Convergence Theorems for Common Fixed Points of a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces

Abstract

We establish some strong convergence theorems for a common fixed point of a finite family of relatively nonexpansive mappings by using a new hybrid iterative method in mathematical programming and the generalized projection method in a Banach space. Our results improve and extend the corresponding results by many others.

1. Introduction

Let 𝐸 be a smooth Banach space and 𝐸^∗ the dual of 𝐸. The function Φ∶𝐸×𝐸→𝑅 is defined by 𝜙𝑦,𝑥=𝑦²−2𝑦,𝐽𝑥+‖𝑥‖², (1) for all 𝑥,𝑦∈𝐸, where 𝐽 is the normalized duality mapping from 𝐸 to 𝐸^∗. Let 𝐶 be a closed convex subset of 𝐸, and let 𝑇 be a mapping from 𝐶 into itself. We denote by 𝐹(𝑇) the set of fixed points of 𝑇. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 (see [1]), if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that the strong lim_𝑛→∞(𝑥_𝑛−𝑇𝑥_𝑛)=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 from 𝐶 into itself is called nonexpansive, if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦, (2) for all 𝑥,𝑦∈𝐶, and relatively nonexpansive (see [2]), if 𝐹(𝑇)=𝐹(𝑇) and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥, (3) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). The iterative methods for approximation of fixed points of nonexpansive mappings, relatively nonexpansive mappings, and other generational nonexpansive mappings have been studied by many researchers; see [3–13].

Actually, Mann [14] firstly introduced Mann iteration process in 1953, which is defined as follows: 𝑥₀𝑥=𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0. (4) It is very useful to approximate a fixed point of a nonexpansive mapping. However, as we all know, it has only weak convergence in a Hilbert space (see [15]). As a matter of fact, the process (3) may fail to converge for a Lipschitz pseudocontractive mapping in a Hilbert space (see [16]). For example, Reich [17] proved that if 𝐸 is a uniformly convex Banach space with Fréchet differentiable norm and if {𝛼_𝑛} is chosen such that ∑^∞_𝑛=0𝛼_𝑛(1−𝛼_𝑛)=∞, then the sequence {𝑥_𝑛} defined by (3) converges weakly to a fixed point of 𝑇.

Some have made attempts to modify the Mann iteration methods, so that strong convergence is guaranteed. Nakajo and Takahashi [18] proposed the following modification of the Mann iteration method for a single nonexpansive mapping 𝑇 in a Hilbert space 𝐻: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑧−𝑦_𝑛≤𝑧−𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛⋂𝑄𝑛}𝑥,𝑛=0,1,2,…, (5) where 𝑃_𝐾 denotes the metric projection from 𝐻 onto a closed convex subset of 𝐻. They proved that if the sequence {𝛼_𝑛} is bounded above from one, then {𝑥_𝑛} defined by (5) converges strongly to 𝑃_𝐹(𝑇)𝑥.

The ideas to generate the process (5) from Hilbert spaces to Banach spaces have been made. By using the properties available on uniformly convex and uniformly smooth Banach spaces, Matsushita and Takahashi [10] presented their idea of the following method for a single relatively nonexpansive mapping 𝑇 in a Banach space 𝐸: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑊_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐻𝑛⋂𝑊𝑛}𝑥,𝑛=0,1,2,…, (6) where 𝐽 is the duality mapping on 𝐸 and Π_𝐹(𝑇)𝑥 is the generalized projection from 𝐶 onto 𝐹(𝑇).

In 2007 and 2008, Plubing and Ungchittrakool [19, 20] improved and generalized the process (6) to the new general process of two relatively nonexpansive mappings in a Banach space: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆𝑥_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑊_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐻𝑛⋂𝑊𝑛}𝑥𝑥,𝑛=0,1,2,…,₀𝑦=𝑥∈𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆𝑥_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+𝛼_𝑛‖𝑥‖²+2𝐽𝑥_𝑛,𝑊−𝐽𝑥,𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐻𝑛⋂𝑊𝑛}𝑥,𝑛=0,1,2,…. (7)(8) They proved that both iterations (7) and (8) converge strongly to a common fixed point of two relatively nonexpansive mappings 𝑆 and 𝑇 provided that the sequences satisfy some appropriate conditions.

Inspired and motivated by these facts, in this paper, we aim to improve and generalize the process (7) and (8) to the new general process of a finite family of relatively nonexpansive mappings in a Banach space. Let 𝐶 be a closed convex subset of a Banach space 𝐸 and let 𝑇₁,𝑇₂,…,𝑇_𝑁∶𝐶→𝐶 be relatively nonexpansive mappings such that 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)≠𝜙. Define {𝑥_𝑛} in the following way: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛,𝑧_𝑛=𝐽⁻¹𝜆_𝑛⁽⁰⁾𝐽𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝐽𝑇_𝑖𝑥_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+𝛼_𝑛‖𝑥‖²+2𝐽𝑥_𝑛,𝑊−𝐽𝑥,𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐻𝑛⋂𝑊𝑛}𝑥,𝑛=0,1,2,…, (9) where Π_{𝐻𝑛⋂𝑊𝑛} is the generalized projection from 𝐶 onto the intersection set 𝐻_𝑛⋂𝑊_𝑛; {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝜆_𝑛⁽⁰⁾},{𝜆_𝑛⁽¹⁾},…,{𝜆_𝑛^(𝑁)} are the sequences in [0,1] with 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1 and ∑^𝑁_𝑖=0𝜆_𝑛^(𝑖)=1 for all 𝑛≥0. We prove, under certain appropriate assumptions on the sequences, that {𝑥_𝑛} defined by (9) converges strongly to 𝑃_𝐹𝑥, where 𝑃_𝐹 is the generalized projection from 𝐶 to 𝐹.

Obviously, the process (9) reduces to become (7) when 𝑁=2,𝛼_𝑛=0 and become (8) when 𝑁=2,𝛽_𝑛=0. So, our results extend and improve the corresponding ones announced by Nakajo and Takahashi [18], Plubtieng and Ungchittrakool [19, 20], Matsushita and Takahashi [10], and Martinez-Yanes and Xu [21].

2. Preliminaries

This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section.

Throughout this paper, let 𝐸 be a real Banach space. Let 𝐽 denote the normalized duality mapping from 𝐸 into 2^𝐸∗ given by 𝐽(𝑥)=𝑓∈𝐸^∗,𝑥,𝑓=‖𝑥‖𝑓,‖𝑥‖=𝑓,∀𝑥∈𝐸, (10) where 𝐸^∗ denotes the dual space of 𝐸 and ⟨⋅,⋅⟩ denotes the generalized duality pairing.

A Banach space 𝐸 is said to be strictly convex if ‖𝑥+𝑦‖/2<1 for ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. It is also said to be uniformly convex if lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 for any two sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝐸 such that ‖𝑥_𝑛‖=‖𝑦_𝑛‖=1 and lim_𝑛→∞(‖𝑥_𝑛+𝑦_𝑛‖/2)=1. Let 𝑈={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸, then the Banach space 𝐸 is said to be smooth provided that lim_𝑡→0((‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡) exists for each 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit is attainted uniformly for each 𝑥,𝑦∈𝑈. It is well known that if 𝐸 is smooth, then the duality mapping 𝐽 is single valued. It is also known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸. Some properties of the duality mapping have been given in [22]. A Banach space 𝐸 is said to have Kadec-Klee property if a sequence {𝑥_𝑛} of 𝐸 satisfying that 𝑥_𝑛⇀𝑥∈𝐸 and ‖𝑥_𝑛‖→‖𝑥‖, then 𝑥_𝑛→𝑥. It is known that if 𝐸 is uniformly convex, then 𝐸 has the Kadec-Klee property; see [22] for more details.

Let 𝐸 be a smooth Banach space. The function Φ∶𝐸×𝐸→𝑅 is defined by 𝜙𝑦,𝑥=𝑦²−2𝑦,𝐽𝑥+‖𝑥‖², (11) for all 𝑥,𝑦∈𝐸. It is obvious from the definition of the function 𝜙 that (1)(‖𝑥‖−‖𝑦‖)²≤𝜙(𝑦,𝑥)≤(‖𝑦‖²+‖𝑥‖²),(2)𝜙(𝑥,𝑦)=𝜙(𝑥,𝑧)+𝜙(𝑧,𝑦)+2⟨𝑥−𝑧,𝐽𝑧−𝐽𝑥⟩,(3)𝜙(𝑥,𝑦)=⟨𝑥,𝐽𝑥−𝐽𝑦⟩+⟨𝑦−𝑥,𝐽𝑦⟩≤‖𝑥‖‖𝐽𝑥−𝐽𝑦‖+‖𝑦−𝑥‖‖𝑦‖,for all 𝑥,𝑦∈𝐸; see [4, 7, 23] for more details.

Lemma 1 (see [4]). If 𝐸 is a strictly convex and smooth Banach space, then for 𝑥,𝑦∈𝐸, 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦.

Lemma 2 (see [23]). Let 𝐸 be a uniformly convex and smooth Banach space and let {𝑦_𝑛},{𝑧_𝑛} be two sequences of 𝐸. If 𝜙(𝑦_𝑛,𝑧_𝑛)→0 and either {𝑦_𝑛} or {𝑧_𝑛} is bounded, then 𝑦_𝑛−𝑧_𝑛→0.

Let 𝐶 be a closed convex subset of 𝐸. Suppose that 𝐸 is reflexive, strictly convex, and smooth. Then, for any 𝑥∈𝐸, there exists a point 𝑥₀∈𝐶 such that 𝜙(𝑥₀,𝑥)=min_𝑦∈𝐶𝜙(𝑦,𝑥). The mapping Π_𝐶∶𝐸→𝐶 defined by Π_𝐶𝑥=𝑥₀ is called the generalized projection (see [4, 7, 23]).

Lemma 3 (see [7]). Let 𝐶 be a 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. (12)

Lemma 4 (see [7]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and let 𝐶 be a closed convex subset of 𝐸 and 𝑥∈𝐸. Then, 𝜙(𝑦,Π_𝐶𝑥)+𝜙(Π_𝐶𝑥,𝑥)≤𝜙(𝑦,𝑥) for all y ∈ C.

Lemma 5 (see [24]). Let 𝐸 be a uniformly convex Banach space and 𝐵_𝑟(0)={𝑥∈𝐸∶‖𝑥‖≤𝑟} a closed ball of 𝐸. Then, there exists a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝜆𝑥+𝜇𝑦+𝜈𝑧²≤𝜆‖𝑥‖²+𝜇𝑦²+𝜈‖𝑧‖²−𝜆𝜇𝑔𝑥−𝑦, (13) for all 𝑥,𝑦,𝑧∈𝐵_𝑟(0) and 𝜆,𝜇,𝜈∈[0,1] with 𝜆+𝜇+𝜈=1.

Lemma 6 (see [19]). Let 𝐸 be a uniformly convex and uniformly smooth Banach space and let 𝐶 be a closed convex subset of 𝐸. Then, for points 𝑤,𝑥,𝑦,𝑧∈𝐸 and a real number 𝑎∈𝑅, the set 𝐾∶={𝑣∈𝐶∶𝜙(𝑣,𝑦)≤𝜙(𝑣,𝑥)+⟨𝑣,𝐽𝑧−𝐽𝑤⟩+𝑎} is closed and convex.

3. Main Results

In this section, we will prove the strong convergence theorem for a common fixed point of a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Let us prove a proposition first.

Proposition 7. Let 𝐸 be a uniformly convex Banach space and 𝐵_𝑟(0)={𝑥∈𝐸∶‖𝑥‖≤𝑟} a closed ball of 𝐸. Then, there exists a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that ^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²≤^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²−1𝑛²𝑔^𝑛𝑛_𝑖=1𝑗=1𝑥_𝑖−𝑥_𝑗, (14) for all n ≥3,𝑥_𝑖∈𝐵_𝑟(0) and 𝜆_𝑖∈[0,1] with ∑^𝑛_𝑖=1𝜆_𝑖=1,𝑖=1,2,…,𝑛.

Proof. If 𝜆₃+𝜆₄⋯+𝜆_𝑛≠0, using Lemma 5 and the convexity of ‖⋅‖², we have ^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²=𝜆₁𝑥₁+𝜆₂𝑥₂+𝜆₃+⋯+𝜆_𝑛×𝜆₃𝑥₃𝜆₃+⋯+𝜆_𝑛𝜆+⋯+^𝑛𝑥_𝑛𝜆₃+⋯+𝜆_𝑛²≤𝜆₁𝑥₁²+𝜆₂𝑥₂²+𝜆₃+⋯+𝜆_𝑛×𝜆₃𝑥₃𝜆₃+⋯+𝜆_𝑛𝜆+⋯+^𝑛𝑥_𝑛𝜆₃+⋯+𝜆_𝑛²−𝜆₁𝜆₂𝑔𝑥₁−𝑥₂≤^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²−𝜆₁𝜆₂𝑔𝑥₁−𝑥₂. (15) If 𝜆₃+𝜆₄⋯+𝜆_𝑛=0, the last inequality above also holds obviously. By the same argument in the proof above, we obtain ^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²≤^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²−𝜆_𝑖𝜆_𝑗𝑔𝑥_𝑖−𝑥_𝑗, (16) for all 𝑖,𝑗∈{1,2,…,𝑛}. Then, 𝑛^2𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²≤𝑛^2𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²−^𝑛𝑛_𝑖=1𝑗=1𝜆_𝑖𝜆_𝑗𝑔𝑥_𝑖−𝑥_𝑗≤𝑛^2𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²−𝑔^𝑛𝑛_𝑖=1𝑗=1𝜆_𝑖𝜆_𝑗𝑥_𝑖−𝑥_𝑗. (17) So, ^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²≤^𝑛_𝑖=1𝜆_𝑖𝑥_𝑖²−1𝑛²𝑔^𝑛𝑛_𝑖=1𝑗=1𝜆_𝑖𝜆_𝑗𝑥_𝑖−𝑥_𝑗. (18)

Theorem 8. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑇₁,𝑇₂,…,𝑇_𝑁∶𝐶→𝐶 relatively nonexpansive mappings such that 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)≠𝜙. The sequence {𝑥_𝑛} is given by (9) with the following restrictions:(a) 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,0≤𝛼_𝑛<1,0≤𝛽_𝑛<1,0<𝛾_𝑛≤1 for all 𝑛≥0; (b) lim_𝑛→∞𝛼_𝑛=0 and limsup_𝑛→∞𝛽_𝑛<1; (c) 𝜆^𝑖_𝑛∈[0,1] with ∑^𝑁_𝑖=0𝜆^𝑖_𝑛=1,𝑖=0,1,2,…,𝑁, for all 𝑛≥0; (d) lim_𝑛→∞𝜆⁰_𝑛=0 and liminf_𝑛→∞𝜆^𝑖_𝑛𝜆^𝑗_𝑛>0,𝑖,𝑗=1,2,…,𝑁; or (d^′) liminf_𝑛→∞𝜆⁰_𝑛𝜆^𝑖_𝑛>0,𝑖=1,2,…,𝑁.Then, the sequence {𝑥_𝑛} converges strongly to Π_𝐹𝑥, where Π_𝐹 is the generalized projection from 𝐶 onto 𝐹.

Proof. We split the proof into seven steps.
Step  1. Show that 𝑃_𝐹 is well defined for every 𝑥∈𝐶.
It is easy to know that 𝐹(𝑇_𝑖), 𝑖=1,2,…,𝑁 are closed convex sets and so is 𝐹. What is more, 𝐹 is nonempty by our assumption. Therefore, 𝑃_𝐹 is well defined for every 𝑥∈𝐶.
Step  2. Show that 𝐻_𝑛 and 𝑊_𝑛 are closed and convex for all 𝑛≥0.
From the definition of 𝑊_𝑛, it is obvious 𝑊_𝑛 is closed and convex for each 𝑛≥0. By Lemma 6, we also know that 𝐻_𝑛 is closed and convex for each 𝑛≥0.
Step  3. Show that 𝐹⊂𝐻_𝑛⋂𝑊_𝑛 for all 𝑛≥0.
Let 𝑢∈𝐹 and let 𝑛≥0. Then, by the convexity of ‖⋅‖², we have 𝜙𝑢,𝑧_𝑛=𝜙𝑢,𝐽⁻¹𝜆_𝑛⁽⁰⁾𝐽𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝐽𝑇_𝑖𝑥_𝑛=‖𝑢‖²−2𝑢,𝜆_𝑛⁽⁰⁾𝐽𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝐽𝑇_𝑖𝑥_𝑛+𝜆_𝑛⁽⁰⁾𝐽𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝐽𝑇_𝑖𝑥_𝑛²≤‖𝑢‖²−2𝜆_𝑛⁽⁰⁾𝑢,𝐽𝑥_𝑛−2^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝑢,𝐽𝑇_𝑖𝑥_𝑛+𝜆_𝑛⁽⁰⁾𝐽𝑥_𝑛²+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝐽𝑇_𝑖𝑥_𝑛²=𝜆_𝑛⁽⁰⁾𝜙𝑢,𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝜙𝑢,𝑇_𝑖𝑥_𝑛≤𝜆_𝑛⁽⁰⁾𝜙𝑢,𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝜙𝑢,𝑥_𝑛=𝜙𝑢,𝑥_𝑛, (19) and then, 𝜙𝑢,𝑦_𝑛=𝜙𝑢,𝐽⁻¹𝛼_𝑛𝐽𝑥+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛=‖𝑢‖²−2𝑢,𝛼_𝑛𝐽𝑥+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛²≤‖𝑢‖²−2𝛼_𝑛⟨𝑢,𝐽𝑥⟩−2𝛽_𝑛𝑢,𝐽𝑥_𝑛−2𝛾_𝑛𝑢,𝐽𝑧_𝑛+𝛼_𝑛‖𝑥‖²+𝛽_𝑛𝑥_𝑛²+𝛾_𝑛𝑧_𝑛²=𝛼_𝑛𝜙(𝑢,𝑥)+𝛽_𝑛𝜙𝑢,𝑥_𝑛+𝛾_𝑛𝜙𝑢,𝑧_𝑛≤𝛼_𝑛𝜙(𝑢,𝑥)+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛=𝜙𝑢,𝑥_𝑛+𝛼_𝑛𝜙(𝑢,𝑥)−𝜙𝑢,𝑥_𝑛≤𝜙𝑢,𝑥_𝑛+𝛼_𝑛‖𝑥‖²+2𝐽𝑥_𝑛−𝐽𝑥,𝑧. (20) Thus, we have 𝑢∈𝐻_𝑛. Therefore, we obtain 𝐹⊂𝐻_𝑛 for all 𝑛≥0.
Next, we prove 𝐹⊂𝑊_𝑛 for all 𝑛≥0. We prove this by induction. For 𝑛=0, we have 𝐹⊂𝐶=𝑊₀. Assume that 𝐹⊂𝑊_𝑛. Since 𝑥_𝑛+1 is the projection of 𝑥 onto 𝐻_𝑛⋂𝑊_𝑛, by Lemma 3, we have 𝑥_𝑛+1−𝑧,𝐽𝑥−𝐽𝑥_𝑛+1≥0, (21) for any 𝑧∈𝐻_𝑛⋂𝑊_𝑛. As 𝐹⊂𝐻_𝑛⋂𝑊_𝑛 by the induction assumption,𝐹⊂𝑊_𝑛 holds, in particular, for all 𝑢∈𝐹. This together with the definition of 𝑊_𝑛+1 implies that 𝐹⊂𝑊_𝑛+1. Hence, 𝐹⊂𝐻_𝑛⋂𝑊_𝑛 for all 𝑛≥0.
Step  4. Show that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞.
In view of (19) and Lemma 4, we have 𝑥_𝑛=𝑃_𝑊𝑛𝑥, which means that, for any 𝑧∈𝑊_𝑛, 𝜙𝑥_𝑛,𝑥≤𝜙(𝑧,𝑥). (22) Since 𝑥_𝑛+1∈𝑊_𝑛 and 𝑢∈𝐹⊂𝑊_𝑛, we obtain 𝜙𝑥_𝑛,𝑥≤𝜙𝑥_𝑛+1,𝜙𝑥,𝑥_𝑛,𝑥≤𝜙(𝑢,𝑥), (23) for all 𝑛≥0. Consequently, lim_𝑛→∞𝜙(𝑥_𝑛,𝑥) exists and {𝑥_𝑛} is bounded. By using Lemma 4, we have 𝜙𝑥_𝑛+1,𝑥_𝑛≤𝜙𝑥_𝑛+1,𝑥−𝜙𝑥_𝑛,𝑥⟶0, (24) as 𝑛→∞. By using Lemma 2, we obtain ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞.
Step  5. Show that ‖𝑥_𝑛−𝑧_𝑛‖→0 as 𝑛→∞.
From 𝑥_𝑛+1=𝑃_{𝐻𝑛⋂𝑊𝑛}𝑥∈𝐻_𝑛, we have 𝜙𝑥_𝑛+1,𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝛼_𝑛‖𝑥‖²+2𝐽𝑥_𝑛−𝐽𝑥,𝑥_𝑛+1⟶0, (25) as 𝑛→∞. By Lemma 2, we also have ‖𝑥_𝑛+1−𝑦_𝑛‖→0, and then, 𝑥_𝑛−𝑦_𝑛≤𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛−𝑥_𝑛+1⟶0, (26) as 𝑛→∞. We observe that 𝜙𝑧_𝑛,𝑥_𝑛=𝜙𝑧_𝑛,𝑦_𝑛+𝜙𝑦_𝑛,𝑥_𝑛+2𝑧_𝑛−𝑦_𝑛,𝐽𝑦_𝑛−𝐽𝑥_𝑛≤𝜙𝑧_𝑛,𝑦_𝑛+𝜙𝑦_𝑛,𝑥_𝑛+2𝑧_𝑛−𝑦_𝑛𝐽𝑦_𝑛−𝐽𝑥_𝑛,𝜙𝑧_𝑛,𝑦_𝑛=𝑧_𝑛²−2𝑧_𝑛,𝛼_𝑛𝐽𝑥+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑧_𝑛²≤𝛼_𝑛𝜙𝑧_𝑛,𝑥+𝛽_𝑛𝜙𝑧_𝑛,𝑥_𝑛. (27) So, 𝜙𝑧_𝑛,𝑥_𝑛≤𝛼_𝑛𝜙𝑧_𝑛,𝑥+𝛽_𝑛𝜙𝑧_𝑛,𝑥_𝑛+𝜙𝑦_𝑛,𝑥_𝑛+2𝑧_𝑛−𝑦_𝑛𝐽𝑦_𝑛−𝐽𝑥_𝑛. (28) Since lim_𝑛→∞𝛼_𝑛=0, limsup_𝑛→∞𝛽_𝑛<1, 𝜙(𝑦_𝑛,𝑥_𝑛)→0, and ‖𝑧_𝑛−𝑦_𝑛‖‖𝐽𝑦_𝑛−𝐽𝑥_𝑛‖→0 as 𝑛→∞, we have 𝜙𝑧_𝑛,𝑥_𝑛≤𝛼_𝑛1−𝛽_𝑛𝜙𝑧_𝑛+1,𝑥1−𝛽_𝑛𝜙𝑦_𝑛,𝑥_𝑛+21−𝛽_𝑛𝑧_𝑛−𝑦_𝑛𝐽𝑦_𝑛−𝐽𝑥_𝑛⟶0, (29) as 𝑛→∞. Using Lemma 2, we obtain ‖𝑥_𝑛−𝑧_𝑛‖→0 as 𝑛→∞.
Step  6. Show that ‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖→0,𝑖=1,2,…,𝑁.
Since {𝑥_𝑛} is bounded and 𝜙(𝑝,𝑇_𝑖𝑥_𝑛)≤𝜙(𝑝,𝑥_𝑛), where 𝑝∈𝐹, 𝑖=1,2,…,𝑁, we also obtain that {𝐽𝑥_𝑛},{𝐽𝑇₁𝑥_𝑛},…,{𝐽𝑇_𝑁𝑥_𝑛} are bounded, and hence, there exists 𝑟>0 such that {𝐽𝑥_𝑛},{𝐽𝑇₁𝑥_𝑛},…,{𝐽𝑇_𝑁𝑥_𝑛}⊂𝐵_𝑟(0). Therefore, Proposition 7 can be applied and we observe that 𝜙𝑝,𝑧_𝑛=𝑝²−2𝑝,𝜆_𝑛⁽⁰⁾𝐽𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝐽𝑇_𝑖𝑥_𝑛+𝜆_𝑛⁽⁰⁾𝐽𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝐽𝑇_𝑖𝑥_𝑛²≤𝑝²−2𝜆_𝑛⁽⁰⁾𝑝,𝐽𝑥_𝑛−2^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝑝,𝐽𝑇_𝑖𝑥_𝑛+𝜆_𝑛⁽⁰⁾𝑥_𝑛²+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝑇_𝑖𝑥_𝑛²−1𝑁²𝑔^𝑁𝑁_𝑖=1𝑗=1𝜆_𝑛^(𝑖)𝜆_𝑛^(𝑗)𝐽𝑇_𝑖𝑥_𝑛−𝐽𝑇_𝑗𝑥_𝑛+2^𝑁_𝑖=1𝜆_𝑛⁽⁰⁾𝜆_𝑛^(𝑖)𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛=𝜆_𝑛⁽⁰⁾𝜙𝑝,𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝜙𝑝,𝑇_𝑖𝑥_𝑛−1𝑁²𝑔^𝑁𝑁_𝑖=1𝑗=1𝜆_𝑛^(𝑖)𝜆_𝑛^(𝑗)𝐽𝑇_𝑖𝑥_𝑛−𝐽𝑇_𝑗𝑥_𝑛+2^𝑁_𝑖=1𝜆_𝑛⁽⁰⁾𝜆_𝑛^(𝑖)𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛≤𝜙𝑝,𝑥_𝑛−1𝑁²𝑔^𝑁𝑁_𝑖=1𝑗=1𝜆_𝑛^(𝑖)𝜆_𝑛^(𝑗)𝐽𝑇_𝑖𝑥_𝑛−𝐽𝑇_𝑗𝑥_𝑛+2^𝑁_𝑖=1𝜆_𝑛⁽⁰⁾𝜆_𝑛^(𝑖)𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛, (30) where 𝑔∶[0,∞)→[0,∞) is a continuous strictly increasing convex function with 𝑔(0)=0. And 𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑧_𝑛=𝑝²−2𝑝,𝐽𝑥_𝑛+𝑥_𝑛²−𝑝²+2𝑝,𝐽𝑧_𝑛−𝑧_𝑛²≤2𝑝𝐽𝑧_𝑛−𝐽𝑥_𝑛+𝑥_𝑛²−𝑧_𝑛²⟶0, (31) as 𝑛→∞. From the properties of the mapping 𝑔, we have lim_𝑛→∞𝜆_𝑛⁽⁰⁾𝜆_𝑛^(𝑖)𝑥_𝑛−𝑇_𝑖𝑥_𝑛=0,lim_𝑛→∞𝜆_𝑛^(𝑖)𝜆_𝑛^(𝑗)𝑇_𝑖𝑥_𝑛−𝑇_𝑗𝑥_𝑛=0, (32) for all 𝑖,𝑗∈{1,2,…,𝑁}. From the condition (d^′), we have ‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖→0 immediately, as 𝑛→∞, 𝑖=1,2,…,𝑁; from the condition (d), we can also have ‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖→0, as 𝑛→∞,𝑖=1,2,…,𝑁. In fact, since liminf_𝑛→∞𝜆_𝑛^(𝑖)𝜆_𝑛^(𝑗)>0, it follows that lim_𝑛→∞𝑇_𝑖𝑥_𝑛−𝑇_𝑗𝑥_𝑛=0, (33) for all 𝑖,𝑗∈{1,2,…,𝑁}. Next, we note by the convexity of ‖⋅‖² and (9) that 𝜙𝑇_𝑗𝑥_𝑛,𝑧_𝑛=𝑇_𝑗𝑥_𝑛²−2𝑇_𝑗𝑥_𝑛,𝜆_𝑛⁽⁰⁾𝐽𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝐽𝑇_𝑖𝑥_𝑛+𝜆_𝑛⁽⁰⁾𝐽𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝐽𝑇_𝑖𝑥_𝑛²≤𝑇_𝑗𝑥_𝑛²−2𝜆_𝑛⁽⁰⁾𝑇_𝑗𝑥_𝑛,𝐽𝑥_𝑛−2^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝑇_𝑗𝑥_𝑛,𝐽𝑇_𝑖𝑥_𝑛+𝜆_𝑛⁽⁰⁾𝑥_𝑛²+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝑇_𝑖𝑥_𝑛²=𝜆_𝑛⁽⁰⁾𝜙𝑇_𝑗𝑥_𝑛,𝑥_𝑛+^𝑁_𝑖=1𝜆_𝑛^(𝑖)𝜙𝑇_𝑗𝑥_𝑛,𝑇_𝑖𝑥_𝑛⟶0, (34) as 𝑛→∞. By Lemma 2, we have lim_𝑛→∞‖𝑇_𝑖𝑥_𝑛−𝑧_𝑛‖=0 and 𝑇_𝑖𝑥_𝑛−𝑥_𝑛≤𝑇_𝑖𝑥_𝑛−𝑧_𝑛+𝑥_𝑛−𝑧_𝑛⟶0, (35) as 𝑛→∞ for all 𝑖∈{1,2,…,𝑁}.
Step  7. Show that 𝑥_𝑛→Π_𝐹𝑥, as 𝑛→∞.
From the result of Step 6, we know that if {𝑥_𝑛𝑘} is a subsequence of {𝑥_𝑛} such that {𝑥_𝑛𝑘}⇀𝑥∈𝐶, then 𝑥∈∩^𝑁_𝑖=1𝐹(𝑇_𝑖)=∩^𝑁_𝑖=1𝐹(𝑇_𝑖). Because 𝐸 is a uniformly convex and uniformly smooth Banach space and {𝑥_𝑛} is bounded, so we can assume {𝑥_𝑛𝑘} is a subsequence of {𝑥_𝑛} such that {𝑥_𝑛𝑘}⇀𝑥∈𝐹 and 𝜔=Π_𝐹𝑥. For any 𝑛≥1, from 𝑥_𝑛+1=Π_{𝐻𝑛⋂𝑊𝑛}𝑥 and 𝜔∈𝐹⊂𝐻_𝑛⋂𝑊_𝑛, we have 𝜙𝑥_𝑛+1,𝑥≤𝜙(𝜔,𝑥). (36) On the other hand, from weakly lower semicontinuity of the norm, we have 𝜙()=‖𝑥‖𝑥,𝑥₂−2⟨𝑥,𝐽𝑥⟩+‖𝑥‖²≤liminf_𝑛→∞𝑥_𝑛𝑘²−2𝑥_𝑛𝑘²,𝐽𝑥+‖𝑥‖²=liminf_𝑛→∞𝜙𝑥_𝑛𝑘,𝑥≤limsup_𝑛→∞𝜙𝑥_𝑛𝑘,𝑥≤𝜙(𝜔,𝑥). (37) From the definition of Π_𝐹𝑥, we obtain 𝑥=𝜔, and hence, lim_𝑛→∞𝜙(𝑥_𝑛𝑘,𝑥)=𝜙(𝜔,𝑥). So, we have lim_𝑘→∞‖𝑥_𝑛𝑘‖=‖𝜔‖. Using the Kadec-klee property of 𝐸, we obtain that {𝑥_𝑛𝑘} converges strongly to Π_𝐹𝑥. Since {𝑥_𝑛𝑘} is an arbitrary weakly convergent sequence of {𝑥_𝑛}, we can conclude that {𝑥_𝑛} converges strongly to Π_𝐹𝑥.

Corollary 9. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑇₁,𝑇₂,…,𝑇_𝑁∶𝐶→𝐶 relatively nonexpansive mappings such that 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)≠𝜙. The sequence {𝑥_𝑛} is given by (9) with the following restrictions: (a) 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,0≤𝛼_𝑛<1,0≤𝛽_𝑛<1,0≤𝛾_𝑛≤1 for all n ≥ 0; (b) lim_𝑛→∞𝛼_𝑛=0 and limsup_𝑛→∞𝛽_𝑛<1; (c) 𝜆^𝑖_𝑛∈[0,1] with ∑^𝑁_𝑖=0𝜆^𝑖_𝑛=1, 𝑖=0,1,2,…,𝑁, for all 𝑛≥0; (d) lim_𝑛→∞𝜆⁰_𝑛=0 and liminf_𝑛→∞𝜆^𝑖_𝑛𝜆^𝑗_𝑛>0,𝑖,𝑗=1,2,…,𝑁; or (d^′) liminf_𝑛→∞𝜆⁰_𝑛𝜆^𝑖_𝑛>0,𝑖=1,2,…,𝑁.Then, the sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥, where 𝑃_𝐹 is the metric projection from 𝐶 onto 𝐹.

Proof. It is true because the generalized projection Π_𝐹 is just the metric projection 𝑃_𝐹 in Hilbert spaces.

Remark 10. The results of Nakajo and Takahashi [18] and Song et al. [11] are the special cases of our results in Corollary 9. And in our results of Theorem 8, if 𝑇₁=𝑇₂=⋯=𝑇_𝑁,𝜆_𝑛⁽⁰⁾ and 𝛼_𝑛=0 for all 𝑛≥0, then, we obtain Theorem 4.1 of Matsushita and Takahashi [10]; if 𝑇₁=𝑇₂=⋯=𝑇_𝑁−1 and 𝛼_𝑛=0 for all 𝑛≥0, then, we obtain Theorem 3.1 of Plubtieng and Ungchittrakool [19]; if 𝑇₁=𝑇₂=⋯=𝑇_𝑁−1 and 𝛽_𝑛=0 for all 𝑛≥0, then, we obtain Theorem 3.2 of Plubtieng and Ungchittrakool [19]. So, our results improve and extend the corresponding results by many others.

Acknowledgments

This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and the National Natural Science Foundation of China (11071169, 11271330).