On Asymptotically Quasi-𝜙-Nonexpansive Mappings in the Intermediate Sense

Abstract

A projection iterative process is investigated for the class of asymptotically quasi-𝜙-nonexpansive mappings in the intermediate sense. Strong convergence theorems of common fixed points of a family of asymptotically quasi-𝜙-nonexpansive mappings in the intermediate sense are established in the framework of Banach spaces.

1. Introduction

Fixed point theory as an important branch of nonlinear analysis theory has been applied in many disciplines, including economics, image recovery, mechanics, quantum physics, and control theory; see, for example, [1–4]. The theory itself is a beautiful mixture of analysis, topology, and geometry. During the four decades, many famous existence theorems of fixed points were established; see, for example, [5–13]. However, from the standpoint of real world applications it is not only to know the existence of fixed points of nonlinear mappings but also to be able to construct an iterative algorithm to approximate their fixed points. The computation of fixed points is important in the study of many real world problems, including inverse problems; for instance, it is not hard to show that the split feasibility problem and the convex feasibility problem in signal processing and image reconstruction can both be formulated as a problem of finding fixed points of certain operators, respectively; see [14–16] for more details and the reference therein. Iterative methods play an important role in the computation of fixed points of nonlinear mappings. Indeed, many well-known problems can be studied by using algorithms which are iterative in their nature.

In this paper, we introduced a new class of nonlinear mappings: asymptotically quasi-𝜙-nonexpansive mappings in the intermediate sense and considered the problem of approximating a common fixed point of a family of the mappings based on a projection iterative process.

The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, strong convergence of a projection iterative algorithm is obtained in a reflexive, strictly convex, and smooth Banach space such that both 𝐸 and 𝐸^∗  enjoy Kadec-Klee property. Some corollaries as the immediate results of main results are given.

2. Preliminaries

Let 𝐻 be a real Hilbert space, 𝐶 a nonempty subset of 𝐻, and 𝑇∶𝐶→𝐶 a mapping. The symbol 𝐹(𝑇) stands for the fixed point set of 𝑇. Recall the following. 𝑇 is said to be nonexpansive if and only if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.1)

𝑇is said to be quasi-nonexpansive if and only if 𝐹(𝑇)≠∅, and 𝑝−𝑇𝑦≤𝑝−𝑦,∀𝑝∈𝐹(𝑇),∀𝑦∈𝐶. (2.2)

We remark here that a nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive: however, the inverse may be not true. See the following example [17].

Example 2.1. Let 𝐻=ℝ¹ and define a mapping by 𝑇∶𝐻→𝐻 by 𝑥𝑇𝑥=21sin𝑥,if𝑥≠0,0,if𝑥=0. (2.3) Then 𝑇 is quasi-nonexpansive but not nonexpansive.

𝑇 is said to be asymptotically nonexpansive if and only if there exists a sequence {𝜇_𝑛}⊂[0,∞) with 𝜇_𝑛→0 as 𝑛→∞ such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤1+𝜇_𝑛𝑥−𝑦,∀𝑥,𝑦∈𝐶,∀𝑛≥1. (2.4) It is easy to see that a nonexpansive mapping is an asymptotically nonexpansive mapping with the sequence {1}. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [7]. Since 1972, a host of authors have studied the convergence of iterative algorithms for such a class of mappings.

𝑇 is said to be asymptotically quasi-nonexpansive if and only if 𝐹(𝑇)≠∅, and there exists a sequence {𝜇_𝑛}⊂[0,∞) with 𝜇_𝑛→0 as 𝑛→∞ such that 𝑝−𝑇^𝑛𝑦≤1+𝜇_𝑛𝑝−𝑦,∀𝑝∈𝐹(𝑇),∀𝑦∈𝐶,∀𝑛≥1. (2.5) It is easy to see that a quasi-nonexpansive mapping is an asymptotically quasi-nonexpansive mapping with the sequence {1}.

𝑇 is said to be asymptotically nonexpansive in the intermediate sense if and only if it is continuous, and the following inequality holds: limsup_𝑛→∞sup_{𝑥,𝑦∈𝐶}𝑇^𝑛𝑥−𝑇^𝑛𝑦−𝑥−𝑦≤0. (2.6)

𝑇 is said to be asymptotically quasi-nonexpansive in the intermediate sense if and only if 𝐹(𝑇)≠∅ and the following inequality holds: limsup_𝑛→∞sup_{𝑝∈𝐹(𝑇),𝑦∈𝐶}𝑝−𝑇^𝑛𝑦−𝑝−𝑦≤0. (2.7)

The class of mappings which are asymptotically nonexpansive in the intermediate sense was considered by Bruck et al. [18]. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense may not be Lipschitz continuous. However, asymptotically nonexpansive mappings are Lipschitz continuous.

In what follows, we always assume that 𝐸 is a Banach space with the dual space 𝐸^∗. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. We use the symbol 𝐽 to stand for the normalized duality mapping from 𝐸 to 2^𝐸∗  defined by 𝐽𝑥=𝑓^∗∈𝐸^∗∶𝑥,𝑓^∗=‖𝑥‖²=𝑓^∗2,∀𝑥∈𝐸, (2.8) where ⟨⋅,⋅⟩ denotes the generalized duality pairing of elements between 𝐸 and 𝐸^∗. It is well known that if 𝐸^∗ is strictly convex, then 𝐽 is single valued; if 𝐸^∗ is reflexive and smooth, then 𝐽 is single valued and demicontinuous; see [19] for more details and the references therein.

It is also well known that if 𝐷 is a nonempty, closed, and convex subset of a Hilbert space 𝐻, and 𝑃_𝐶∶𝐻→𝐷 is the metric projection from 𝐻 onto 𝐷, then 𝑃_𝐷 is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [20] introduced a generalized projection operator in Banach spaces which is an analogue of the metric projection in Hilbert spaces.

Recall that a Banach space 𝐸 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1, and 𝑥≠𝑦. It is said to be uniformly convex if lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 for any two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝐸 such that ‖𝑥_𝑛‖=‖𝑦_𝑛‖=1 and lim_𝑛→∞‖(𝑥_𝑛+𝑦_𝑛)/2‖=1. Let 𝑈_𝐸={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. Then the Banach space 𝐸 is said to be smooth provided lim_𝑡→0((‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡) exists for all 𝑥,𝑦∈𝑈_𝐸. It is also said to be uniformly smooth if the limit is attained uniformly for all 𝑥,𝑦∈𝑈_𝐸.

Recall that a Banach space 𝐸 enjoys Kadec-Klee property if for any sequence {𝑥_𝑛}⊂𝐸 and 𝑥∈𝐸 with 𝑥_𝑛⇀𝑥, and ‖𝑥_𝑛‖→‖𝑥‖, then ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞. For more details on Kadec-Klee property, the readers can refer to [19, 21] and the references therein. It is well known that if 𝐸 is a uniformly convex Banach spaces, then 𝐸 enjoys Kadec-Klee property.

Let 𝐸 be a smooth Banach space. Consider the functional defined by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐸. (2.9) Notice that, in a Hilbert space 𝐻, (2.9) is reduced to 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖² for all 𝑥,𝑦∈𝐻. The generalized projection Π_𝐶∶𝐸→𝐶 is a mapping that assigns to an arbitrary point 𝑥∈𝐸, the minimum point of the functional 𝜙(𝑥,𝑦); that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the following minimization problem: 𝜙(𝑥,𝑥)=min_𝑦∈𝐶𝜙𝑦,𝑥. (2.10) The existence and uniqueness of the operator Π_𝐶 follow from the properties of the functional 𝜙(𝑥,𝑦) and the strict monotonicity of the mapping 𝐽; see, for example, [19, 20]. In Hilbert spaces, Π_𝐶=𝑃_𝐶. It is obvious from the definition of the function 𝜙 that 𝑦−‖𝑥‖²≤𝜙𝑦,𝑥≤𝑦+‖𝑥‖²,∀𝑥,𝑦∈𝐸,𝜙𝑥,𝑦=𝜙(𝑥,𝑧)+𝜙𝑧,𝑦+2𝑥−𝑧,𝐽𝑧−𝐽𝑦,∀𝑥,𝑦,𝑧∈𝐸. (2.11)(2.12)

Remark 2.2. If 𝐸 is a reflexive, strictly convex, and smooth Banach space, then, for all 𝑥,𝑦∈𝐸, 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦. It is sufficient to show that if 𝜙(𝑥,𝑦)=0, then 𝑥=𝑦. From (2.11), we have ‖𝑥‖=‖𝑦‖. This implies that ⟨𝑥,𝐽𝑦⟩=‖𝑥‖²=‖𝐽𝑦‖². From the definition of 𝐽, we see that 𝐽𝑥=𝐽𝑦. It follows that 𝑥=𝑦; see [20] for more details.

Next, we recall the following.

(1) A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [22] if and only if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇).

(2) 𝑇 is said to be relatively nonexpansive if and only if 𝐹(𝑇)=𝐹(𝑇)≠∅,𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,∀𝑝∈𝐹(𝑇). (2.13) The asymptotic behavior of relatively nonexpansive mappings was studied in [23, 24].

(3) 𝑇 is said to be relatively asymptotically nonexpansive if and only if 𝐹(𝑇)=𝐹(𝑇)≠∅,𝜙𝑝,𝑇^𝑛𝑥≤1+𝜇_𝑛𝜙𝑝,𝑥,∀𝑥∈𝐶,∀𝑝∈𝐹(𝑇),∀𝑛≥1, (2.14) where {𝜇_𝑛}⊂[0,∞) is a sequence such that 𝜇_𝑛→1 as 𝑛→∞.

Remark 2.3. The class of relatively asymptotically nonexpansive mappings was first considered in Su and Qin [25]; see also, Agarwal et al. [26], and Qin et al. [27].
(4) 𝑇 is said to be quasi-𝜙-nonexpansive if and only if 𝐹(𝑇)≠∅,𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,∀𝑝∈𝐹(𝑇). (2.15)

(5) 𝑇 is said to be asymptotically quasi-𝜙-nonexpansive if and only if there exists a sequence {𝜇_𝑛}⊂[0,∞) with 𝜇_𝑛→0 as 𝑛→∞ such that 𝐹(𝑇)≠∅,𝜙𝑝,𝑇^𝑛𝑥≤1+𝜇_𝑛𝜙𝑝,𝑥,∀𝑥∈𝐶,∀𝑝∈𝐹(𝑇),∀𝑛≥1. (2.16)

Remark 2.4. The class of quasi-𝜙-nonexpansive mappings and the class of asymptotically quasi-𝜙-nonexpansive mappings were first considered in Zhou et al. [28]; see also Qin et al. [29], Qin, and Agarwal [30], Qin et al. [31], Qin et al. [32], and Qin et al. [33].

Remark 2.5. The class of quasi-𝜙-nonexpansive mappings and the class of asymptotically quasi-𝜙-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-𝜙-nonexpansive mappings and asymptotically quasi-𝜙-nonexpansive do not require 𝐹(𝑇)=𝐹(𝑇).

Remark 2.6. The class of quasi-𝜙-nonexpansive mappings and the class of asymptotically quasi-𝜙-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.

In this paper, based on asymptotically (quasi-) nonexpansive mappings in the intermediate sense which was first considered by Bruck et al. [18], we introduce and consider the following new nonlinear mapping: asymptotically (quasi-) 𝜙-nonexpansive mappings in the intermediate sense.

(6) 𝑇 is said to be an asymptotically 𝜙-nonexpansive mapping in the intermediate sense if and only if limsup_𝑛→∞sup_{𝑥,𝑦∈𝐶}𝜙𝑇^𝑛𝑥,𝑇^𝑛𝑦−𝜙𝑥,𝑦≤0. (2.17)

(7) 𝑇 is said to be an asymptotically quasi-𝜙-nonexpansive mapping in the intermediate sense if and only if 𝐹(𝑇)≠∅, and limsup_𝑛→∞sup_{(𝑇)𝑝∈𝐹,𝑥∈𝐶}𝜙𝑝,𝑇^𝑛𝑥−𝜙𝑝,𝑥≤0. (2.18)

Remark 2.7. The class of asymptotically (quasi-)  𝜙-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically (quasi-) nonexpansive mappings in the intermediate sense in the framework of Banach spaces.

Let 𝐸=ℝ¹ and 𝐶=[0,1]. Define the following mapping 𝑇∶𝐶→𝐶 by 1𝑇𝑥=21𝑥,𝑥∈0,2,10,𝑥∈2,1. (2.19) Then 𝑇 is an asymptotically 𝜙-nonexpansive mapping in the intermediate sense with the fixed point set {0}. We also have the following: 𝜙𝑇^𝑛𝑥,𝑇^𝑛𝑦=𝑇^𝑛𝑥−𝑇^𝑛𝑦²=12^2𝑛𝑥−𝑦²≤𝑥−𝑦²1=𝜙𝑥,𝑦,∀𝑥,𝑦∈0,2,𝜙𝑇^𝑛𝑥,𝑇^𝑛𝑦=𝑇^𝑛𝑥−𝑇^𝑛𝑦²=0≤𝑥−𝑦²1=𝜙𝑥,𝑦,∀𝑥,𝑦∈2,𝜙𝑇,1_𝑛𝑥,𝑇^𝑛𝑦=𝑇^𝑛𝑥−𝑇^𝑛𝑦²=12^𝑛𝑥−0²≤12^𝑛+1𝑥−𝑦2^𝑛𝑦²≤+1𝑥−𝑦2^𝑛2≤𝑥−𝑦²+𝜉_𝑛=𝜙𝑥,𝑦+𝜉_𝑛1,∀𝑥∈0,21,∀𝑦∈2,1, (2.20) where 𝜉_𝑛=1/2^2𝑛+1/2^𝑛−1. Hence, we have 𝜙𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝜙𝑥,𝑦+𝜉_𝑛,∀𝑥,𝑦∈[0,1]. (2.21) (8) The mapping 𝑇 is said to be asymptotically regular on 𝐶 if and only if lim_𝑛→∞sup_𝑥∈𝐶𝑇^𝑛+1𝑥−𝑇^𝑛𝑥=0. (2.22)

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

Lemma 2.8   (see [20]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth Banach space 𝐸, and 𝑥∈𝐸. Then 𝑥₀=Π_𝐶𝑥 if and only if 𝑥₀−𝑦,𝐽𝑥−𝐽𝑥₀≥0,∀𝑦∈𝐶. (2.23)

Lemma 2.9 (see [20]). Let 𝐸 be a reflexive, strictly, convex, and smooth Banach space, 𝐶 a nonempty, closed, and convex subset of 𝐸, and 𝑥∈𝐸. Then 𝜙𝑦,Π_𝐶𝑥+𝜙Π_𝐶𝑥,𝑥≤𝜙𝑦,𝑥,∀𝑦∈𝐶. (2.24)

3. Main Results

Theorem 3.1. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space such that both 𝐸 and 𝐸^∗ have Kadec-Klee property. Let 𝐶 be a nonempty, bounded closed, and convex subset of 𝐸. Let Δ be an index set, and 𝑇_𝑖∶𝐶→𝐶 a closed asymptotically quasi-𝜙-nonexpansive mapping in the intermediate sense, for every 𝑖∈Δ. Assume that ⋂_𝑖∈Δ𝐹(𝑇_𝑖) is nonempty, and 𝑇_𝑖 is asymptotically regular, for every 𝑖∈Δ. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀𝐶∈𝐸,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_(1,𝑖)𝐶=𝐶,₁=_𝑖∈Δ𝐶_(1,𝑖),𝑥₁=Π_𝐶1𝑥₀,𝐶_{(𝑛+1,𝑖)}=𝑢∈𝐶_(𝑛,𝑖)∶𝜙𝑥_𝑛,𝑇^𝑛_𝑖𝑥_𝑛≤2𝑥_𝑛−𝑢,𝐽𝑥_𝑛−𝐽𝑇^𝑛_𝑖𝑥_𝑛+𝜉_(𝑛,𝑖),𝐶_𝑛+1=_𝑖∈Δ𝐶_{(𝑛+1,𝑖)},𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.1) where 𝜉_(𝑛,𝑖)=max{0,sup_{𝑝∈𝐹(𝑇𝑖),𝑥∈𝐶}(𝜙(𝑝,𝑇^𝑛_𝑖𝑥)−𝜙(𝑝,𝑥))}. Then {𝑥_𝑛} converges strongly to Π_{⋂𝑖∈Δ𝐹(𝑇𝑖)}𝑥₀, where Π_{⋂𝑖∈Δ𝐹(𝑇𝑖)} stands for the generalized projection from 𝐸 onto ⋂_𝑖∈Δ𝐹(𝑇_𝑖).

Proof. The proof is split into the following 5 steps.
Step 1. It show that ⋂_𝑖∈Δ𝐹(𝑇) is closed and convex.
Since 𝑇_𝑖 is closed, we can easily conclude that 𝐹(𝑇_𝑖) is closed. The proof that ⋂_𝑖∈Δ𝐹(𝑇) is closed. We only prove that ⋂_𝑖∈Δ𝐹(𝑇_𝑖) is convex. Let 𝑝_1,𝑖,𝑝_2,𝑖∈𝐹(𝑇_𝑖) and 𝑝_𝑖=𝑡_𝑖𝑝_1,𝑖+(1−𝑡_𝑖)𝑝_2,𝑖, where 𝑡_𝑖∈(0,1), for every 𝑖∈Δ. We see that 𝑝_𝑖=𝑇_𝑖𝑝_𝑖. Indeed, we see from the definition of 𝑇_𝑖 that 𝜙𝑝_1,𝑖,𝑇^𝑛_𝑖𝑝_𝑖≤𝜙𝑝_1,𝑖,𝑝_𝑖+𝜉_(𝑛,𝑖),𝜙𝑝_2,𝑖,𝑇^𝑛_𝑖𝑝_𝑖≤𝜙𝑝_(2,𝑖),𝑝_𝑖+𝜉_(𝑛,𝑖). (3.2) In view of (2.12), we obtain (3.2) that 𝜙𝑝_𝑖,𝑇^𝑛_𝑖𝑝_𝑖≤2𝑝_𝑖−𝑝_1,𝑖,𝐽𝑝_𝑖−𝐽𝑇^𝑛_𝑖𝑝_𝑖+𝜉_(𝑛,𝑖),𝜙𝑝_𝑖,𝑇^𝑛_𝑖𝑝_𝑖≤2𝑝_𝑖−𝑝_2,𝑖,𝐽𝑝_𝑖−𝐽𝑇^𝑛_𝑖𝑝_𝑖+𝜉_(𝑛,𝑖). (3.3) Multiplying 𝑡_𝑖 and (1−𝑡_𝑖) on the both sides of (3.3), respectively, yields that 𝜙(𝑝_𝑖,𝑇^𝑛_𝑖𝑝_𝑖)≤𝜉_(𝑛,𝑖). This implies that lim_𝑛→∞𝜙𝑝_𝑖,𝑇^𝑛_𝑖𝑝_𝑖=0. (3.4) In light of (2.11), we arrive at lim_𝑛→∞𝑇^𝑛_𝑖𝑝_𝑖=𝑝_𝑖. (3.5) It follows that lim_𝑛→∞𝐽𝑇^𝑛_𝑖𝑝_𝑖=𝐽𝑝_𝑖. (3.6) Since 𝐸^∗  is reflexive, we may, without loss of generality, assume that 𝐽(𝑇^𝑛_𝑖𝑝_𝑖)⇀𝑒^∗,𝑖∈𝐸^∗. In view of the reflexivity of 𝐸, we have 𝐽(𝐸)=𝐸^∗. This shows that there exists an element 𝑒^𝑖∈𝐸 such that 𝐽𝑒^𝑖=𝑒^∗,𝑖. It follows that 𝜙𝑝_𝑖,𝑇^𝑛_𝑖𝑝_𝑖=𝑝_𝑖²−2𝑝_𝑖,𝐽𝑇^𝑛_𝑖𝑝_𝑖+𝑇^𝑛_𝑖𝑝_𝑖²=𝑝_𝑖²−2𝑝_𝑖,𝐽𝑇^𝑛_𝑖𝑝_𝑖+𝐽𝑇^𝑛_𝑖𝑝_𝑖². (3.7) Taking liminf_𝑛→∞ on the both sides of the equality above, we obtain that 0≥𝑝_𝑖²−2𝑝_𝑖,𝑒^∗,𝑖+𝑒^∗,𝑖2=𝑝_𝑖²−2𝑝_𝑖,𝐽𝑒^𝑖+𝐽𝑒^𝑖2=𝑝_𝑖²−2𝑝_𝑖,𝐽𝑒^𝑖+𝑒^𝑖2=𝜙𝑝_𝑖,𝑒^𝑖. (3.8) This implies that 𝑝_𝑖=𝑒^𝑖, that is, 𝐽𝑝_𝑖=𝑒^∗,𝑖. It follows that 𝐽(𝑇^𝑛_𝑖𝑝_𝑖)⇀𝐽𝑝_𝑖∈𝐸^∗. In view of Kadec-Klee property of 𝐸^∗, we obtain from (3.6) that lim_𝑛→∞‖𝐽(𝑇^𝑛_𝑖𝑝_𝑖)−𝐽𝑝_𝑖‖=0. Since 𝐽⁻¹∶𝐸^∗→𝐸 is demicontinuous, we see that 𝑇^𝑛_𝑖𝑝_𝑖⇀𝑝_𝑖. By virtue of Kadec-Klee property of 𝐸, we see from (3.5) that 𝑇^𝑛_𝑖𝑝_𝑖→𝑝_𝑖 as 𝑛→∞. Hence 𝑇_𝑖𝑇^𝑛_𝑖𝑝_𝑖=𝑇_𝑖^𝑛+1𝑝_𝑖→𝑝_𝑖, as 𝑛→∞. In view of the closedness of 𝑇_𝑖, we can obtain that 𝑝_𝑖∈𝐹(𝑇_𝑖), for every 𝑖∈Δ. This shows, for every 𝑖∈Δ, that 𝐹(𝑇_𝑖) is convex. This proves that ⋂_𝑖∈Δ𝐹(𝑇_𝑖) is convex. This completes Step 1.
Step 2. It show that 𝐶_𝑛 is closed and convex, ∀𝑛≥1.
It suffices to show, for any fixed but arbitrary 𝑖∈Δ, that 𝐶_𝑛,𝑖 is closed and convex, for every 𝑛≥1. This can be proved by induction on 𝑛. It is obvious that 𝐶_(1,𝑖)=𝐶 is closed and convex. Assume that 𝐶_(𝑗,𝑖) is closed and convex for some 𝑗≥1. We next prove that 𝐶_{(𝑗+1,𝑖)} is closed and convex for the same 𝑗. This completes the proof that 𝐶_𝑛 is closed and convex. The closedness of 𝐶_{(𝑗+1,𝑖)} is clear. We only prove the convexness. Indeed, ∀𝑎_𝑖,𝑏_𝑖∈𝐶_{(𝑗+1,𝑖)}, we see that 𝑎_𝑖,𝑏_𝑖∈𝐶_(𝑗,𝑖), and 𝜙𝑥_𝑗,𝑇^𝑗_𝑖𝑥_𝑗≤2𝑥_𝑗−𝑎_𝑖,𝐽𝑥_𝑗−𝐽𝑇^𝑗_𝑖𝑥_𝑗+𝜉_(𝑗,𝑖),𝜙𝑥_𝑗,𝑇^𝑗_𝑖𝑥_𝑗≤2𝑥_𝑗−𝑏_𝑖,𝐽𝑥_𝑗−𝐽𝑇^𝑗_𝑖𝑥_𝑗+𝜉_(𝑗,𝑖). (3.9) In view of (3.9), we find that 𝜙𝑥_𝑗,𝑇^𝑗_𝑖𝑥_𝑗≤2𝑥_𝑗−𝑐_𝑖,𝐽𝑥_𝑗−𝐽𝑇^𝑗_𝑖𝑥_𝑗+𝜉_(𝑗,𝑖), (3.10) where 𝑐_𝑖=𝑡_𝑖𝑎_𝑖+(1−𝑡_𝑖)𝑏_𝑖∈𝐶_(𝑗,𝑖), 𝑡_𝑖∈(0,1). It follows that 𝐶_{(𝑗+1,𝑖)} is convex. This in turn implies that 𝐶_𝑛=⋂_𝑖∈Λ𝐶_(𝑛,𝑖) is closed, and convex. This completes Step 2.
Step 3. It show that ⋂_𝑖∈Δ𝐹(𝑇_𝑖)⊂𝐶_𝑛, ∀𝑛≥1.
It is obvious that ⋂_𝑖∈Δ𝐹(𝑇_𝑖)⊂𝐶=𝐶₁. Suppose that ⋂_𝑖∈Δ𝐹(𝑇_𝑖)⊂𝐶_𝑗 for some 𝑗≥1. For any 𝑢∈⋂_𝑖∈Δ𝐹(𝑇_𝑖)⊂𝐶_𝑗, we see that 𝜙𝑢,𝑇^𝑗_𝑖𝑥_𝑗≤𝜙𝑢,𝑥_𝑗+𝜉_(𝑗,𝑖). (3.11) On the other hand, we obtain from (2.12) that 𝜙𝑢,𝑇^𝑗_𝑖𝑥_𝑗=𝜙𝑢,𝑥_𝑗+𝜙𝑥_𝑗,𝑇^𝑗_𝑖𝑥_𝑗+2𝑢−𝑥_𝑗,𝐽𝑥_𝑗−𝐽𝑇^𝑗_𝑖𝑥_𝑗. (3.12) Combining (3.11) with (3.12), we arrive at 𝜙𝑥_𝑗,𝑇^𝑗_𝑖𝑥_𝑗≤2𝑥_𝑗−𝑢,𝐽𝑥_𝑗−𝐽𝑇^𝑗_𝑖𝑥_𝑗+𝜉_(𝑗,𝑖), (3.13) which implies that 𝑢∈𝐶_{(𝑗+1,𝑖)}. This proves that ⋂_𝑖∈Δ𝐹(𝑇_𝑖)⊂𝐶_𝑛, ∀𝑛≥1. This completes Step 3.
Step  4. It show that 𝑥_𝑛→𝑥, where 𝑥∈⋂_𝑖∈Δ𝐹(𝑇_𝑖), as 𝑛→∞.
Since {𝑥_𝑛} is bounded and the space is reflexive, we may assume that 𝑥_𝑛⇀𝑥. Since 𝐶_𝑛 is closed and convex, we see that 𝑥∈𝐶_𝑛. On the other hand, we see from the weakly lower semicontinuity of the norm that 𝜙𝑥,𝑥₀=‖𝑥‖²−2𝑥,𝐽𝑥₀+𝑥₀²≤liminf_𝑛→∞𝑥_𝑛²−2𝑥_𝑛,𝐽𝑥₀+𝑥₀²=liminf_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤limsup_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥,𝑥₀, (3.14) which implies that 𝜙(𝑥_𝑛,𝑥₀)→𝜙(𝑥,𝑥₀) as 𝑛→∞. Hence, ‖𝑥_𝑛‖→‖𝑥‖ as 𝑛→∞. In view of Kadec-Klee property of 𝐸, we see that 𝑥_𝑛→𝑥 as 𝑛→∞. On the other hand, we see from 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1 that 𝑥_𝑛+1∈𝐶_{(𝑛+1,𝑖)}. It follows that 𝜙𝑥_𝑛,𝑇^𝑛_𝑖𝑥_𝑛≤2𝑥_𝑛−𝑥_𝑛+1,𝐽𝑥_𝑛−𝐽𝑇^𝑛_𝑖𝑥_𝑛+𝜉_(𝑛,𝑖), (3.15) from which it follows that 𝜙(𝑥_𝑛,𝑇^𝑛_𝑖𝑥_𝑛)→0 as 𝑛→∞. In view of (2.11), we see that ‖𝑥_𝑛‖−‖𝑇^𝑛_𝑖𝑥_𝑛‖→0 as 𝑛→∞. This in turn implies that ‖𝑇^𝑛_𝑖𝑥_𝑛‖→‖𝑥‖ as 𝑛→∞. Hence, ‖𝐽(𝑇^𝑛_𝑖𝑥_𝑛)‖→‖𝐽𝑥‖ as 𝑛→∞. This shows that {𝐽(𝑇^𝑛_𝑖𝑥_𝑛)} is bounded. Since 𝐸 is reflexive, we see that 𝐸^∗ is also reflexive. We may, without loss of generality, assume that 𝐽(𝑇^𝑛_𝑖𝑥_𝑛)⇀𝑓^∗,𝑖∈𝐸^∗. In view of the reflexivity of 𝐸, we have 𝐽(𝐸)=𝐸^∗. This shows that there exists an element 𝑓^𝑖∈𝐸 such that 𝐽𝑓^𝑖=𝑓^∗,𝑖. It follows that 𝜙𝑥_𝑛,𝑇^𝑛_𝑖𝑥_𝑛=𝑥_𝑛²−2⟨𝑥_𝑛,𝐽𝑇^𝑛_𝑖𝑥_𝑛⟩+𝑇^𝑛_𝑖𝑥_𝑥²=𝑥_𝑛²−2𝑥_𝑛,𝐽𝑇^𝑛_𝑖𝑥_𝑛+𝐽𝑇^𝑛_𝑖𝑥_𝑛². (3.16) Taking liminf_𝑛→∞ on the both sides of the equality above, we obtain that 0≥‖𝑥‖²−2⟨𝑥,𝑓^∗,𝑖⟩+𝑓^∗,𝑖2=‖𝑥‖²−2⟨𝑥,𝐽𝑓^𝑖⟩+𝐽𝑓^𝑖2=‖𝑥‖²−2⟨𝑥,𝐽𝑓^𝑖⟩+𝑓^𝑖2=𝜙𝑥,𝑓^𝑖. (3.17) This implies that 𝑥=𝑓^𝑖, that is, 𝐽𝑥=𝑓^∗,𝑖. It follows that 𝐽(𝑇^𝑛_𝑖𝑥_𝑛)⇀𝐽𝑥∈𝐸^∗. In view of Kadec-Klee property of 𝐸^∗, we obtain that lim_𝑛→∞‖𝐽(𝑇^𝑛_𝑖𝑥_𝑛)−𝐽𝑥‖=0. Since 𝐽⁻¹∶𝐸^∗→𝐸 is demicontinuous, we see that 𝑇^𝑛_𝑖𝑥_𝑛⇀𝑥. In the light of Kadec-Klee property of 𝐸, we see that 𝑇^𝑛_𝑖𝑥_𝑛→𝑥, ∀𝑖∈Δ, as 𝑛→∞. On the other hand, we have 𝑇_𝑖^𝑛+1𝑥_𝑛−𝑥≤𝑇_𝑖^𝑛+1𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛+𝑇^𝑛_𝑖𝑥_𝑛−𝑥. (3.18) It follows from the asymptotic regularity of 𝑇 that 𝑇_𝑖^𝑛+1𝑥_𝑛→𝑥 as 𝑛→∞. That is, 𝑇_𝑖𝑇^𝑛_𝑖𝑥_𝑛→𝑥. From the closedness of 𝑇_𝑖, we obtain that 𝑥=𝑇_𝑖𝑥. This implies that 𝑥∈⋂_𝑖∈Δ𝐹(𝑇_𝑖).
Step 5. It show that 𝑥=Π_{⋂𝑖∈Δ𝐹(𝑇𝑖)}𝑥₀.
In view of 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we see from Lemma 2.8 that 𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛≥0,∀𝑧∈𝐶_𝑛. (3.19) Since ⋂_𝑖∈Δ𝐹(𝑇_𝑖)⊂𝐶_𝑛, we arrive at 𝑥_𝑛−𝑤,𝐽𝑥₀−𝐽𝑥_𝑛≥0,∀𝑤∈_𝑖∈Δ𝐹𝑇_𝑖. (3.20) Letting 𝑛→∞ in the above, we arrive at 𝑥−𝑤,𝐽𝑥₀−𝐽𝑥≥0,∀𝑤∈_𝑖∈Δ𝐹𝑇_𝑖. (3.21) It follows from Lemma 2.8 that 𝑥=Π_{⋂𝑖∈Δ𝐹(𝑇𝑖)}𝑥₀. This completes the proof of Theorem 3.1.

Remark 3.2. The space in Theorem 3.1 can be applicable to 𝐿^𝑝, 𝑝>1. Since the class of asymptotically quasi-𝜙-nonexpansive mappings includes the class of asymptotically quasi-𝜙-nonexpansive mappings as a special case, we see that Theorem 3.1 still holds for the class of asymptotically quasi-𝜙-nonexpansive mappings.

For a single mapping, we can easily conclude the following.

Corollary 3.3. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space such that both 𝐸 and 𝐸^∗  have Kadec-Klee property. Let 𝐶 be a nonempty, bounded, closed, and convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a closed asymptotically quasi-𝜙-nonexpansive mapping in the intermediate sense. Assume that 𝐹(𝑇) is nonempty, and 𝑇 is asymptotically regular. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀𝐶∈𝐸,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑥=𝐶,₁=Π_𝐶1𝑥₀,𝐶_𝑛+1=𝑢∈𝐶_𝑛∶𝜙𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤2𝑥_𝑛−𝑢,𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛+𝜉_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.22) where 𝜉_𝑛=max{0,sup_{𝑝∈𝐹(𝑇),𝑥∈𝐶}(𝜙(𝑝,𝑇^𝑛𝑥)−𝜙(𝑝,𝑥))}. Then {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₀, where Π_𝐹(𝑇) stands for the generalized projection from 𝐸 onto 𝐹(𝑇).

In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following.

Corollary 3.4. Let 𝐶 be a nonempty, bounded, closed, and convex subset of a Hilbert space 𝐸. Let Δ be an index set, and 𝑇_𝑖∶𝐶→𝐶 a closed asymptotically quasi-nonexpansive mapping in the intermediate sense, for every 𝑖∈Δ. Assume that ⋂_𝑖∈Δ𝐹(𝑇_𝑖) is nonempty, and 𝑇_𝑖 is asymptotically regular, for every 𝑖∈Δ. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀𝐶∈𝐸,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_(1,𝑖)𝐶=𝐶,₁=_𝑖∈Δ𝐶_(1,𝑖),𝑥₁=𝑃_𝐶1𝑥₀,𝐶_{(𝑛+1,𝑖)}=𝑢∈𝐶_(𝑛,𝑖)∶𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛²≤2𝑥_𝑛−𝑢,𝑥_𝑛−𝑇^𝑛_𝑖𝑥_𝑛+𝜉_(𝑛,𝑖),𝐶_𝑛+1=_𝑖∈Δ𝐶_{(𝑛+1,𝑖)},𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.23) where 𝜉_(𝑛,𝑖)=max{0,sup_{𝑝∈𝐹(𝑇𝑖),𝑥∈𝐶}(‖𝑝−𝑇^𝑛_𝑖𝑥‖²−‖𝑝−𝑥‖²)}. Then {𝑥_𝑛} converges strongly to 𝑃_{⋂𝑖∈Δ𝐹(𝑇𝑖)}𝑥₀, where 𝑃_{⋂𝑖∈Δ𝐹(𝑇𝑖)} stands for the metric projection from 𝐸 onto ⋂_𝑖∈Δ𝐹(𝑇_𝑖).

For a single mapping, we can easily conclude the following.

Corollary 3.5. Let 𝐶 be a nonempty, bounded, closed, and convex subset of a Hilbert space 𝐸. Let 𝑇 be a closed asymptotically quasi-nonexpansive mapping in the intermediate sense. Assume that 𝐹(𝑇) is nonempty, and 𝑇 is asymptotically regular. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀𝐶∈𝐸,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,₁𝑥=𝐶,₁=𝑃_𝐶1𝑥₀,𝐶_𝑛+1=𝑢∈𝐶_𝑛∶𝑥_𝑛−𝑇^𝑛𝑥_𝑛²≤2𝑥_𝑛−𝑢,𝑥_𝑛−𝑇^𝑛𝑥_𝑛+𝜉_𝑛,𝐶_𝑛+1=_𝑖∈Δ𝐶_{(𝑛+1,𝑖)},𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.24) where 𝜉_𝑛=max{0,sup_{𝑝∈𝐹(𝑇),𝑥∈𝐶}(‖𝑝−𝑇^𝑛𝑥‖²−‖𝑝−𝑥‖²)}. Then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥₀, where 𝑃_𝐹(𝑇) stands for the metric projection from 𝐸 onto 𝐹(𝑇).