Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings

Abstract

Suppose that 𝐾 is nonempty closed convex subset of a uniformly convex and smooth Banach space 𝐸 with 𝑃 as a sunny nonexpansive retraction and 𝐹∶=𝐹(𝑇₁)∩𝐹(𝑇₂)={𝑥∈𝐾∶𝑇₁𝑥=𝑇₂𝑥=𝑥}≠∅. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two weakly inward nonself asymptotically nonexpansive mappings with respect to 𝑃 with two sequences {𝑘_𝑛^(𝑖)}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛^(𝑖)−1)<∞(𝑖=1,2), respectively. For any given 𝑥₁∈𝐾, suppose that {𝑥_𝑛} is a sequence generated iteratively by 𝑥_𝑛+1=(1−𝛼_𝑛)(𝑃𝑇₁)^𝑛𝑦_𝑛+𝛼_𝑛(𝑃𝑇₂)^𝑛𝑦_𝑛, 𝑦_𝑛=(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛(𝑃𝑇₁)^𝑛𝑥_𝑛, 𝑛∈ℕ, where {𝛼_𝑛} and {𝛽_𝑛} are sequences in [𝑎,1−𝑎] for some 𝑎∈(0,1). Under some suitable conditions, the strong and weak convergence theorems of {𝑥_𝑛} to a common fixed point of 𝑇₁ and 𝑇₂ are obtained.

1. Introduction

Let 𝐸 be a real Banach space with 𝐾, its nonempty subset. Let 𝑇∶𝐾→𝐾 be a mapping. A point 𝑥∈𝐾 is called a fixed point of𝑇 if and only if 𝑇𝑥=𝑥. In this paper, ℕ stands for the set of natural numbers. We will also denote by 𝐹(𝑇) the set of fixed points of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐾∶𝑇𝑥=𝑥} and by 𝐹∶=𝐹(𝑇₁)∩𝐹(𝑇₂), the set of common fixed points of two mappings 𝑇₁ and 𝑇₂. 𝑇 is called asymptotically nonexpansive if for a sequence  {𝑘_𝑛}⊂[1,∞) with lim_𝑛→∞𝑘_𝑛=1, ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾 and all 𝑛∈ℕ. 𝑇 is called uniformly 𝐿-Lipschitzian if for some 𝐿>0, ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖ for all 𝑛∈ℕ and all 𝑥,𝑦∈𝐾. 𝑇 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾. Let 𝑃∶𝐸→𝐾 be a nonexpansive retraction of 𝐸 into 𝐾. A nonself-mapping 𝑇∶𝐾→𝐸 is called asymptotically nonexpansive (according to Chidume et al. [1]) if for a sequence  {𝑘_𝑛}⊂[1,∞) with lim_𝑛→∞𝑘_𝑛=1, we have ‖𝑇(𝑃𝑇)^𝑛−1𝑥−𝑇(𝑃𝑇)^𝑛−1𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾 and 𝑛∈ℕ. 𝑇 is called uniformly 𝐿-Lipschitzian if for some 𝐿>0, ‖𝑇(𝑃𝑇)^𝑛−1𝑥−𝑇(𝑃𝑇)^𝑛−1𝑦‖≤𝐿‖𝑥−𝑦‖ for all 𝑛∈ℕ and all 𝑥,𝑦∈𝐾.

In what follows, we fix 𝑥₁∈𝐾 as a starting point of the process under consideration, and take {𝛼_𝑛}, {𝛽_𝑛} sequences in (0,1).

Agarwal et al. [2] recently introduced the iteration process 𝑥_𝑛+1=1−𝛼_𝑛𝑇^𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑛∈ℕ. (1.1) They showed that their process is independent of Mann and Ishikawa and converges faster than both of these. See Proposition  3.1 [2].

Obviously the above process deals with one self-mapping only. The case of two mappings in iteration processes has also remained under study since Das and Debata [3] gave and studied a two mappings scheme. Also see, for example, Takahashi and Tamura [4] and Khan and Takahashi [5]. Note that two mappings case, that is, approximating the common fixed points, has its own importance as it has a direct link with the minimization problem, see, for example, Takahashi [6].

Being an important generalization of the class of nonexpansive self-mappings, the class of asymptotically nonexpansive self-mappings was introduced by Goebel and Kirk [7] whereas the concept of asymptotically nonexpansive nonself-mappings was introduced by Chidume et al. [1] in 2003 as a generalization of asymptotically nonexpansive self-mappings. Actually they studied the iteration process 𝑥_𝑛+1=𝑃1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇(𝑃𝑇)^𝑛−1𝑥_𝑛,𝑛∈ℕ. (1.2)

Nonself asymptotically nonexpansive mappings have been studied by many authors [8–11]. Wang [10] studied the process 𝑥_𝑛+1=𝑃1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛,𝑦_𝑛=𝑃1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛,𝑛∈ℕ. (1.3)

Very recently, Thianwan [12] considered a new iterative scheme (called projection type Ishikawa iteration) as follows: 𝑥_𝑛+1=𝑃1−𝛼_𝑛𝑦_𝑛+𝛼_𝑛𝑇₁𝑃𝑇₁^𝑛−1𝑦_𝑛,𝑦_𝑛=𝑃1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇₂𝑃𝑇₂^𝑛−1𝑥_𝑛,𝑛∈ℕ. (1.4)

As a matter of fact, if 𝑇 is a self-mapping, then 𝑃 is an identity mapping. In addition, if 𝑇∶𝐾→𝐸 is asymptotically nonexpansive and 𝑃∶𝐸→𝐾 is a nonexpansive retraction, then 𝑃𝑇∶𝐾→𝐾 is asymptotically nonexpansive. Indeed, for all 𝑥,𝑦∈𝐾 and 𝑛∈ℕ, it follows that (𝑃𝑇)^𝑛𝑥−(𝑃𝑇)^𝑛𝑦=𝑃𝑇(𝑃𝑇)^𝑛−1𝑥−𝑃𝑇(𝑃𝑇)^𝑛−1𝑦≤𝑇(𝑃𝑇)^𝑛−1𝑥−𝑇(𝑃𝑇)^𝑛−1𝑦≤𝑘_𝑛𝑥−𝑦. (1.5) The converse, however, may not be true. Therefore, Zhou et al. [13] introduced the following generalized definition recently.

Definition 1.1 (see [13]). Let 𝐾 be a nonempty subset of real normed linear space 𝐸. Let 𝑃∶𝐸→𝐾 be the nonexpansive retraction of 𝐸 into 𝐾.(i)A nonself-mapping 𝑇∶K→𝐸 is called asymptotically nonexpansive with respect to 𝑃 if there exists sequences {𝑘_𝑛}∈[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that (𝑃𝑇)^𝑛𝑥−(𝑃𝑇)^𝑛𝑦≤𝑘_𝑛𝑥−𝑦,∀𝑥,𝑦∈𝐾,𝑛∈ℕ. (1.6)(ii)A nonself-mapping 𝑇∶𝐾→𝐸 is said to be uniformly 𝐿-Lipschitzian with respect to 𝑃 if there exists a constant 𝐿≥0 such that (𝑃𝑇)^𝑛𝑥−(𝑃𝑇)^𝑛𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐾,𝑛∈ℕ. (1.7)

Futhermore, by studying the following iterative process 𝑥₁∈𝐾,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑃𝑇₁^𝑛𝑥_𝑛+𝛾_𝑛𝑃𝑇₂^𝑛𝑥_𝑛,𝑛∈ℕ, (1.8) where {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} are three sequences in [𝑎,1−𝑎] for some 𝑎∈(0,1), satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, Zhou et al. [13] obtained some strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings with respect to 𝑃 in uniformly convex Banach spaces. As a consequence, the main results of Chidume et al. [1] were deduced.

Incorporating the ideas of Agarwal et al. [2], Thianwan [12], and Zhou et al. [13], a new two-step iterative scheme for two nonself asymptotically nonexpansive mappings is introduced and studied in this paper. Our process reads as follows.

Let 𝐾 be a nonempty closed convex subset of a real normed linear space 𝐸 with retraction 𝑃. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two nonself asymptotically nonexpansive mappings with respect to 𝑃: 𝑥₁𝑥∈𝐾,_𝑛+1=1−𝛼_𝑛𝑃𝑇₁^𝑛𝑦_𝑛+𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑃𝑇₁^𝑛𝑥_𝑛,𝑛∈ℕ, (1.9) where {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1]. Following the method of Agarwal et al. [2], it is not difficult to see that our process is able to compute common fixed points at a rate better than (1.3) and (1.4).

Under suitable conditions, the sequence  {𝑥_𝑛} defined by (1.9) can also be generalized to iterative sequence with errors. Thus all the results proved in this paper can also be proved for the iterative process with errors. In this case our main iterative process (1.9) looks like 𝑥₁𝑥∈𝐾,_𝑛+1=𝛼_𝑛𝑃𝑇₁^𝑛𝑦_𝑛+𝛽_𝑛𝑃𝑇₂^𝑛𝑦_𝑛+𝛾_𝑛𝑢_𝑛,𝑦_𝑛=𝛼^′_𝑛𝑥_𝑛+𝛽^′_𝑛𝑃𝑇₁^𝑛𝑥_𝑛+𝛾^′_𝑛𝑣_𝑛,𝑛∈ℕ, (1.10) where {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛼^′_𝑛}, {𝛽^′_𝑛}, {𝛾^′_𝑛} are real sequences in [0,1] satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1=𝛼^′_𝑛+𝛽^′_𝑛+𝛾^′_𝑛 and {𝑢_𝑛}, {𝑣_𝑛} are bounded sequences in 𝐾. Observe that the iterative process (1.10) with errors reduces to the iterative process (1.9) when 𝛾_𝑛=𝛾^′_𝑛=0.

2. Preliminaries

For the sake of convenience, we restate the following concepts and results.

Let 𝐸 be a Banach space with its dimension greater than or equal to 2. The modulus of 𝐸 is the function 𝛿_𝐸(𝜀)∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜀)1=inf1−2𝑥+𝑦∶‖𝑥‖=1,𝑦=1,𝜀=𝑥−𝑦. (2.1)

A Banach space 𝐸 is uniformly convex if and only if 𝛿_𝐸(𝜀)>0 for all 𝜀∈(0,2].

Let 𝐸 be a Banach space and 𝑆(𝐸)={𝑥∈𝐸∶‖𝑥‖=1}. The space 𝐸 is said to be smooth if lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (2.2) exists for all 𝑥,𝑦∈𝑆(𝐸).

A subset 𝐾 of 𝐸 is said to be a retract if there exists a continuous mapping 𝑃∶𝐸→𝐾 such that 𝑃𝑥=𝑥 for all 𝑥∈𝐾. A mapping 𝑃∶𝐸→𝐸 is said to be a retraction if 𝑃²=𝑃. Let 𝐶 and 𝐾 be subsets of a Banach space 𝐸. A mapping 𝑃 from 𝐶 into 𝐾 is called sunny if 𝑃(𝑃𝑥+𝑡(𝑥−𝑃𝑥))=𝑃𝑥 for 𝑥∈𝐶 with 𝑃𝑥+𝑡(𝑥−𝑃𝑥)∈𝐶 and 𝑡≥0.

Note that, if 𝑃 is a retraction, then 𝑃𝑧=𝑧 for every 𝑧∈𝑅(𝑃) (the range of 𝑃). It is well-known that every closed convex subset of a uniformly convex Banach space is a retract.

For any 𝑥∈𝐾, the inward set 𝐼_𝐾(𝑥) is defined as follows: 𝐼_𝐾(𝑥)=𝑦∈𝐸∶𝑦=𝑥+𝜆(𝑧−𝑥),𝑧∈𝐾,𝜆≥0. (2.3) A mapping 𝑇∶𝐾→𝐸 is said to satisfy the inward condition if 𝑇𝑥∈𝐼_𝐾(𝑥) for all 𝑥∈𝐾. 𝑇 is said to be weakly inward if 𝑇𝑥∈cl𝐼_𝐾(𝑥) for each 𝑥∈𝐾, where cl𝐼_𝐾(𝑥) is the closure of 𝐼_𝐾(𝑥).

A Banach space 𝐸 is said to satisfy Opial's condition if, for any sequence  {𝑥_𝑛} in 𝐸, 𝑥_𝑛⇀𝑥 implies that limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦 (2.4) for all 𝑦∈𝐸 with 𝑦≠𝑥, where 𝑥_𝑛⇀𝑥 means that {𝑥_𝑛} converges weakly to 𝑥.

Recall that the mapping 𝑇∶𝐾→𝐾 with 𝐹(𝑇)≠∅ is said to satisfy condition (𝐴) [14] if there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0, 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that ‖𝑥−𝑇𝑥‖≥𝑓(𝑑(𝑥,𝐹(𝑇))) for all 𝑥∈𝐾, where 𝑑(𝑥,𝐹(𝑇))=inf{‖𝑥−𝑝‖∶𝑝∈𝐹(𝑇)}. Khan and Fukhar-ud-din [15] modified condition (𝐴) for two mappings as follows: Two mappings 𝑇₁, 𝑇₂∶𝐾→𝐾 are said to satisfy condition (𝐴^′) [15] if there is a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0, 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that 12𝑥−𝑇₁𝑥+𝑥−𝑇₂𝑥≥𝑓(𝑑(𝑥,𝐹)) (2.5) for all 𝑥∈𝐾, where 𝑑(𝑥,𝐹)=inf{‖𝑥−𝑝‖∶𝑝∈𝐹∶=𝐹(𝑇₁)∩𝐹(𝑇₂)}.

Note that condition (𝐴^′) reduces to condition (𝐴) when 𝑇₁=𝑇₂. It is also well-known that condition (𝐴) is weaker than demicompactness or semicompactness, see [14].

A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is said to be demiclosed at 𝑝 if whenever {𝑥_𝑛} is a sequence in 𝐷(𝑇) such that {𝑥_𝑛} converges weakly to 𝑥^∗∈𝐷(𝑇) and {𝑇𝑥_𝑛} converges strongly to 𝑝, then 𝑇𝑥^∗=𝑝.

We need the following lemmas for our main results.

Lemma 2.1 (see [16]). If {𝑟_𝑛}, {𝑡_𝑛} are two sequences of nonnegative real numbers such that 𝑟_𝑛+1≤1+𝑡_𝑛𝑟_𝑛,𝑛∈ℕ (2.6) and ∑^∞_𝑛=1𝑡_𝑛<∞, then lim_𝑛→∞𝑟_𝑛 exists.

Lemma 2.2 (see [17]). Suppose that 𝐸 is a uniformly convex Banach space and 0<𝑝≤𝑡_𝑛≤𝑞<1 for all 𝑛∈ℕ. Also, suppose that {𝑥_𝑛} and {𝑦_𝑛} are sequences of 𝐸  such that limsup_𝑛→∞𝑥_𝑛≤𝑟,limsup_𝑛→∞𝑦_𝑛≤𝑟,lim_𝑛→∞1−𝑡_𝑛𝑥_𝑛+𝑡_𝑛𝑦_𝑛=𝑟 (2.7) hold for some 𝑟≥0. Then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 2.3 (see [18]). Let 𝐸 be real smooth Banach space, let 𝐾 be nonempty closed convex subset of 𝐸 with 𝑃 as a sunny nonexpansive retraction, and let 𝑇∶𝐾→𝐸 be a mapping satisfying weakly inward condition. Then 𝐹(𝑃𝑇)=𝐹(𝑇).

Lemma 2.4 (see [1]). Let 𝐸 be a uniformly convex Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇 be a nonself asymptotically nonexpansive mapping. Then 𝐼−𝑇 is demiclosed with respect to zero, that is, 𝑥_𝑛⇀𝑥 and 𝑥_𝑛−𝑇𝑥_𝑛→0 imply that 𝑇𝑥=𝑥.

3. Main Results

3.1. Convergence Theorems in Real Banach Spaces

In this section, we prove the strong convergence of the iteration scheme (1.9) to a common fixed point of nonself asymptotically nonexpansive mappings 𝑇₁ and 𝑇₂ with respect to 𝑃 in real Banach spaces. Let 𝑇₁, 𝑇₂∶𝐾→𝐸 be two nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequences {𝑘_𝑛^(𝑖)}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛^(𝑖)−1)<∞(𝑖=1,2), respectively. Put 𝑘_𝑛=max{𝑘_𝑛⁽¹⁾,𝑘_𝑛⁽²⁾}, then obviously ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. From now on we will take this sequence  {𝑘_𝑛} for both 𝑇₁ and 𝑇₂.

We first prove the following lemmas.

Lemma 3.1. Let 𝐸 be a real normed linear space and 𝐾 a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is defined by (1.9) and 𝐹≠∅. Then, (i)lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for all 𝑝∈𝐹;(ii)there exists a constant 𝑀>0 such that ‖𝑥_𝑛+𝑚−𝑝‖≤𝑀‖𝑥_𝑛−𝑝‖  for all 𝑚,𝑛∈ℕ and 𝑝∈𝐹.

Proof. (i) Let 𝑝∈𝐹. From (1.9), we have 𝑦_𝑛−𝑝=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑃𝑇₁^𝑛𝑥_𝑛≤−𝑝1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑃𝑇₁^𝑛𝑥_𝑛≤−𝑝1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑘_𝑛𝑥_𝑛=−𝑝1+𝛽_𝑛𝑘_𝑛−1𝑥_𝑛≤𝑘−𝑝1+_𝑛−1𝑥_𝑛−𝑝=𝑘_𝑛𝑥_𝑛−𝑝. (3.1) By (3.1) and (1.9), we obtain 𝑥_𝑛+1−𝑝=1−𝛼_𝑛𝑃𝑇₁^𝑛𝑦_𝑛−𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛=−𝑝1−𝛼_𝑛𝑃𝑇₁^𝑛𝑦_𝑛−𝑝−𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛≤−𝑝1−𝛼_𝑛𝑘_𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑘_𝑛𝑦_𝑛−𝑝=𝑘_𝑛𝑦_𝑛−𝑝≤𝑘²_𝑛𝑥_𝑛=𝑘−𝑝1+²_𝑛−1𝑥_𝑛−𝑝. (3.2) Note that ∑^∞_𝑛=1(𝑘_𝑛−1)<∞ is equivalent to ∑^∞_𝑛=1(𝑘²_𝑛−1)<∞. Thus, by (3.2) and Lemma 2.1, lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for all 𝑝∈𝐹(𝑇).
(ii) From (3.2), we have 𝑥_𝑛+1−𝑝≤𝑘²_𝑛𝑥_𝑛−𝑝. (3.3) It is well known that 1+𝑥≤𝑒^𝑥 for all 𝑥≥0. Using it for the above inequality, we have 𝑥_𝑛+𝑚−𝑝≤1+𝑘²_{𝑛+𝑚−1}−1𝑥_{𝑛+𝑚−1}−𝑝≤𝑒^{𝑘2𝑛+𝑚−1−1}𝑥_{𝑛+𝑚−1}−𝑝≤𝑒^{𝑘2𝑛+𝑚−1−1}1+𝑘²_{𝑛+𝑚−2}−1𝑥_{𝑛+𝑚−2}−𝑝≤𝑒^{𝑘2𝑛+𝑚−1−1+𝑘2𝑛+𝑚−2−1}𝑥_{𝑛+𝑚−2}⋮−𝑝≤𝑒^{∑𝑛+𝑚−1𝑗=𝑛(𝑘2𝑗−1)}𝑥_𝑛𝑥−𝑝≤𝑀_𝑛−𝑝, (3.4) where 𝑀=𝑒^{∑𝑛+𝑚−1𝑗=𝑛(𝑘2𝑗−1)}. That is, ‖𝑥_𝑛+𝑚−𝑝‖≤𝑀‖𝑥_𝑛−𝑝‖  for all 𝑚,𝑛∈ℕ and 𝑝∈𝐹.

Theorem 3.2. Let 𝐸 be a real Banach space and 𝐾 a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is defined by (1.9) and  𝐹≠∅.  Then, {𝑥_𝑛} converges strongly to a common fixed point of  𝑇₁ and 𝑇₂ if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, where 𝑑(𝑥_𝑛,𝐹)=inf{‖𝑥−𝑝‖∶𝑝∈𝐹}.

Proof. The necessity of the conditions is obvious. Thus, we need only prove the sufficiency. Suppose that liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0. From (3.2), we have 𝑑𝑥_𝑛+1,𝐹≤1+𝑘²_𝑛−1𝑑𝑥_𝑛,𝐹. (3.5) As ∑^∞_𝑛=1(𝑘_𝑛−1)<∞, therefore lim_𝑛→∞𝑑(𝑥_𝑛,𝐹) exists by Lemma 2.1. But by hypothesis liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, therefore we must have lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.
Next we show that {𝑥_𝑛} is a Cauchy sequence. Let 𝜀>0. Since lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, therefore there exists a constant 𝑛₀ such that for all 𝑛≥𝑛₀, we have 𝑑𝑥_𝑛<𝜀,𝐹,4𝑀 (3.6) where 𝑀>0 is the constant in Lemma 3.1(ii). So we can find 𝑝^′∈𝐹 such that 𝑥_𝑛0−𝑝^′<𝜀.2𝑀 (3.7) Using Lemma 3.1(ii), we have for all 𝑛≥𝑛₀ and 𝑚∈ℕ that 𝑥_𝑛+𝑚−𝑥_𝑛≤𝑥_𝑛+𝑚−𝑝^′+𝑥_𝑛−𝑝^′≤𝑀𝑥_𝑛0−𝑝^′+𝑀𝑥_𝑛0−𝑝^′=2𝑀𝑥_𝑛0−𝑝^′<𝜀. (3.8) Hence {𝑥_𝑛} is a Cauchy sequence in a closed subset 𝐾 of a Banach space 𝐸, therefore it must converge to a point in 𝐾. Let lim_𝑛→∞𝑥_𝑛=𝑞. Now, lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0 gives that 𝑑(𝑞,𝐹)=0. Since the set of fixed points of asymptotically nonexpansive mappings is closed, we have 𝑞∈𝐹. This completes the proof of the theorem.

On the lines similar to this theorem, we can also prove the following theorem which addresses the error terms.

Theorem 3.3. Let 𝐸 be a real Banach space and 𝐾 a nonempty closed convex subset of 𝐸 which is also a nonexpansive retract of 𝐸. Let 𝑇₁, 𝑇₂∶𝐾→𝐸 be two asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is defined by (1.10) with ∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞ and 𝐹≠∅, Then, {𝑥_𝑛} converges strongly to a common fixed point of  𝑇₁ and 𝑇₂ if and only if  liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, where 𝑑(𝑥_𝑛,𝐹)=inf{‖𝑥−𝑝‖∶𝑝∈𝐹}.

3.2. Convergence Theorems in Real Uniformly Convex Banach Spaces

In this section, we prove the strong and weak convergence of the sequence defined by the iteration scheme (1.9) to a common fixed point of nonself asymptotically nonexpansive mappings 𝑇₁ and 𝑇₂ with respect to 𝑃 in real uniformly convex and smooth Banach space. We first prove the following lemma.

Lemma 3.4. Let 𝐾 be a nonempty closed convex subset of a real uniformly convex Banach space 𝐸. Let 𝑇₁, 𝑇₂∶𝐾→𝐸 be two nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is defined by(1.9), where {𝛼_n} and {𝛽_n} are sequences in [𝑎,1−𝑎] for some 𝑎∈(0,1). If 𝐹≠∅, then lim_𝑛→∞𝑥_𝑛−𝑃𝑇₁𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝑃𝑇₂𝑥_𝑛=0. (3.9)

Proof. By Lemma 3.1(i), lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Assume that lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝑐. If 𝑐=0, the conclusion is obvious. Suppose 𝑐>0. Taking limsup on both sides in that inequality (3.1), we have limsup_𝑛→∞𝑦_𝑛−𝑝≤limsup_𝑛→∞𝑥_𝑛−𝑝=lim_𝑛→∞𝑥_𝑛−𝑝=𝑐. (3.10) Thus ‖(𝑃𝑇₁)^𝑛𝑦_𝑛−𝑝‖≤𝑘_𝑛‖𝑦_𝑛−𝑝‖ for all 𝑛∈ℕ implies that limsup_𝑛→∞𝑃𝑇₁^𝑛𝑦_𝑛−𝑝≤𝑐. (3.11) Similarly, limsup_𝑛→∞𝑃𝑇₂^𝑛𝑦_𝑛−𝑝≤𝑐. (3.12) Further, 𝑐=lim_𝑛→∞𝑥_𝑛+1−𝑝=lim_𝑛→∞1−𝛼_𝑛𝑃𝑇₁^𝑛𝑦_𝑛+𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛−𝑝=lim_𝑛→∞1−𝛼_𝑛𝑃𝑇₁^𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛−𝑝≤lim_𝑛→∞1−𝛼_𝑛limsup_𝑛→∞𝑃𝑇₁^𝑛𝑦_𝑛−𝑝+𝛼_𝑛limsup_𝑛→∞𝑃𝑇₂^𝑛𝑦_𝑛−𝑝≤lim_𝑛→∞1−𝛼_𝑛𝑐+𝛼_𝑛𝑐=𝑐 (3.13) gives that lim_𝑛→∞1−𝛼_𝑛𝑃𝑇₁^𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛−𝑝=𝑐. (3.14) Hence, using (3.11), (3.12), (3.14), and Lemma 2.2, we obtain lim_𝑛→∞𝑃𝑇₂^𝑛𝑦_𝑛−𝑃𝑇₁^𝑛𝑦_𝑛=0. (3.15) Noting that 𝑥_𝑛+1−𝑝=1−𝛼_𝑛𝑃𝑇₁^𝑛𝑦_𝑛+𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛≤−𝑝𝑃𝑇₁^𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛−𝑃𝑇₁^𝑛𝑦_𝑛≤𝑘_𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛−𝑃𝑇₁^𝑛𝑦_𝑛 (3.16) which yields that 𝑐≤liminf_𝑛→∞𝑦_𝑛−𝑝. (3.17) By (3.10) and (3.17), we obtain lim_𝑛→∞𝑦_𝑛−𝑝=𝑐. (3.18) Moreover, ‖(𝑃𝑇₁)^𝑛𝑥_𝑛−𝑝‖≤𝑘_𝑛‖𝑥_𝑛−𝑝‖ for all 𝑛∈ℕ implies that limsup_𝑛→∞𝑃𝑇₁^𝑛𝑥_𝑛−𝑝≤𝑐. (3.19) Hence 𝑐=lim_𝑛→∞𝑦_𝑛−𝑝=lim_𝑛→∞1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑃𝑇₁^𝑛𝑥_𝑛−𝑝=lim_𝑛→∞1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑃𝑇₁^𝑛𝑥_𝑛−𝑝≤lim_𝑛→∞1−𝛽_𝑛limsup_𝑛→∞𝑥_𝑛−𝑝+𝛽_𝑛limsup_𝑛→∞𝑃𝑇₁^𝑛𝑥_𝑛−𝑝≤lim_𝑛→∞1−𝛽_𝑛𝑐+𝛽_𝑛𝑐=𝑐 (3.20) gives that lim_𝑛→∞1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑃𝑇₁^𝑛𝑥_𝑛−𝑝=𝑐. (3.21) Again by Lemma 2.2, we obtain lim_𝑛→∞𝑃𝑇₁^𝑛𝑥_𝑛−𝑥_𝑛=0. (3.22) In addition, from 𝑦_𝑛=(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛(𝑃𝑇₁)^𝑛𝑥_𝑛, we have 𝑦_𝑛−𝑥_𝑛=𝛽_𝑛𝑃𝑇₁^𝑛𝑥_𝑛−𝑥_𝑛. (3.23) Hence by (3.22), lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (3.24) Also 𝑃𝑇₂^𝑛𝑦_𝑛−𝑥_𝑛≤𝑃𝑇₂^𝑛𝑦_𝑛−𝑃𝑇₁^𝑛𝑦_𝑛+𝑃𝑇₁^𝑛𝑦_𝑛−𝑃𝑇₁^𝑛𝑥_𝑛+𝑃𝑇₁^𝑛𝑥_𝑛−𝑥_𝑛≤𝑃𝑇₂^𝑛𝑦_𝑛−𝑃𝑇₁^𝑛𝑦_𝑛+𝑘_𝑛𝑦_𝑛−𝑥_𝑛+𝑃𝑇₁^𝑛𝑥_𝑛−𝑥_𝑛 (3.25) implies by (3.15), (3.22), and (3.24) that lim_𝑛→∞𝑃𝑇₂^𝑛𝑦_𝑛−𝑥_𝑛=0. (3.26) Using (3.24) and (3.26), we obtain 𝑃𝑇₂^𝑛𝑥_𝑛−𝑥_𝑛≤𝑃𝑇₂^𝑛𝑥_𝑛−𝑃𝑇₂^𝑛𝑦_𝑛+𝑃𝑇₂^𝑛𝑦_𝑛−𝑥_𝑛≤𝑘_𝑛𝑥_𝑛−𝑦_𝑛+𝑃𝑇₂^𝑛𝑦_𝑛−𝑥_𝑛, (3.27) so that lim_𝑛→∞𝑃𝑇₂^𝑛𝑥_𝑛−𝑥_𝑛=0. (3.28) Then 𝑃𝑇₁^𝑛𝑦_𝑛−𝑥_𝑛≤𝑃𝑇₁^𝑛𝑦_𝑛−𝑃𝑇₁^𝑛𝑥_𝑛+𝑃𝑇₁^𝑛𝑥_𝑛−𝑥_𝑛≤𝑘_𝑛𝑦_𝑛−𝑥_𝑛+𝑃𝑇₁^𝑛𝑥_𝑛−𝑥_𝑛 (3.29) gives lim_𝑛→∞𝑃𝑇₁^𝑛𝑦_𝑛−𝑥_𝑛=0. (3.30) From (3.15), and (3.30), we have 𝑥_𝑛+1−𝑥_𝑛=1−𝛼_𝑛𝑃𝑇₁^𝑛𝑦_𝑛+𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛−𝑥_𝑛≤𝑃𝑇₁^𝑛𝑦_𝑛−𝑥_𝑛+𝛼_𝑛𝑃𝑇₂^𝑛𝑦_𝑛−𝑃𝑇₁^𝑛𝑦_𝑛⟶0as𝑛⟶∞. (3.31) Thus from ‖𝑥_𝑛+1−𝑦_𝑛‖≤‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝑥_𝑛−𝑦_𝑛‖, we get lim_𝑛→∞𝑥_𝑛+1−𝑦_𝑛=0. (3.32) From (3.30), (3.31) and 𝑥_𝑛+1−𝑃𝑇₁^𝑛𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑃𝑇₁^𝑛𝑦_𝑛, (3.33) we have lim_𝑛→∞𝑥_𝑛+1−𝑃𝑇₁^𝑛𝑦_𝑛=0. (3.34) Now we make use of the fact that every nonself asymptotically nonexpansive mapping with respect to 𝑃 must be uniformly 𝐿-Lipschitzian with respect to 𝑃 combined with (3.22), (3.32), and (3.34), where 𝐿=sup_𝑛∈ℕ{𝑘_𝑛}≥1, to reach at 𝑥_𝑛−𝑃𝑇₁𝑥_𝑛≤𝑥_𝑛−𝑃𝑇₁^𝑛𝑥_𝑛+𝑃𝑇₁^𝑛𝑥_𝑛−𝑃𝑇₁^𝑛𝑦_𝑛−1+𝑃𝑇₁^𝑛𝑦_𝑛−1−𝑃𝑇₁𝑥_𝑛≤𝑥_𝑛−𝑃𝑇₁^𝑛𝑥_𝑛+𝑘_𝑛𝑥_𝑛−𝑦_𝑛−1+𝐿𝑃𝑇₁^𝑛−1𝑦_𝑛−1−𝑥_𝑛. (3.35) Thus lim_𝑛→∞𝑥_𝑛−𝑃𝑇₁𝑥_𝑛=0. (3.36) From (3.24), (3.26), and (3.31), we have 𝑥_𝑛+1−𝑃𝑇₂^𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑃𝑇₂^𝑛𝑦_𝑛+𝑃𝑇₂^𝑛𝑦_𝑛−𝑃𝑇₂^𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑃𝑇₂^𝑛𝑦_𝑛+𝑘_𝑛𝑦_𝑛−𝑥_𝑛⟶0as𝑛→∞, (3.37) and so lim_𝑛→∞𝑥_𝑛+1−𝑃𝑇₂^𝑛𝑥_𝑛=0. (3.38) Again making use of the fact that every nonself asymptotically nonexpansive mapping with respect to 𝑃 must be uniformly 𝐿-Lipschitzian with respect to 𝑃 and (3.28), (3.31) and (3.38), we have 𝑥_𝑛+1−𝑃𝑇₂𝑥_𝑛+1≤𝑥_𝑛+1−𝑃𝑇₂^𝑛+1𝑥_𝑛+1+𝑃𝑇₂^𝑛+1𝑥_𝑛+1−𝑃𝑇₂^𝑛+1𝑥_𝑛+𝑃𝑇₂^𝑛+1𝑥_𝑛−𝑃𝑇₂𝑥_𝑛+1≤𝑥_𝑛+1−𝑃𝑇₂^𝑛+1𝑥_𝑛+1+𝑘_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝐿𝑃𝑇₂^𝑛𝑥_𝑛−𝑥_𝑛+1. (3.39) This gives, lim_𝑛→∞𝑥_𝑛−𝑃𝑇₂𝑥_𝑛=0. (3.40) This completes the proof of the lemma.

Theorem 3.5. Let 𝐾 be a nonempty closed convex subset of a real uniformly convex and smooth Banach space 𝐸 with 𝑃 as a sunny nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two weakly inward and nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is defined by  (1.9), where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in [𝑎,1−𝑎] for some 𝑎∈(0,1). If one of 𝑇₁ and 𝑇₂ is completely continuous and 𝐹≠∅, then{x_n} converges strongly to a common fixed point of 𝑇₁ and 𝑇₂.

Proof. By Lemma 3.1(i), lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for any 𝑝∈𝐹. It is sufficient to show that {𝑥_𝑛} has a subsequence which converges strongly to a common fixed point of 𝑇₁ and 𝑇₂. By Lemma 3.4, lim_𝑛→∞‖𝑥_𝑛−(𝑃𝑇₁)𝑥_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−(𝑃𝑇₂)𝑥_𝑛‖=0. Suppose that 𝑇₁ is completely continuous. Noting that 𝑃 is nonexpansive, we conclude that there exists subsequence  {𝑃𝑇₁𝑥_𝑛𝑗} of {𝑃𝑇₁𝑥_𝑛} such that 𝑃𝑇₁𝑥_𝑛𝑗→𝑝. Thus ‖𝑥_𝑛𝑗−𝑝‖≤‖𝑥_𝑛𝑗−𝑃𝑇₁𝑥_𝑛𝑗‖+‖𝑃𝑇₁𝑥_𝑛𝑗−𝑝‖ implies 𝑥_𝑛𝑗→𝑝 as 𝑗→∞. Again lim_𝑗→∞‖𝑥_𝑛𝑗−(𝑃𝑇₁)𝑥_𝑛𝑗‖=0 yields by continuity of 𝑃 and 𝑇₁ that 𝑝=𝑃𝑇₁𝑝. Similarly 𝑝=𝑃𝑇₂𝑝. By Lemma 2.3, 𝑝=𝑇₁𝑝=𝑇₂𝑝. Since 𝐹 is closed, so 𝑝∈𝐹. Thus {𝑥_𝑛} converges strongly to a common fixed point 𝑝 of 𝑇₁ and 𝑇₂. This completes the proof.

Theorem 3.6. Let 𝐾 be a nonempty closed convex subset of a real uniformly convex and smooth Banach space 𝐸 with 𝑃 as a sunny nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two weakly inward and nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is defined by  (1.9), where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in [𝑎,1−𝑎] for some 𝑎∈(0,1). If 𝑇₁ and 𝑇₂ satisfy condition (𝐴^′) and 𝐹≠∅, then {𝑥_𝑛} converges strongly to a common fixed point of 𝑇₁ and 𝑇₂.

Proof. By Lemma 3.1(i), lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, and so, lim_𝑛→∞𝑑(𝑥_𝑛,𝐹) exists for all  𝑝∈𝐹. Also, by Lemma 3.4  lim_𝑛→∞‖𝑥_𝑛−(𝑃𝑇₁)𝑥_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−(𝑃𝑇₂)𝑥_𝑛‖=0. It follows from condition (𝐴^′) and Lemma 2.3 that lim_𝑛→∞𝑓𝑑𝑥_𝑛,𝐹≤lim_𝑛→∞12𝑥_𝑛−𝑃𝑇₁𝑥_𝑛+𝑥_𝑛−𝑃𝑇₂𝑥_𝑛=0. (3.41) That is, lim_𝑛→∞𝑓𝑑𝑥_𝑛,𝐹=0. (3.42) Since 𝑓∶[0,∞)→[0,∞) is a nondecreasing function satisfying 𝑓(0)=0, 𝑓(𝑡)>0 for all 𝑡∈(0,∞), therefore we have lim_𝑛→∞𝑑𝑥_𝑛,𝐹=0. (3.43) From Theorem 3.2, we obtain that {𝑥_𝑛} is a Cauchy sequence in 𝐾. Since 𝐾 is a closed subset of a complete space, there exists a 𝑞∈𝐾 such that 𝑥_𝑛→𝑞 as 𝑛→∞. Then, lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0 yields that 𝑑(𝑞,𝐹)=0. Further, it follows from the closedness of 𝐹 that 𝑞∈𝐹. This completes the proof.

Theorem 3.7. Let 𝐾 be a nonempty closed convex subset of a real uniformly convex and smooth Banach space 𝐸 satisfying Opial's condition with 𝑃 as a sunny nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two weakly inward and nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is defined by (1.9), where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in [𝑎,1−𝑎] for some 𝑎∈(0,1). If 𝐹≠∅, then {𝑥_𝑛} converges weakly to a common fixed point of 𝑇₁ and 𝑇₂.

Proof. Let 𝑝∈𝐹. By Lemma 3.1(i), lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists, and {𝑥_𝑛} is bounded. Note that 𝑃𝑇₁ and 𝑃𝑇₂ are self-mappings from 𝐾 into itself. We now prove that {𝑥_𝑛} has a unique weak subsequential limit in 𝐹. Suppose that subsequences {𝑥_𝑛𝑘} and {𝑥_𝑛𝑗} of {𝑥_𝑛} converge weakly to 𝑝₁ and 𝑝₂, respectively. By Lemma 3.4, we have lim_𝑛→∞‖𝑥_𝑛𝑘−(𝑃𝑇_𝑖)𝑥_𝑛𝑘‖=0, (𝑖=1,2). Lemma 2.4 guarantees that (𝐼−𝑃𝑇₁)𝑝₁=0, that is, (𝑃𝑇₁)𝑝₁=𝑝₁. Similarly, (𝑃𝑇₂)𝑝₁=𝑝₁. Again in the same way, we can prove that 𝑝₂∈𝐹. Lemma 2.3 now assures that 𝑝₁,𝑝₂∈𝐹. For uniqueness, assume that 𝑝₁≠𝑝₂, then by Opial’s condition, we have lim_𝑛→∞𝑥_𝑛−𝑝₁=lim_𝑘→∞𝑥_𝑛𝑘−𝑝₁<lim_𝑘→∞𝑥_𝑛𝑘−𝑝₂=lim_𝑗→∞𝑥_𝑛𝑗−𝑝₂<lim_𝑗→∞𝑥_𝑛𝑗−𝑝₁=lim_𝑛→∞𝑥_𝑛−𝑝₁, (3.44) which is a contradiction and hence 𝑝₁=𝑝₂. As a result, {𝑥_𝑛} converges weakly to a common fixed point of 𝑇₁ and 𝑇₂.

In a way similar to the above, we can also prove the results involving error terms as follows.

Theorem 3.8. Let 𝐾 be a nonempty closed convex subset of a real uniformly convex and smooth Banach space 𝐸 with 𝑃 as a sunny nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two weakly inward and nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is the sequence defined by (1.10) satisfying the following conditions:(i)∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞;(ii){𝛼_𝑛} and {𝛼^′_𝑛} are two sequences in [𝑎,1−𝑎] for some 𝑎∈(0,1).If one of 𝑇₁ and 𝑇₂ is completely continuous and 𝐹≠∅, then {𝑥_𝑛} converges strongly to  a common fixed point of 𝑇₁ and 𝑇₂.

Theorem 3.9. Let 𝐾 be a nonempty closed convex subset of a real uniformly convex and smooth Banach space 𝐸 with 𝑃 as a sunny nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two weakly inward and nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is the sequence defined by (1.10) satisfying the following conditions:(i)∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞;(ii){𝛼_𝑛} and {𝛼^′_𝑛} are two sequences in [𝑎,1−𝑎] for some 𝑎∈(0,1). If 𝑇₁ and 𝑇₂ satisfy condition (𝐴^′) and 𝐹≠∅, then {𝑥_𝑛} converges strongly to a common fixed point of 𝑇₁ and 𝑇₂.

Theorem 3.10. Let 𝐾 be a nonempty closed convex subset of a real uniformly convex and smooth Banach space 𝐸 satisfying Opial's condition with 𝑃 as a sunny nonexpansive retraction. Let 𝑇₁,𝑇₂∶𝐾→𝐸 be two weakly inward and nonself asymptotically nonexpansive mappings with respect to 𝑃 with sequence  {𝑘_𝑛}⊂[1,∞) satisfying ∑^∞_𝑛=1(𝑘_𝑛−1)<∞. Suppose that {𝑥_𝑛} is the sequence defined by (1.10) satisfying the following conditions:(i)∑^∞_𝑛=1𝛾_𝑛<∞, ∑^∞_𝑛=1𝛾^′_𝑛<∞;(ii){𝛼_𝑛} and {𝛼^′_𝑛} are two sequences in [𝑎,1−𝑎] for some 𝑎∈(0,1).If 𝐹≠∅, then {𝑥_𝑛} converges weakly to a common fixed point of 𝑇₁ and 𝑇₂.

Acknowledgments

The authors are extremely grateful to the referees for useful suggestions that improved the content of the paper. This paper was supported by Ataturk University Rectorship under "The Scientific and Research Project of Ataturk University," Project no: 2010/276.