m-1}{M}_1\hfill \\ \le e^{\left({\theta }_{n+m-1}+{\theta }_{n+m-2}\right)M_0}\left|\right|x_{n+m-2}-p\left|\right|+e^{{\theta }_{n+m-1}{M}_0}{M}_1\left({\theta }_{n+m-1}+{\theta }_{n+m-2}\right)\hfill \\ \cdots \cdots \hfill \\ \le e^{M_0{\mathrm{\Sigma }}_{j=1}^{\infty }{\theta }_j}\left|\right|x_n-p\left|\right|+e^{M_0{\mathrm{\Sigma }}_{j=1}^{\infty }{\theta }_j}\cdot M_1\cdot \sum _{j=n}^{n+m-1}{\theta }_j\hfill \\ \le M_2\left|\right|x_n-p\left|\right|+M_1{M}_2\sum _{j=n}^{n+m-1}{\theta }_j,\hfill \end{array}$$

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}: C → X be a finite family of nonself uniformly quasi-Lipschitzian mappings with L _i > 0, i.e.,

$$\left|\right|T_i{\left(P{T}_i\right)}^{n-1}x-p_i\left|\right|\le L_i\left|\right|x-p_i\left|\right|,$$

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

$$\begin{array}{cc}\hfill \left|\right|y_{1n}-p\left|\right|& \le {\alpha }_{1n}\left|\right|x_n-p\left|\right|+{\beta }_{1n}L\left|\right|x_n-p\left|\right|+{\gamma }_{1n}\left|\right|u_{1n}-p\left|\right|\hfill \\ \le {\alpha }_{1n}\left|\right|x_n-p\left|\right|+{\beta }_{1n}L\left|\right|x_n-p\left|\right|+{\gamma }_{1n}\left[\left|\right|u_{1n}-x_n\left|\right|+\left|\right|x_n-p\left|\right|\right]\hfill \\ \le \left(1+L\right)\left|\right|x_n-p\left|\right|+M^{\prime },\hfill \end{array}$$

where

$$M^{\prime }=\underset{1\le i\le k}{max}\left\{\left|\right|u_{in}-x_n\left|\right|\right\}.$$

The same to the proof of Lemma 2.1, we get that

$$\left|\right|x_{n+1}-p\left|\right|\le \left[1+{\theta }_n{M}_0\right]\left|\right|x_n-p\left|\right|+{\theta }_n{M}_1^{\prime },$$

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 lim_{n → ∞}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 n ≥ N₀, we have

$$d\left(x_n,F\right)\le \frac{\epsilon }{4M_2}\mathsf{\text{and}}\sum _{i=N_0}^{\infty }{\theta }_i\le \frac{\epsilon }{4M_1{M}_2}.$$

In particular, there exists a p₁ ∈ F and a constant N₁ > N₀, such that

$$\left|\right|x_{N_1}-p_1\left|\right|\le \frac{\epsilon }{4M_2}.$$

It follows from (2.2) that when n > N₁, we have

$$\begin{array}{cc}\hfill \left|\right|x_{n+m}-x_n\left|\right|& \le \left|\right|x_{n+m}-p_1\left|\right|+\left|\right|x_n-p_1\left|\right|\hfill \\ \le 2M_2\left|\right|x_{N_1}-p_1\left|\right|+M_1{M}_2\left(\sum _{j=N_1}^{n+m-1}{\theta }_j+\sum _{j=N_1}^{n-1}{\theta }_j\right)\hfill \\ \le 2M_2\frac{\epsilon }{4M_2}+M_1{M}_2\left(\frac{\epsilon }{4M_1{M}_2}+\frac{\epsilon }{4M_1{M}_2}\right)=\epsilon .\hfill \end{array}$$

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

$$|d\left(p,F\right)-d\left(x_n,F\right)|\le \left|\right|x_n-p\left|\right|,$$

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}: C → X 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}: C → X 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.

• (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, 4–9]).

• (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 [4–6]. 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}: 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 $F={\bigcap }_{i=1}^k\mathsf{\text{Fix}}\left(T_i\right)\ne \varnothing$, and closed. If for any given 1 ≤ l ≤ k,

• (i) $\underset{n\to \infty }{lim}||x_n-T_l{x}_n||=\mathsf{\text{0}}$;

• (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 $\underset{n\to \infty }{lim}||x_n-T{x}_n||=0$, 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

• (i) $\underset{n\to \infty }{lim}||x_n-T_i{x}_n||=0$ for all 1 ≤ i ≤ k;

• (ii) for some 1 ≤ l ≤ k, 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 $\underset{n\to \infty }{lim}||x_n-T_l{x}_n||=\mathsf{\text{0}}$, 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

$$\left|\right|x^{*}-T_i{x}^{*}\left|\right|=\underset{n_j\to \infty }{lim}\left|\right|x_{n_j}-T_i{x}_{n_j}\left|\right|=\underset{n_j\to \infty }{lim}\left|\right|x_{n_j}-T_l{x}_{n_j}\left|\right|=0.$$

This implies that x* ∈ F. From Theorem 2.1, it follows that

$$\left|\right|x_{n+1}-x^{*}\left|\right|\le \left[1+{\theta }_n{M}_0\right]\left|\right|x_n-x^{*}\left|\right|+{\theta }_n{M}_1^{\prime }.$$

Since ${\sum }_{n=1}^{\infty }{\theta }_n<\infty$, it implies from Lemma 1.1 that there exist a constant b ≥ 0 such that

$$\underset{n\to \infty }{lim}\left|\right|x_n-x^{*}\left|\right|=b.$$

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}.

□