Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

Abstract

Let 𝐶 be a nonempty closed convex subset of a real uniformly smooth Banach space 𝑋, {𝑇_𝑘}^∞𝑘=1∶𝐶→𝐶 an infinite family of nonexpansive mappings with the nonempty set of common fixed points ⋂^∞𝑘=1Fix(𝑇_𝑘), and 𝑓∶𝐶→𝐶 a contraction. We introduce an explicit iterative algorithm 𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝐿_𝑛𝑥_𝑛, where 𝐿_𝑛=∑^𝑛𝑘=1𝜔_𝑘/s_𝑛𝑇_𝑘,𝑆_𝑛=∑^𝑛𝑘=1𝜔_𝑘, and 𝑤_𝑘>0 with ∑^∞𝑘=1𝜔_𝑘=1. Under certain appropriate conditions on {𝛼_𝑛}, we prove that {𝑥_𝑛} converges strongly to a common fixed point 𝑥^∗ of {𝑇_𝑘}^∞𝑘=1, which solves the following variational inequality: ⟨𝑥^∗−𝑓(𝑥^∗),𝐽(𝑥^∗−𝑝)⟩≤0,𝑝∈⋂^∞𝑘=1Fix(𝑇_𝑘), where 𝐽 is the (normalized) duality mapping of 𝑋. This algorithm is brief and needs less computational work, since it does not involve 𝑊-mapping.

1. Introduction

Let 𝑋 be a real Banach space, 𝐶 a nonempty closed convex subset of 𝑋, and 𝑋^∗ the dual space of 𝑋. The (normalized) duality mapping 𝐽∶𝑋→2^𝑋∗ is defined by 𝐽(𝑥)=𝑥^∗∈𝑋^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²,‖𝑥^∗‖=‖𝑥‖,∀𝑥∈𝑋.(1.1) If 𝑋 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity mapping. It is well known that if 𝑋 is smooth, then 𝐽 is single valued.

Recall that a mapping 𝑓∶𝐶→𝐶 is a contraction, if there exists a constant 𝛼∈[0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.2) We use Π_𝐶 to denote the collection of all contractions on 𝐶, that is, Π_𝐶={𝑓∶𝑓isacontractionon𝐶}.(1.3) A mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive, if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.4) We use Fix(𝑇) to denote the set of fixed points of 𝑇, namely, Fix(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping ([1–11]). Browder [1] first considered the following approximation in a Hilbert space. Fix 𝑢∈𝐶 and define a contraction 𝐹_𝑡 from 𝐶 into itself by𝐹_𝑡𝑥=𝑡𝑢+(1−𝑡)𝑇𝑥,𝑥∈𝐶,(1.5) where 𝑡∈(0,1). Banach contraction mapping principle guarantees that 𝐹_𝑡 has a unique fixed point in 𝐶. Denote by 𝑧_𝑡∈𝐶 the unique fixed point of 𝐹_𝑡, that is,𝑧_𝑡=𝑡𝑢+(1−𝑡)𝑇𝑧_𝑡.(1.6) In the case of 𝑇 having fixed points, Browder [1] proved the following.

Theorem 1.1. In a Hilbert space, as 𝑡→0, 𝑧_𝑡 defined in (1.6) converges strongly to a fixed point of 𝑇 that is closest to 𝑢, that is, the nearest point projection of 𝑢 onto Fix(𝑇).

Halpern [3] introduced an iteration process (discretization of (1.6)) in a Hilbert as follows:𝑧_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑧_𝑛,𝑛≥0,(1.7) where 𝑢,𝑧₀∈𝐶 are arbitrary (but fixed) and {𝛼_𝑛} is a sequence in (0,1). Lions [4] proved the following.

Theorem 1.2. In a Hilbert space, if {𝛼_𝑛} satisfies the following conditions: (K1)lim_𝑛→∞𝛼_𝑛=0;(K2)∑^∞𝑛=0𝛼_𝑛=∞;(K3)lim_𝑛→∞|𝛼_𝑛−𝛼_𝑛−1|/𝛼^2𝑛+1=0. Then {𝑧_𝑛} converges strongly to the nearest point projection of 𝑢 onto Fix(𝑇).

The Banach space versions of Theorems 1.1 and 1.2 were obtained by Reich [5]. He proved the following.

Theorem 1.3. In a uniformly smooth Banach space 𝑋, both 𝑧_𝑡 defined in (1.6) and {𝑧_𝑛} defined in (1.7) converge strongly to a same fixed point of T. If one defines 𝑄∶𝐶→Fix(𝑇) by 𝑄(𝑢)=lim_𝑡→0𝑧_𝑡,(1.8) then 𝑄 is the sunny nonexpansive retraction from 𝐶 onto Fix(𝑇). Namely, Q satisfies the property: ‖𝑄𝑥−𝑄𝑦‖²≤⟨𝑥−𝑦,𝐽(𝑄𝑥−𝑄𝑦)⟩,𝑥,𝑦∈𝐶,(1.9) where 𝐽 is the duality mapping of 𝑋.

Moudafi [6] introduced a viscosity approximation method and proved the strong convergence of both the implicit and explicit methods in Hilbert spaces. Xu [7] extended Moudafi's results in Hilbert spaces. Given a real number 𝑡∈(0,1) and a contraction 𝑓∈Π_𝐶, define a contraction 𝑇^𝑓𝑡∶𝐶→𝐶 by𝑇^𝑓𝑡𝑥=𝑡𝑓(𝑥)+(1−𝑡)𝑇𝑥,𝑥∈𝐶.(1.10) Let 𝑥_𝑡∈𝐶 be the unique fixed point of 𝑇^𝑓𝑡. Thus,𝑥_𝑡=𝑡𝑓𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡.(1.11) Corresponding explicit iterative process is defined by𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,(1.12) where 𝑥₀∈𝐶 is arbitrary (but fixed) and {𝛼_𝑛} is a sequence in (0,1). It was proved by Xu [7] that under certain appropriate conditions on {𝛼_𝑛}, both 𝑥_𝑡 defined in (1.11) and {𝑥_𝑛} defined in (1.12) converged strongly to 𝑥^∗∈𝐶, which is the unique solution of the variational inequality: ⟨(𝐼−𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,𝑥∈Fix(𝑇).(1.13) Xu [7] also extended Moudafi's results to the setting of Banach spaces and proved the strong convergence of both the implicit method (1.11) and explicit method (1.12) in uniformly smooth Banach spaces.

In order to deal with some problems involving the common fixed points of infinite family of nonexpansive mappings, 𝑊-mapping is often used, see [12–20]. Let {𝑇_𝑘}^∞𝑘=1∶𝐶→𝐶 be an infinite family of nonexpansive mappings and let {𝜉_𝑘}^∞𝑘=1 be a real number sequence such that 0<𝜉_𝑘<1 for every 𝑘∈ℕ. For any 𝑛∈ℕ, we define a mapping 𝑊_𝑛 of 𝐶 into itself as follows:𝑈𝑈_𝑛,𝑛+1=𝐼,_𝑛,𝑛𝑈=𝜉_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜉_𝑛𝐼,_{𝑛,𝑛−1}𝑈=⋮𝜉_𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+1−𝜉_𝑛−1𝐼,_𝑛,𝑘𝑈=𝜉_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜉_𝑘𝐼,_{𝑛,𝑘−1}𝑈=⋮𝜉_𝑘−1𝑇_𝑘−1𝑈_𝑛,𝑘+1−𝜉_𝑘−1𝐼,_𝑛,2𝑊=𝜉₂𝑇₂𝑈_𝑛,3+1−𝜉₂𝐼,_𝑛=𝑈_𝑛,1=𝜉₁𝑇₁𝑈_𝑛,2+1−𝜉₁𝐼.(1.14) Such 𝑊_𝑛 is called the 𝑊-mapping generated by {𝑇_𝑘}^∞𝑘=1 and {𝜉_𝑘}^∞𝑘=1, see [12, 13].

Yao et al. [10] introduced the following iterative algorithm for infinite family of nonexpansive mappings. Let 𝑋 be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm and 𝐶 a nonempty closed convex subset of 𝑋. Sequence {𝑥_𝑛} is defined by𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑊_𝑛𝑥_𝑛,𝑛≥0,(1.15) where 𝑢,𝑥₀∈𝐶 are arbitrary (but fixed) and {𝛼_𝑛}⊂(0,1). It was proved that under certain appropriate conditions on {𝛼_𝑛}, the sequence {𝑥_𝑛} generated by (1.15) converges strongly to a common fixed point of {𝑇_𝑘}^∞𝑘=1 [13].

Since 𝑊-mapping contains many composite operations of {𝑇_𝑘}, it is complicated and needs large computational work. In this paper, we introduce a new iterative algorithm for solving the common fixed point problem of infinite family of nonexpansive mappings. Let 𝑋 be a real uniformly smooth Banach space, 𝐶 a nonempty closed convex subset of 𝑋, {𝑇_𝑘}^∞𝑘=1∶𝐶→𝐶 an infinite family of nonexpansive mappings with the nonempty set of common fixed points ⋂^∞𝑘=1Fix(𝑇_𝑘), and 𝑓∈Π_𝐶. Given any 𝑥₀∈𝐶, define a sequence {𝑥_𝑛} by𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐿_𝑛𝑥_𝑛,𝑛≥0,(1.16) where {𝛼_𝑛}⊂(0,1), 𝐿_𝑛=∑^𝑛𝑘=1(𝜔_𝑘/𝑠_𝑛)𝑇_𝑘, 𝑆_𝑛=∑^𝑛𝑘=1𝜔_𝑘 and 𝑤_𝑘>0 with ∑^∞𝑘=1𝜔_𝑘=1. Under certain appropriate conditions on {𝛼_𝑛}, we prove that {𝑥_𝑛} converges strongly to 𝑥^∗∈⋂^∞𝑘=1Fix(𝑇_𝑘), which solves the following variational inequality: ⟨𝑥^∗−𝑓𝑥^∗,𝐽𝑥^∗−𝑝⟩≤0,𝑝∈^∞_𝑘=1Fix𝑇_𝑘,(1.17) where 𝐽 is the duality mapping of 𝑋. Because 𝐿_𝑛 doesn't contain many composite operations of {𝑇_𝑘}, this algorithm is brief and needs less computational wok.

We will use 𝑀 to denote a constant, which may be different in different places.

2. Preliminaries

Let 𝐵={𝑥∈𝑋∶‖𝑥‖=1} denotes the unit sphere of 𝑋. A Banach space 𝑋 is said to be strictly convex, if ‖(𝑥+𝑦)/2‖<1 holds for all 𝑥,𝑦∈𝐵, 𝑥≠𝑦. A Banach space 𝑋 is said to be uniformly convex if for each 𝜀∈(0,2], there exists a constant 𝛿>0 such that for any x,𝑦∈𝐵, ‖𝑥−𝑦‖≥𝜀 implies ‖(𝑥+𝑦)/2‖≤1−𝛿. It is known that a uniformly convex Banach space is reflexive and strictly convex, see [21].

The norm of 𝑋 is said to be Gâteaux differentiable iflim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.1) exists for each 𝑥,𝑦∈𝐵 and in this case 𝑋 is said to be smooth. The norm of 𝑋 is said to be uniformly Gâteaux differentiable if for each 𝑦∈𝐵, the limit (2.1) is attained uniformly for 𝑥∈𝐵. The norm of 𝑋 is said to be Frêchet differentiable, if for each 𝑥∈𝐵, the limit (2.1) is attained uniformly for 𝑦∈𝐵. The norm of 𝑋 is said to be uniformly Frêchet differentiable, if the limit (2.1) is attained uniformly for 𝑥,𝑦∈𝐵 and in this case 𝑋 is said to be uniformly smooth.

Let 𝐷 be a nonempty subset of 𝐶. A mapping 𝑄∶𝐶→𝐷 is said to be sunny [22] if𝑄(𝑥+𝑡(𝑥−𝑄(𝑥)))=𝑄(𝑥),∀𝑥∈𝐶,𝑡≥0,(2.2) whenever 𝑥+𝑡(𝑥−𝑄(𝑥))∈𝐶. A mapping 𝑄∶𝐶→𝐷 is called a retraction if 𝑄𝑥=𝑥 for all 𝑥∈𝐷. Furthermore, 𝑄 is sunny nonexpansive retraction from 𝐶 onto 𝐷 if 𝑄 is a retraction from 𝐶 onto 𝐷 which is also sunny and nonexpansive.

A subset 𝐷 of 𝐶 is called a sunny nonexpansive retraction of 𝐶 if there exits a sunny nonexpansive retraction from 𝐶 onto 𝐷.

Lemma 2.1 (see [22]). Let 𝐶 be a closed convex subset of a smooth Banach space 𝑋. Let 𝐷 be a nonempty subset of 𝐶 and 𝑄∶𝐶→𝐷 be a retraction. Then the following are equivalent.(a)𝑄 is sunny and nonexpansive.(b)‖𝑄𝑥−𝑄𝑦‖²≤⟨𝑥−𝑦,𝐽(𝑄𝑥−𝑄𝑦)⟩, for all 𝑥,𝑦∈𝐶.(c)⟨𝑥−𝑄𝑥,𝐽(𝑦−𝑄𝑥)⟩≤0, for all 𝑥∈𝐶,𝑦∈𝐷.

Lemma 2.2 (see [23]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1≤1−𝛾_𝑛𝑠_𝑛+𝛾_𝑛𝛽_𝑛+𝛿_𝑛,𝑛≥0,(2.3) where {𝛾_𝑛}⊂(0,1), {𝛽_𝑛} and {𝛿_𝑛} satisfy the following conditions: (A1)∑^∞𝑛=0𝛾_𝑛=∞;(A2)limsup_𝑛→∞𝛽_𝑛≤0;(A3)𝛿_𝑛≥0 (𝑛≥0), ∑^∞𝑛=1𝛿_𝑛<∞. Then lim_𝑛→∞𝑠_𝑛=0.

Lemma 2.3 (see [24]). In a Banach space 𝑋, the following inequality holds: ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩,∀𝑥,𝑦∈𝑋,(2.4) where 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).

Lemma 2.4 (see [25]). Let 𝐶 be a closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇_𝑛∶𝑛∈ℕ} be a sequence of nonexpansive mappings on 𝐶. Suppose ⋂^∞𝑛=1Fix(𝑇_𝑛) is nonempty. Let {𝜆_𝑛} be a sequence of positive numbers with ∑^∞𝑛=1𝜆_𝑛=1. Then a mapping 𝑆 on 𝐶 defined by 𝑆𝑥=∑^∞𝑛=1𝜆_𝑛𝑇_𝑛𝑥 for 𝑥∈𝐶 is well defined, nonexpansive and Fix(𝑆)=⋂^∞𝑛=1Fix(𝑇_𝑛) holds.

Lemma 2.5 (see [7]). Let 𝑋 be a uniformly smooth Banach space, 𝐶 a closed convex subset of 𝑋, 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅, and 𝑓∈Π_𝐶. Then {𝑥_𝑡} defined by 𝑥_𝑡=𝑡𝑓𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡(2.5) converges strongly to a point in Fix(𝑇). If we define a mapping 𝑄∶Π_𝐶→Fix(𝑇) by 𝑄(𝑓)∶=lim_𝑡→0𝑥_𝑡,𝑓∈Π_𝐶,(2.6) then 𝑄(𝑓) solves the variational inequality: ⟨(𝐼−𝑓)𝑄(𝑓),𝐽(𝑄(𝑓)−𝑝)⟩≤0,𝑓∈Π_𝐶,𝑝∈Fix(𝑇).(2.7)

Lemma 2.6. Let 𝑋 be a Banach space, {𝑥_𝑘} a bounded sequence of 𝑋, and {𝜔_𝑘} a sequence of positive numbers with ∑^∞𝑘=1𝜔_𝑘=1. Then ∑^∞𝑘=1𝜔_𝑘𝑥_𝑘 is convergent in 𝑋.

Lemma 2.7. Let X be Banach space, {𝑇_𝑘∶𝑘∈ℕ} a sequence of nonexpansive mappings on X with ⋂^∞𝑘=1Fix(𝑇_𝑘)≠∅, and {𝜔_𝑘} a sequence of positive numbers with ∑^∞𝑘=1𝜔_𝑘=1. Let 𝑇=∑^∞𝑘=1𝜔_𝑘𝑇_𝑘, 𝐿_𝑚=∑^𝑚𝑘=1(𝜔_𝑘/𝑆_𝑚)𝑇_𝑘, and 𝑆_𝑚=∑^𝑚𝑘=1𝜔_𝑘. Then 𝐿_𝑚 uniformly converges to 𝑇 in each bounded subset 𝑆 of 𝑋.

Proof. Forall𝑥∈𝑆, we observe that ‖‖𝐿_𝑚𝑥−𝑇𝑥‖‖=‖‖‖‖^𝑚_𝑘=1𝜔_𝑘𝑆_𝑚𝑇_𝑘𝑥−^∞_𝑘=1=‖‖‖‖_𝑚𝜔_𝑘𝑇_𝑘𝑥‖‖‖‖_𝑘=1𝜔_𝑘−𝜔_𝑘𝑆_𝑚𝑆_𝑚𝑇_𝑘𝑥−^∞_𝑘=_𝑚+1≤‖‖‖‖_𝑚𝜔_𝑘𝑇_𝑘𝑥‖‖‖‖_𝑘=11−𝑆_𝑚𝑆_𝑚𝜔_𝑘𝑇_𝑘𝑥‖‖‖‖+‖‖‖‖^∞_𝑘=_𝑚+1≤1−𝑆_𝑚𝜔_𝑘𝑇_𝑘𝑥‖‖‖‖𝑆_{𝑚𝑚𝑘=1}𝜔_𝑘‖‖𝑇_𝑘𝑥‖‖+^∞_𝑘=_𝑚+1≤1−𝑆_𝑚𝜔_𝑘‖‖𝑇_𝑘𝑥‖‖𝑆_𝑚𝑀+𝑀^∞_𝑘=_𝑚+1𝜔_𝑘,(2.8) where 𝑀=sup_{𝑥∈𝑆,𝑘≥1}‖𝑇_𝑘𝑥‖<∞. Taking 𝑚→∞ in above last inequality, we have that lim_𝑚→∞‖‖𝐿_𝑚𝑥−𝑇𝑥‖‖=0(2.9) holds uniformly for 𝑥∈𝑆 and this completes the proof.

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real uniformly smooth Banach space 𝑋, {𝑇_𝑘}^∞𝑘=1∶𝐶→𝐶 an infinite family of nonexpansive mappings with ⋂^∞𝑘=1Fix(𝑇_𝑘)≠∅, and {𝜔_𝑘} a sequence of positive numbers with ∑^∞𝑘=1𝜔_𝑘=1. Let 𝐿_𝑛=∑^𝑛𝑘=1(𝜔_𝑘/𝑆_𝑛)𝑇_𝑘, 𝑆_𝑛=∑^𝑛𝑘=1𝜔_𝑘, and 𝑓∈Π_𝐶 with coefficient 𝛼∈[0,1). Given any 𝑥₀∈𝐶, let {𝑥_𝑛} be a sequence generated by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐿_𝑛𝑥_𝑛,𝑛≥0,(3.1) where {𝛼_𝑛}⊂(0,1) satisfies the following conditions: (A1)lim_𝑛→∞𝛼_𝑛=0;(A2)∑^∞𝑛=0𝛼_𝑛=∞;(A3)either ∑^∞𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1. Then {𝑥_𝑛} converges strongly to 𝑥^∗∈⋂^∞𝑘=1Fix(𝑇_𝑘), which solves the following variational inequality: ⟨𝑥^∗−𝑓𝑥^∗,𝐽𝑥^∗−𝑝⟩≤0,𝑝∈^∞_𝑘=1Fix𝑇_𝑘.(3.2)

Proof. Step 1. We show that {𝑥_𝑛} is bounded.
Noticing nonexpansiveness of 𝐿_𝑛, take a 𝑝∈⋂^∞𝑘=1Fix(𝑇_𝑘) to derive that ‖‖𝑥_𝑛+1−p‖‖≤=𝛼‖‖𝛼_𝑛𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐿_𝑛𝑥_𝑛−𝑝‖‖=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑝‖‖+1−𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−𝑝‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑓(𝑝)+𝑓(𝑝)−𝑝‖‖+1−𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−𝑝‖‖≤𝛼𝛼‖‖_𝑛𝑓𝑥_𝑛−𝑓(𝑝)‖‖+𝛼_𝑛‖𝑓(𝑝)−𝑝‖+1−𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−𝑝‖‖=1−‖‖(𝑥1_𝑛−−𝛼𝑝)‖‖𝛼+_𝑛𝛼_𝑛‖𝑓(𝑝)−𝑝‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖=1−(1−𝛼)𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖(𝑓(𝑝)−𝑝)‖‖‖𝑥_𝑛−𝑝‖‖+(1−𝛼)𝛼_𝑛‖(𝑓(𝑝)−𝑝)‖≤max‖(𝑓(𝑝)−𝑝)‖1−𝛼1−𝛼,‖‖𝑥^𝑛−𝑝‖‖.(3.3) By induction, we obtain ‖‖𝑥_𝑛+1−𝑝‖‖≤max‖(𝑓(𝑝)−𝑝)‖1−𝛼,‖‖𝑥⁰−𝑝‖‖,𝑛≥0,(3.4) and {𝑥_𝑛} is bounded, so are {𝑇_𝑘𝑥_𝑛}, {𝐿_𝑛𝑥_𝑛}, and {𝑓(𝑥_𝑛)}.Step 2. We prove that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
By (3.1), We have ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖==‖‖‖‖𝛼𝛼_𝑛𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐿_𝑛𝑥_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−1−𝛼_𝑛−1𝐿_𝑛−1𝑥_𝑛−1‖‖+𝑓1−𝑥_𝑛𝛼_𝑛−𝛼_𝑛𝑓𝑥_𝑛−1+𝛼_𝑛𝑓𝑥_𝑛−1−𝛼_𝑛−1𝑓𝑥_𝑛−1≤𝛼𝛼_𝑛𝐿_𝑛𝑥_𝑛−1−𝛼_𝑛𝐿_𝑛−1𝑥_𝑛−1+1−𝛼_𝑛𝐿_𝑛−1𝑥_𝑛−1−1−𝛼_𝑛−1𝐿_𝑛−1𝑥_𝑛−1‖‖+1‖‖−𝑥_𝑛𝛼_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝑓𝑥_𝑛−1‖‖=𝛼𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−𝐿_𝑛−1𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝐿_𝑛−1𝑥_𝑛−1‖‖+1‖‖−𝑥_𝑛𝛼_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝑓𝑥_𝑛−1‖‖+‖‖𝐿_𝑛−1𝑥_𝑛−1‖‖≤𝛼𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−𝐿_𝑛−1𝑥_𝑛−1‖‖+1‖‖−𝑥_𝑛𝛼_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝑓𝑥_𝑛−1‖‖+‖‖𝐿_𝑛−1𝑥_𝑛−1‖‖≤1−(1−𝛼‖‖)𝐿𝛼_𝑛𝑛𝑥_𝑛−𝐿_𝑛𝑥_𝑛−1‖‖+1−𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−1−𝐿_𝑛−1𝑥_𝑛−1‖‖+||𝛼_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1−𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−1−𝐿_𝑛−1𝑥_𝑛−1‖‖≤1−−(1𝛼−𝛼_𝑛−1||)𝛼‖‖_𝑛𝑓𝑥_𝑛−1‖‖+‖‖𝐿_𝑛−1𝑥_𝑛−1‖‖+||𝛼_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1−𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−1−𝐿_𝑛−1𝑥_𝑛−1‖‖−𝛼_𝑛−1||𝑀,(3.5) where 𝑀=sup_𝑛≥1(‖𝑓(𝑥_𝑛−1)‖+‖𝐿_𝑛−1𝑥_𝑛−1‖). At the same time, we observe that ^∞_𝑛=1‖‖𝐿_𝑛𝑥_𝑛−1−𝐿_𝑛−1𝑥_𝑛−1‖‖=^∞_𝑛=1‖‖‖‖^𝑛_𝑘=1𝜔_𝑘𝑆_𝑛𝑇_𝑘𝑥_𝑛−1−^𝑛−1_𝑘=1𝜔_𝑘𝑆_𝑛−1𝑇_𝑘𝑥_𝑛−1=_∞‖‖‖‖_𝑛=1‖‖‖‖𝜔_𝑛𝑆_𝑛𝑇_𝑛𝑥_𝑛−1+^𝑛−1_𝑘=1−𝜔_𝑛𝜔_𝑘𝑆_𝑛𝑆_𝑛−1𝑇_𝑘𝑥_𝑛−1≤_∞‖‖‖‖_𝑛=1‖‖‖𝜔𝑆^𝑛_𝑛𝑇_𝑛𝑥_𝑛−1‖‖‖+‖‖‖‖^𝑛−1_𝑘=1𝜔_𝑛𝜔_𝑘𝑆_𝑛𝑆_𝑛−1𝑇_𝑘𝑥_𝑛−1≤_∞‖‖‖‖_𝑛=1𝜔_𝑛𝑆_𝑛‖‖𝑇_𝑛𝑥_𝑛−1‖‖+^∞_{𝑛=1𝑛−1𝑘=1}𝜔_𝑛𝜔_𝑘𝑆_𝑛𝑆_𝑛−1≤_∞‖‖𝑇_𝑘𝑥_𝑛−1‖‖_𝑛=1𝜔_𝑛𝑆_𝑛𝑀+^∞_𝑛=1𝜔_𝑛𝑆_𝑛=_∞𝑀_𝑛=12𝑀𝑆_𝑛𝜔_𝑛,(3.6) where 𝑀=sup_{𝑘≥1,𝑛≥1}‖𝑇_𝑘𝑥_𝑛−1‖. Applying Lemma 2.6 and compatibility test of series, we have ^∞_𝑛=1‖‖𝐿_𝑛𝑥_𝑛−1−𝐿_𝑛−1𝑥_𝑛−1‖‖<∞.(3.7) Put 𝛾_𝑛=(1−𝛼)𝛼_𝑛,𝛽_𝑛=||𝛼^𝑛−𝛼_𝑛−1||𝑀(1−𝛼)𝛼_𝑛𝛿_𝑛,=1−𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−1−𝐿_𝑛−1𝑥_𝑛−1‖‖.(3.8) It follows that ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤1−𝛾_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝛾_𝑛𝛽_𝑛+𝛿_𝑛.(3.9) It is easily seen from (A2), (A3), and (3.7) that ^∞_𝑛=1𝛾_𝑛=∞,limsup_𝑛→∞𝛽_𝑛≤0,^∞_𝑛=1𝛿_𝑛<∞.(3.10) Applying Lemma 2.2 to (3.9), we obtain lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=0.(3.11)Step 3. We show that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.
Indeed we observe that ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖≤=‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐿_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖‖−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑇𝑥_𝑛‖‖+1−𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖‖.(3.12) Hence, by (3.11), (A1), and Lemma 2.7, we have lim_𝑛→∞‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖=0.(3.13)Step 4. We prove that limsup_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝐽𝑥_𝑛−𝑥^∗≤0,(3.14) where 𝑥^∗=lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 being the fixed point of the contraction 𝑥⟼𝑡𝑓(𝑥)+(1−𝑡)𝑇𝑥.(3.15) From Lemma 2.5, we have 𝑥^∗∈Fix(𝑇) and (𝐼−𝑓)𝑥^∗,𝐽𝑥^∗−𝑝≤0,𝑝∈Fix(𝑇).(3.16) By Lemma 2.4, we have 𝑥^∗∈⋂^∞𝑘=1Fix(𝑇_𝑘) and (𝐼−𝑓)𝑥^∗,𝐽𝑥^∗−𝑝≤0,𝑝∈^∞_𝑘=1Fix𝑇_𝑘.(3.17) By 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑇𝑥_𝑡, we have 𝑥_𝑡−𝑥_𝑛=𝑡𝑓𝑥_𝑡−𝑥_𝑛+(1−𝑡)𝑇𝑥_𝑡−𝑥_𝑛.(3.18) It follows from Lemma 2.3 that ‖‖𝑥_𝑡−𝑥_𝑛‖‖²=‖‖𝑡(𝑓(𝑥_𝑡)−𝑥_𝑛)+(1−𝑡)(𝑇𝑥_𝑡−𝑥_𝑛)‖‖²≤≤((11−−𝑡𝑡))²₂‖‖𝑇𝑥_𝑡−𝑥_𝑛‖‖²+2𝑡⟨𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛⟩+2𝑡⟨𝑥_𝑡‖‖𝑇𝑥_𝑡−𝑇𝑥_𝑛‖‖+‖‖𝑇𝑥_𝑛−𝑥_𝑛‖‖²+2𝑡⟨𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩≤(1−𝑡)₂−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛⟩+2𝑡⟨𝑥_𝑡‖‖𝑥_𝑡−𝑥_𝑛‖‖+‖‖𝑇𝑥_𝑛−𝑥_𝑛‖‖²+2𝑡⟨𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩≤(1−𝑡)₂−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛⟩+2𝑡‖‖𝑥_𝑡‖‖𝑥_𝑡−𝑥_𝑛‖‖²+𝑏_𝑛(𝑡)+2𝑡⟨𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩−𝑥_𝑛‖‖²,(3.19) where 𝑏_𝑛(𝑡)=‖𝑇𝑥_𝑛−𝑥_𝑛‖(2‖𝑥_𝑡−𝑥_𝑛‖+‖𝑇𝑥_𝑛−𝑥_𝑛‖)→0 (𝑛→∞). It follows from above last inequality that ⟨𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩≤𝑡2‖‖𝑥^𝑡−𝑥_𝑛‖‖²+12𝑡𝑏^𝑛(𝑡).(3.20) Taking 𝑛→∞ in (3.20) yields limsup_𝑛→∞⟨𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩≤𝑡2𝑀,(3.21) where 𝑀≥‖𝑥_𝑡−𝑥_𝑛‖² for all 𝑛≥1 and 𝑡∈(0,1). Taking 𝑡→0 in (3.21), we have limsup_𝑡→0limsup_𝑛→∞⟨𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩≤0.(3.22) Noticing the fact that two limits are interchangeable due to the fact the duality mapping 𝐽 is norm-to-norm uniformly continuous on bounded sets, it follows from (3.22), we have limsup_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝐽𝑥_𝑛−𝑥^∗==lliimmssuup_𝑛→∞p_𝑡→0limsup_𝑡→0⟨𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩≤0.limsup_𝑛→∞⟨𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛⟩(3.23) Hence (3.14) holds.Step 5. Finally, we prove that 𝑥_𝑛→𝑥^∗ (𝑛→∞).
Indeed we observe that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝐿_𝑛𝑥_𝑛−𝑥^∗‖‖²=‖‖𝛼_𝑛(𝑓(𝑥_𝑛)−𝑓(𝑥^∗))+(1−𝛼_𝑛)(𝐿_𝑛𝑥_𝑛−𝑥^∗)+𝛼_𝑛(𝑓(𝑥^∗)−𝑥^∗)‖‖²≤‖‖𝛼_𝑛(𝑓(𝑥_𝑛)−𝑓(𝑥^∗))+(1−𝛼_𝑛)(𝐿_𝑛𝑥_𝑛−𝑥^∗)‖‖²≤+𝛼2_𝑛⟨𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗⟩‖‖𝑓𝑥_𝑛−𝑓𝑥^∗‖‖+1−𝛼_𝑛‖‖𝐿_𝑛𝑥_𝑛−𝑥^∗‖‖²≤+𝛼2𝛼𝛼_𝑛𝑛⟨𝑓𝑥^∗−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗⟩‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²≤+12𝛼−_𝑛(⟨1𝑓−𝑥𝛼^∗)𝛼−_𝑛𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗⟩≤1−(1−𝛼)𝛼_𝑛²‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛⟨𝑓𝑥^∗−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗⟩‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛⟨𝑓𝑥^∗−𝑥^∗,𝐽𝑥_𝑛+1−𝑥^∗⟩.(3.24) By view of (3.14) and condition (A2), it follows from Lemma 2.2 that 𝑥_𝑛→𝑥^∗(𝑛→∞). This completes the proof.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a real uniformly smooth Banach space 𝑋, {𝑇_𝑘}^∞𝑘=1∶𝐶→𝐶 an infinite family of nonexpansive mappings with ⋂^∞𝑘=1Fix(𝑇_𝑘)≠∅, {𝜔_𝑘} a sequence of positive numbers with ∑^∞𝑘=1𝜔_𝑘=1. Let 𝐿_𝑛=∑^𝑛𝑘=1(𝜔_𝑘/𝑆_𝑛)𝑇_𝑘, 𝑆_𝑛=∑^𝑛𝑘=1𝜔_𝑘, and 𝑢∈𝐶. Given any 𝑥₀∈𝐶, let {𝑥_𝑛} be a sequence generated by 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝐿_𝑛𝑥_𝑛,𝑛≥0,(3.25) where {𝛼_𝑛}⊂(0,1) satisfies the following conditions: (A1)lim_𝑛→∞𝛼_𝑛=0;(A2)∑^∞𝑛=0𝛼_𝑛=∞;(A3)either∑^∞𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛=)1. Then {𝑥_𝑛} converges strongly to 𝑥^∗∈⋂^∞𝑘=1Fix(𝑇_𝑘), which solves the following variational inequality: 𝑥^∗−𝑢,𝐽𝑥^∗−𝑝≤0,𝑝∈^∞_𝑘=1Fix𝑇_𝑘.(3.26)

Acknowledgment

This paper is supported by the Fundamental Research Funds for the Central Universities (ZXH2011D005).