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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 142626, 13 pages
http://dx.doi.org/10.1155/2011/142626
Research Article

Hybrid Projection Algorithms for Asymptotically Strict Quasi-ϕ-Pseudocontractions

1School of Mathematics and Information Sciences, North China University of Water Conservancy and Electric Power, Zhengzhou 450011, China
2Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
3Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 27 November 2010; Revised 5 March 2011; Accepted 31 March 2011

Academic Editor: Paul Eloe

Copyright © 2011 Xiaolong Qin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑇𝐶𝐶 a nonlinear mapping. Denote by 𝐹(𝑇) the set of fixed points of 𝑇. Recall that 𝑇 is said to be nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦,𝑥,𝑦𝐶.(1.1) We remark that the mapping 𝑇 is said to be quasinonexpansive if 𝐹(𝑇) and (1.1) holds for all 𝑥𝐶 and 𝑦𝐹(𝑇). 𝑇 is said to be asymptotically nonexpansive if there exists a sequence {𝜇𝑛}[0,) with 𝜇𝑛0 as 𝑛 such that 𝑇𝑛𝑥𝑇𝑛𝑦1+𝜇𝑛𝑥𝑦,𝑥,𝑦𝐶.(1.2) We remark that the mapping 𝑇 is said to be asymptotically quasinonexpansive if 𝐹(𝑇) and (1.2) holds for all 𝑥𝐶 and 𝑦𝐹(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 𝑇𝑥𝑇𝑦2𝑥𝑦2+𝜅(𝐼𝑇)𝑥(𝐼𝑇)𝑦2,𝑥,𝑦𝐶.(1.3) We remark that the mapping 𝑇 is said to be a strict quasipseudocontraction if 𝐹(𝑇) and (1.3) holds for all 𝑥𝐶 and 𝑦𝐹(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 𝐼𝑇 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 {𝜇𝑛}[0,) with 𝜇𝑛0 as 𝑛 such that 𝑇𝑛𝑥𝑇𝑛𝑦21+𝜇𝑛𝑥𝑦2+𝜅(𝐼𝑇𝑛)𝑥(𝐼𝑇𝑛)𝑦2,𝑥,𝑦𝐶.(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 𝑥𝐶 and 𝑦𝐹(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 𝑇𝐶𝐶 be an asymptotically 𝜅-strict pseudocontraction for some 0𝜅<1. Assume that the fixed-point set 𝐹(𝑇) of 𝑇 is nonempty and bounded. Let {𝑥𝑛} be the sequence generated by the following (CQ) algorithm: 𝑥0𝑦𝐶chosenarbitrarily,𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑇𝑛𝑥𝑛,𝐶𝑛=𝑦𝑧𝐶𝑛𝑧2𝑥𝑛𝑧2+𝜅𝛼𝑛1𝛼𝑛𝑥𝑛𝑇𝑥𝑛2+𝜃𝑛,𝑄𝑛=𝑧𝐶𝑥𝑛𝑧,𝑥0𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝑄𝑛𝑥0,(1.5) where 𝜃𝑛=Δ2𝑛1𝛼𝑛𝜇𝑛𝑜𝑟𝑛,Δ𝑛𝑥=sup𝑛.𝑧𝑝𝐹(𝑇)(1.6) Assume that the control sequence {𝛼𝑛} is chosen so that limsup𝑛𝛼𝑛<1𝜅 then {𝑥𝑛} converges strongly to 𝑃𝐹(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, 524]. 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 2𝐸 defined by 𝑓𝐽𝑥=𝐸𝑥,𝑓=𝑥2=𝑓2,𝑥𝐸,(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 𝑃𝐶𝐻𝐶 is the metric projection of 𝐻 onto 𝐶, then 𝑃𝐶 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 Π𝐶 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,𝑦𝐸 with 𝑥=𝑦=1 and 𝑥𝑦. It is said to be uniformly convex if lim𝑛𝑥𝑛𝑦𝑛=0 for any two sequences {𝑥𝑛}and{𝑦𝑛} in 𝐸 such that 𝑥𝑛=𝑦𝑛=1 and lim𝑛(𝑥𝑛+𝑦𝑛)/2=1. 𝐸 is said to have Kadec-Klee property if a sequence {𝑥𝑛} of 𝐸 satisfying that 𝑥𝑛𝑥 and 𝑥𝑛𝑥, then 𝑥𝑛𝑥. It is known that if 𝐸 is uniformly convex, then 𝐸 enjoys Kadec-Klee property; see [25, 27] for more details. Let 𝑈𝐸={𝑥𝐸𝑥=1} be the unit sphere of 𝐸 then the Banach space 𝐸 is said to be smooth provided lim𝑡0𝑥+𝑡𝑦𝑥𝑡(2.2) exists for each 𝑥,𝑦𝑈𝐸. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦𝑈𝐸. 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)=𝑥22𝑥,𝐽𝑦+𝑦2,𝑥,𝑦𝐸.(2.3) Observe that, in a Hilbert space 𝐻, (2.3) is reduced to 𝜙(𝑥,𝑦)=𝑥𝑦2 for all 𝑥,𝑦𝐻. The generalized projection Π𝐶𝐸𝐶 is a mapping that assigns to an arbitrary point 𝑥𝐸 the minimum point of the functional 𝜙(𝑥,𝑦), that is, Π𝐶𝑥=𝑥, where 𝑥 is the solution to the following minimization problem: 𝜙𝑥,𝑥=min𝑦𝐶𝜙(𝑦,𝑥).(2.4) The existence and uniqueness of the operator Π𝐶 follow from the properties of the functional 𝜙(𝑥,𝑦) and the strict monotonicity of the mapping 𝐽; see, for example, [2629]. In Hilbert spaces, Π𝐶=𝑃𝐶. It is obvious from the definition of the function 𝜙 that ()𝑦𝑥2)𝜙(𝑦,𝑥)(𝑦+𝑥2,𝑥,𝑦𝐸,(2.5)𝜙(𝑥,𝑦)=𝜙(𝑥,𝑧)+𝜙(𝑧,𝑦)+2𝑥𝑧,𝐽𝑧𝐽𝑦,𝑥,𝑦,𝑧𝐸.(2.6)

Remark 2.1. If 𝐸 is a reflexive, strictly convex, and smooth Banach space, then, for all 𝑥,𝑦𝐸, 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=𝑥2=𝐽𝑦2. 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 𝑇𝐶𝐶 a mapping.(1)A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [30] if 𝐶 contains a sequence {𝑥𝑛} which converges weakly to 𝑝 such that lim𝑛𝑥𝑛𝑇𝑥𝑛=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),𝑥𝐶,𝑝𝐹(𝑇).(2.7) The asymptotic behavior of a relatively nonexpansive mapping was studied in [3032].(3)𝑇 is said to be relatively asymptotically nonexpansive [6, 11] if 𝐹(𝑇)=𝐹(𝑇),𝜙(𝑝,𝑇𝑛𝑥)1+𝜇𝑛𝜙(𝑝,𝑥),𝑥𝐶,𝑝𝐹(𝑇),(2.8) where {𝜇𝑛}[0,) is a sequence such that 𝜇𝑛1 as 𝑛.(4)𝑇 is said to be 𝜙-nonexpansive [14, 16, 17] if 𝜙(𝑇𝑥,𝑇𝑦)𝜙(𝑥,𝑦),𝑥,𝑦𝐶.(2.9)(5)𝑇 is said to be quasi-𝜙-nonexpansive [14, 16, 17] if 𝐹(𝑇),𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥),𝑥𝐶,𝑝𝐹(𝑇).(2.10)(6)𝑇 is said to be asymptotically 𝜙-nonexpansive [14] if there exists a real sequence {𝜇𝑛}[0,) with 𝜇𝑛0 as 𝑛 such that 𝜙(𝑇𝑛𝑥,𝑇𝑛𝑦)1+𝜇𝑛𝜙(𝑥,𝑦),𝑥,𝑦𝐶.(2.11)(7)𝑇 is said to be asymptotically quasi-𝜙-nonexpansive [14] if there exists a real sequence {𝜇𝑛}[0,) with 𝜇𝑛0 as 𝑛 such that 𝐹(𝑇),𝜙(𝑝,𝑇𝑛𝑥)1+𝜇𝑛𝜙(𝑝,𝑥),𝑥𝐶,𝑝𝐹(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),𝑥𝐶,𝑝𝐹(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, lim𝑛sup𝑥𝐾𝑇𝑛+1𝑥𝑇𝑛𝑥=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 𝑇𝐶𝐶 is said to be an asymptotically strict quasi-𝜙-pseudocontraction if 𝐹(𝑇), and there exists a sequence {𝜇𝑛}[0,) with 𝜇𝑛0 as 𝑛 and a constant 𝜅[0,1) such that 𝜙(𝑝,𝑇𝑛𝑥)1+𝜇𝑛𝜙(𝑝,𝑥)+𝜅𝜙(𝑥,𝑇𝑛𝑥),𝑥𝐶,𝑝𝐹(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|𝑥𝑛|2<}, and let 𝐵𝐸 be the closed unit ball in 𝐸. Define a mapping 𝑇𝐵𝐸𝐵𝐸 by 𝑇𝑥1,𝑥2=,0,𝑥21,𝑎2𝑥2,𝑎3𝑥3,,(2.16) where {𝑎𝑖} is a sequence of real numbers such that 𝑎2>0, 0<𝑎𝑗<1, where 𝑖2, and Π𝑖=2𝑎𝑗=1/2. Then 𝜙(𝑝,𝑇𝑛𝑥)=𝑝𝑇𝑛𝑥2Π2𝑛𝑖=2𝑎𝑗𝑝𝑥2+𝜅𝑥𝑇𝑛𝑥2Π=2𝑛𝑖=2𝑎𝑗𝜙(𝑝,𝑥)+𝜅𝜙(𝑥,𝑇𝑛𝑥),𝑥𝐵𝐸,𝑛2,(2.17) where 𝑝=(0,0,) is a fixed point of 𝑇 and 𝜅[0,1) is a real number. In view of lim𝑛2(Π𝑛𝑖=2𝑎𝑗)=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 {𝑥𝑛}, {𝑦𝑛} be two sequences of 𝐸. If 𝜙(𝑥𝑛,𝑦𝑛)0 and either {𝑥𝑛} or {𝑦𝑛} is bounded, then 𝑥𝑛𝑦𝑛0.

Lemma 2.7 (see [26]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑥𝐸 then 𝑥0=Π𝐶𝑥 if and only if 𝑥0𝑦,𝐽𝑥𝐽𝑥00,𝑦𝐶.(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 𝑥𝐸 then 𝜙𝑦,Π𝐶𝑥Π+𝜙𝐶𝑥,𝑥𝜙(𝑦,𝑥),𝑦𝐶.(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 𝑇𝐶𝐶 be a closed and asymptotically strict quasi -𝜙-pseudocontraction with a sequence {𝜇𝑛}[0,) such that 𝜇𝑛0 as 𝑛. Assume that 𝑇 is uniformly asymptotically regular on 𝐶 and 𝐹(𝑇) is nonempty and bounded. Let {𝑥𝑛} be a sequence generated in the following manner: 𝑥0𝐶𝐸chosenarbitrarily,1𝑥=𝐶,1=Π𝐶1𝑥0,𝐶𝑛+1=𝑢C𝑛𝑥𝜙𝑛,𝑇𝑛𝑥𝑛2𝑥1𝜅𝑛𝑢,𝐽𝑥𝑛𝐽𝑇𝑛𝑥𝑛+𝜇𝑛𝑀𝑛,𝑥1𝜅𝑛+1=Π𝐶𝑛+1𝑥0,𝑛0,(Υ) where 𝑀𝑛=sup{𝜙(𝑝,𝑥𝑛)𝑝𝐹(𝑇)} then the sequence {𝑥𝑛} converges strongly to 𝑥=Π𝐹(𝑇)𝑥0.

Proof. The proof is split into five steps.
Step 1. Show that 𝐹(𝑇) is closed and convex.
Let {𝑝𝑛} be a sequence in 𝐹(𝑇) such that 𝑝𝑛𝑝 as 𝑛. We see that 𝑝𝐹(𝑇). Indeed, we obtain from the definition of 𝑇 that 𝜙𝑝𝑛,𝑇𝑛𝑝1+𝜇𝑛𝜙𝑝𝑛,𝑝+𝜅𝜙(𝑝,𝑇𝑛𝑝).(3.1) In view of (2.6), we see that 𝜙𝑝𝑛,𝑇𝑛𝑝𝑝=𝜙𝑛,𝑝+𝜙(𝑝,𝑇𝑛𝑝)+2𝑝𝑛𝑝,𝐽𝑝𝐽𝑇𝑛𝑝.(3.2) It follows that 𝜙𝑝𝑛,𝑝+𝜙(𝑝,𝑇𝑛𝑝)+2𝑝𝑛𝑝,𝐽𝑝𝐽𝑇𝑛𝑝1+𝜇𝑛𝜙𝑝𝑛,𝑝+𝜅𝜙(𝑝,𝑇𝑛𝑝),(3.3) which implies that 𝜙(𝑝,𝑇𝑛𝜇𝑝)𝑛𝜙𝑝1𝜅𝑛+2,𝑝1𝜅𝑝𝑝𝑛,𝐽𝑝𝐽𝑇𝑛𝑝,(3.4) from which it follows that lim𝑛𝜙(𝑝,𝑇𝑛𝑝)=0.(3.5) From Lemma 2.6, we see that 𝑇𝑛𝑝𝑝 as 𝑛. This implies that 𝑇𝑇𝑛𝑝=𝑇𝑛+1𝑝𝑝 as 𝑛. From the closedness of 𝑇, we obtain that 𝑝𝐹(𝑇). This proves the closedness of 𝐹(𝑇).
Next, we show the convexness of 𝐹(𝑇). Let 𝑝1,𝑝2𝐹(𝑇) and 𝑝𝑡=𝑡𝑝1+(1𝑡)𝑝2, where 𝑡(0,1). We see that 𝑝𝑡=𝑇𝑝𝑡. Indeed, we have from the definition of 𝑇 that 𝜙𝑝1,𝑇𝑛𝑝𝑡1+𝜇𝑛𝜙𝑝1,𝑝𝑡𝑝+𝜅𝜙𝑡,𝑇𝑛𝑝𝑡,𝜙𝑝2,𝑇𝑛𝑝𝑡1+𝜇𝑛𝜙𝑝2,𝑝𝑡𝑝+𝜅𝜙𝑡,𝑇𝑛𝑝𝑡.(3.6) By virtue of (2.6), we obtain that 𝜙𝑝𝑡,𝑇𝑛𝑝𝑡𝜇𝑛𝜙𝑝1𝜅1,𝑝𝑡+21𝜅𝑝𝑡𝑝1,𝐽𝑝𝑡𝐽𝑇𝑛𝑝𝑡𝜙𝑝,(3.7)𝑡,𝑇𝑛𝑝𝑡𝜇n𝜙𝑝1𝜅2,𝑝𝑡+21𝜅𝑝𝑡𝑝2,𝐽𝑝𝑡𝐽𝑇𝑛𝑝𝑡.(3.8) Multiplying 𝑡 and (1𝑡) on both the sides of (3.7) and (3.8), respectively, yields that 𝜙𝑝𝑡,𝑇𝑛𝑝𝑡𝑡𝜇𝑛𝜙𝑝1𝜅1,𝑝𝑡+(1𝑡)𝜇𝑛𝜙𝑝1𝜅2,𝑝𝑡.(3.9) It follows that lim𝑛𝜙𝑝𝑡,𝑇𝑛𝑝𝑡=0.(3.10) In view of Lemma 2.6, we see that 𝑇𝑛𝑝𝑡𝑝𝑡 as 𝑛. This implies that 𝑇𝑇0𝑥0𝑎35𝑐𝑝𝑡=𝑇𝑛+1𝑝𝑡𝑝𝑡 as 𝑛. From the closedness of 𝑇, we obtain that 𝑝𝑡𝐹(𝑇). This proves that 𝐹(𝑇) is convex. This completes Step 1.
Step 2. Show that 𝐶𝑛 is closed and convex for each 𝑛1.
It is not hard to see that 𝐶𝑛 is closed for each 𝑛1. Therefore, we only show that 𝐶𝑛 is convex for each 𝑛1. It is obvious that 𝐶1=𝐶 is convex. Suppose that 𝐶 is convex for some . Next, we show that 𝐶+1 is also convex for the same . Let 𝑎,𝑏𝐶+1 and 𝑐=𝑡𝑎+(1𝑡)𝑏, where 𝑡(0,1). It follows that 𝜙𝑥,𝑇𝑥2𝑥1𝜅𝑎,𝐽𝑥𝐽𝑇𝑥+𝜇𝑀,𝜙𝑥1𝜅,𝑇𝑥2𝑥1𝜅𝑏,𝐽𝑥𝐽𝑇𝑥+𝜇𝑀,1𝜅(3.11) where 𝑎,𝑏𝐶. From the above two inequalities, we can get that 𝜙𝑥,𝑇𝑥2𝑥1𝜅𝑐,𝐽𝑥𝐽𝑇𝑥+𝜇𝑀,1𝜅(3.12) where 𝑐𝐶. It follows that 𝐶+1 is closed and convex. This completes Step 2.
Step 3. Show that 𝐹(𝑇)𝐶𝑛 for each 𝑛1.
It is obvious that 𝐹(𝑇)𝐶=𝐶1. Suppose that 𝐹(𝑇)𝐶 for some . For any 𝑧𝐹(𝑇)𝐶, we see that 𝜙𝑧,𝑇𝑥1+𝜇𝜙𝑧,𝑥𝑥+𝜅𝜙,𝑇𝑥.(3.13) On the other hand, we obtain from (2.6) that 𝜙𝑧,𝑇𝑥=𝜙𝑧,𝑥𝑥+𝜙,𝑇𝑥+2𝑧𝑥,𝐽𝑥𝐽𝑇𝑥.(3.14) Combining (3.13) with (3.14), we arrive at 𝜙𝑥,𝑇𝑥𝜇𝜙1𝜅𝑧,𝑥+2𝑥1𝜅𝑧,𝐽𝑥𝐽𝑇𝑥𝜇𝑀+21𝜅𝑥1𝜅𝑧,𝐽𝑥𝐽𝑇𝑥,(3.15) which implies that 𝑧𝐶+1. This shows that 𝐹(𝑇)𝐶+1. This completes Step 3.
Step 4. Show that the sequence {𝑥𝑛} is bounded.
In view of 𝑥𝑛=Π𝐶𝑛𝑥0, we see that 𝑥𝑛𝑧,𝐽𝑥0𝐽𝑥𝑛0,𝑧𝐶𝑛.(3.16) In view of 𝐹(𝑇)𝐶𝑛, we arrive at 𝑥𝑛𝑤,𝐽𝑥0𝐽𝑥𝑛0,𝑤𝐹(𝑇).(3.17) It follows from Lemma 2.8 that 𝜙𝑥𝑛,𝑥0Π=𝜙𝐶𝑛𝑥0,𝑥0Π𝜙𝐹(𝑇)𝑥0,𝑥0Π𝜙𝐹(𝑇)𝑥0,𝑥𝑛Π𝜙𝐹(𝑇)𝑥0,𝑥0.(3.18) This implies that the sequence {𝜙(𝑥𝑛,𝑥0)} is bounded. It follows from (2.5) that the sequence {𝑥𝑛} is also bounded. This completes Step 4.
Step 5. Show that 𝑥𝑛𝑥, where 𝑥=Π𝐹(𝑇)𝑥0, as 𝑛.
Since {𝑥𝑛} is bounded and the space is reflexive, we may assume that 𝑥𝑛𝑥 weakly. Since 𝐶𝑛 is closed and convex, we see that 𝑥𝐶𝑛. On the other hand, we see from the weakly lower semicontinuity of the norm that 𝜙𝑥,𝑥0=𝑥22𝑥,𝐽𝑥0+𝑥02liminf𝑛𝑥𝑛22𝑥𝑛,𝐽𝑥0𝑥+02=liminf𝑛𝜙𝑥𝑛,𝑥0limsup𝑛𝜙𝑥𝑛,𝑥0𝜙𝑥,𝑥0,(3.19) which implies that 𝜙(𝑥𝑛,𝑥0)𝜙(𝑥,𝑥0) as 𝑛. Hence, 𝑥𝑛𝑥 as 𝑛. In view of Kadec-Klee property of 𝐸, we see that 𝑥𝑛𝑥 as 𝑛.
Now, we are in a position to show that 𝑥𝐹(𝑇). Notice that lim𝑛𝑥𝑛+1𝑥𝑛=0. On the other hand, we see from 𝑥𝑛+1=Π𝐶𝑛+1𝑥0𝐶𝑛+1𝐶𝑛 that 𝜙𝑥𝑛,𝑇𝑛𝑥𝑛2𝑥1𝜅𝑛𝑥𝑛+1,𝐽𝑥𝑛𝐽𝑇𝑛𝑥𝑛+𝜇𝑛𝑀𝑛,1𝜅(3.20) from which it follows that 𝜙(𝑥𝑛,𝑇𝑛𝑥𝑛)0 as 𝑛. In view of Lemma 2.6, we arrive at lim𝑛𝑇𝑛𝑥𝑛𝑥𝑛=0.(3.21) Note that 𝑥𝑛𝑥 as 𝑛 in view of 𝑇𝑛𝑥𝑛𝑥𝑇𝑛𝑥𝑛𝑥𝑛+𝑥𝑛𝑥.(3.22) It follows from (3.21) that 𝑇𝑛𝑥𝑛𝑥as𝑛.(3.23) On the other hand, we have 𝑇𝑛+1𝑥𝑛𝑥𝑇𝑛+1𝑥𝑛𝑇𝑛𝑥𝑛+𝑇𝑛𝑥𝑛𝑥.(3.24) It follows from the uniformly asymptotic regularity of 𝑇 and (3.23) that 𝑇𝑛+1𝑥𝑛𝑥as𝑛(3.25) that is, 𝑇𝑇𝑛𝑥𝑛𝑥. From the closedness of 𝑇, we obtain that 𝑥=𝑇𝑥.
Finally, we show that 𝑥=Π𝐹(𝑇)𝑥0 which completes the proof. Indeed, we obtain from 𝑥𝑛=Π𝐶𝑛𝑥0 that 𝑥𝑛𝑤,𝐽𝑥0𝐽𝑥𝑛0,𝑤𝐶𝑛.(3.26) In particular, we have 𝑥𝑛𝑤,𝐽𝑥0𝐽𝑥𝑛0,𝑤𝐹(𝑇).(3.27) Taking the limit as 𝑛 in (3.27), we obtain that 𝑥𝑤,𝐽𝑥0𝐽𝑥0,𝑤𝐹(𝑇).(3.28) Hence, we obtain from Lemma 2.7 that 𝑥=Π𝐹(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 𝑇𝐶𝐶 be a closed and asymptotically quasi -𝜙-snonexpansive mapping with a sequence {𝜇𝑛}[0,) such that 𝜇𝑛0 as 𝑛. Assume that 𝑇 is uniformly asymptotically regular on 𝐶 and 𝐹(𝑇) is nonempty and bounded. Let {𝑥𝑛} be a sequence generated in the following manner: 𝑥0𝐶𝐸chosenarbitrarily,1𝑥=𝐶,1=Π𝐶1𝑥0,𝐶𝑛+1=𝑢𝐶𝑛𝑥𝜙𝑛,𝑇𝑛𝑥𝑛𝑥2𝑛𝑢,𝐽𝑥𝑛𝐽𝑇𝑛𝑥𝑛+𝜇𝑛𝑀𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0,𝑛0,(3.29) where 𝑀={𝜙(𝑝,𝑥𝑛)𝑝𝐹(𝑇)} then the sequence {𝑥𝑛} converges strongly to 𝑥=Π𝐹(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 𝑇𝐶𝐶 be a closed and asymptotically strict quasipseudocontraction with a sequence {𝜇𝑛}[0,) such that 𝜇𝑛0 as 𝑛. Assume that 𝑇 is uniformly asymptotically regular on 𝐶 and 𝐹(𝑇) is nonempty and bounded. Let {𝑥𝑛} be a sequence generated by the following manner: 𝑥0𝐶𝐻chosenarbitrarily,1𝑥=𝐶,1=𝑃𝐶1𝑥0,𝐶𝑛+1=𝑢𝐶𝑛𝑥𝑛𝑇𝑛𝑥𝑛22𝑥1𝜅𝑛𝑢,𝑥𝑛𝑇𝑛𝑥𝑛+𝜇𝑛𝑀𝑛,𝑥1𝜅𝑛+1=𝑃𝐶𝑛+1𝑥0,𝑛0,(3.30) where 𝑀𝑛=sup{𝑝𝑥𝑛2𝑝𝐹(𝑇)} then the sequence {𝑥𝑛} converges strongly to 𝑥=𝑃𝐹(𝑇)𝑥0.

Theorem 3.4. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝑇𝐶𝐶 be a closed and asymptotically quasinonexpansive mapping with a sequence {𝜇𝑛}[0,) such that 𝜇𝑛0 as 𝑛. Assume that 𝑇 is uniformly asymptotically regular on 𝐶 and 𝐹(𝑇) is nonempty and bounded. Let {𝑥𝑛} be a sequence generated by the following manner: 𝑥0𝐶𝐻chosenarbitrarily,1𝑥=𝐶,1=𝑃𝐶1𝑥0,𝐶𝑛+1=𝑢𝐶𝑛𝑥𝑛𝑇𝑛𝑥𝑛2𝑥2𝑛𝑢,𝑥𝑛𝑇𝑛𝑥𝑛+𝜇𝑛𝑀𝑛,𝑥𝑛+1=𝑃𝐶𝑛+1𝑥0,𝑛0,(3.31) where 𝑀𝑛={𝑝𝑥𝑛2𝑝𝐹(𝑇)} then the sequence {𝑥𝑛} converges strongly to 𝑥=𝑃𝐹(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).

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