On Unification of the Strong Convergence Theorems for a Finite Family of Total Asymptotically Nonexpansive Mappings in Banach Spaces

Abstract

We unify all known iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically I-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced by (C. E. Chidume and E. U. Ofoedu, 2009). We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically I-nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed-convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.

1. Introduction

Let 𝐾 be a nonempty subset of a real normed linear space 𝑋, and let 𝑇∶𝐾→𝐾 be a mapping. Denote by 𝐹(𝑇) the set of fixed points of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐾∶𝑇𝑥=𝑥}. Throughout this paper, we always assume that 𝑋 is a real Banach space and 𝐹(𝑇)≠∅. Now let us recall some known definitions.

Definition 1.1. A mapping 𝑇∶𝐾→𝐾 is said to be(i)nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾,(ii)asymptotically nonexpansive if there exists a sequence {𝜆_𝑛}⊂[1,∞) with lim_𝑛→∞𝜆_𝑛=1 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝜆_𝑛‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾 and 𝑛∈ℕ,(iii) asymptotically nonexpansive in the intermediate sense, if it is continuous and the following inequality holds:limsup_𝑛→∞sup_{𝑥,𝑦∈𝐾}𝑇^𝑛𝑥−𝑇^𝑛𝑦−𝑥−𝑦≤0. (1.1)

Remark 1.2. Observe that if we define 𝑎_𝑛∶=sup_{𝑥,𝑦∈𝐾}𝑇^𝑛𝑥−𝑇^𝑛𝑦−𝑥−𝑦,𝜎_𝑛∶=max0,𝑎_𝑛, (1.2) then 𝜎_𝑛→0 as 𝑛→∞, and (1.1) reduces to 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝑥−𝑦+𝜎_𝑛,∀𝑥,𝑦∈𝐾,𝑛≥1. (1.3)

In [1, 2], Browder studied the iterative construction for fixed points of nonexpansive mappings on closed and convex subsets of a Hilbert space. Note that for the past 30 years or so, the study of the iterative processes for the approximation of fixed points of nonexpansive mappings and fixed points of some of their generalizations have been flourishing areas of research for many mathematicians (see for more details [3, 4]).

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [5] as a generalization of the class of nonexpansive mappings. They proved that if 𝐾 is a nonempty closed-convex bounded subset of a uniformly convex real Banach space and 𝑇 is an asymptotically nonexpansive self-mapping of 𝐾, then 𝑇 has a fixed point.

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [6]. It is known [7] that if 𝐾 is a nonempty closed-convex bounded subset of a uniformly convex Banach space 𝑋 and 𝑇∶𝐾→𝐾 is an asymptotically nonexpansive mapping in the intermediate sense, then 𝑇 has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings (see, e.g., [8]).

The iterative approximation problems for nonexpansive mapping, asymptotically nonexpansive mapping, and asymptotically nonexpansive mapping in the intermediate sense were studied extensively in [5–20].

There are many different types of concepts which generalize a notion of nonexpansive mapping. One of such concepts is a total asymptotically nonexpansive mapping [21], and second one is an asymptotically 𝐼-nonexpansive mapping [22]. Let us recall some notions.

Definition 1.3. Let 𝐾 be a nonempty closed subset of a real normed linear space 𝑋.𝑇∶𝐾→𝐾 is called a total asymptotically nonexpansive mapping if there exist nonnegative real sequence {𝜇_𝑛} and {𝜆_𝑛} with 𝜇_𝑛,𝜆_𝑛→0 as 𝑛→∞ and strictly increasing continuous function 𝜙∶ℝ⁺→ℝ⁺ with 𝜙(0)=0 such that for all 𝑥,𝑦∈𝐾, 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝑥−𝑦+𝜇_𝑛𝜙𝑥−𝑦+𝜆_𝑛,𝑛≥1. (1.4)

Remark 1.4. If 𝜙(𝜉)=𝜉, then (1.4) reduces to 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤1+𝜇_𝑛𝑥−𝑦+𝜆_𝑛,𝑛≥1. (1.5) In addition, if 𝜆_𝑛=0 for all 𝑛≥1, then total asymptotical nonexpansive mappings coincide with asymptotically nonexpansive mappings. If 𝜇_𝑛=0 and 𝜆_𝑛=0 for all 𝑛≥1, we obtain from (1.5) the class of mappings that includes the class of nonexpansive mappings. If 𝜇_𝑛=0 and 𝜆_𝑛=𝜎_𝑛=max{0,𝑎_𝑛}, where 𝑎_𝑛∶=sup_{𝑥,𝑦∈𝐾}(‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖−‖𝑥−𝑦‖) for all 𝑛≥1, then (1.5) reduces to (1.3) which has been studied as mappings asymptotically nonexpansive in the intermediate sense.

The idea of the definition of a total asymptotically nonexpansive mappings is to unify various definitions of classes of mappings associated with the class of asymptotically nonexpansive mappings and to prove a general convergence theorems applicable to all these classes of nonlinear mappings.

Alber et al. [21] studied methods of approximation of fixed points of total asymptotically nonexpansive mappings. C. E. Chidume and E. U. Ofoedu [23] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings in Banach spaces. Recently, C. E. Chidume and E. U. Ofoedu [24] constructed a new iterative sequence much simpler than other types of approximation of common fixed points of finite families of total asymptotically nonexpansive mappings.

On the other hand, in [22] an asymptotically 𝐼-nonexpansive mapping was introduced.

Definition 1.5. Let 𝑇∶𝐾→𝐾, 𝐼∶𝐾→𝐾 be two mappings of a nonempty subset 𝐾 of a real normed linear space 𝑋, then 𝑇 is said to be(i)𝐼-nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝐼𝑥−𝐼𝑦‖ for all 𝑥,𝑦∈𝐾,(ii)asymptotically 𝐼-nonexpansive, if there exists a sequence {𝜆_𝑛}⊂[1,∞) with lim_𝑛→∞𝜆_𝑛=1 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝜆_𝑛‖𝐼^𝑛𝑥−𝐼^𝑛𝑦‖ for all 𝑥,𝑦∈𝐾 and 𝑛≥1.

Best approximation properties of 𝐼-nonexpansive mappings were investigated in [22, 25]. In [26], strong convergence of Mann iterations of 𝐼-nonexpansive mapping has been proved. In [27], the weak convergence of three-step Noor iterative scheme for an 𝐼-nonexpansive mapping in a Banach space has been established. In [28], the weakly convergence theorem for asymptotically 𝐼-nonexpansive mapping defined in Hilbert space was proved. Recently, in [29–31], the weak and strong convergence of explicit and implicit iteration process to a common fixed point of a finite family of asymptotically 𝐼-nonexpansive mappings have been studied.

In this paper, we introduce a new type of concept of a generalization of nonexpansive mapping’s nation, which is a combination of Definitions 1.3 and 1.5.

Definition 1.6. Let 𝑇∶𝐾→𝐾, 𝐼∶𝐾→𝐾 be two mappings of a nonempty subset 𝐾 of a real normed linear space 𝑋, then 𝑇 is said to be a total asymptotically 𝐼-nonexpansive mapping if there exist nonnegative real sequences {𝜇_𝑛} and {𝜆_𝑛} with 𝜇_𝑛,𝜆_𝑛→0 as 𝑛→∞ and the strictly increasing continuous function 𝜙∶ℝ⁺→ℝ⁺ with 𝜙(0)=0 such that for all 𝑥,𝑦∈𝐾, 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝐼^𝑛𝑥−𝐼^𝑛𝑦+𝜇_𝑛𝜙𝐼^𝑛𝑥−𝐼^𝑛𝑦+𝜆_𝑛,𝑛≥1. (1.6)

Now let us provide an example of a total asymptotically 𝐼-nonexpansive mapping, which is not asymptotically nonexpansive mapping.

Example 1.7. Let us consider the space ℓ₁, and let 𝐵₁={𝐱∈ℓ₁∶‖𝐱‖₁≤1}. Define a nonlinear operator 𝑇∶ℓ₁→ℓ₁ by 𝑇_𝛼𝑥₁,𝑥₂,…,𝑥_𝑛,…=0,𝛼𝑥₁,𝛼𝑥₂,…,𝛼𝑥_𝑛,…,𝛼∈(0,1). (1.7) Let ‖𝐱‖₁≤1, then from 𝑇_𝛼(𝐱)₁=𝛼‖𝐱‖₁−𝑥₁+𝑥₁≤𝛼1−𝑥₁+𝑥₁≤1, (1.8) one gets 𝑇(𝐵₁)⊂𝐵₁.
One can find that 𝑇^𝑘_𝛼𝑥₁,𝑥₂,…,𝑥_𝑛,…=0,…,0_𝑘,𝛼^𝑘𝑥₁,𝛼^𝑘𝑥₂,…,𝛼^𝑘𝑥_𝑛,…. (1.9) Hence, 𝑇^𝑘_𝛼(𝐱)−𝑇^𝑘_𝛼(𝐲)₁=𝛼^𝑘‖𝐱−𝐲‖₁+𝑥₁−𝑦₁−𝑥₁−𝑦₁. (1.10) From 𝐱,𝐲∈𝐵₁, we have 𝑥₁−𝑦₁≤𝑥₁−𝑦₁≤𝑥−𝑦₁. (1.11) So, it follows from (1.10) and (1.11) that 𝑇^𝑘_𝛼(𝐱)−𝑇^𝑘_𝛼(𝐲)₁≤𝛼^𝑘‖𝐱−𝐲‖₁+‖𝐱−𝐲‖₁∀𝐱,𝐲∈𝐵₁,𝑘∈ℕ. (1.12)
Now consider a new Banach space ℝ×ℓ₁ with a norm ‖𝕏‖=|𝑥|+‖𝐱‖₁, where 𝕏=(𝑥,𝐱) and define a new mapping 𝑆∶ℝ×ℓ₁→ℝ×ℓ₁ by 𝑆(𝑥,𝐱)=𝑥,𝑇_𝛼(𝐱). (1.13) Let 𝐾=[0,1]×𝐵₁, then it is clear that 𝑆(𝐾)⊂𝐾. One can see that 𝑆^𝑘(𝑥,𝐱)=(𝑥,𝑇^𝑘_𝛼(𝐱)). Therefore, using (1.14), we obtain 𝑆^𝑘(𝕏)−𝑆^𝑘(𝕐)=𝑥−𝑦+𝑇^𝑘_𝛼(𝐱)−𝑇^𝑘_𝛼(𝐲)₁≤𝑥−𝑦+𝛼^𝑘‖𝐱−𝐲‖₁+‖𝐱−𝐲‖₁≤‖𝕏−𝕐‖+𝛼^𝑘‖𝕏−𝕐‖+‖𝕏−𝕐‖. (1.14) We let 𝜙(𝑡)=𝑡+√𝑡 and 𝜇_𝑘=𝛼^𝑘. It is clear that 𝜙(0)=0 and 𝜙 is strictly increasing, and moreover, (1.14) implies 𝑆^𝑘(𝕏)−𝑆^𝑘(𝕏)≤‖𝐼𝕏−𝐼𝕐‖+𝜇_𝑘𝜙(‖𝐼𝕏−𝐼𝕐‖), (1.15) that is 𝑆 is a totally asymptotically 𝐼-nonexpansive mapping. Here, 𝐼 is the identity mapping of ℝ×ℓ₁.
Now we are going to show that 𝑆 is not asymptotically nonexpansive. Namely, we will establish that for any sequence of positive numbers {𝜆_𝑛} with 𝜆_𝑛→0 and any 𝑘∈ℕ, one can find 𝕏₀,𝕐₀ such that 𝑆^𝑘𝕏₀−𝑆^𝑘𝕐₀>1+𝜆_𝑘𝕏₀−𝕐₀. (1.16) In fact, choose 𝕏₀, 𝕐₀ as follows: 𝕏₀=0,𝐱₀,𝕐₀=0,𝐲₀, (1.17) where 𝐱₀=𝑥₀,0,…,0,…,𝐲₀=𝑥₀4,0,…,0,…,0<𝑥₀<4𝛼^2𝑘91+𝜆_𝑘². (1.18)
From (1.10), one finds that 𝑆^𝑘𝕏₀−𝑆^𝑘𝕐₀=𝛼^𝑘√𝑥₀−√𝑥₀2=𝛼^𝑘√𝑥₀2,𝕏₀−𝕐₀=3𝑥₀4. (1.19) The last equalities with (1.18) imply that 𝑆^𝑘𝕏₀−𝑆^𝑘𝕐₀𝕏₀−𝕐₀=2𝛼^𝑘3√𝑥₀>1+𝜆_𝑘. (1.20) This yields the required assertion. Note that 𝑆 has infinitely many fixed points in 𝐾, that is, Fix(𝑆)={(𝑥,𝟎)∶𝑥∈[0,1]}.

Example 1.8. Let us consider the Banach space ℝ×ℓ₁ defined as before, and let 𝑓 be a mapping of a segment 𝐶⊂ℝ to itself, that is, 𝑓∶𝐶→𝐶 with 𝑓(0)=0 and 𝑓^𝑛(𝑥)−𝑓^𝑛𝑦≤𝑥−𝑦+𝑐_𝑛,𝑐_𝑛>0,𝑛∈ℕ, (1.21) where 𝑐_𝑛→0. Note that such kind of functions do exist. One can take (see for more details [8]) 𝐶=[−1/𝜋,1/𝜋] and 𝑓_𝜅(𝑥)=1𝜅𝑥sin𝑥,𝑥≠0,𝜅∈(0,1).0,𝑥=0. (1.22)
Define a new mapping 𝑆_𝑓∶𝐶×𝐵₁→𝐶×𝐵₁ by 𝑆_𝑓(𝑥,𝐱)=𝑓(𝑥),𝑇_𝛼(𝐱), (1.23) here 𝑇 is defined as above (see (1.7)). Using the same argument as the above Example 1.7, we can establish that 𝑆^𝑘_𝑓(𝕏)−𝑆^𝑘_𝑓(𝕏)≤‖𝕏−𝕐‖+𝜇_𝑘𝜙(‖𝕏−𝕐‖)+𝑐_𝑘∀𝑘∈ℕ. (1.24) Moreover, such a mapping is not asymptotically nonexpansive. Note that the mapping 𝑆_𝑓𝜅 with the function 𝑓_𝜅 has a unique fixed point in 𝐶×𝐵₁.

Remark 1.9. To the best our knowledge, we should stress that the constructed examples are currently only unique examples of totaly asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Before, no such examples were known in the literature.

The aim of the present paper is unification of all known iterative methods by introducing a new iterative scheme for approximation of common fixed points of finite families of total asymptotically 𝐼-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced in [24] and allows us to construct more simpler methods than [23, 24].

Namely, let 𝐾 be a nonempty closed-convex subset of a real Banach space 𝑋 and {𝑇_𝑖}^𝑚_𝑖=1: 𝐾→𝐾 be a finite family of total asymptotically 𝐼_𝑖-nonexpansive mappings, that is, 𝑇^𝑛_𝑖𝑥−𝑇^𝑛_𝑖𝑦≤𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦+𝜇_𝑖𝑛𝜙_𝑖𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦+𝜆_𝑖𝑛, (1.25) and {𝐼_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 is a finite family of total asymptotically nonexpansive mappings, that is, 𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦≤𝑥−𝑦+𝜇_𝑖𝑛𝜑_𝑖𝑥−𝑦+𝜆_𝑖𝑛, (1.26) here 𝜙_𝑖,𝜑_𝑖∶ℝ⁺→ℝ⁺ are the strictly increasing continuous functions with 𝜙_𝑖(0)=𝜑_𝑖(0)=0 for all 𝑖=1,𝑚, and {𝜇_𝑖𝑛}^∞_𝑛=1,{𝜆_𝑖𝑛}^∞_𝑛=1,{𝜇_𝑖𝑛}^∞_𝑛=1,{𝜆_𝑖𝑛}^∞_𝑛=1 are nonnegative real sequences with 𝜇_𝑖𝑛,𝜆_𝑖𝑛,𝜇_𝑖𝑛,𝜆_𝑖𝑛→0 as 𝑛→∞ for all 𝑖=1,𝑚. Then for given sequences {𝛼_𝑗𝑛}^∞_𝑛=1,{𝛽_𝑗𝑛}^∞_𝑛=1 in (0,1), where 𝑗=0,𝑚, we will consider the following explicit iterative process:𝑥₀𝑥∈𝐾,_𝑛+1=𝛼_0𝑛𝑥_𝑛+^𝑚_𝑖=1𝛼_𝑖𝑛𝑇^𝑛_𝑖𝑦_𝑛,𝑦_𝑛=𝛽_0𝑛𝑥_𝑛+^𝑚_𝑖=1𝛽_𝑖𝑛𝐼^𝑛_𝑖𝑥_𝑛, (1.27) such that ∑^𝑚_𝑗=0𝛼_𝑗𝑛=1 and ∑^𝑚_𝑗=0𝛽_𝑗𝑛=1.

C. E. Chidume and E. U. Ofoedu [24] have considered only a particular case of the explicit iterative process (1.27), in which {𝐼_𝑖}^𝑚_𝑖=1 is to be taken as the identity mappings. One of the main results of ([24], see Theorem 3.5, page 11) was correct, while the provided proof of that result was wrong. Since, in their proof, they used Lemma 2.3, but which actually is not applicable in that situation, the sequence {𝑡_𝑛}^∞_𝑛=1 tends to 0. As a counterexample, we can consider the following one: let 𝑥∈𝑋,‖𝑥‖=𝑑>0, and let the sequences 𝑥_𝑛, 𝑦_𝑛, and 𝑡_𝑛 be defined as follows:𝑥_𝑛=𝑥,𝑦_𝑛=−𝑥,𝑡_𝑛=1𝑛,∀𝑛∈ℕ. (1.28) It is then clear thatlim_𝑛→∞𝑡_𝑛𝑥_𝑛+1−𝑡_𝑛𝑦_𝑛=‖𝑥‖lim_𝑛→∞21−𝑛=𝑑. (1.29) However,lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=2𝑑>0. (1.30)

In this paper, we shall provide a correct proof of Theorem 3.5 page 11 in [24]. As we already mentioned in Lemma 2.3 is not applicable the main result of [24]. Therefore, we first will generalize Lemma 2.3 to the case of finite number of sequences. Such a generalization gives us a possibility to prove the mentioned result. On other hand, the provided generalization presents an independent interest as well. Moreover, we extend and unify the main result of [24] for a finite family of total asymptotically𝐼_𝑖-nonexpansive mappings {𝑇_𝑖}^𝑚_𝑖=1. Namely, we shall prove the strong convergence of the explicit iterative process (1.27) to a common fixed point of the finite family of total asymptotically𝐼_𝑖-nonexpansive mappings {𝑇_𝑖}^𝑚_𝑖=1 and the finite family of total asymptotically nonexpansive mappings {𝐼_𝑖}^𝑚_𝑖=1. Here, we stress that Lemmas 3.1 and 3.2 play a crucial role. All presented results here extend, generalize, unify, and improve the corresponding main results of [21, 24, 29–33].

2. Preliminaries

Throughout this paper, we always assume that 𝑋 is a real Banach space. The following lemmas play an important role in proving our main results.

Lemma 2.1 (see [16]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} be three sequences of nonnegative real numbers with ∑^∞_𝑛=1𝑏_𝑛<∞, ∑^∞_𝑛=1𝑐_𝑛<∞. If the following condition is satisfied:(i)𝑎_𝑛+1≤(1+𝑏_𝑛)𝑎_𝑛+𝑐_𝑛,𝑛≥1, then the limit lim_𝑛→∞𝑎_𝑛 exists.

Lemma 2.2 (see [34]). Let 𝑋 be a uniformly convex Banach space and 𝑡∈(0,1). Suppose that {𝑥_𝑛},{𝑦_𝑛} are two sequences in 𝑋 such that lim_𝑛→∞𝑡𝑥_𝑛+(1−𝑡)𝑦_𝑛=𝑑,limsup_𝑛→∞𝑥_𝑛≤𝑑,limsup_𝑛→∞𝑦_𝑛≤𝑑 (2.1) hold some 𝑑≥0, then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 2.3 (see [14]). Let 𝑋 be a uniformly convex Banach space, and let 𝑏, 𝑐 be two constants with 0<𝑏<𝑐<1. Suppose that {𝑡_𝑛} is a sequence in [𝑏,𝑐] and {𝑥_𝑛},{𝑦_𝑛} are two sequences in 𝑋 such that lim_𝑛→∞𝑡_𝑛𝑥_𝑛+1−𝑡_𝑛𝑦_𝑛=𝑑,limsup_𝑛→∞𝑥_𝑛≤𝑑,limsup_𝑛→∞𝑦_𝑛≤𝑑, (2.2) hold some 𝑑≥0, then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

3. Main Results

In this section, we shall prove our main results. To formulate ones, we need some auxiliary results.

First we are going to generalize Lemmas 2.2 and 2.3 for 𝑚 number of sequences {𝑧_𝑖𝑛}^∞_𝑛=1 from the uniformly convex Banach space 𝑋, where 𝑖=1,𝑚.

Lemma 3.1. Let 𝑋 be a uniformly convex Banach space and 𝛼_𝑖∈(0,1),𝑖=1,𝑚 any constants with ∑^𝑚_𝑖=1𝛼_𝑖=1. Suppose that {𝑧_𝑖𝑛}^∞_𝑛=1,𝑖=1,𝑚 are sequences in 𝑋 such that lim_𝑛→∞^𝑚_𝑖=1𝛼_𝑖𝑧_𝑖𝑛=𝑑,limsup_𝑛→∞𝑧_𝑖𝑛≤𝑑,∀𝑖=1,𝑚 (3.1) hold some 𝑑≥0, then lim_𝑛→∞‖𝑧_𝑖𝑛‖=𝑑 and lim_𝑛→∞‖𝑧_𝑖𝑛−𝑧_𝑗𝑛‖=0 for any 𝑖,𝑗=1,𝑚.

Proof. Let us first prove lim_𝑛→∞‖𝑧_𝑖𝑛‖=𝑑 for any 𝑖=1,𝑚. Indeed, it follows from (3.1) that 𝑑=lim_𝑛→∞^𝑚_𝑘=1𝛼_𝑘𝑧_𝑘𝑛=liminf_𝑛→∞^𝑚_𝑘=1𝛼_𝑘𝑧_𝑘𝑛≤liminf_𝑛→∞^𝑚_𝑘=1𝛼_𝑘𝑧_𝑘𝑛≤𝛼_𝑖liminf_𝑛→∞𝑧_𝑖𝑛+_𝑘≠𝑖𝛼_𝑘limsup_𝑛→∞𝑧_𝑘𝑛≤𝛼_𝑖liminf_𝑛→∞𝑧_𝑖𝑛+1−𝛼_𝑖𝑑. (3.2) We then get that liminf_𝑛→∞‖𝑧_𝑖𝑛‖≥𝑑, which means lim_𝑛→∞‖𝑧_𝑖𝑛‖=𝑑.
Now we prove the statement lim_𝑛→∞‖𝑧_𝑖𝑛−𝑧_𝑗𝑛‖=0 by means of mathematical induction with respect to 𝑚. For 𝑚=2, the statement immediately follows from Lemma 2.2. Assume that the statement is true, for 𝑚=𝑘−1. Let us prove for 𝑚=𝑘. To do this, denote 𝑡_𝑛=11−𝛼_𝑘^𝑘−1_𝑖=1𝛼_𝑖𝑧_𝑖𝑛. (3.3) Since (1/1−𝛼_𝑘)∑^𝑘−1_𝑖=1𝛼_𝑖=1, we get limsup_𝑛→∞‖𝑡_𝑛‖≤𝑑. On the other hand, one has 𝑑=liminf_𝑛→∞^𝑘_𝑖=1𝛼_𝑖𝑧_𝑖𝑛=liminf_𝑛→∞1−𝛼_𝑘𝑡_𝑛+𝛼_𝑘𝑧_𝑘𝑛≤1−𝛼_𝑘liminf_𝑛→∞𝑡_𝑛+𝛼_𝑘limsup_𝑛→∞𝑧_𝑘𝑛≤1−𝛼_𝑘liminf_𝑛→∞𝑡_𝑛+𝛼_𝑘𝑑. (3.4) We then obtain liminf_𝑛→∞‖𝑡_𝑛‖≥𝑑 which means lim_𝑛→∞‖𝑡_𝑛‖=𝑑. In this case, according to the assumption of induction with the sequence 𝑡_𝑛, we can conclude that lim_𝑛→∞‖𝑧_𝑖𝑛−𝑧_𝑗𝑛‖=0, if 1≤𝑖,𝑗≤𝑘−1.
Since lim_𝑛→∞‖(1−𝛼_𝑘)𝑡_𝑛+𝛼_𝑘𝑧_𝑘𝑛‖=𝑑 due to Lemma 2.2, one gets lim_𝑛→∞𝑡_𝑛−𝑧_𝑘𝑛=0. (3.5) If 1≤𝑗≤𝑘−1, then the following inequality 𝑧_𝑗𝑛−𝑧_𝑘𝑛≤𝑧_𝑗𝑛−𝑡_𝑛+𝑡_𝑛−𝑧_𝑘𝑛≤11−𝛼_𝑘^𝑘−1_𝑖=1𝛼_𝑖𝑧_𝑖𝑛−𝑧_𝑗𝑛+𝑡_𝑛−𝑧_𝑘𝑛 (3.6) implies that lim_𝑛→∞‖𝑧_𝑗𝑛−𝑧_𝑘𝑛‖=0. This completes the proof.

Lemma 3.2. Let 𝑋 be a uniformly convex Banach space, and let 𝛼_∗,𝛼^∗ be two constants with 0<𝛼_∗<𝛼^∗<1. Suppose that {𝛼_𝑖𝑛}^∞_𝑛=1⊂[𝛼_∗,𝛼^∗], 𝑖=1,𝑚 are any sequences with ∑^𝑚_𝑖=1𝛼_𝑖𝑛=1 for all 𝑛∈ℕ. Suppose that {𝑧_𝑖𝑛}^∞_𝑛=1, 𝑖=1,𝑚 are sequences in 𝑋 such that lim_𝑛→∞^𝑚_𝑖=1𝛼_𝑖𝑛𝑧_𝑖𝑛=𝑑,limsup_𝑛→∞𝑧_𝑖𝑛≤𝑑,∀𝑖=1,𝑚 (3.7) hold for some 𝑑≥0, then lim_𝑛→∞‖𝑧_𝑖𝑛‖=𝑑 and lim_𝑛→∞‖𝑧_𝑖𝑛−𝑧_𝑗𝑛‖=0 for any 𝑖,𝑗=1,𝑚.

Proof. Analogously as in the proof of Lemma 3.1, it is easy to show that lim_𝑛→∞‖𝑧_𝑖𝑛‖=𝑑. Therefore, let us prove the statement lim_𝑛→∞‖𝑧_𝑖𝑛−𝑧_𝑗𝑛‖=0 for any 𝑖,𝑗=1,𝑚. Suppose to the contrary, that there exist two numbers 𝑖₀,𝑗₀ such that limsup_𝑛→∞𝑧_𝑖0𝑛−𝑧_𝑗0𝑛=𝛽_𝑖0,𝑗0>0 (3.8) then there exists a subsequence {𝑧_{𝑖0𝑛𝑘}−𝑧_{𝑗0𝑛𝑘}}^∞_𝑘=1 of {𝑧_𝑖0𝑛−𝑧_𝑗0𝑛}^∞_𝑛=1 such that lim_𝑘→∞‖𝑧_{𝑖0𝑛𝑘}−𝑧_{𝑗0𝑛𝑘}‖=𝛽_𝑖0𝑗0.
Let us consider the subsequences {𝛼_𝑖𝑛𝑘}^∞_𝑘=1 of {𝛼_𝑖𝑛}^∞_𝑛=1, here 𝑖=1,𝑚. Since {𝛼_𝑖𝑛𝑘}^∞_𝑘=1⊂[𝛼_∗,𝛼^∗], there exists a subsequence {𝑛_𝑘𝑙}^∞_𝑙=1 of {𝑛_𝑘}^∞_𝑘=1 such that lim_𝑙→∞𝛼_{𝑖𝑛𝑘𝑙}=𝛼_𝑖 for all 𝑖=1,𝑚. Since ∑^𝑚_𝑖=1𝛼_𝑖𝑛=1, for all 𝑛∈ℕ, one gets ∑^𝑚_𝑖=1𝛼_𝑖=1, and 𝛼_𝑖∈[𝛼_∗,𝛼^∗], for all 𝑖=1,𝑚. We know that 𝑑=lim_𝑙→∞^𝑚_𝑖=1𝛼_{𝑖𝑛𝑘𝑙}𝑧_{𝑖𝑛𝑘𝑙}=liminf_𝑙→∞^𝑚_𝑖=1𝛼_{𝑖𝑛𝑘𝑙}−𝛼_𝑖𝑧_{𝑖𝑛𝑘𝑙}+𝛼_𝑖𝑧_{𝑖𝑛𝑘𝑙}≤liminf_𝑙→∞^𝑚_𝑖=1𝛼_{𝑖𝑛𝑘𝑙}−𝛼_𝑖𝑧_{𝑖𝑛𝑘𝑙}+^𝑚_𝑖=1𝛼_𝑖𝑧_{𝑖𝑛𝑘𝑙}≤^𝑚_𝑖=1limsup_𝑙→∞𝛼_{𝑖𝑛𝑘𝑙}−𝛼_𝑖𝑧_{𝑖𝑛𝑘𝑙}+liminf_𝑙→∞^𝑚_𝑖=1𝛼_𝑖𝑧_{𝑖𝑛𝑘𝑙}. (3.9) It then follows that liminf_𝑙→∞‖∑^𝑚_𝑖=1𝛼_𝑖𝑧_{𝑖𝑛𝑘𝑙}‖≥𝑑. On the other hand, we have limsup_𝑙→∞^𝑚_𝑖=1𝛼_𝑖𝑧_{𝑖𝑛𝑘𝑙}≤^𝑚_𝑖=1𝛼_𝑖limsup_𝑙→∞𝑧_{𝑖𝑛𝑘𝑙}≤𝑑. (3.10) Therefore, lim_𝑙→∞‖∑^𝑚_𝑖=1𝛼_𝑖𝑧_{𝑖𝑛𝑘𝑙}‖=𝑑. Consequently, Lemma 3.1 implies that lim_𝑙→∞‖𝑧_{𝑖0𝑛𝑘𝑙}−𝑧_{𝑗0𝑛𝑘𝑙}‖=0. However, it contradicts to lim_𝑙→∞𝑧_{𝑖0𝑛𝑘𝑙}−𝑧_{𝑗0𝑛𝑘𝑙}=lim_𝑘→∞𝑧_{𝑖0𝑛𝑘}−𝑧_{𝑗0𝑛𝑘}=𝛽_𝑖0𝑗0>0. (3.11) This completes the proof.

Proposition 3.3. Let 𝑋 be a real Banach space, and let 𝐾 be a nonempty closed-convex subset of 𝑋. Let {𝑇_𝑖}^𝑚_𝑖=1: 𝐾→𝐾 be a finite family of total asymptotically 𝐼_𝑖-nonexpansive mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1, {𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚, and let {𝐼_𝑖}^𝑚_𝑖=1: 𝐾→𝐾 be a finite family of total asymptotically nonexpansive mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1, {𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚. Suppose that there exist 𝑀_𝑖,𝑀^∗_𝑖,𝑁_𝑖,𝑁^∗_𝑖>0, 𝑖=1,𝑚 such that 𝜙_𝑖(𝜉_𝑖)≤𝑀^∗_𝑖𝜉_𝑖, for all 𝜉_𝑖≥𝑀_𝑖 and 𝜑_𝑖(𝜁_𝑖)≤𝑁^∗_𝑖𝜁_𝑖 for all 𝜁_𝑖≥𝑁_𝑖, where 𝑖=1,𝑚, then the following holds for any 𝑥,𝑦∈𝐾 and for any 𝑖=1,𝑚: 𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦≤1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑥−𝑦+𝜇_𝑖𝑛𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛,𝑇^𝑛_𝑖𝑥−𝑇^𝑛_𝑖𝑦≤1+𝜇_𝑖𝑛𝑀^∗_𝑖1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑥−𝑦+𝜇_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖+𝜇_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝜆_𝑖𝑛. (3.12)(3.13)

Proof. Since 𝜙_𝑖,𝜑_𝑖: ℝ⁺→ℝ⁺ are the strictly increasing continuous functions, where 𝑖=1,𝑚, it follows that 𝜙_𝑖(𝜉_𝑖)≤𝜙_𝑖(𝑀_𝑖) and 𝜑_𝑖(𝜁_𝑖)≤𝜙_𝑖(𝑁_𝑖) whenever 𝜉_𝑖≤𝑀_𝑖 and 𝜁_𝑖≤𝑁_𝑖, where 𝑖=1,𝑚. By the hypothesis of Proposition 3.3, for all 𝜉_𝑖,𝜁_𝑖≥0 and 𝑖=1,𝑚, we then get 𝜙_𝑖𝜉_𝑖≤𝜙_𝑖𝑀_𝑖+𝑀^∗_𝑖𝜉_𝑖,𝜑_𝑖𝜁_𝑖≤𝜑_𝑖𝑁_𝑖+𝑁^∗_𝑖𝜁_𝑖. (3.14)(3.15) Since {𝑇_𝑖}^𝑚_𝑖=1∶𝐾→𝐾, {𝐼_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 are total asymptotically 𝐼_𝑖-nonexpansive and total asymptotically nonexpansive mappings, respectively, from (3.14) and (3.15), one gets 𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦≤𝑥−y+𝜇_𝑖𝑛𝜑_𝑖𝑥−𝑦+𝜆_𝑖𝑛≤𝑥−𝑦+𝜇_𝑖𝑛𝜑_𝑖𝑁_𝑖+𝑁^∗_𝑖𝑥−𝑦+𝜆_𝑖𝑛=1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑥−𝑦+𝜇_𝑖𝑛𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛,𝑇^𝑛_𝑖𝑥−𝑇^𝑛_𝑖𝑦≤𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦+𝜇_𝑖𝑛𝜙_𝑖𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦+𝜆_𝑖𝑛≤𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦+𝜇_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝑀^∗_𝑖𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦+𝜆_𝑖𝑛=1+𝜇_𝑖𝑛𝑀^∗_𝑖𝐼^𝑛_𝑖𝑥−𝐼^𝑛_𝑖𝑦+𝜇_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝜆_𝑖𝑛≤1+𝜇_𝑖𝑛𝑀^∗_𝑖1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑥−𝑦+𝜇_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖+𝜇_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝜆_𝑖𝑛. (3.16)

Lemma 3.4. Let 𝑋 be a uniformly convex real Banach space, and let 𝐾 be a nonempty closed-convex subset of 𝑋. Let {𝑇_𝑖}^𝑚_𝑖=1: 𝐾→𝐾 be a finite family of total asymptotically 𝐼_𝑖-nonexpansive mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1, {𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚, and let {𝐼_𝑖}^𝑚_𝑖=1: 𝐾→𝐾 be a finite family of total asymptotically nonexpansive mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1, {𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚, such that 𝐹∶=⋂^𝑚_𝑖=1(𝐹(𝑇_𝑖)∩𝐹(𝐼_𝑖))≠∅. Suppose that ∑^∞_𝑛=1𝜇_𝑖𝑛<∞, ∑^∞_𝑛=1𝜆_𝑖𝑛<∞, ∑^∞_𝑛=1𝜇_𝑖𝑛<∞, ∑^∞_𝑛=1𝜆_𝑖𝑛<∞ for all 𝑖=1,𝑚, and there exist 𝑀_𝑖,𝑀^∗_𝑖,𝑁_𝑖,𝑁^∗_𝑖>0, 𝑖=1,𝑚 such that 𝜙_𝑖(𝜉_𝑖)≤𝑀^∗_𝑖𝜉_𝑖, for all 𝜉_𝑖≥𝑀_𝑖 and 𝜑_𝑖(𝜁_𝑖)≤𝑁^∗_𝑖𝜁_𝑖 for all 𝜁_𝑖≥𝑁_𝑖, where 𝑖=1,𝑚. If {𝑥_𝑛} is the explicit iterative sequence defined by (1.27), then for each 𝑝∈𝐹, the limit lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists.

Proof. Since 𝐹≠∅, for any given 𝑝∈𝐹, it follows from (1.27) and (3.13) that 𝑥_𝑛+1−𝑝=1−^𝑚_𝑖=1𝛼_𝑖𝑛𝑥_𝑛−𝑝+^𝑚_𝑖=1𝛼_𝑖𝑛𝑇^𝑛_𝑖𝑦_𝑛≤−𝑝1−^𝑚_𝑖=1𝛼_𝑖𝑛𝑥_𝑛−𝑝+^𝑚_𝑖=1𝛼_𝑖𝑛𝑇^𝑛_𝑖𝑦_𝑛≤−𝑝1−^𝑚_𝑖=1𝛼_𝑖𝑛𝑥_𝑛+−𝑝_𝑚𝑖=1𝛼_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑦_𝑛+−𝑝_𝑚𝑖=1𝛼_𝑖𝑛𝜇_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖𝜑_𝑖𝑁_𝑖+𝛼_𝑖𝑛𝜆_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖+^𝑚_𝑖=1𝛼_𝑖𝑛𝜇_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝛼_𝑖𝑛𝜆_𝑖𝑛. (3.17) Again from (1.27) and (3.12), we derive that ‖𝑦_𝑛−𝑝‖=1−^𝑚_𝑖=1𝛽_𝑖𝑛𝑥_𝑛−𝑝+^𝑚_𝑖=1𝛽_𝑖𝑛𝐼^𝑛_𝑖𝑥_𝑛≤−𝑝1−^𝑚_𝑖=1𝛽_𝑖𝑛𝑥_𝑛−𝑝+^𝑚_𝑖=1𝛽_𝑖𝑛𝐼^𝑛_𝑖𝑥_𝑛=−𝑝1+^𝑚_𝑖=1𝜇_𝑖𝑛𝛽_𝑖𝑛𝑁^∗_𝑖𝑥_𝑛+−𝑝_𝑚𝑖=1𝜇_𝑖𝑛𝛽_𝑖𝑛𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛𝛽_𝑖𝑛. (3.18)
Then from (3.17) and (3.18), one finds 𝑥_𝑛+1−𝑝≤1+𝑏_𝑛𝑥_𝑛−𝑝+𝑐_𝑛. (3.19) Here 𝑏_𝑛=^𝑚_𝑖=1𝜇_𝑖𝑛𝛼_𝑖𝑛𝑀^∗_𝑖+𝜇_𝑖𝑛𝛼_𝑖𝑛𝑁^∗_𝑖+𝛼^𝑚_{𝑖𝑛𝑖=1}𝜇_𝑖𝑛𝛽_𝑖𝑛𝑁^∗_𝑖+^𝑚_𝑖=1𝜇_𝑖𝑛𝜇_𝑖𝑛𝛼_𝑖𝑛𝑀^∗_𝑖𝑁^∗_𝑖+^𝑚_𝑖=1𝜇_𝑖𝑛𝛼_𝑖𝑛𝑀^∗_𝑖⋅^𝑚_𝑖=1𝜇_𝑖𝑛𝛽_𝑖𝑛𝑁^∗_𝑖+^𝑚_𝑖=1𝜇_𝑖𝑛𝛼_𝑖𝑛𝑁^∗_𝑖⋅^𝑚_𝑖=1𝜇_𝑖𝑛𝛽_𝑖𝑛𝑁^∗_𝑖,𝑐_𝑛=^𝑚_𝑖=1𝜇_𝑖𝑛𝛽_𝑖𝑛𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛𝛽_𝑖𝑛⋅^𝑚_𝑖=1𝛼_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖1+𝜇_𝑖𝑛𝑁^∗_𝑖+^𝑚_𝑖=1𝜇_𝑖𝑛𝛼_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛𝛼_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖+^𝑚_𝑖=1𝜇_𝑖𝑛𝛼_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝜆_𝑖𝑛𝛼_𝑖𝑛. (3.20) Denoting 𝑎_𝑛=‖𝑥_𝑛−𝑝‖ in (3.19), one gets 𝑎_𝑛+1≤1+𝑏_𝑛𝑎_𝑛+𝑐_𝑛. (3.21) Since ∑^∞_𝑛=1𝑏_𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑛<∞, it follows from Lemma 2.1 the existence of the limit lim_𝑛→∞𝑎_𝑛. This means the limit lim_𝑛→∞𝑥_𝑛−𝑝=𝑑 (3.22) exists, where 𝑑≥0 is a constant. This completes the proof.

Now we prove the following result.

Theorem 3.5. Let 𝑋 be a uniformly convex real Banach space, and let 𝐾 be a nonempty closed-convex subset of 𝑋. Let {𝑇_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 be a finite family of total asymptotically 𝐼_𝑖-nonexpansive continuous mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1,{𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚, and let {𝐼_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 be a finite family of total asymptotically nonexpansive continuous mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1, {𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚, such that 𝐹∶=⋂^𝑚_𝑖=1(𝐹(𝑇_𝑖)∩𝐹(𝐼_𝑖))≠∅. Suppose that ∑^∞_𝑛=1𝜇_𝑖𝑛<∞, ∑^∞_𝑛=1𝜆_𝑖𝑛<∞, ∑^∞_𝑛=1𝜇_𝑖𝑛<∞, ∑^∞_𝑛=1𝜆_𝑖𝑛<∞ for all 𝑖=1,𝑚, and there exist 𝑀_𝑖,𝑀^∗_𝑖,𝑁_𝑖,𝑁^∗_𝑖>0,𝑖=1,𝑚 such that 𝜙_𝑖(𝜉_𝑖)≤𝑀^∗_𝑖𝜉_𝑖, for all 𝜉_𝑖≥𝑀_𝑖 and 𝜑_𝑖(𝜁_𝑖)≤𝑁^∗_𝑖𝜁_𝑖 for all 𝜁_𝑖≥𝑁_𝑖, where 𝑖=1,𝑚, then the explicit iterative sequence {𝑥_𝑛} defined by (1.27) converges strongly to a common fixed point in 𝐹 if and only if liminf_𝑛→∞𝑑𝑥_𝑛,𝐹=0. (3.23)

Proof. The necessity of condition (3.23) is obvious. Let us prove the sufficiency part of the theorem.
Since {𝑇_𝑖}^𝑚_𝑖=1,{𝐼_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 are continuous mappings, the sets 𝐹(𝑇_𝑖) and 𝐹(𝐼_𝑖) are closed. Hence, 𝐹=⋂^𝑚_𝑖=1(𝐹(𝑇_𝑖)∩𝐹(𝐼_𝑖)) is a nonempty closed set.
For any given 𝑝∈𝐹, we have (see (3.19)) 𝑥_𝑛+1−𝑝≤1+𝑏_𝑛𝑥_𝑛−𝑝+𝑐_𝑛. (3.24) Hence, one finds 𝑑𝑥_𝑛+1,𝐹≤1+𝑏_𝑛𝑑𝑥_𝑛,𝐹+𝑐_𝑛. (3.25) From (3.25) due to Lemma 2.1, we obtain the existence of the limit lim_𝑛→∞𝑑(𝑥_𝑛,𝐹). By condition (3.23), one gets lim_𝑛→∞𝑑𝑥_𝑛,𝐹=liminf_𝑛→∞𝑑𝑥_𝑛,𝐹=0. (3.26)
Let us prove that the sequence {𝑥_𝑛} converges strongly to a common fixed point in 𝐹. We first show that {𝑥_𝑛} is Cauchy sequence in 𝑋. In fact, due to 1+𝑡≤exp(𝑡) for all 𝑡>0, and from (3.24), we obtain 𝑥_𝑛+1−𝑝≤exp𝑏_𝑛𝑥_𝑛−𝑝+𝑐_𝑛. (3.27) Thus, for any positive integers 𝑚,𝑛, from (3.27) with ∑^∞_𝑛=1𝑏_𝑛<∞,∑^∞_𝑛=1𝑐_𝑛<∞, we find 𝑥_𝑛+𝑚−𝑝≤exp𝑏_{𝑛+𝑚−1}𝑥_{𝑛+𝑚−1}−𝑝+𝑐_{𝑛+𝑚−1}≤exp𝑏_{𝑛+𝑚−1}+𝑏_{𝑛+𝑚−2}𝑥_{𝑛+𝑚−2}−𝑝+𝑐_{𝑛+𝑚−1}+𝑐_{𝑛+𝑚−2}≤⋯≤exp^{𝑛+𝑚−1}_𝑖=𝑛𝑏_𝑖𝑥_𝑛−𝑝+^{𝑛+𝑚−1}_𝑖=𝑛𝑐_𝑖≤exp^∞_𝑖=𝑛𝑏_𝑖𝑥_𝑛−𝑝+^∞_𝑖=𝑛𝑐_𝑖. (3.28) Therefore, we get 𝑥_𝑛+𝑚−𝑥_𝑛≤𝑥_𝑛+𝑚−𝑝+𝑥_𝑛≤−𝑝1+exp^∞_𝑖=𝑛𝑏_𝑖𝑥_𝑛−𝑝+exp^∞_𝑖=𝑛𝑏_𝑖^∞_𝑖=𝑛𝑐_𝑖≤𝑊𝑥_𝑛−𝑝+^∞_𝑖=𝑛𝑐_𝑖, (3.29) for all 𝑝∈𝐹, where 0<𝑊−1=exp(∑^∞_𝑖=𝑛𝑏_𝑖)<∞. Taking infimum over 𝑝∈𝐹 in (3.29) gives 𝑥_𝑛+𝑚−𝑥_𝑛≤𝑊𝑑𝑥_𝑛,𝐹+^∞_𝑖=𝑛𝑐_𝑖. (3.30)
Since lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0 and ∑^∞_𝑖=1𝑐_𝑖<∞, given 𝜀>0, there exists an integer 𝑁₀>0 such that for all 𝑛>𝑁₀, we have 𝑑(𝑥_𝑛,𝐹)<(𝜀/2𝑊) and ∑^∞_𝑖=𝑛𝑐_𝑖<(𝜀/2𝑊). Consequently, for all integers 𝑛≥𝑁₀ and 𝑚≥1 and from (3.30), we derive 𝑥_𝑛+𝑚−𝑥_𝑛≤𝜀, (3.31) which means that {𝑥_𝑛} is Cauchy sequence in 𝑋, and since 𝑋 is complete, there exists 𝑥^∗∈𝑋 such that the sequence {𝑥_𝑛} converges strongly to 𝑥^∗.
Now we show that 𝑥^∗ is a common fixed point in 𝐹. Suppose for contradiction that 𝑥^∗∉𝐹. Since 𝐹 is closed subset of 𝑋, we have that 𝑑(𝑥^∗,𝐹)>0. However, for all 𝑝∈𝐹, we have 𝑥^∗−𝑝≤𝑥_𝑛−𝑥^∗+𝑥_𝑛−𝑝. (3.32) This implies that 𝑑𝑥^∗,𝐹≤𝑥_𝑛−𝑥^∗+𝑑𝑥_𝑛,𝐹, (3.33) so that as 𝑛→∞ we obtain 𝑑(𝑥^∗,𝐹)=0 which contradicts 𝑑(𝑥^∗,𝐹)>0. Hence, 𝑥^∗ is a common fixed point in 𝐹. This proves the required assertion.

To formulate and prove the main result, we need once more an auxiliary result.

Proposition 3.6. Let 𝑋 be a uniformly convex real Banach space, and let 𝐾 be a nonempty closed-convex subset of 𝑋. Let {𝑇_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 be a finite family of total asymptotically 𝐼_𝑖-nonexpansive continuous mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1,{𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚, and let {𝐼_𝑖}^𝑚_𝑖=1: 𝐾→𝐾 be a finite family of total asymptotically nonexpansive continuous mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1, {𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚, such that 𝐹∶=⋂^𝑚_𝑖=1(𝐹(𝑇_𝑖)∩𝐹(𝐼_𝑖))≠∅. Suppose that ∑^∞_𝑛=1𝜇_𝑖𝑛<∞, ∑^∞_𝑛=1𝜆_𝑖𝑛<∞,∑^∞_𝑛=1𝜇_𝑖𝑛<∞, ∑^∞_𝑛=1𝜆_𝑖𝑛<∞ for all 𝑖=1,𝑚, and {𝛼_𝑗𝑛}^∞_𝑛=1,{𝛽_𝑗𝑛}^∞_𝑛=1 are sequences with {𝛼_𝑗𝑛}^∞_𝑛=1⊂[𝛼_∗,𝛼^∗] and {𝛽_𝑗𝑛}^∞_𝑛=1⊂[𝛽_∗,𝛽^∗], for all 𝑗=0,𝑚, here 0<𝛼_∗<𝛼^∗<1, 0<𝛽_∗<𝛽^∗<1, and there exist 𝑀_𝑖,𝑀^∗_𝑖,𝑁_𝑖,𝑁^∗_𝑖>0, 𝑖=1,𝑚 such that 𝜙_𝑖(𝜉_𝑖)≤𝑀^∗_𝑖𝜉_𝑖, for all 𝜉_𝑖≥𝑀_𝑖 and 𝜑_𝑖(𝜁_𝑖)≤𝑁^∗_𝑖𝜁_𝑖 for all 𝜁_𝑖≥𝑁_𝑖, where 𝑖=1,𝑚. then the explicit iterative sequence {𝑥_𝑛} defined by (1.27) satisfies the following: lim_𝑛→∞𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝐼^𝑛_𝑖𝑥_𝑛=0, (3.34)(3.35) for all 𝑖=1,𝑚.

Proof. According to Lemma 3.4 for any 𝑝∈𝐹, we have lim_𝑛→∞‖𝑥_𝑛−𝑝‖=𝑑. It follows from (1.27) that 𝑥_𝑛+1−𝑝=𝛼_0𝑛𝑥_𝑛−𝑝+^𝑚_𝑖=1𝛼_𝑖𝑛𝑇^𝑛_𝑖𝑦_𝑛−𝑝⟶𝑑, (3.36) as 𝑛→∞. By means of ∑^∞_𝑛=1𝜇_𝑖𝑛<∞, ∑^∞_𝑛=1𝜆_𝑖𝑛<∞,∑^∞_𝑛=1𝜇_𝑖𝑛<∞,∑^∞_𝑛=1𝜆_𝑖𝑛<∞, for all 𝑖=1,𝑚, from (3.18), one yields that limsup_𝑛→∞𝑦_𝑛−𝑝≤limsup_𝑛→∞1+^𝑚_𝑖=1𝜇_𝑖𝑛𝛽_𝑖𝑛𝑁^∗_𝑖𝑥_𝑛−𝑝+limsup_𝑛→∞^𝑚_𝑖=1𝜇_𝑖𝑛𝛽_𝑖𝑛𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛𝛽_𝑖𝑛=limsup_𝑛→∞𝑥_𝑛−𝑝=𝑑, (3.37) and from (3.13), (3.37), we have limsup_𝑛→∞𝑇^𝑛_𝑖𝑦_𝑛−𝑝≤limsup_𝑛→∞1+𝜇_𝑖𝑛𝑀^∗_𝑖1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑦_𝑛−𝑝+limsup_𝑛→∞𝜇_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖𝜑_𝑖𝑁_𝑖+limsup_𝑛→∞𝜆_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖+𝜇_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝜆_𝑖𝑛≤𝑑, (3.38) for all 𝑖=1,𝑚. Now using limsup_𝑛→∞𝑥_𝑛−𝑝=𝑑, (3.39) with (3.38) and applying Lemma 3.2 to (3.36), one finds lim_𝑛→∞𝑥_𝑛−𝑇^𝑛_𝑖𝑦_𝑛=0, (3.40) for all 𝑖=1,𝑚. Now from (1.27) and (3.40), we infer that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞^𝑚_𝑖=1𝛼_𝑖𝑛𝑇^𝑛_𝑖𝑦_𝑛−𝑥_𝑛=0. (3.41) On the other hand, from (3.13), we have 𝑥_𝑛−𝑝≤𝑥_𝑛−𝑇^𝑛_𝑖𝑦_𝑛+𝑇^𝑛_𝑖𝑦_𝑛≤𝑥−𝑝_𝑛−𝑇^𝑛_𝑖𝑦_𝑛+1+𝜇_𝑖𝑛𝑀^∗_𝑖1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑦_𝑛−𝑝+𝜇_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖+𝜇_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝜆_𝑖𝑛, (3.42) which implies 𝑥_𝑛−𝑝−𝑥_𝑛−𝑇^𝑛_𝑖𝑦_𝑛≤1+𝜇_𝑖𝑛𝑀^∗_𝑖1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑦_𝑛−𝑝+𝜇_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛1+𝜇_𝑖𝑛𝑀^∗_𝑖+𝜇_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝜆_𝑖𝑛. (3.43) The last inequality with (3.22), (3.40) yields liminf_𝑛→∞𝑦_𝑛−𝑝=𝑑. (3.44) Combining (3.44) with (3.37), we get lim_𝑛→∞𝑦_𝑛−𝑝=𝑑. (3.45) Again from (1.27), we can see that 𝑦_𝑛−𝑝=𝛽_0𝑛𝑥_𝑛−𝑝+^𝑚_𝑖=1𝛽_𝑖𝑛𝐼^𝑛_𝑖𝑥_𝑛−𝑝⟶𝑑,𝑛⟶∞. (3.46) From (3.12) and (3.22), one finds limsup_𝑛→∞𝐼^𝑛_𝑖𝑥_𝑛−𝑝≤limsup_𝑛→∞1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑥_𝑛−𝑝+𝜇_𝑖𝑛𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛=𝑑 (3.47) for all 𝑖=1,𝑚. Now applying Lemma 3.2 to (3.46), we obtain lim_𝑛→∞𝑥_𝑛−𝐼^𝑛_𝑖𝑥_𝑛=0, (3.48) for all 𝑖=1,𝑚. We then have lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=lim_𝑛→∞^𝑚_𝑖=1𝛽_𝑖𝑛𝐼^𝑛_𝑖𝑥_𝑛−𝑥_𝑛=0. (3.49) Consider 𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛≤𝑥_𝑛−𝑇^𝑛_𝑖𝑦_𝑛+𝑇^𝑛_𝑖𝑦_𝑛−𝑇^𝑛_𝑖𝑥_𝑛≤𝑥_𝑛−𝑇^𝑛_𝑖𝑦_𝑛+1+𝜇_𝑖𝑛𝑀^∗_𝑖1+𝜇_𝑖𝑛𝑁^∗_𝑖𝑦_𝑛−𝑥_𝑛+1+𝜇_𝑖𝑛𝑀^∗_𝑖𝜇_𝑖𝑛𝜑_𝑖𝑁_𝑖+𝜆_𝑖𝑛+𝜇_𝑖𝑛𝜙_𝑖𝑀_𝑖+𝜆_𝑖𝑛, (3.50) for all 𝑖=1,𝑚. Then from (3.40) and (3.49), we get lim_𝑛→∞𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛=0, (3.51) for all 𝑖=1,𝑚.

Now we are ready to formulate a main result concerning strong convergence of the sequence {𝑥_𝑛}.

Theorem 3.7. Let 𝑋 be a uniformly convex real Banach space, and let 𝐾 be a nonempty closed-convex subset of 𝑋. Let {𝑇_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 be a finite family of total asymptotically 𝐼_𝑖-nonexpansive continuous mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1,{𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚, and let {𝐼_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 be a finite family of total asymptotically nonexpansive continuous mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1, {𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚, such that 𝐹∶=⋂^𝑚_𝑖=1(𝐹(𝑇_𝑖)∩𝐹(𝐼_𝑖))≠∅. Suppose that ∑^∞_𝑛=1𝜇_𝑖𝑛<∞, ∑^∞_𝑛=1𝜆_𝑖𝑛<∞, ∑^∞_𝑛=1𝜇_𝑖𝑛<∞, ∑^∞_𝑛=1𝜆_𝑖𝑛<∞ for all 𝑖=1,𝑚, and {𝛼_𝑗𝑛}^∞_𝑛=1,{𝛽_𝑗𝑛}^∞_𝑛=1 are sequences with {𝛼_𝑗𝑛}^∞_𝑛=1⊂[𝛼_∗,𝛼^∗] and {𝛽_𝑗𝑛}^∞_𝑛=1⊂[𝛽_∗,𝛽^∗], for all 𝑗=0,𝑚, here 0<𝛼_∗<𝛼^∗<1, 0<𝛽_∗<𝛽^∗<1, and there exist 𝑀_𝑖,𝑀^∗_𝑖,𝑁_𝑖,𝑁^∗_𝑖>0, 𝑖=1,𝑚 such that 𝜙_𝑖(𝜉_𝑖)≤𝑀^∗_𝑖𝜉_𝑖, for all 𝜉_𝑖≥𝑀_𝑖 and 𝜑_𝑖(𝜁_𝑖)≤𝑁^∗_𝑖𝜁_𝑖 for all 𝜁_𝑖≥𝑁_𝑖, where 𝑖=1,𝑚. If at least one mapping of the mappings {𝑇_𝑖}^𝑚_𝑖=1 and {𝐼_𝑖}^𝑚_𝑖=1 is compact, then the explicitly iterative sequence {𝑥_𝑛} defined by (1.27) converges strongly to a common fixed point of {𝑇_𝑖}^𝑚_𝑖=1 and {𝐼_𝑖}^𝑚_𝑖=1.

Proof. Without any loss of generality, we may assume that 𝑇₁ is compact. This means that there exists a subsequence {𝑇^𝑛𝑘₁𝑥_𝑛𝑘}^∞_𝑘=1 of {𝑇^𝑛₁𝑥_𝑛}^∞_𝑛=1 such that {𝑇^𝑛𝑘₁𝑥_𝑛𝑘}^∞_𝑘=1 converges strongly to 𝑥^∗∈𝐾. Then from (3.34), we have that {𝑥_𝑛𝑘}^∞_𝑘=1 converges strongly to 𝑥^∗. Also from (3.34), we obtain that {𝑇^𝑛𝑘_𝑖𝑥_𝑛𝑘}^∞_𝑘=1 converges strongly to 𝑥^∗, for all 𝑖=2,𝑚. Since {𝑇_𝑖}^𝑚_𝑖=1 are continuous mappings, so {𝑇^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘}^∞_𝑘=1 converges strongly to 𝑇_𝑖𝑥^∗, for all 𝑖=1,𝑚. On the other hand, from (3.35) and continuousness of {𝐼_𝑖}^𝑚_𝑖=1, we obtain that {𝐼^𝑛𝑘_𝑖𝑥_𝑛𝑘}^∞_𝑘=1 converges strongly to 𝑥^∗, and {𝐼^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘}^∞_𝑘=1 converges strongly to 𝐼_𝑖𝑥^∗, for all 𝑖=1,𝑚. Due to (3.41), {‖𝑥_𝑛𝑘+1−𝑥_𝑛𝑘‖} converges to 0, as 𝑘→∞. Then, {𝑥_𝑛𝑘+1}^∞_𝑘=1 converges strongly to 𝑥^∗ and moreover, (3.13) and (3.12) imply that {‖𝑇^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+1−𝑇^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘‖} and {‖𝐼^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+1−𝐼^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘‖} converge to 0, as 𝑘→∞, for all 𝑖=1,𝑚. From (3.34), (3.35), it yields that ‖𝑥_𝑛𝑘+1−𝑇^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+1‖ and ‖𝑥_𝑛𝑘+1−𝐼^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+1‖ converge to 0 as 𝑘→∞, for all 𝑖=1,𝑚. Observe that 𝑥^∗−𝑇_𝑖𝑥^∗≤𝑥^∗−𝑥_𝑛𝑘+1+𝑥_𝑛𝑘+1−𝑇^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+1+𝑇^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+1−𝑇^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+𝑇^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘−𝑇_𝑖𝑥^∗,𝑥^∗−𝐼_𝑖𝑥^∗≤𝑥^∗−𝑥_𝑛𝑘+1+𝑥_𝑛𝑘+1−𝐼^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+1+𝐼^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+1−𝐼^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘+𝐼^𝑛𝑘_𝑖⁺¹𝑥_𝑛𝑘−𝐼_𝑖𝑥^∗, (3.52) for all 𝑖=1,𝑚. Taking limit as 𝑘→∞, we have that 𝑥^∗=𝑇_𝑖𝑥^∗ and 𝑥^∗=𝐼_𝑖𝑥^∗, for all 𝑖=1,𝑚, which means 𝑥^∗∈𝐹. However, by Lemma 3.4, the limit lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖ exists, then lim_𝑛→∞𝑥_𝑛−𝑥^∗=lim_{𝑛𝑘→∞}𝑥_𝑛𝑘−𝑥^∗=0, (3.53) which means {𝑥_𝑛} converges strongly to 𝑥^∗∈𝐹. This completes the proof.

Remark 3.8. If one has that all 𝐼_𝑖 are identity mappings, then the obtained results recover and correctly prove the main result of [24].

Remark 3.9. Suppose that we are given two families {𝑇_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 and {𝑆_𝑖}^𝑚_𝑖=1∶𝐾→𝐾 of total asymptotically nonexpansive continuous mappings such that ⋂^𝑚_𝑖=1(𝐹(𝑇_𝑖)∩𝐹(𝑆_𝑖))≠∅. Define the following explicit iterative process: 𝑥₀𝑥∈𝐾,_𝑛+1=𝛼_0𝑛𝑥_𝑛+^𝑚_𝑖=1𝛼_𝑖𝑛𝑇^𝑛_𝑖𝑦_𝑛+𝛼_𝑚+1,𝑛𝑢_𝑛,𝑦_𝑛=𝛽_0𝑛𝑥_𝑛+^𝑚_𝑖=1𝛽_𝑖𝑛𝑆^𝑛_𝑖𝑥_𝑛+𝛽_𝑚+1,𝑛𝑣_𝑛, (3.54) such that ∑^𝑚+1_𝑗=0𝛼_𝑗𝑛=1 and ∑^𝑚+1_𝑗=0𝛽_𝑗𝑛=1.

Under suitable conditions, by the same argument and methods used above, one can prove, with either little mirror or no modifications, the strong convergence of the explicit iterative process defined by (3.54) to a common fixed point of the given families.

Remark 3.10. Let {𝑇_𝑖}^𝑚_𝑖=1: 𝐾→𝐾 be a finite family of total asymptotically nonexpansive continuous mappings with sequences {𝜇_𝑖𝑛}^∞_𝑛=1, {𝜆_𝑖𝑛}^∞_𝑛=1, where 𝑖=1,𝑚. It is clear for each operator 𝑇_𝑖 that one has 𝑇^𝑛_𝑖𝑥−𝑇^𝑛_𝑖𝑦≤𝑇^𝑛_𝑖𝑥−𝑇^𝑛_𝑖𝑦+𝜇_𝑖n𝑇^𝑛_𝑖𝑥−𝑇^𝑛_𝑖𝑦, (3.55) and this means that 𝑇_𝑖 is total asymptotically 𝑇_𝑖-nonexpansive mappings with sequence {𝜇_𝑖𝑛}^∞_𝑛=1 and the function 𝜙(𝜆)=𝜆. Hence, our iteration scheme can be written as follows: 𝑥₀𝑥∈𝐾,_𝑛+1=𝛼_0𝑛𝑥_𝑛+^𝑚_𝑖=1𝛼_𝑖𝑛𝑇^𝑛_𝑖𝑦_𝑛,𝑦_𝑛=𝛽_0𝑛𝑥_𝑛+^𝑚_𝑖=1𝛽_𝑖𝑛𝑇^𝑛_𝑖𝑥_𝑛, (3.56) where {𝛼_𝑗𝑛}^∞_𝑛=1,{𝛽_𝑗𝑛}^∞_𝑛=1 in (0,1), (𝑗=0,𝑚) with ∑^𝑚_𝑗=0𝛼_𝑗𝑛=1, ∑^𝑚_𝑗=0𝛽_𝑗𝑛=1.

The defined scheme is a new iterative method generalizing one given in [24]. So, according to our main results for the defined sequence {𝑥_𝑛} (see (3.56)), we obtain strong convergence theorems. On the other hand, playing with numbers {𝛼_𝑗𝑛}^∞_𝑛=1,{𝛽_𝑗𝑛}^∞_𝑛=1 and by means of the defined method, one may introduce lots of different schemes. All of the them strongly converge to a common fixed point of {𝑇_𝑖}^𝑚_𝑖=1. Moreover, the recursion formula (3.56) is much simpler than the others studied earlier for this problem [21, 23, 29, 30, 32, 35–38]. Therefore, all presented results here generalize, unify, and extend the corresponding main results of the mentioned papers. Note that one can consider the method (1.27) with errors, and all the theorems could be carried over for such iteration scheme as well with little or no modifications.

We stress that all the theorems of this paper carry over to the class of total asymptotically quasi-𝐼-nonexpansive mappings (see [24]), [39] with little or no modifications.

Acknowledgments

A part of this work was done at the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy. The first named author (F. Mukhamedov) thanks the ICTP for providing financial support during his visit as a Junior Associate at the centre. The authors also acknowledge the Malaysian Ministry of Science, Technology and Innovation Grant no. 01-01-08-SF0079.