Hybrid Projection Algorithms for Asymptotically Strict Quasi-ϕ-Pseudocontractions

Abstract

A new nonlinear mapping is introduced. Hybrid projection algorithms are considered for the class of new nonlinear mappings. Strong convergence theorems are established in a real Banach space.

1. Introduction

Let 𝐸 be a real Banach space, 𝐶 a nonempty subset of 𝐸, and 𝑇∶𝐶→𝐶 a nonlinear mapping. Denote by 𝐹(𝑇) the set of fixed points of 𝑇. Recall that 𝑇 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.1) We remark that the mapping 𝑇 is said to be quasinonexpansive if 𝐹(𝑇)≠∅ and (1.1) holds for all 𝑥∈𝐶 and 𝑦∈𝐹(𝑇). 𝑇 is said to be asymptotically nonexpansive if there exists a sequence {𝜇_𝑛}⊂[0,∞) with 𝜇_𝑛→0 as 𝑛→∞ such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤1+𝜇_𝑛‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.2) We remark that the mapping 𝑇 is said to be asymptotically quasinonexpansive if 𝐹(𝑇)≠∅ and (1.2) holds for all 𝑥∈𝐶 and 𝑦∈𝐹(𝑇). The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that if 𝐶 is a nonempty bounded closed convex subset of a uniformly convex Banach space 𝐸, then every asymptotically nonexpansive selfmapping 𝑇 has a fixed point in 𝐶. Further, the set 𝐹(𝑇) of fixed points of 𝑇 is closed and convex. Since 1972, many authors have studied the weak and strong convergence problems of iterative algorithms for the class of mappings.

Recall that 𝑇 is said to be a strict pseudocontraction if there exists a constant 𝜅∈[0,1) such that ‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²+𝜅‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖²,∀𝑥,𝑦∈𝐶.(1.3) We remark that the mapping 𝑇 is said to be a strict quasipseudocontraction if 𝐹(𝑇)≠∅ and (1.3) holds for all 𝑥∈𝐶 and 𝑦∈𝐹(𝑇).

The class of strict pseudocontractions was introduced by Browder and Petryshyn [2]. In 2007, Marino and Xu [3] proved that the fixed point set of strict pseudocontractions is closed and convex. They also proved that 𝐼−𝑇 is demiclosed at the origin in real Hilbert spaces. A strong convergence theorem of hybrid projection algorithms for strict pseudocontractions was established; see [3] for more details.

Recall that 𝑇 is said to be an asymptotically strict pseudocontraction if there exist a constant 𝜅∈[0,1) and a sequence {𝜇_𝑛}⊂[0,∞) with 𝜇_𝑛→0 as 𝑛→∞ such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖²≤1+𝜇_𝑛‖𝑥−𝑦‖²+𝜅‖(𝐼−𝑇^𝑛)𝑥−(𝐼−𝑇^𝑛)𝑦‖²,∀𝑥,𝑦∈𝐶.(1.4) We remark that the mapping 𝑇 is said to be an asymptotically strict quasipseudocontraction if 𝐹(𝑇)≠∅ and (1.4) holds for all 𝑥∈𝐶 and 𝑦∈𝐹(𝑇).

The class of asymptotically strict pseudocontractions was introduced by Qihou [4] in 1996. Kim and Xu [5] proved that the fixed-point set of asymptotically strict pseudocontractions is closed and convex. They also obtained a strong convergence theorem for the class of asymptotically strict pseudocontractions by hybrid projection algorithms. To be more precise, they proved the following theorem.

Theorem KX. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be an asymptotically 𝜅-strict pseudocontraction for some 0≤𝜅<1. Assume that the fixed-point set 𝐹(𝑇) of 𝑇 is nonempty and bounded. Let {𝑥_𝑛} be the sequence generated by the following (CQ) algorithm: 𝑦𝑥₀∈𝐶chosenarbitrarily,_𝑛𝐶_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇^𝑛𝑥_𝑛,=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑄_𝑛𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²+𝜅−𝛼_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖²+𝜃_𝑛,=𝑧∈𝑥𝐶∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,(1.5) where 𝜃_𝑛=Δ^2𝑛1−𝛼_𝑛𝜇_𝑛⟶𝑜𝑟𝑛⟶∞,Δ_𝑛=sup‖‖𝑥_𝑛−𝑧‖‖∶𝑝∈𝐹(𝑇).(1.6) Assume that the control sequence {𝛼_𝑛} is chosen so that limsup_𝑛→∞𝛼_𝑛<1−𝜅 then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

It is well known that, in an infinite dimensional Hilbert space, the normal Mann iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Hybrid projection algorithms are popular tool to prove strong convergence of iterative sequences without compactness assumptions. Recently, hybrid projection algorithms have received rapid developments; see, for example, [3, 5–24]. In this paper, we will introduce a new mapping, asymptotically strict quasi-𝜙-pseudocontractions, and give a strong convergence theorem by a simple hybrid projection algorithm in a real Banach space.

2. Preliminaries

Let 𝐸 be a Banach space with the dual space 𝐸^∗. We denote by 𝐽 the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽𝑥=𝑓^∗∈𝐸^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖²=‖𝑓^∗‖²,∀𝑥∈𝐸,(2.1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing of elements between 𝐸 and 𝐸^∗; see [25]. It is well known that if 𝐸^∗ is strictly convex, then 𝐽 is single valued, and if 𝐸^∗ is uniformly convex, then 𝐽 is uniformly continuous on bounded subsets of 𝐸.

It is also well known that if 𝐶 is a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑃_𝐶∶𝐻→𝐶 is the metric projection of 𝐻 onto 𝐶, then 𝑃_𝐶 is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [26] recently introduced a generalized projection operator Π_𝐶 in a Banach space 𝐸 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. 𝐸 is said to have Kadec-Klee property if a sequence {𝑥_𝑛} of 𝐸 satisfying that 𝑥_𝑛⇀𝑥 and ‖𝑥_𝑛‖→‖𝑥‖, then 𝑥_𝑛→𝑥. It is known that if 𝐸 is uniformly convex, then 𝐸 enjoys Kadec-Klee property; see [25, 27] for more details. 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(2.2) exists for each 𝑥,𝑦∈𝑈_𝐸. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦∈𝑈_𝐸. It is well known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.

Let 𝐸 be a smooth Banach space. Consider the functional defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸.(2.3) Observe that, in a Hilbert space 𝐻, (2.3) 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∈𝐶𝜙(𝑦,𝑥).(2.4) 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, [26–29]. In Hilbert spaces, Π_𝐶=𝑃_𝐶. It is obvious from the definition of the function 𝜙 that 𝜙(𝑥,(𝑦‖)𝑦‖=−𝜙(‖𝑥𝑥,‖𝑧))²+≤𝜙(𝜙𝑧(,𝑦𝑦,)𝑥+)≤2⟨(𝑥‖−𝑦‖𝑧,+𝐽‖𝑧𝑥−‖𝐽)²𝑦,⟩,∀𝑥∀,𝑥𝑦,∈𝑦,𝑧𝐸,∈𝐸.((22..56))

Remark 2.1. 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 (2.5), 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 [25, 27] for more details.

Now, we give some definitions for our main results in this paper.

Let 𝐶 be a closed convex subset of a real Banach space 𝐸 and 𝑇∶𝐶→𝐶 a mapping.(1)A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [30] 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 [15, 31, 32] if 𝐹(𝑇)=𝐹(𝑇)≠∅,𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.7) The asymptotic behavior of a relatively nonexpansive mapping was studied in [30–32].(3)𝑇 is said to be relatively asymptotically nonexpansive [6, 11] if 𝐹(𝑇)=𝐹(𝑇)≠∅,𝜙(𝑝,𝑇^𝑛𝑥)≤1+𝜇_𝑛𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇),(2.8) where {𝜇_𝑛}⊂[0,∞) is a sequence such that 𝜇_𝑛→1 as 𝑛→∞.(4)𝑇 is said to be 𝜙-nonexpansive [14, 16, 17] if 𝜙(𝑇𝑥,𝑇𝑦)≤𝜙(𝑥,𝑦),∀𝑥,𝑦∈𝐶.(2.9)(5)𝑇 is said to be quasi-𝜙-nonexpansive [14, 16, 17] if 𝐹(𝑇)≠∅,𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.10)(6)𝑇 is said to be asymptotically 𝜙-nonexpansive [14] if there exists a real sequence {𝜇_𝑛}⊂[0,∞) with 𝜇_𝑛→0 as 𝑛→∞ such that 𝜙(𝑇^𝑛𝑥,𝑇^𝑛𝑦)≤1+𝜇_𝑛𝜙(𝑥,𝑦),∀𝑥,𝑦∈𝐶.(2.11)(7)𝑇 is said to be asymptotically quasi-𝜙-nonexpansive [14] if there exists a real sequence {𝜇_𝑛}⊂[0,∞) with 𝜇_𝑛→0 as 𝑛→∞ such that 𝐹(𝑇)≠∅,𝜙(𝑝,𝑇^𝑛𝑥)≤1+𝜇_𝑛𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.12)(8)𝑇 is said to be a strict quasi-𝜙-pseudocontraction if 𝐹(𝑇)≠∅, and there exists a constant 𝜅∈[0,1) such that 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥)+𝜅𝜙(𝑥,𝑇𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.13) We remark that 𝑇 is said to be a quasistrict pseudocontraction in [13]. (9)𝑇 is said to be asymptotically regular on 𝐶 if, for any bounded subset 𝐾 of 𝐶, lim_𝑛→∞sup_𝑥∈𝐾‖‖𝑇^𝑛+1𝑥−𝑇^𝑛𝑥‖‖=0.(2.14)

Remark 2.2. 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 mappings do not require 𝐹(𝑇)=𝐹(𝑇), where 𝐹(𝑇) denotes the asymptotic fixed-point set of 𝑇.

Remark 2.3. In the framework of Hilbert spaces, quasi-𝜙-nonexpansive mappings and asymptotically quasi-𝜙-nonexpansive mappings are reduced to quasinonexpansive mappings and asymptotically quasinonexpansive mappings.

In this paper, we introduce a new nonlinear mapping: asymptotically strict quasi-𝜙-pseudocontractions.

Definition 2.4. Recall that a mapping 𝑇∶𝐶→𝐶 is said to be an asymptotically strict quasi-𝜙-pseudocontraction if 𝐹(𝑇)≠∅, and there exists a sequence {𝜇_𝑛}⊂[0,∞) with 𝜇_𝑛→0 as 𝑛→∞ and a constant 𝜅∈[0,1) such that 𝜙(𝑝,𝑇^𝑛𝑥)≤1+𝜇_𝑛𝜙(𝑝,𝑥)+𝜅𝜙(𝑥,𝑇^𝑛𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.15)

Remark 2.5. In the framework of Hilbert spaces, asymptotically strict quasi-𝜙-pseudocontractions are asymptotically strict quasipseudocontractions.

Next, we give an example which is an asymptotically strict quasi-𝜙-pseudocontraction.

Let 𝐸=𝑙₂∶={𝑥={𝑥₁,𝑥₂,…}∶∑^∞𝑛=1|𝑥_𝑛|²<∞}, and let 𝐵_𝐸 be the closed unit ball in 𝐸. Define a mapping 𝑇∶𝐵_𝐸→𝐵_𝐸 by 𝑇𝑥₁,𝑥₂,…=0,𝑥²¹,𝑎₂𝑥₂,𝑎₃𝑥₃,…,(2.16) where {𝑎_𝑖} is a sequence of real numbers such that 𝑎₂>0, 0<𝑎_𝑗<1, where 𝑖≠2, and Π^∞𝑖=2𝑎_𝑗=1/2. Then 𝜙(𝑝,𝑇^𝑛𝑥)=‖𝑝−𝑇^𝑛𝑥‖²≤2Π^𝑛𝑖=2𝑎_𝑗‖𝑝−𝑥‖²+𝜅‖𝑥−𝑇^𝑛𝑥‖²=2Π^𝑛𝑖=2𝑎_𝑗𝜙(𝑝,𝑥)+𝜅𝜙(𝑥,𝑇^𝑛𝑥),∀𝑥∈𝐵_𝐸,𝑛≥2,(2.17) where 𝑝=(0,0,…) is a fixed point of 𝑇 and 𝜅∈[0,1) is a real number. In view of lim_𝑛→∞2(Π^𝑛𝑖=2𝑎_𝑗)=1, we see that 𝑇 is an asymptotically strict quasi-𝜙-pseudocontraction.

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

Lemma 2.6 (see [29]). Let 𝐸 be a uniformly convex and smooth Banach space, and let {𝑥_𝑛}, {𝑦_𝑛} be two sequences of 𝐸. If 𝜙(𝑥_𝑛,𝑦_𝑛)→0 and either {𝑥_𝑛} or {𝑦_𝑛} is bounded, then 𝑥_𝑛−𝑦_𝑛→0.

Lemma 2.7 (see [26]). Let 𝐶 be a nonempty closed 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.(2.18)

Lemma 2.8 (see [26]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑥∈𝐸 then 𝜙𝑦,Π_𝐶𝑥+𝜙Π_𝐶𝑥,𝑥≤𝜙(𝑦,𝑥),∀𝑦∈𝐶.(2.19)

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty closed and convex subset of a uniformly convex and smooth Banach space 𝐸. Let 𝑇∶𝐶→𝐶 be a closed and asymptotically strict quasi -𝜙-pseudocontraction with a sequence {𝜇_𝑛}⊂[0,∞) such that 𝜇_𝑛→0 as 𝑛→∞. Assume that 𝑇 is uniformly asymptotically regular on 𝐶 and 𝐹(𝑇) is nonempty and bounded. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀∈𝐸c𝐶hosenarbitrarily,₁𝑥=𝐶,₁𝐶_𝑛+1=Π_𝐶1𝑥₀,=𝑢∈C_𝑛∶𝜙𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤21−𝜅𝑥^𝑛−𝑢,𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛+𝜇_𝑛𝑀_𝑛𝑥1−𝜅_𝑛+1,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(Υ) where 𝑀_𝑛=sup{𝜙(𝑝,𝑥_𝑛)∶𝑝∈𝐹(𝑇)} then the sequence {𝑥_𝑛} converges strongly to 𝑥=Π_𝐹(𝑇)𝑥₀.

Proof. The proof is split into five steps.
Step 1. Show that 𝐹(𝑇) is closed and convex.
Let {𝑝_𝑛} be a sequence in 𝐹(𝑇) such that 𝑝_𝑛→𝑝 as 𝑛→∞. We see that 𝑝∈𝐹(𝑇). Indeed, we obtain from the definition of 𝑇 that 𝜙𝑝_𝑛,𝑇^𝑛𝑝≤1+𝜇_𝑛𝜙𝑝_𝑛,𝑝+𝜅𝜙(𝑝,𝑇^𝑛𝑝).(3.1) In view of (2.6), we see that 𝜙𝑝_𝑛,𝑇^𝑛𝑝=𝜙𝑝_𝑛,𝑝+𝜙(𝑝,𝑇^𝑛𝑝)+2⟨𝑝_𝑛−𝑝,𝐽𝑝−𝐽𝑇^𝑛𝑝⟩.(3.2) It follows that 𝜙𝑝_𝑛,𝑝+𝜙(𝑝,𝑇^𝑛𝑝)+2⟨𝑝_𝑛−𝑝,𝐽𝑝−𝐽𝑇^𝑛𝑝⟩≤1+𝜇_𝑛𝜙𝑝_𝑛,𝑝+𝜅𝜙(𝑝,𝑇^𝑛𝑝),(3.3) which implies that 𝜙(𝑝,𝑇^𝑛𝑝)≤𝜇^𝑛1−𝜅𝜙𝑝^𝑛,𝑝+21−𝜅⟨𝑝−𝑝^𝑛,𝐽𝑝−𝐽𝑇^𝑛𝑝⟩,(3.4) from which it follows that lim_𝑛→∞𝜙(𝑝,𝑇^𝑛𝑝)=0.(3.5) From Lemma 2.6, we see that 𝑇^𝑛𝑝→𝑝 as 𝑛→∞. This implies that 𝑇𝑇^𝑛𝑝=𝑇^𝑛+1𝑝→𝑝 as 𝑛→∞. From the closedness of 𝑇, we obtain that 𝑝∈𝐹(𝑇). This proves the closedness of 𝐹(𝑇).
Next, we show the convexness of 𝐹(𝑇). Let 𝑝₁,𝑝₂∈𝐹(𝑇) and 𝑝_𝑡=𝑡𝑝₁+(1−𝑡)𝑝₂, where 𝑡∈(0,1). We see that 𝑝_𝑡=𝑇𝑝_𝑡. Indeed, we have from the definition of 𝑇 that 𝜙𝜙𝑝𝑝₁₂,𝑇^𝑛𝑝_𝑡≤1+𝜇_𝑛𝜙𝑝₁,𝑝_𝑡+𝜅𝜙𝑝_𝑡,𝑇^𝑛𝑝_𝑡,,𝑇^𝑛𝑝_𝑡≤1+𝜇_𝑛𝜙𝑝₂,𝑝_𝑡+𝜅𝜙𝑝_𝑡,𝑇^𝑛𝑝_𝑡.(3.6) By virtue of (2.6), we obtain that 𝜙𝑝_𝑡,𝑇^𝑛𝑝_𝑡≤𝜇^𝑛1−𝜅𝜙𝑝¹,𝑝_𝑡+21−𝜅⟨𝑝^𝑡𝜙𝑝−𝑝₁,𝐽𝑝_𝑡−𝐽𝑇^𝑛𝑝_𝑡⟩,(3.7)_𝑡,𝑇^𝑛𝑝_𝑡≤𝜇ⁿ1−𝜅𝜙𝑝²,𝑝_𝑡+21−𝜅⟨𝑝^𝑡−𝑝₂,𝐽𝑝_𝑡−𝐽𝑇^𝑛𝑝_𝑡⟩.(3.8) Multiplying 𝑡 and (1−𝑡) on both the sides of (3.7) and (3.8), respectively, yields that 𝜙𝑝_𝑡,𝑇^𝑛𝑝_𝑡≤𝑡𝜇^𝑛1−𝜅𝜙𝑝¹,𝑝_𝑡+(1−𝑡)𝜇^𝑛1−𝜅𝜙𝑝²,𝑝_𝑡.(3.9) It follows that lim_𝑛→∞𝜙𝑝_𝑡,𝑇^𝑛𝑝_𝑡=0.(3.10) In view of Lemma 2.6, we see that 𝑇^𝑛𝑝_𝑡→𝑝_𝑡 as 𝑛→∞. This implies that 𝑇𝑇^{0𝑥0𝑎35𝑐}𝑝_𝑡=𝑇^𝑛+1𝑝_𝑡→𝑝_𝑡 as 𝑛→∞. From the closedness of 𝑇, we obtain that 𝑝_𝑡∈𝐹(𝑇). This proves that 𝐹(𝑇) is convex. This completes Step 1.Step 2. Show that 𝐶_𝑛 is closed and convex for each 𝑛≥1.
It is not hard to see that 𝐶_𝑛 is closed for each 𝑛≥1. Therefore, we only show that 𝐶_𝑛 is convex for each 𝑛≥1. It is obvious that 𝐶₁=𝐶 is convex. Suppose that 𝐶_ℎ is convex for some ℎ∈ℕ. Next, we show that 𝐶_ℎ+1 is also convex for the same ℎ. Let 𝑎,𝑏∈𝐶_ℎ+1 and 𝑐=𝑡𝑎+(1−𝑡)𝑏, where 𝑡∈(0,1). It follows that 𝜙𝑥_ℎ,𝑇^ℎ𝑥_ℎ≤21−𝜅𝑥^ℎ−𝑎,𝐽𝑥_ℎ−𝐽𝑇^ℎ𝑥_ℎ+𝜇_ℎ𝑀_ℎ𝜙𝑥1−𝜅_ℎ,,𝑇^ℎ𝑥_ℎ≤21−𝜅𝑥^ℎ−𝑏,𝐽𝑥_ℎ−𝐽𝑇^ℎ𝑥_ℎ+𝜇_ℎ𝑀_ℎ1−𝜅,(3.11) where 𝑎,𝑏∈𝐶_ℎ. From the above two inequalities, we can get that 𝜙𝑥_ℎ,𝑇^ℎ𝑥_ℎ≤21−𝜅𝑥^ℎ−𝑐,𝐽𝑥_ℎ−𝐽𝑇^ℎ𝑥_ℎ+𝜇_ℎ𝑀_ℎ1−𝜅,(3.12) where 𝑐∈𝐶_ℎ. It follows that 𝐶_ℎ+1 is closed and convex. This completes Step 2.Step 3. Show that 𝐹(𝑇)⊂𝐶_𝑛 for each 𝑛≥1.
It is obvious that 𝐹(𝑇)⊂𝐶=𝐶₁. Suppose that 𝐹(𝑇)⊂𝐶_ℎ for some ℎ∈ℕ. For any 𝑧∈𝐹(𝑇)⊂𝐶_ℎ, we see that 𝜙𝑧,𝑇^ℎ𝑥_ℎ≤1+𝜇_ℎ𝜙𝑧,𝑥_ℎ+𝜅𝜙𝑥_ℎ,𝑇^ℎ𝑥_ℎ.(3.13) On the other hand, we obtain from (2.6) that 𝜙𝑧,𝑇^ℎ𝑥_ℎ=𝜙𝑧,𝑥_ℎ+𝜙𝑥_ℎ,𝑇^ℎ𝑥_ℎ+2𝑧−𝑥_ℎ,𝐽𝑥_ℎ−𝐽𝑇^ℎ𝑥_ℎ.(3.14) Combining (3.13) with (3.14), we arrive at 𝜙𝑥_ℎ,𝑇^ℎ𝑥_ℎ≤𝜇^ℎ1−𝜅𝜙𝑧,𝑥^ℎ+1−2𝜅𝑥^ℎ≤𝜇_ℎ−𝑧,𝐽𝑥_ℎ−𝐽𝑇^ℎ𝑥_ℎ𝑀_ℎ1−𝜅+1−2𝜅𝑥^ℎ−𝑧,𝐽𝑥_ℎ−𝐽𝑇^ℎ𝑥_ℎ,(3.15) which implies that 𝑧∈𝐶_ℎ+1. This shows that 𝐹(𝑇)⊂𝐶_ℎ+1. This completes Step 3.Step 4. Show that the sequence {𝑥_𝑛} is bounded.
In view of 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we see that ⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0,∀𝑧∈𝐶_𝑛.(3.16) In view of 𝐹(𝑇)⊂𝐶_𝑛, we arrive at ⟨𝑥_𝑛−𝑤,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0,∀𝑤∈𝐹(𝑇).(3.17) It follows from Lemma 2.8 that 𝜙𝑥_𝑛,𝑥₀=𝜙Π_𝐶𝑛𝑥₀,𝑥₀≤𝜙Π_𝐹(𝑇)𝑥₀,𝑥₀−𝜙Π_𝐹(𝑇)𝑥₀,𝑥_𝑛≤𝜙Π_𝐹(𝑇)𝑥₀,𝑥₀.(3.18) This implies that the sequence {𝜙(𝑥_𝑛,𝑥₀)} is bounded. It follows from (2.5) that the sequence {𝑥_𝑛} is also bounded. This completes Step 4.Step 5. Show that 𝑥_𝑛→𝑥, where 𝑥=Π_𝐹(𝑇)𝑥₀, as 𝑛→∞.
Since {𝑥_𝑛} is bounded and the space is reflexive, we may assume that 𝑥_𝑛→𝑥 weakly. 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 𝜙𝑥,𝑥₀=‖‖𝑥‖‖²−2𝑥,𝐽𝑥₀+‖‖𝑥₀‖‖²=≤lliimmiinnf_𝑛→∞f_𝑛→∞‖‖𝑥^𝑛‖‖²−2⟨𝑥_𝑛,𝐽𝑥₀⟩+‖‖𝑥₀‖‖²≤limsup_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤𝜙𝜙𝑥_𝑛,𝑥₀𝑥,𝑥₀,(3.19) 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→∞.
Now, we are in a position to show that 𝑥∈𝐹(𝑇). Notice that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. On the other hand, we see from 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛 that 𝜙𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤21−𝜅𝑥^𝑛−𝑥_𝑛+1,𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛+𝜇_𝑛𝑀_𝑛1−𝜅,(3.20) from which it follows that 𝜙(𝑥_𝑛,𝑇^𝑛𝑥_𝑛)→0 as 𝑛→∞. In view of Lemma 2.6, we arrive at lim_𝑛→∞‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖=0.(3.21) Note that 𝑥_𝑛→𝑥 as 𝑛→∞ in view of ‖‖𝑇^𝑛𝑥_𝑛−𝑥‖‖≤‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑥‖‖.(3.22) It follows from (3.21) that 𝑇^𝑛𝑥_𝑛⟶𝑥as𝑛⟶∞.(3.23) On the other hand, we have ‖‖𝑇^𝑛+1𝑥_𝑛−𝑥‖‖≤‖‖𝑇^𝑛+1𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖+‖‖𝑇^𝑛𝑥_𝑛−𝑥‖‖.(3.24) It follows from the uniformly asymptotic regularity of 𝑇 and (3.23) that 𝑇^𝑛+1𝑥_𝑛⟶𝑥as𝑛⟶∞(3.25) that is, 𝑇𝑇^𝑛𝑥_𝑛→𝑥. From the closedness of 𝑇, we obtain that 𝑥=𝑇𝑥.
Finally, we show that 𝑥=Π_𝐹(𝑇)𝑥₀ which completes the proof. Indeed, we obtain from 𝑥_𝑛=Π_𝐶𝑛𝑥₀ that ⟨𝑥_𝑛−𝑤,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0,∀𝑤∈𝐶_𝑛.(3.26) In particular, we have 𝑥^𝑛−𝑤^′,𝐽𝑥₀−𝐽𝑥_𝑛≥0,∀𝑤^∈𝐹(𝑇).(3.27) Taking the limit as 𝑛→∞ in (3.27), we obtain that 𝑥−𝑤^′,𝐽𝑥₀−𝐽𝑥≥0,∀𝑤^∈𝐹(𝑇).(3.28) Hence, we obtain from Lemma 2.7 that 𝑥=Π_𝐹(𝑇)𝑥₀. This completes the proof.

As applications of Theorem 3.1, we have the following.

Corollary 3.2. Let 𝐶 be a nonempty closed and convex subset of a uniformly convex and smooth Banach space 𝐸. Let 𝑇∶𝐶→𝐶 be a closed and asymptotically quasi -𝜙-snonexpansive mapping with a sequence {𝜇_𝑛}⊂[0,∞) such that 𝜇_𝑛→0 as 𝑛→∞. Assume that 𝑇 is uniformly asymptotically regular on 𝐶 and 𝐹(𝑇) is nonempty and bounded. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₀∈𝐸c𝐶hosenarbitrarily,₁𝑥=𝐶,₁𝐶_𝑛+1=Π_𝐶1𝑥₀,=𝑢∈𝐶_𝑛∶𝜙𝑥𝑥_𝑛,𝑇^𝑛_𝑛+1𝑥_𝑛≤2𝑥_𝑛−𝑢,𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛+𝜇_𝑛𝑀_𝑛,=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.29) where 𝑀={𝜙(𝑝,𝑥_𝑛)∶𝑝∈𝐹(𝑇)} then the sequence {𝑥_𝑛} converges strongly to 𝑥=Π_𝐹(𝑇)𝑥₀.

Proof. Putting 𝜅=0 in Theorem 3.1, we can conclude the desired conclusion easily.

Next, we give two theorems in the framework of real Hilbert spaces.

Theorem 3.3. Let 𝐶 a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a closed and asymptotically strict quasipseudocontraction with a sequence {𝜇_𝑛}⊂[0,∞) such that 𝜇_𝑛→0 as 𝑛→∞. Assume that 𝑇 is uniformly asymptotically regular on 𝐶 and 𝐹(𝑇) is nonempty and bounded. Let {𝑥_𝑛} be a sequence generated by the following manner: 𝑥₀∈𝐻c𝐶hosenarbitrarily,₁𝑥=𝐶,₁𝐶_𝑛+1=𝑃_𝐶1𝑥₀,=𝑢∈𝐶_𝑛∶‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖²≤21−𝜅𝑥^𝑛−𝑢,𝑥_𝑛−𝑇^𝑛𝑥_𝑛+𝜇_𝑛𝑀_𝑛𝑥1−𝜅_𝑛+1,=𝑃_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.30) where 𝑀_𝑛=sup{‖𝑝−𝑥_𝑛‖²∶𝑝∈𝐹(𝑇)} then the sequence {𝑥_𝑛} converges strongly to 𝑥=𝑃_𝐹(𝑇)𝑥₀.

Theorem 3.4. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a closed and asymptotically quasinonexpansive mapping with a sequence {𝜇_𝑛}⊂[0,∞) such that 𝜇_𝑛→0 as 𝑛→∞. Assume that 𝑇 is uniformly asymptotically regular on 𝐶 and 𝐹(𝑇) is nonempty and bounded. Let {𝑥_𝑛} be a sequence generated by the following manner: 𝑥₀∈𝐻c𝐶hosenarbitrarily,₁𝑥=𝐶,₁𝐶_𝑛+1=𝑃_𝐶1𝑥₀,=𝑢∈𝐶_𝑛∶‖‖𝑥𝑥^𝑛−_𝑛+1𝑇^𝑛𝑥_𝑛‖‖²≤2𝑥_𝑛−𝑢,𝑥_𝑛−𝑇^𝑛𝑥_𝑛+𝜇_𝑛𝑀_𝑛,=𝑃_𝐶𝑛+1𝑥₀,∀𝑛≥0,(3.31) where 𝑀_𝑛={‖𝑝−𝑥_𝑛‖²∶𝑝∈𝐹(𝑇)} then the sequence {𝑥_𝑛} converges strongly to 𝑥=𝑃_𝐹(𝑇)𝑥₀.

Acknowledgments

The authors are grateful to the editor and the referees for their valuable comments and suggestions which improve the contents of the paper. The first author was partially supported by Natural Science Foundation of Zhejiang Province (Y6110270).