An implicit iterative algorithm with errors for two families of generalized asymptotically nonexpansive mappings

Abstract

In this paper, an implicit iterative algorithm with errors is considered for two families of generalized asymptotically nonexpansive mappings. Strong and weak convergence theorems of common fixed points are established based on the implicit iterative algorithm.

Mathematics Subject Classification (2000) 47H09 · 47H10 · 47J25

1 Introduction

In nonlinear analysis theory, due to applications to complex real-world problems, a growing number of mathematical models are built up by introducing constraints which can be expressed as subproblems of a more general problem. These constraints can be given by fixed-point problems, see, for example, [1–3]. Study of fixed points of nonlinear mappings and its approximation algorithms constitutes a topic of intensive research efforts. Many well-known problems arising in various branches of science can be studied by using algorithms which are iterative in their nature. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonexpansive mappings, see, for example, [3–5].

For iterative algorithms, the most oldest and simple one is Picard iterative algorithm. It is known that T enjoys a unique fixed point, and the sequence generated in Picard iterative algorithm can converge to the unique fixed point. However, for more general nonexpansive mappings, Picard iterative algorithm fails to convergence to fixed points of nonexpansive even that it enjoys a fixed point.

Recently, Mann-type iterative algorithm and Ishikawa-type iterative algorithm (implicit and explicit) have been considered for the approximation of common fixed points of nonlinear mappings by many authors, see, for example, [6–24]. A classical convergence theorem of nonexpansive mappings has been established by Xu and Ori [23]. In 2006, Chang et al. [6] considered an implicit iterative algorithm with errors for asymptotically nonexpansive mappings in a Banach space. Strong and weak convergence theorems are established. Recently, Cianciaruso et al. [9] considered an Ishikawa-type iterative algorithm for the class of asymptotically nonexpansive mappings. Strong and weak convergence theorems are also established. In this paper, based on the class of generalized asymptotically nonexpansive mappings, an Ishikawa-type implicit iterative algorithm with errors for two families of mappings is considered. Strong and weak convergence theorems of common fixed points are established. The results presented in this paper mainly improve the corresponding results announced in Chang et al. [6], Chidume and Shahzad [7], Cianciaruso et al. [9], Guo and Cho [10], Khan et al. [12], Plubtieng et al. [14], Qin et al. [15], Shzhzad and Zegeye [18], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], Zhou and Chang [24].

2 Preliminaries

Let C be a nonempty closed convex subset of a Banach space E. Let T : C → C be a mapping. Throughout this paper, we use F(T) to denote the fixed point set of T.

Recall the following definitions.

T is said to be nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\forall x,y\in C.$$

T is said to be asymptotically nonexpansive if there exists a positive sequence {h _n } ⊂ [1, ∞) with h _n → 1 as n → ∞ such that

$$\parallel T^{n}x-T^{n}y\parallel \le h_n\parallel x-y\parallel ,\forall x,y\in C,n\ge 1.$$

It is easy to see that every nonexpansive mapping is asymptotically nonexpansive with the asymptotical sequence {1}. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [25] in 1972. It is known that if C is a nonempty bounded closed convex subset of a uniformly convex Banach space E, then every asymptotically nonexpansive mapping on C has a fixed point. Further, the set F(T) of fixed points of T is closed and convex. Since 1972, a host of authors have studied weak and strong convergence problems of implicit iterative processes for such a class of mappings.

T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

$$\underset{n\to \infty }{lim\; sup}\underset{x,y\in C}{sup}\left(\parallel T^{n}x-T^{n}y\parallel -\parallel x-y\parallel \right)\le 0.$$

(2.1)

Putting ξ _n = max{0, sup_x,y∈C(||Tⁿx - Tⁿy|| - ||x - y||)}, we see that ξ _n → 0 as n → ∞. Then, (2.1) is reduced to the following:

$$\parallel T^{n}x-T^{n}y\parallel \le \parallel x-y\parallel +{\xi }_n,\forall x,y\in C,n\ge 1.$$

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [26] (see also Kirk [27]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that if C is a nonempty closed convex and bounded subset of a real Hilbert space, then every asymptotically nonexpansive self mapping in the intermediate sense has a fixed point; see [28] more details.

T is said to be generalized asymptotically nonexpansive if it is continuous and there exists a positive sequence {h _n } ⊂ [1, ∞) with h _n → 1 as n → ∞ such that

$$\underset{n\to \infty }{lim\; sup}\underset{x,y\in C}{sup}\left(\parallel T^{n}x-T^{n}y\parallel -h_n\parallel x-y\parallel \right)\le 0.$$

(2.2)

Putting ξ _n = max{0, sup_x,y∈C(||Tⁿx - Tⁿy|| - h _n ||x - y||)}, we see that ξ _n → 0 as n → ∞. Then, (2.2) is reduced to the following:

$$\parallel T^{n}x-T^{n}y\parallel \le h_n\parallel x-y\parallel +{\xi }_n,\forall x,y\in C,n\ge 1.$$

We remark that if h _n ≡ 1, then the class of generalized asymptotically nonexpansive mappings is reduced to the class of asymptotically nonexpansive mappings in the intermediate.

In 2006, Chang et al. [6] considered the following implicit iterative algorithms for a finite family of asymptotically nonexpansive mappings {T₁, T₂,..., T _N } with {α _n } a real sequence in (0, 1), {u _n } a bounded sequence in C and an initial point x₀ ∈ C:

$$\begin{array}{ll}\hfill x_1& ={\alpha }_1{x}_0+\left(1-{\alpha }_1\right)T_1{x}_1+u_1,\\ \hfill x_2& ={\alpha }_2{x}_1+\left(1-{\alpha }_2\right)T_2{x}_2+u_2,\\ \cdots \\ \hfill x_N& ={\alpha }_N{x}_{N-1}+\left(1-{\alpha }_N\right)T_N{x}_N+u_N,\\ \hfill x_{N+1}& ={\alpha }_{N+1}{x}_N+\left(1-{\alpha }_{N+1}\right)T_1^n{x}_{N+1}+u_{N+1},\\ \cdots \\ \hfill x_{2N}& ={\alpha }_{2N}{x}_{2N-1}+\left(1-{\alpha }_{2N}\right)T_N^2{x}_{2N}+u_{2N},\\ \hfill x_{2N+1}& ={\alpha }_{2N+1}{x}_{2N}+\left(1-{\alpha }_{2N+1}\right)T_1^3{x}_{2N+1}+u_{2N+1},\\ \cdots .\end{array}$$

The above table can be rewritten in the following compact form:

$$x_n={\alpha }_n{x}_{n-1}+\left(1-{\alpha }_n\right)T_{i\left(n\right)}^{j\left(n\right)}{x}_n+u_n,\forall n\ge 1,$$

where for each n ≥ 1 fixed, j(n) - 1 denotes the quotient of the division of n by N and i(n) the rest, i.e., n = (j(n) - 1)N + i(n).

Based on the implicit iterative algorithm, they obtained, under the assumption that C + C ⊂ C, weak and strong convergence theorems of common fixed points for a finite family of asymptotically nonexpansive mappings {T₁, T₂,..., T _N }; see [6] for more details.

Recently, Cianciaruso et al. [9] considered a Ishikawa-like iterative algorithm for the class of asymptotically nonexpansive mappings in a Banach space. To be more precise, they introduced and studied the following implicit iterative algorithm with errors.

$$\left\{\begin{array}{c}\hfill y_n=\left(1-{\beta }_n-{\delta }_n\right)x_n+{\beta }_n{T}_{i\left(n\right)}^{j\left(n\right)}{x}_n+{\delta }_n{v}_n,\hfill \\ \hfill x_n=\left(1-{\alpha }_n-{\gamma }_n\right)x_{n-1}+{\alpha }_n{T}_{i\left(n\right)}^{j\left(n\right)}{y}_n+{\gamma }_n{u}_n,\forall n\ge 1,\hfill \end{array}\right.$$

(2.3)

where {α _n }, {β _n }, {γ _n }, and {δ _n } are real number sequences in [0,1], {u _n } and {v _n } are bounded sequence in C. Weak and strong convergence theorems are established in a uniformly convex Banach space; see [29] for more details.

In this paper, motivated and inspired by the results announced in Chang et al. [6], Chidume and Shahzad [7], Cianciaruso et al. [9], Guo and Cho [10], Plubtieng et al. [14], Qin et al. [15], Shzhzad and Zegeye [18], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], Zhou and Chang [24], we consider the following Ishikawa-like implicit iteration algorithm with errors for two finite families of generalized asymptotically nonexpansive mappings {T₁, T₂,..., T _N } and {S₁, S₂,..., S _N }.

$$\begin{array}{c}{x}_0\in C\mathrm{,}\\ x_1={\alpha }_1{x}_0+{\beta }_1{T}_1({{\alpha }^{\prime }}_1{x}_1+{{\beta }^{\prime }}_1{S}_1{x}_1+{{\gamma }^{\prime }}_1{v}_1)+{\gamma }_1{u}_1\mathrm{,}\\ x_2={\alpha }_2{x}_1+{\beta }_2{T}_2({{\alpha }^{\prime }}_2{x}_2+{{\beta }^{\prime }}_2{S}_2{x}_2+{{\gamma }^{\prime }}_2{v}_2)+{\gamma }_2{u}_2\mathrm{,}\\ \cdots \\ x_N={\alpha }_N{x}_{N-1}+{\beta }_N{T}_N({{\alpha }^{\prime }}_N{x}_N+{{\beta }^{\prime }}_N{S}_N{x}_N+{{\gamma }^{\prime }}_N{v}_N)+{\gamma }_N{u}_N\mathrm{,}\\ x_{N+1}={\alpha }_{N+1}{x}_N+{\beta }_{N+1}{T}_{N+1}({{\alpha }^{\prime }}_{N+1}{x}_{N+1}+{{\beta }^{\prime }}_{N+1}{S}_{N+1}{x}_{N+1}\\ +{{\gamma }^{\prime }}_{N+1}{v}_{N+1})+{\gamma }_{N+1}{u}_{N+1}\mathrm{,}\\ \cdots \\ x_{2N}={\alpha }_{2N}{x}_{2N-1}+{\beta }_{2N}{T}_{2N}({{\alpha }^{\prime }}_{2N}{x}_{2N}+{{\beta }^{\prime }}_{2N}{S}_{2N}{x}_{2N}+{{\gamma }^{\prime }}_{2N}{v}_{2N})\\ +{\gamma }_{2N}{u}_{2N}\mathrm{,}\\ x_{2N+1}={\alpha }_{2N+1}{x}_{2N}+{\beta }_{2N+1}{T}_{2N+1}({{\alpha }^{\prime }}_{2N+1}{x}_{2N+1}+{{\beta }^{\prime }}_{2N+1}{S}_{2N+1}{x}_{2N+1}\\ +{{\gamma }^{\prime }}_{2N+1}{v}_{2N+1})+{\gamma }_{2N+1}{u}_{2N+1}\mathrm{,}\\ \cdots \mathrm{,}\end{array}$$

where {α _n }, {β _n }, {γ _n }, $\left\{{\alpha }_n^{\prime }\right\}$, $\left\{{\beta }_n^{\prime }\right\}$, and $\left\{{\gamma }_n^{\prime }\right\}$ are sequences in [0,1] such that ${\alpha }_n+{\beta }_n+{\gamma }_n={\alpha }_n^{\prime }+{\beta }_n^{\prime }+{\gamma }_n^{\prime }=1$ for each n ≥ 1. We have rewritten the above table in the following compact form:

$$x_n={\alpha }_n{x}_{n-1}+{\beta }_n{T}_{i\left(n\right)}^{j\left(n\right)}\left({\alpha }_n^{\prime }{x}_n+{\beta }_n^{\prime }{S}_{i\left(n\right)}^{j\left(n\right)}{x}_n+{\gamma }_n^{\prime }{v}_n\right)+{\gamma }_n{u}_n,n\ge 1,$$

where for each n ≥ 1 fixed, j(n) - 1 denotes the quotient of the division of n by N and i(n) the rest, i.e., n = (j(n) - 1)N + i(n).

Putting $y_n={\alpha }_n^{\prime }{x}_n+{\beta }_n^{\prime }{S}_n{x}_n+{\gamma }_n^{\prime }{v}_n$, we have the following composite iterative algorithm:

$$\left\{\begin{array}{c}\hfill y_n={{\alpha }^{\prime }}_n{x}_n+{{\beta }^{\prime }}_n{S}_{i\left(n\right)}^{j\left(n\right)}{x}_n+{{\gamma }^{\prime }}_n{v}_n,\hfill \\ \hfill x_n={\alpha }_n{x}_{n-1}+{\beta }_n{T}_{i\left(n\right)}^{j\left(n\right)}{y}_n+{\gamma }_n{u}_n,n\ge 1.\hfill \end{array}\right.$$

(2.4)

We remark that the implicit iterative algorithm (2.4) is general which includes (2.3) as a special case.

Now, we show that (2.4) can be employed to approximate fixed points of generalized asymptotically nonexpansive mappings which is assumed to be Lipschitz continuous. Let T _i be a $L_t^i$-Lipschitz generalized asymptotically nonexpansive mapping with a sequence $\left\{h_n^i\right\}\subset \left[1,\infty \right)$ such that $h_n^i\to 1$ as n → ∞ and S _i be a $L_s^i$-Lipschitz generalized asymptotically nonexpansive mapping with sequences $\left\{k_n^i\right\}\subset \left[1,\infty \right)$ such that $k_n^i\to 1$ as n → ∞ for each 1 ≤ i ≤ N. Define a mapping W _n : C → C by

$$W_n\left(x\right)={\alpha }_n{x}_{n-1}+{\beta }_n{T}_{i\left(n\right)}^{j\left(n\right)}\left({\alpha }_n^{\prime }x+{\beta }_n^{\prime }{S}_{i\left(n\right)}^{j\left(n\right)}x+{\gamma }_n^{\prime }{v}_n\right)+{\gamma }_n{u}_n,\forall n\ge 1.$$

It follows that

$$\begin{array}{c}\parallel W_n\left(x\right)-W_n\left(y\right)\parallel \\ \le {\beta }_n\parallel T_{i\left(n\right)}^{j\left(n\right)}\left({{\alpha }^{\prime }}_{n}x+{{\beta }^{\prime }}_n{S}_{i\left(n\right)}^{j\left(n\right)}x+{{\gamma }^{\prime }}_n{v}_n\right)-T_{i\left(n\right)}^{j\left(n\right)}\left({{\alpha }^{\prime }}_{n}y+{{\beta }^{\prime }}_n{S}_{i\left(n\right)}^{j\left(n\right)}y+{{\gamma }^{\prime }}_n{v}_n\right)\parallel \\ \le {\beta }_{n}L\left({{\alpha }^{\prime }}_n\parallel x-y\parallel +{{\beta }^{\prime }}_n\parallel S_{i\left(n\right)}^{j\left(n\right)}x-S_{i\left(n\right)}^{j\left(n\right)}y\parallel \right)\\ \le {\beta }_{n}L\left({{\alpha }^{\prime }}_n+{{\beta }^{\prime }}_{n}L\right)\parallel x-y\parallel ,\forall x,y\in C,\end{array}$$

where

$$L=max\left\{L_t^1,\dots ,L_t^N,L_s^1,\dots L_s^N\right\}.$$

(2.5)

If ${\beta }_{n}L\left({\alpha }_n^{\prime }+{\beta }_n^{\prime }L\right)<1$ for all n ≥ 1, then W _n is a contraction. By Banach contraction mapping principal, we see that there exists a unique fixed point x _n ∈ C such that

$$\begin{array}{ll}\hfill x_n=W_n\left(x_n\right)& ={\alpha }_n{x}_{n-1}+{\beta }_n{T}_{i\left(n\right)}^{j\left(n\right)}\left({{\alpha }^{\prime }}_{n}x+{{\beta }^{\prime }}_n{S}_{i\left(n\right)}^{j\left(n\right)}x+{{\gamma }^{\prime }}_n{v}_n\right)\\ +{\gamma }_n{u}_n,\forall n\ge 1.\end{array}$$

That is, the implicit iterative algorithm (2.4) is well defined.

The purpose of this paper is to establish strong and weak convergence theorem of fixed points of generalized asymptotically nonexpansive mappings based on (2.4).

Next, we recall some well-known concepts.

Let E be a real Banach space and U _E = {x ∈ E : ||x|| = 1}. E is said to be uniformly convex if for any ε ∈ (0, 2] there exists δ > 0 such that for any x, y ∈ U _E ,

$$\parallel x-y\parallel \ge \epsilon \text{implies}∥\frac{x+y}{2}∥\le 1-\delta .$$

It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall that E is said to satisfy Opial's condition[30] if for each sequence {x _n } in E, the condition that the sequence x _n → x weakly implies that

$$\underset{n\to \infty }{lim\; inf}\parallel x_n-x\parallel <\underset{n\to \infty }{lim\; inf}\parallel x_n-y\parallel$$

for all y ∈ E and y ≠ x. It is well known [30] that each l^p (1 ≤ p < ∞) and Hilbert spaces satisfy Opial's condition. It is also known [29] that any separable Banach space can be equivalently renormed to that it satisfies Opial's condition.

Recall that a mapping T : C → C is said to be demiclosed at the origin if for each sequence {x _n } in C, the condition x _n → x₀ weakly and Tx _n → 0 strongly implies Tx₀ = 0. T is said to be semicompact if any bounded sequence {x _n } in C satisfying lim_n→∞||x _n - Tx _n || = 0 has a convergent subsequence.

In order to prove our main results, we also need the following lemmas.

Lemma 2.1. [20]Let {a _n }, {b _n } and {c _n } be three nonnegative sequences satisfying the following condition:

$$a_{n+1}\le \left(1+b_n\right)a_n+c_n,\forall n\ge n_0,$$

where n₀is some nonnegative integer. If${\sum }_{n=0}^{\infty }{c}_n<\infty$and${\sum }_{n=0}^{\infty }{b}_n<\infty$, then lim_n→∞a _n exists.

Lemma 2.2. [17]Let E be a uniformly convex Banach space and 0 < λ ≤ t _n ≤ η < 1 for all n ≥ 1. Suppose that {x _n } and {y _n } are sequences of E such that

$$\underset{n\to \infty }{lim\; sup}\parallel x_n\parallel \le r,\underset{n\to \infty }{lim\; sup}\parallel y_n\parallel \le r$$

and

$$\underset{n\to \infty }{lim}\parallel t_n{x}_n+\left(1-t_n\right)y_n\parallel =r$$

hold for some r ≥ 0. Then lim_n→∞||x _n - y _n || = 0.

The following lemma can be obtained from Qin et al. [31] or Sahu et al. [32] immediately.

Lemma 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a Lipschitz generalized asymptotically nonexpansive mapping. Then I - T is demiclosed at origin.

3 Main results

Now, we are ready to give our main results in this paper.

Theorem 3.1. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T _i : C → C be a uniformly$L_t^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{h_n^i\right\}\subset \left[1,\infty \right)$, where$h_n^i\to 1$as n → ∞ and S _i : C → C be a uniformly$L_s^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{k_n^i\right\}\subset \left[1,\infty \right)$, where$k_n^i\to 1$as n → ∞ for each 1 ≤ i ≤ N. Assume that$\mathcal{F}={\bigcap }_{i=1}^{N}F\left(T_i\right)\bigcap {\bigcap }_{i=1}^{N}F\left(S_i\right)\ne \varnothing$. Let {u _n }, {v _n } be bounded sequences in C and e _n = max{h _n , k _n }, where$h_n=sup\left\{h_n^i:1\le i\le N\right\}$and$k_n=sup\left\{k_n^i:1\le i\le N\right\}$. Let {α _n }, {β _n }, {γ _n }, $\left\{{\alpha }_n^{\prime }\right\}$, $\left\{{\beta }_n^{\prime }\right\}$and$\left\{{\gamma }_n^{\prime }\right\}$be sequences in [0,1] such that${\alpha }_n+{\beta }_n+{\gamma }_n={\alpha }_n^{\prime }+{\beta }_n^{\prime }+{\gamma }_n^{\prime }=1$for each n ≥ 1. Let {x _n } be a sequence generated in (2.4). Put${\mu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel T_i^{n}x-T_i^{n}y\parallel -h_n^i\parallel x-y\parallel \right)\right\}$and${\nu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel S_i^{n}x-S_i^{n}y\parallel -k_n^i\parallel x-y\parallel \right)\right\}$. Let ξ _n = max{μ _n , ν _n }, where${\mu }_n=max\left\{{\mu }_n^i:1\le i\le N\right\}$and${\nu }_n=max\left\{{\nu }_n^i:1\le i\le N\right\}$. Assume that the following restrictions are satisfied:

• (a) ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$ and ${\sum }_{n=1}^{\infty }{{\gamma }^{\prime }}_n<\infty$;

• (b) ${\sum }_{n=1}^{\infty }\left(e_n-1\right)<\infty$ and ${\sum }_{n=1}^{\infty }{\xi }_n<\infty$;

• (c) ${\beta }_{n}L\left({\alpha }_n^{\prime }+{\beta }_n^{\prime }L\right)<1$, where L is defined in (2.5);

• (d) there exist constants λ, η ∈ (0, 1) such that λ ≤ α _n , ${\alpha }_n^{\prime }\le \eta$.

Then

$$\underset{n\to \infty }{lim}\parallel x_n-T_r{x}_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-S_r{x}_n\parallel =0,\forall r\in \left\{1,2,\dots ,N\right\}.$$

Proof. Fixing $f\in \mathcal{F}$, we see that

$$\begin{array}{ll}\hfill ∥y_n-f∥& =\parallel {{\alpha }^{\prime }}_n{x}_n+{{\beta }^{\prime }}_n{S}_{i\left(n\right)}^{j\left(n\right)}{x}_n+{{\gamma }^{\prime }}_n{v}_n-f\parallel \\ \le {{\alpha }^{\prime }}_n\parallel x_n-f\parallel +{{\beta }^{\prime }}_n\parallel S_{i\left(n\right)}^{j\left(n\right)}{x}_n-f\parallel +{{\gamma }^{\prime }}_n\parallel v_n-f\parallel \\ \le {{\alpha }^{\prime }}_n\parallel x_n-f\parallel +{{\beta }^{\prime }}_n{e}_{j\left(n\right)}\parallel x_n-f\parallel +{{\beta }^{\prime }}_n{\xi }_{j\left(n\right)}+{{\gamma }^{\prime }}_n\parallel v_n-f\parallel \\ \le e_{j\left(n\right)}\parallel x_n-f\parallel +{{\beta }^{\prime }}_n{\xi }_{j\left(n\right)}+{{\gamma }^{\prime }}_n\parallel v_n-f\parallel \end{array}$$

(3.1)

and

$$\begin{array}{c}\parallel x_n-f\parallel \\ =\parallel {\alpha }_n{x}_{n-1}+{\beta }_n{T}_{i\left(n\right)}^{j\left(n\right)}{y}_n+{\gamma }_n{u}_n-f\parallel \\ \le {\alpha }_n\parallel x_{n-1}-f\parallel +{\beta }_n\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-f\parallel +{\gamma }_n\parallel u_n-f\parallel \\ \le {\alpha }_n\parallel x_{n-1}-f\parallel +{\beta }_n{e}_{j\left(n\right)}\parallel y_n-f\parallel +{\beta }_n{\xi }_{j\left(n\right)}+{\gamma }_n\parallel u_n-f\parallel .\\ \le {\alpha }_n\parallel x_{n-1}-f\parallel +\left(1-{\alpha }_n\right)e_{j\left(n\right)}\parallel y_n-f\parallel +{\beta }_n{\xi }_{j\left(n\right)}+{\gamma }_n\parallel u_n-f\parallel .\end{array}$$

(3.2)

Substituting (3.1) into (3.2), we see that

$$\begin{array}{c}\parallel x_n-f\parallel \\ \le {\alpha }_n\parallel x_{n-1}-f\parallel +\left(1-{\alpha }_n\right)e_{j\left(n\right)}\left(e_{j\left(n\right)}\parallel x_n-f\parallel +{{\beta }^{\prime }}_n{\xi }_{j\left(n\right)}+{{\gamma }^{\prime }}_n\parallel v_n-f\parallel \right)\\ +{\beta }_n{\xi }_{j\left(n\right)}+{\gamma }_n\parallel u_n-f\parallel .\\ \le {\alpha }_n\parallel x_{n-1}-f\parallel +\left(1-{\alpha }_n\right)e_{j\left(n\right)}^2\parallel x_n-f\parallel +\left(1+e_{j\left(n\right)}\right){\xi }_{j\left(n\right)}\\ +e_{j\left(n\right)}{{\gamma }^{\prime }}_n\parallel v_n-f\parallel +{\gamma }_n\parallel u_n-f\parallel .\end{array}$$

Notice that ${\sum }_{n=1}^{\infty }\left(e_n-1\right)<\infty$. We see from the restrictions (b) and (d) that there exists a positive integer n₀ such that

$$\left(1-{\alpha }_n\right)e_{j\left(n\right)}^2\le R<1,\forall n\ge n_0,$$

where $R=\left(1-\lambda \right)\left(1+\frac{\lambda }{2-2\lambda }\right)$. It follows that

$$\begin{array}{ll}\hfill ∥x_n-f∥& \le \left(1+\frac{\left(1-{\alpha }_n\right)\left(e_{j\left(n\right)}^2-1\right)}{1-\left(1-{\alpha }_n\right)e_{j\left(n\right)}^2}\right)\parallel x_{n-1}-f\parallel \\ +\frac{\left(1+e_{j\left(n\right)}\right){\xi }_{j\left(n\right)}+e_{j\left(n\right)}{{\gamma }^{\prime }}_n\parallel v_n-f\parallel +{\gamma }_n\parallel u_n-f\parallel }{1-\left(1-{\alpha }_n\right)e_{j\left(n\right)}^2}\\ \le \left(1+\frac{\left(1+M_1\right)\left(e_{j\left(n\right)}-1\right)}{1-R}\right)\parallel x_{n-1}-f\parallel \\ +\frac{\left(1+M_1\right){\xi }_{j\left(n\right)}+M_1{M}_2{{\gamma }^{\prime }}_n+M_3{\gamma }_n}{1-R},\end{array}$$

(3.3)

where M₁ = sup_n≥1{e _n }, M₂ = sup_n≥1{||v _n - f||}, and M₃ = sup_n≥1{||u _n - f||}. In view of Lemma 2.1, we see that lim_n→∞||x _n - f|| exists for each $f\in \mathcal{F}$. This implies that the sequence {x _n } is bounded. Next, we assume that lim_n→∞||x _n - f|| = d > 0. From (3.1), we see that

$$\begin{array}{c}\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-f+{\gamma }_n\left(u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\right)\parallel \\ \le \parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-f\parallel +{\gamma }_n\parallel u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\parallel \\ \le e_{j\left(n\right)}\parallel y_n-f\parallel +{\xi }_{j\left(n\right)}+{\gamma }_n\parallel u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\parallel \\ \le e_{j\left(n\right)}^2\parallel x_n-f\parallel +e_{j\left(n\right)}{\xi }_{j\left(n\right)}+e_{j\left(n\right)}{{\gamma }^{\prime }}_n\parallel v_n-f\parallel +{\xi }_{j\left(n\right)}\\ +{\gamma }_n\parallel u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\parallel .\end{array}$$

This implies from the restrictions (a) and (b) that

$$\underset{n\to \infty }{lim\; sup}\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-f+{\gamma }_n\left(u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\right)\parallel \le d.$$

Notice that

$$\parallel x_{n-1}-f+{\gamma }_n\left(u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\right)\parallel \le \parallel x_{n-1}-f\parallel +{\gamma }_n\parallel u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\parallel .$$

This shows from the restriction (a) that

$$\underset{n\to \infty }{lim\; sup}\parallel x_{n-1}-f+{\gamma }_n\left(u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\right)\parallel \le d.$$

On the other hand, we have

$$\begin{array}{ll}\hfill d& =\underset{n\to \infty }{lim}\parallel x_n-f\parallel \\ =\underset{n\to \infty }{lim}\parallel {\alpha }_n\left(x_{n-1}-f+{\gamma }_n\left(u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\right)\right)\\ +\left(1-{\alpha }_n\right)\left(T_{i\left(n\right)}^{j\left(n\right)}{y}_n-f+{\gamma }_n\left(u_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\right)\right)\parallel .\end{array}$$

It follows from Lemma 2.2 that

$$\underset{n\to \infty }{lim}\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-x_{n-1}\parallel =0.$$

(3.4)

Notice that

$$\parallel x_n-x_{n-1}\parallel \le {\beta }_n\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-x_{n-1}\parallel +{\gamma }_n\parallel u_n-x_{n-1}\parallel .$$

It follows from (3.4) and the restriction (a) that

$$\underset{n\to \infty }{lim}\parallel x_n-x_{n-1}\parallel =0.$$

(3.5)

This implies that

$$\underset{n\to \infty }{lim}\parallel x_n-x_{n+l}\parallel =0,\forall l=1,2,\dots ,N.$$

(3.6)

Notice that

$$\parallel x_n-f+{\gamma }_n^{\prime }\left(v_n-S_{i\left(n\right)}^{j\left(n\right)}{x}_n\right)\parallel \le \parallel x_n-f\parallel +{\gamma }_n^{\prime }\parallel v_n-S_{i\left(n\right)}^{j\left(n\right)}{x}_n\parallel$$

and

$$\begin{array}{c}\parallel S_{i\left(n\right)}^{j\left(n\right)}{x}_n-f+{{\gamma }^{\prime }}_n\left(v_n-S_{i\left(n\right)}^{j\left(n\right)}{x}_n\right)\parallel \\ \le \parallel S_{i\left(n\right)}^{j\left(n\right)}{x}_n-f\parallel +{{\gamma }^{\prime }}_n\parallel v_n-S_{i\left(n\right)}^{j\left(n\right)}{x}_n\parallel \\ \le e_{j\left(n\right)}\parallel x_n-f\parallel +{\xi }_{j\left(n\right)}+{{\gamma }^{\prime }}_n\parallel v_n-S_{i\left(n\right)}^{j\left(n\right)}{x}_n\parallel .\end{array}$$

which in turn imply that

$$\underset{n\to \infty }{lim\; sup}\parallel x_n-f+{\gamma }_n^{\prime }\left(v_n-S_{i\left(n\right)}^{j\left(n\right)}{x}_n\right)\parallel \le d$$

and

$$\underset{n\to \infty }{lim\; sup}\parallel S_{i\left(n\right)}^{j\left(n\right)}{x}_n-f+{\gamma }_n^{\prime }\left(v_n-S_{i\left(n\right)}^{j\left(n\right)}{x}_n\right)\parallel \le d.$$

On the other hand, we have

$$\begin{array}{ll}\hfill \parallel x_n-f\parallel & =\parallel {\alpha }_n{x}_{n-1}+{\beta }_n{T}_{i\left(n\right)}^{j\left(n\right)}{y}_n+{\gamma }_n{u}_n-f\parallel \\ \le {\alpha }_n\parallel x_{n-1}-f\parallel +{\beta }_n\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-f\parallel +{\gamma }_n\parallel u_n-f\parallel \\ \le {\alpha }_n\parallel x_{n-1}-T_{i\left(n\right)}^{k\left(n\right)}{y}_n\parallel +\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-f\parallel +{\gamma }_n\parallel u_n-f\parallel \\ \le {\alpha }_n\parallel x_{n-1}-T_{i\left(n\right)}^{k\left(n\right)}{y}_n\parallel +e_{j\left(n\right)}\parallel y_n-f\parallel +{\xi }_{j\left(n\right)}+{\gamma }_n\parallel u_n-f\parallel ,\end{array}$$

from which it follows that lim inf_n→∞||y _n - f|| ≥ d. In view of (3.1), we see that lim sup_n→∞||y _n - f|| ≤ d. This proves that

$$\underset{n\to \infty }{lim}\parallel y_n-f\parallel =d.$$

Notice that

$$\begin{array}{ll}\hfill \underset{n\to \infty }{lim}\parallel y_n-f\parallel & =\underset{n\to \infty }{lim}\parallel {{\alpha }^{\prime }}_n\left(x_n-f+{{\gamma }^{\prime }}_n\left(v_n-S_{i\left(n\right)}^{j\left(n\right)}{x}_n\right)\right)\\ +\left(1-{{\alpha }^{\prime }}_n\right)\left(S_{i\left(n\right)}^{j\left(n\right)}{x}_n-f+{{\gamma }^{\prime }}_n\left(v_n-S_{i\left(n\right)}^{j\left(n\right)}{x}_n\right)\right)\parallel .\end{array}$$

This implies from Lemma 2.2 that

$$\underset{n\to \infty }{lim}\parallel S_{i\left(n\right)}^{j\left(n\right)}{x}_n-x_n\parallel =0.$$

(3.7)

On the other hand, we have

$$\begin{array}{c}\parallel T_{i\left(n\right)}^{j\left(n\right)}{x}_n-x_n\parallel \\ \le \parallel T_{i\left(n\right)}^{j\left(n\right)}{x}_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\parallel +\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-x_{n-1}\parallel +\parallel x_{n-1}-x_n\parallel \\ \le \parallel T_{i\left(n\right)}^{j\left(n\right)}{x}_n-T_{i\left(n\right)}^{j\left(n\right)}{y}_n\parallel +\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-x_{n-1}\parallel +\parallel x_{n-1}-x_n\parallel \\ \le L\parallel x_n-y_n\parallel +\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-x_{n-1}\parallel +\parallel x_{n-1}-x_n\parallel \\ \le L{{\beta }^{\prime }}_n\parallel S_{i\left(n\right)}^{j\left(n\right)}{x}_n-x_n\parallel +L{{\gamma }^{\prime }}_n\parallel v_n-x_n\parallel +\parallel T_{i\left(n\right)}^{j\left(n\right)}{y}_n-x_{n-1}\parallel \\ +\parallel x_{n-1}-x_n\parallel .\end{array}$$

This combines with (3.4), (3.5), and (3.7) gives that

$$\underset{n\to \infty }{lim}\parallel T_{i\left(n\right)}^{j\left(n\right)}{x}_n-x_n\parallel =0.$$

(3.8)

Since n = (j(n) - 1)N + i(n), where i(n) ∈ {1, 2,..., N}, we see that

$$\begin{array}{c}\parallel x_n-S_{i(n)}{x}_n\parallel \le \parallel x_n-S_{i(n)}^{j(n)}{x}_n\parallel +\parallel S_{i(n)}^{j(n)}{x}_n-S_{i(n)}{x}_n\parallel \\ \le \parallel x_n-S_{i(n)}^{j(n)}{x}_n\parallel +L\parallel S_{i(n)}^{j(n)-1}{x}_n-x_n\parallel \\ \le \parallel x_n-S_{i(n)}^{j(n)}{x}_n\parallel +L(\parallel S_{i(n)}^{j(n)-1}{x}_n-S_{i(n-N)}^{j(n)-1}{x}_{n-N}\parallel \\ +\parallel S_{i(n-N)}^{j(n)-1}{x}_{n-N}-x_{n-N}\parallel +\parallel x_{n-N}-x_n\parallel ).\end{array}$$

(3.9)

Notice that

$$j\left(n-N\right)=j\left(n\right)-1andi\left(n-N\right)=i\left(n\right).$$

This in turn implies that

$$\begin{array}{ll}\hfill \parallel S_{i\left(n\right)}^{j\left(n\right)-1}{x}_n-S_{i\left(n-N\right)}^{j\left(n\right)-1}{x}_{n-N}\parallel & =\parallel S_{i\left(n\right)}^{j\left(n\right)-1}{x}_n-S_{i\left(n\right)}^{j\left(n\right)-1}{x}_{n-N}\parallel \\ \le L\parallel x_n-x_{n-N}\parallel \end{array}$$

(3.10)

and

$$\parallel S_{i\left(n-N\right)}^{j\left(n\right)-1}{x}_{n-N}-x_{n-N}\parallel =\parallel S_{i\left(n-N\right)}^{j\left(n-N\right)}{x}_{n-N}-x_{n-N}\parallel .$$

(3.11)

Substituting (3.10) and (3.11) into (3.9) yields that

$$\begin{array}{c}\parallel x_n-S_{i(n)}{x}_n\parallel \le \parallel x_n-S_{i(n)}^{j(n)}{x}_n\parallel +L(L\parallel x_n-x_{n-N}\parallel \\ +\parallel S_{i(n-N)}^{j(n-N)}{x}_{n-N}-x_{n-N}\parallel ).\end{array}$$

It follows from (3.6) and (3.7) that

$$\underset{n\to \infty }{lim}\parallel x_n-S_{i\left(n\right)}{x}_n\parallel =0.$$

(3.12)

In particular, we see that

$$\left\{\begin{array}{c}\hfill {lim}_{j\to \infty }\parallel x_{jN+1}-S_1{x}_{jN+1}\parallel =0,\hfill \\ \hfill {lim}_{j\to \infty }\parallel x_{jN+2}-S_2{x}_{jN+2}\parallel =0,\hfill \\ \hfill ⋮\hfill \\ \hfill {lim}_{j\to \infty }\parallel x_{jN+N}-S_N{x}_{jN+N}\parallel =0.\hfill \end{array}\right.$$

For any r, s = 1, 2,..., N, we obtain that

$$\begin{array}{c}\parallel x_{jN+s}-S_r{x}_{jN+s}\parallel \\ \le \parallel x_{jN+s}-x_{jN+r}\parallel +\parallel x_{jN+r}-S_r{x}_{jN+r}\parallel +\parallel S_r{x}_{jN+r}-S_r{x}_{jN+s}\parallel \\ \le \left(1+L\right)\parallel x_{jN+s}-x_{jN+r}\parallel +\parallel x_{jN+r}-S_r{x}_{jN+r}\parallel .\end{array}$$

Letting j → ∞, we arrive at

$$\underset{j\to \infty }{lim}\parallel x_{jN+s}-S_r{x}_{jN+s}\parallel =0,$$

which is equivalent to

$$\underset{n\to \infty }{lim}\parallel x_n-S_r{x}_n\parallel =0.$$

(3.13)

Notice that

$$\begin{array}{c}\parallel x_n-T_{i(n)}{x}_n\parallel \le \parallel x_n-T_{i(n)}^{j(n)}{x}_n\parallel +\parallel T_{i(n)}^{j(n)}{x}_n-T_{i(n)}{x}_n\parallel \\ \le \parallel x_n-T_{i(n)}^{j(n)}{x}_n\parallel +L\parallel T_{i(n)}^{j(n)-1}{x}_n-x_n\parallel \\ \le \parallel x_n-T_{i(n)}^{j(n)}{x}_n\parallel +L(\parallel T_{i(n)}^{j(n)-1}{x}_n-T_{i(n-N)}^{j(n)-1}{x}_{n-N}\parallel \\ +\parallel T_{i(n-N)}^{j(n)-1}{x}_{n-N}-x_{n-N}\parallel +\parallel x_{n-N}-x_n\parallel ).\end{array}$$

(3.14)

On the other hand, we have

$$\begin{array}{ll}\hfill \parallel T_{i\left(n\right)}^{j\left(n\right)-1}{x}_n-T_{i\left(n-N\right)}^{j\left(n\right)-1}{x}_{n-N}\parallel & =\parallel T_{i\left(n\right)}^{j\left(n\right)-1}{x}_n-T_{i\left(n\right)}^{j\left(n\right)-1}{x}_{n-N}\parallel \\ \le L\parallel x_n-x_{n-N}\parallel \end{array}$$

(3.15)

and

$$\parallel T_{i\left(n-N\right)}^{j\left(n\right)-1}{x}_{n-N}-x_{n-N}\parallel =\parallel T_{i\left(n-N\right)}^{j\left(n-N\right)}{x}_{n-N}-x_{n-N}\parallel .$$

(3.16)

Substituting (3.15) and (3.16) into (3.14) yields that

$$\begin{array}{c}\parallel x_n-T_{i(n)}{x}_n\parallel \le \parallel x_n-T_{i(n)}^{k(n)}{x}_n\parallel +L(L\parallel x_n-x_{n-N}\parallel \\ +\parallel T_{i(n-N)}^{k(n-N)}{x}_{n-N}-x_{n-N}\parallel ).\end{array}$$

It follows from (3.6) and (3.8) that

$$\underset{n\to \infty }{lim}\parallel x_n-T_{i\left(n\right)}{x}_n\parallel =0.$$

(3.17)

In particular, we see that

$$\left\{\begin{array}{c}\hfill {lim}_{k\to \infty }\parallel x_{jN+1}-T_1{x}_{jN+1}\parallel =0,\hfill \\ \hfill {lim}_{k\to \infty }\parallel x_{jN+2}-T_2{x}_{jN+2}\parallel =0,\hfill \\ \hfill ⋮\hfill \\ \hfill {lim}_{k\to \infty }\parallel x_{jN+N}-T_N{x}_{jN+N}\parallel =0.\hfill \end{array}\right.$$

For any r, s = 1, 2,..., N, we obtain that

$$\begin{array}{c}\parallel x_{jN+s}-T_r{x}_{jN+s}\parallel \\ \le \parallel x_{jN+s}-x_{jN+r}\parallel +\parallel x_{jN+r}-T_r{x}_{jN+r}\parallel +\parallel T_r{x}_{jN+r}-T_r{x}_{jN+s}\parallel \\ \le \left(1+L\right)\parallel x_{jN+s}-x_{jN+r}\parallel +\parallel x_{jN+r}-T_r{x}_{jN+r}\parallel .\end{array}$$

Letting j → ∞, we arrive

$$\underset{j\to \infty }{lim}\parallel x_{jN+s}-T_r{x}_{jN+s}\parallel =0,$$

which is equivalent to

$$\underset{n\to \infty }{lim}\parallel x_n-T_r{x}_n\parallel =0.$$

(3.18)

This completes the proof. □

Now, we are in a position to give weak convergence theorems.

Theorem 3.2. Let E be a real Hilbert space and C be a nonempty closed convex subset of E. Let T _i : C → C be a uniformly$L_t^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{h_n^i\right\}\subset \left[1,\infty \right)$, where$h_n^i\to 1$as n → ∞ and S _i : C → C be a uniformly$L_s^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{k_n^i\right\}\subset \left[1,\infty \right)$, where$k_n^i\to 1$as n → ∞ for each 1 ≤ i ≤ N. Assume that$\mathcal{F}={\bigcap }_{i=1}^{N}F\left(T_i\right)\bigcap {\bigcap }_{i=1}^{N}F\left(S_i\right)\ne \varnothing$. Let {u _n }, {v _n } be bounded sequences in C and e _n = max{h _n , k _n }, where$h_n=sup\left\{h_n^i:1\le i\le N\right\}$and$k_n=sup\left\{k_n^i:1\le i\le N\right\}$. Let {α _n }, {β _n }, {γ _n }, $\left\{{\alpha }_n^{\prime }\right\}$, $\left\{{\beta }_n^{\prime }\right\}$and$\left\{{\gamma }_n^{\prime }\right\}$be sequences in [0,1] such that${\alpha }_n+{\beta }_n+{\gamma }_n={\alpha }_n^{\prime }+{\beta }_n^{\prime }+{\gamma }_n^{\prime }=1$for each n ≥ 1. Let {x _n } be a sequence generated in (2.4). Put${\mu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel T_i^{n}x-T_i^{n}y\parallel -h_n^i\parallel x-y\parallel \right)\right\}$and${\nu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel S_i^{n}x-S_i^{n}y\parallel -k_n^i\parallel x-y\parallel \right)\right\}$. Let ξ _n = max{μ _n , ν _n }, where${\mu }_n=max\left\{{\mu }_n^i:1\le i\le N\right\}$and${\nu }_n=max\left\{{\nu }_n^i:1\le i\le N\right\}$. Assume that the following restrictions are satisfied:

• (a) ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$ and ${\sum }_{n=1}^{\infty }{{\gamma }^{\prime }}_n<\infty$;

• (b) ${\sum }_{n=1}^{\infty }\left(e_n-1\right)<\infty$ and ${\sum }_{n=1}^{\infty }{\xi }_n<\infty$;

• (c) ${\beta }_{n}L\left({\alpha }_n^{\prime }+{\beta }_n^{\prime }L\right)<1$, where L is defined in (2.5);

• (d) there exist constants λ, η ∈ (0, 1) such that λ ≤ α _n , ${\alpha }_n^{\prime }\le \eta$.

Then the sequence {x _n } converges weakly to some point in$\mathcal{F}$.

Proof. Since E is a Hilbert space and {x _n } is bounded, we can obtain that there exists a subsequence $\left\{x_{n_p}\right\}$ of {x _n } converges weakly to x* ∈ C. It follows from (3.13) and (3.18) that

$$\underset{p\to \infty }{lim}\parallel x_{n_p}-T_r{x}_{n_p}\parallel =\underset{p\to \infty }{lim}\parallel x_{n_p}-S_r{x}_{n_p}\parallel =0.$$

Since I - S _r and I - T _r are demiclosed at origin by Lemma 2.3, we see that $x^{*}\in \mathcal{F}$. Next, we show that the whole sequence {x _n } converges weakly to x*. Suppose the contrary. Then there exists some subsequence $\left\{x_{n_q}\right\}$ of {x _n } such that $\left\{x_{n_q}\right\}$ converges weakly to x** ∈ C. In the same way, we can show that $x^{**}\in \mathcal{F}$. Notice that we have proved that lim_n→∞||x _n - f|| exists for each $f\in \mathcal{F}$. By virtue of Opial's condition of E, we have

$$\begin{array}{ll}\hfill \underset{p\to \infty }{lim\; inf}\parallel x_{n_p}-x^{*}\parallel & <\underset{p\to \infty }{lim\; inf}\parallel x_{n_p}-x^{**}\parallel \\ =\underset{q\to \infty }{lim\; inf}\parallel x_{n_q}-x^{**}\parallel \\ <\underset{q\to \infty }{lim\; inf}\parallel x_{n_q}-x^{*}\parallel .\end{array}$$

This is a contradiction. Hence, x* = x**. This completes the proof. □

Remark 3.3. Theorem 3.2 which includes the corresponding results announced in Chang et al. [6], Chidume and Shahzad [7], Guo and Cho [10], Plubtieng et al. [14], Qin et al. [15], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], and Zhou and Chang [24] as special cases mainly improves the results of Cianciaruso et al. [9] in the following aspects.

• (1) Extend the mappings from one family of mappings to two families of mappings;

• (2) Extend the mappings from the class of asymptotically nonexpansive mappings to the class of generalized asymptotically nonexpansive mappings.

If S _r = I for each r ∈ {1, 2,..., N} and ${\gamma }_n^{\prime }=0$, then Theorem 3.2 is reduced to the following.

Corollary 3.4. Let E be a real Hilbert space and C be a nonempty closed convex subset of E. Let T _i : C → C be a uniformly$L_t^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{h_n^i\right\}\subset \left[1,\infty \right)$, where$h_n^i\to 1$as n → ∞ for each 1 ≤ i ≤ N. Assume that$\mathcal{F}={\bigcap }_{i=1}^{N}F\left(T_i\right)\ne \varnothing$. Let {u _n } be a bounded sequence in C and$h_n=sup\left\{h_n^i:1\le i\le N\right\}$. Let {α _n }, {β _n } and {γ _n } be sequences in [0,1] such that α _n + β _n + γ _n = 1 for each n ≥ 1. Let {x _n } be a sequence generated in the following process:

$$x_0\in C,x_n={\alpha }_n{x}_{n-1}+{\beta }_n{T}_{i\left(n\right)}^{j\left(n\right)}{x}_n+{\gamma }_n{u}_n,n\ge 1.$$

(3.19)

Put${\mu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel T_i^{n}x-T_i^{n}y\parallel -h_n^i\parallel x-y\parallel \right)\right\}$. Let${\mu }_n=max\left\{{\mu }_n^i:1\le i\le N\right\}$. Assume that the following restrictions are satisfied:

• (a) ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$;

• (b) ${\sum }_{n=1}^{\infty }\left(h_n-1\right)<\infty$ and ${\sum }_{n=1}^{\infty }{\mu }_n<\infty$;

• (c) β _n L < 1, where $L=max\left\{L_t^i:1\le i\le N\right\}$;

• (d) there exist constants λ, η ∈ (0, 1) such that λ ≥ α _n , ${\alpha }_n^{\prime }\le \eta$.

Then the sequence {x _n } converges weakly to some point in$\mathcal{F}$.

Next, we are in a position to state strong convergence theorems in a Banach space.

Theorem 3.5. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T _i : C → C be a uniformly$L_t^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{h_n^i\right\}\subset \left[1,\infty \right)$, where$h_n^i\to 1$as n → ∞ and S _i : C → C be a uniformly$L_s^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{k_n^i\right\}\subset \left[1,\infty \right)$, where$k_n^i\to 1$as n → ∞ for each 1 ≤ i ≤ N. Assume that$\mathcal{F}={\bigcap }_{i=1}^{N}F\left(T_i\right)\bigcap {\bigcap }_{i=1}^{N}F\left(S_i\right)\ne \varnothing$. Let {u _n }, {v _n } be bounded sequences in C and e _n = max{h _n , k _n }, where$h_n=sup\left\{h_n^i:1\le i\le N\right\}$and$k_n=sup\left\{k_n^i:1\le i\le N\right\}$. Let {α _n }, {β _n }, {γ _n }, $\left\{{\alpha }_n^{\prime }\right\}$, $\left\{{\beta }_n^{\prime }\right\}$and$\left\{{\gamma }_n^{\prime }\right\}$be sequences in [0,1] such that${\alpha }_n+{\beta }_n+{\gamma }_n={\alpha }_n^{\prime }+{\beta }_n^{\prime }+{\gamma }_n^{\prime }=1$for each n ≥ 1. Let {x _n } be a sequence generated in (2.4). Put${\mu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel T_i^{n}x-T_i^{n}y\parallel -h_n^i\parallel x-y\parallel \right)\right\}$and${\nu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel S_i^{n}x-S_i^{n}y\parallel -k_n^i\parallel x-y\parallel \right)\right\}$. Let ξ _n = max{μ _n , ν _n }, where${\mu }_n=max\left\{{\mu }_n^i:1\le i\le N\right\}$and${\nu }_n=max\left\{{\nu }_n^i:1\le i\le N\right\}$. Assume that the following restrictions are satisfied:

• (a) ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$ and ${\sum }_{n=1}^{\infty }{{\gamma }^{\prime }}_n<\infty$;

• (b) ${\sum }_{n=1}^{\infty }\left(e_n-1\right)<\infty$ and ${\sum }_{n=1}^{\infty }{\xi }_n<\infty$;

• (c) ${\beta }_{n}L\left({\alpha }_n^{\prime }+{\beta }_n^{\prime }L\right)<1$, where L is defined in (2.5);

• (d) there exist constants λ, η ∈ (0, 1) such that λ ≥ α_n, ${\alpha }_n^{\prime }\le \eta$.

If one of mappings in {T₁, T₂,..., T _N } or one of mappings in {S₁, S₂,..., S _N } are semicompact, then the sequence {x _n } converges strongly to some point in$\mathcal{F}$.

Proof. Without loss of generality, we may assume that S₁ are semicompact. It follows from (3.13) that

$$\underset{n\to \infty }{lim}\parallel x_n-S_1{x}_n\parallel =0.$$

By the semicompactness of S₁, we see that there exists a subsequence $\left\{x_{n_p}\right\}$ of {x _n } such that $x_{n_p}\to w\in C$ strongly. From (3.13) and (3.18), we have

$$\parallel w-S_{r}w\parallel \le \parallel w-x_{n_p}\parallel +\parallel x_{n_p}-S_r{x}_{n_p}\parallel +\parallel S_r{x}_{n_p}-S_{r}w\parallel$$

and

$$\parallel w-T_{r}w\parallel \le \parallel w-x_{n_p}\parallel +\parallel x_{n_p}-T_r{x}_{n_p}\parallel +\parallel T_r{x}_{n_p}-T_{r}w\parallel$$

Since S _r and T _r are Lipshcitz continuous, we obtain that $w\in \mathcal{F}$. From Theorem 3.1, we know that lim_n→∞||x _n - f|| exists for each $f\in \mathcal{F}$. That is, lim_n→∞||x _n - w|| exists. From $x_{n_p}\to w$, we have

$$\underset{n\to \infty }{lim}\parallel x_n-w\parallel =0.$$

This completes the proof of Theorem 3.5. □

If S _r = I for each r ∈ {1, 2,..., N} and ${\gamma }_n^{\prime }=0$, then Theorem 3.5 is reduced to the following.

Corollary 3.6. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T _i : C → C be a uniformly$L_t^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{h_n^i\right\}\subset \left[1,\infty \right)$, where$h_n^i\to 1$as n → ∞ for each 1 ≤ i ≤ N. Assume that$\mathcal{F}={\bigcap }_{i=1}^{N}F\left(T_i\right)\ne \varnothing$. Let {u _n } be a bounded sequence in C and$h_n=sup\left\{h_n^i:1\le i\le N\right\}$. Let {α _n }, {β _n } and {γ _n } be sequences in [0,1] such that α _n + β _n + γ _n = 1 for each n ≥ 1. Let {x _n } be a sequence generated in (3.19). Put${\mu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel T_i^{n}x-T_i^{n}y\parallel -h_n^i\parallel x-y\parallel \right)\right\}$. Let${\mu }_n=max\left\{{\mu }_n^i:1\le i\le N\right\}$. Assume that the following restrictions are satisfied:

• (a) ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$;

• (b) ${\sum }_{n=1}^{\infty }\left(h_n-1\right)<\infty$ and ${\sum }_{n=1}^{\infty }{\mu }_n<\infty$;

• (c) β _n L < 1, where $L=max\left\{L_t^i:1\le i\le N\right\}$;

• (d) there exist constants λ, η ∈ (0, 1) such that λ ≥ α_n, ${\alpha }_n^{\prime }\le \eta$.

If one of mappings in {T₁, T₂,..., T _N } is semicompact, then the sequence {x _n } converges strongly to some point in$\mathcal{F}$.

Theorem 3.7. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T _i : C → C be a uniformly$L_t^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{h_n^i\right\}\subset \left[1,\infty \right)$, where$h_n^i\to 1$as n → ∞ and S _i : C → C be a uniformly$L_s^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{k_n^i\right\}\subset \left[1,\infty \right)$, where$k_n^i\to 1$as n → ∞ for each 1 ≤ i ≤ N. Assume that$\mathcal{F}={\bigcap }_{i=1}^{N}F\left(T_i\right)\bigcap {\bigcap }_{i=1}^{N}F\left(S_i\right)\ne \varnothing$. Let {u _n }, {v _n } be bounded sequences in C and e _n = max{h _n , k _n }, where$h_n=sup\left\{h_n^i:1\le i\le N\right\}$and$k_n=sup\left\{k_n^i:1\le i\le N\right\}$. Let {α _n }, {β _n }, {γ _n }, $\left\{{\alpha }_n^{\prime }\right\}$, $\left\{{\beta }_n^{\prime }\right\}$and$\left\{{\gamma }_n^{\prime }\right\}$be sequences in [0,1] such that${\alpha }_n+{\beta }_n+{\gamma }_n={\alpha }_n^{\prime }+{\beta }_n^{\prime }+{\gamma }_n^{\prime }=1$for each n ≥ 1. Let {x _n } be a sequence generated in (2.4). Put${\mu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel T_i^{n}x-T_i^{n}y\parallel -h_n^i\parallel x-y\parallel \right)\right\}$and${\nu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel S_i^{n}x-S_i^{n}y\parallel -k_n^i\parallel x-y\parallel \right)\right\}$. Let ξ _n = max{μ _n , ν _n }, where${\mu }_n=max\left\{{\mu }_n^i:1\le i\le N\right\}$and${\nu }_n=max\left\{{\nu }_n^i:1\le i\le N\right\}$. Assume that the following restrictions are satisfied:

• (a) ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$ and ${\sum }_{n=1}^{\infty }{{\gamma }^{\prime }}_n<\infty$;

• (b) ${\sum }_{n=1}^{\infty }\left(e_n-1\right)<\infty$ and ${\sum }_{n=1}^{\infty }{\xi }_n<\infty$;

• (c) ${\beta }_{n}L\left({\alpha }_n^{\prime }+{\beta }_n^{\prime }L\right)<1$, where L is defined in (2.5);

• (d) there exist constants λ, η ∈ (0, 1) such that λ ≥ α _n ,.${\alpha }_n^{\prime }\le \eta$.

If there exists a nondecreasing function g : [0, ∞) → [0, ∞) with g(0) = 0 and g(m) > 0 for all m ∈ (0, ∞) such that

$$\underset{1\le r\le N}{max}\left\{\parallel x-S_{r}x\parallel \right\}+\underset{1\le r\le N}{max}\left\{\parallel x-T_{r}x\parallel \right\}\ge g\left(dist\left(x,\mathcal{F}\right)\right),\forall x\in C,$$

then the sequence {x _n } converges strongly to some point in$\mathcal{F}$.

Proof. In view of (3.13) and (3.18) that $g\left(dist\left(x_n,\mathcal{F}\right)\right)\to 0$, which implies $dist\left(x_n,\mathcal{F}\right)\to 0$. Next, we show that the sequence {x _n } is Cauchy. In view of (3.3), we obtain by putting

$$a_n=\frac{\left(1+M_1\right)\left(e_{j\left(n\right)}-1\right)}{1-R}and{b}_n=\frac{\left(1+M_1\right){\xi }_{j\left(n\right)}+M_1{M}_2{{\gamma }^{\prime }}_n+M_3{\gamma }_n}{1-R}$$

that

$$\parallel x_n-f\parallel \le \left(1+a_n\right)\parallel x_{n-1}-f\parallel +b_n.$$

It follows, for any positive integers m, n, where m > n > n₀, that

$$\parallel x_m-p\parallel \le B\parallel x_n-p\parallel +B{\sum }_{i=n+1}^{\infty }{b}_i+b_m,$$

where $B=exp\left\{{\sum }_{n=1}^{\infty }{a}_n\right\}$. It follows that

$$\begin{array}{ll}\hfill \parallel x_n-x_m\parallel & \le \parallel x_n-f\parallel +\parallel x_m-f\parallel \\ \le \left(1+B\right)\parallel x_n-f\parallel +B{\sum }_{i=n+1}^{\infty }{b}_i+b_m.\end{array}$$

Taking the infimum over all $f\in \mathcal{F}$, we arrive at

$$\parallel x_n-x_m\parallel \le \left(1+B\right)dist\left(x_n,\mathcal{F}\right)+B{\sum }_{i=n+1}^{\infty }{b}_i+b_m.$$

In view of ${\sum }_{n=1}^{\infty }{b}_n<\infty$ and $dist\left(x_n,\mathcal{F}\right)\to 0$, we see that {x _n } is a Cauchy sequence in C and so {x _n } converges strongly to some x* ∈ C. Since T _r and S _r are Lipschitz for each r ∈ {1, 2,..., N}, we see that $\mathcal{F}$ is closed. This in turn implies that $x^{*}\in \mathcal{F}$. This completes the proof. □

If S _r = I for each r ∈ {1, 2,..., N} and ${\gamma }_n^{\prime }=0$, then Theorem 3.7 is reduced to the following.

Corollary 3.8. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T _i : C → C be a uniformly$L_t^i$-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence$\left\{h_n^i\right\}\subset \left[1,\infty \right)$, where$h_n^i\to 1$as n → ∞ for each 1 ≤ i ≤ N. Assume that$\mathcal{F}={\bigcap }_{i=1}^{N}F\left(T_i\right)\ne \varnothing$. Let {u _n } be a bounded sequence in C and$h_n=sup\left\{h_n^i:1\le i\le N\right\}$. Let {α _n }, {β _n } and {γ _n } be sequences in [0,1] such that α _n + β _n + γ _n = 1 for each n ≥ 1. Let {x _n } be a sequence generated in (3.19). Put${\mu }_n^i=max\left\{0,{sup}_{x,y\in C}\left(\parallel T_i^{n}x-T_i^{n}y\parallel -h_n^i\parallel x-y\parallel \right)\right\}$. Let${\mu }_n=max\left\{{\mu }_n^i:1\le i\le N\right\}$. Assume that the following restrictions are satisfied:

• (a) ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$;

• (b) ${\sum }_{n=1}^{\infty }\left(h_n-1\right)<\infty$ and ${\sum }_{n=1}^{\infty }{\mu }_n<\infty$;

• (c) β _n L < 1, where $L=max\left\{L_t^i:1\le i\le N\right\}$;

• (d) there exist constants λ, η ∈ (0, 1) such that, λ ≥ α _n , ${\alpha }_n^{\prime }\le \eta$.

If there exists a nondecreasing function g : [0, ∞) → [0, ∞) with g(0) = 0 and g(m) > 0 for all m ∈ (0, ∞) such that

$$\underset{1\le r\le N}{max}\left\{\parallel x-T_{r}x\parallel \right\}\ge g\left(dist\left(x,\mathcal{F}\right)\right),\forall x\in C,$$

then the sequence {x _n } converges strongly to some point in$\mathcal{F}$.