Viscosity Method for Hierarchical Fixed Point Problems with an Infinite Family of Nonexpansive Nonself-Mappings

Abstract

A viscosity method for hierarchical fixed point problems is presented to solve variational inequalities, where the involved mappings are nonexpansive nonself-mappings. Solutions are sought in the set of the common fixed points of an infinite family of nonexpansive nonself-mappings. The results generalize and improve the recent results announced by many other authors.

1. Introduction and Preliminaries

Let 𝑋 a real Banach space and 𝐽 be the normalized duality mapping from 𝑋 into 2^𝑋∗ given by 𝐽(𝑥)=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖𝑥^∗,‖𝑥‖=𝑥^∗ (1) for all 𝑥∈𝑋, where 𝑋^∗ denotes the dual space of 𝑋 and ⟨⋅,⋅⟩ the generalized duality pairing between 𝑋 and 𝑋^∗. If 𝑋=𝐻 is a Hilbert space, then 𝐽 becomes the identity mapping on 𝐻. A point 𝑥∈𝐶 is a fixed point of 𝑇∶𝐶⊂𝑋→𝑋 provided 𝑇𝑥=𝑥. Denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}.

Let 𝑋 be a normed linear space with dim𝑋≥2. The modulus of smoothness of 𝑋 is the function 𝜌_𝑋∶[0,+∞)→[0,+∞) defined by 𝜌_𝑋(𝜏)+∶=sup𝑥+𝑦𝑥−𝑦2−1∶‖𝑥‖=1,𝑦=𝜏. (2) The space 𝑋 is said to be smooth if 𝜌_𝑋(𝜏)>0,forall𝜏>0. It is well known that if 𝑋 is smooth then 𝐽 is single valued. A Banach space 𝑋 is said to be strictly convex if ‖𝑥‖=‖𝑦‖=1,  𝑥≠𝑦, implies ‖𝑥+𝑦‖/2<1.

Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝑋. Recall the following concepts.

Definition 1. (i) A mapping 𝑓∶𝐶→𝐶 is a 𝜌-contraction if 𝜌∈[0,1) and if the following property is satisfied 𝑓(𝑥)−𝑓𝑦≤𝜌𝑥−𝑦,∀𝑥,𝑦∈𝐶. (3)(ii) A mapping 𝑇∶𝐶→𝐸 is nonexpansive provided 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (4)(iii)A mapping 𝑆∶𝐶→𝑋 is(a)accretive if for any 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑆𝑥−𝑆𝑦,𝑗𝑥−𝑦≥0; (5) (b)𝛽-strongly accretive if for any 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑆𝑥−𝑆𝑦,𝑗𝑥−𝑦≥𝛽𝑥−𝑦², (6) for some real constant 𝛽>0.
Noting that if 𝑆∶𝐶→𝑋 is nonexpansive, then 𝐼−𝑆 is accretive; if 𝑓∶𝐶→𝐶 is a 𝜌-contraction, then 𝐼−𝑓 is (1−𝜌)-strongly accretive. particulary, if 𝑋=𝐻 is a Hilbert space, then (strongly) accretive mappings become (strongly) monotone mappings.

Definition 2. Let 𝐶 and 𝐷 be nonempty subsets of a Banach space 𝑋 such that 𝐶 is nonempty closed convex and 𝐷⊂𝐶. (i)A mapping 𝑄∶𝐶→𝐷 is called sunny, if 𝑄(𝑄𝑥+𝑡(𝑥−𝑄𝑥))=𝑄𝑥 for each 𝑥∈𝐶 and 𝑡≥0 with 𝑄(𝑄𝑥+𝑡(𝑥−𝑄𝑥))∈𝐶.(ii)A mapping 𝑄∶𝐶→𝐷 is called a retraction from 𝐶 to 𝐷 if 𝑄 is continuous and 𝐹(𝑄)=𝐷.(iii)A subset 𝐷 of 𝐶⊂𝐸 is said to be a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction 𝑄 of 𝐶 onto 𝐷. For details, see [1–3].
Note that if 𝑋=𝐻 is a Hilbert space, 𝑄 becomes the projection on 𝐶, denoted by 𝑃_𝐶.
Let 𝑃∶𝐶→𝐶 a nonexpansive self-mapping on 𝐶 and {𝑇_𝑛} be a countable family of nonexpansive nonself-mappings of 𝐶 into 𝑋 such that ℱ=⋂^∞_𝑛=1𝐹(𝑇_𝑛)≠∅. Then we consider the following problem: find hierarchically a common fixed point of the infinite family {𝑇_𝑛} with respect to a nonexpansive mapping 𝑃; namely, find 𝑥^∗∈ℱ, such that 𝑥^∗−𝑃𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈ℱ. (7)

Particularly,if {𝑇_𝑛} is a finite family of nonexpansive nonself-mappings, problem (7) has been studied by Ceng and Petruşel [4]. If 𝑋=𝐻 and {𝑇_𝑛} is an infinite family of nonexpansive self-mappings, Problem (7) reduces to the following problem: find hierarchically a common fixed point of {𝑇_𝑛} with respect to a nonexpansive mapping 𝑃, namely, find 𝑥^∗∈ℱ, such that 𝑥^∗−𝑃𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈ℱ, (8) which was studied by Zhang et al. [5]. If 𝑋=𝐻 is a Hilbert space and 𝑇_𝑛=𝑇,forall𝑛≥1, where 𝑇 is a nonexpansive mapping on 𝐶, then problem (7) reduces to the following problem: finding hierarchically a fixed point of 𝑇 with respect to another nonexpansive mapping 𝑃; namely, find 𝑥^∗∈𝐹(𝑇) such that 𝑥^∗−𝑃𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐹(𝑇). (9)

Problem (7) includes many problems as special cases, so it is very important in the area of optimization and related fields, such as signal processing and image reconstruction (see [6–9]).

In 2007, Moudafi [10] introduced the following Krasnoselski-Mann's algorithm in Hilbert spaces: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝜎_𝑛𝑃𝑥_𝑛+1−𝜎_𝑛𝑇𝑥_𝑛,∀𝑛≥0, (10) where {𝛼_𝑛} and {𝜎_𝑛} are two real sequences in (0,1) and 𝑇 and 𝑃 are two nonexpansive mappings of 𝐶 into itself. Furthermore, he established a weak convergence result for Algorithm (10) for solving problem (9).

Subsequently, Yao and Liou [11] derived a weak convergence result of algorithm (10) under the restrictions on parameters weaker than those in [10, Theorem 2.1].

Recently, Marino and Xu [12] introduced the following explicit hierarchical fixed point algorithm in Hilbert spaces: 𝑥_𝑛+1=𝜆_𝑛𝑓𝑥_𝑛+1−𝜆_𝑛𝛼_𝑛𝑉𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥0, (11) where 𝑓 is a contraction on 𝐶 and 𝑉,𝑇 are two nonexpansive mappings of 𝐶 into itself and proved that the sequence {𝑥_𝑛} generated by (11) converges strongly to a solution of problem (9).

Very recently, Zhang et al. [5] introduced the following iterative algorithm in order to find hierarchically a fixed point of Problem (8): 𝑥₀𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑃𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛, (12) where 𝑓∶𝐶→𝐶 is a contraction, 𝑃∶𝐶→𝐶 is a nonexpansive mapping, {𝑇_𝑛}∶𝐶→𝐶 is a countable family of nonexpansive mappings, and 𝑇∶𝐶→𝐶 is a mapping defined by 𝑇=^∞_𝑛=1𝜆_𝑛𝑇_𝑛,𝜆_𝑛≥0(𝑛=1,2,…)with^∞_𝑛=1𝜆_𝑛=1. (13) Under suitable conditions on parameters {𝛼_𝑛} and {𝛽_𝑛}, they established some strong and weak convergence theorems. Note that, in [5], {𝑇_𝑛} is an infinite family of self-mappings and 𝑃 is also a self-mapping. And they obtained the results in the setting of Hilbert spaces.

Motivated and inspired by the above researches, in a reflexive Banach space which admits a weakly sequentially continuous duality mapping 𝐽, we propose and analyze an iteration process for a countable family of nonexpansive nonself-mappings {𝑇_𝑛}∶𝐶→𝑋 and 𝑆∶𝐶→𝑋 is a nonexpansive nonself-mapping as follows: 𝑥₀𝑥∈𝐶,_𝑛+1=𝑄𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛,𝑛≥0, (14) where 𝑄 is a sunny nonexpansive retraction of 𝑋 onto 𝐶 and establishes a convergence theorem. particularly, if 𝑋=𝐻 is a Hilbert space, we obtain some convergence results.

To prove the main results, we need the following lemmas.

Lemma 3 (see [1]). Let 𝐶 be a nonempty and convex subset of a smooth Banach space 𝑋, 𝐷⊂𝐶, 𝐽∶𝑋→𝑋^∗ the normalized duality mapping of 𝑋, and 𝑄∶𝐶→𝐷 a retraction. Then the following conditions are equivalent:(i)⟨𝑥−𝑄𝑥,𝐽(𝑦−𝑄𝑥)⟩≤0, for all 𝑥∈𝐶 and 𝑦∈𝐷;(ii)𝑄 is both sunny and nonexpansive.

Lemma 4 (see [13, Lemma 3.1, 3.3]). Let 𝑋 be a real smooth and strictly convex Banach space and 𝐶 a nonempty closed and convex subset of 𝑋 which is also a sunny nonexpansive retract of 𝑋. Assuming that 𝑇∶𝐶→𝑋 is a nonexpansive mapping and 𝑄 is a sunny nonexpansive retraction of 𝑋 onto 𝐶, then 𝐹(𝑇)=𝐹(𝑄𝑇).

Lemma 5 (see [1]). Let 𝑋 be a real Banach space and 𝐽∶𝑋→2^𝑋∗ the normalized duality mapping. Then for any 𝑥,𝑦∈𝑋, the following hold: (i)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩,  for all𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦);(ii)‖𝑥‖²+2⟨𝑦,𝑗(𝑥)⟩≤‖𝑥+𝑦‖²,   for all 𝑗(𝑥)∈𝐽(𝑥).

Lemma 6 (see [14]). Let {𝑎_𝑛} and {𝑏_𝑛} be two sequences of nonnegative real numbers satisfying ^∞_𝑛=0𝑏_𝑛𝑎<∞,_𝑛+1≤𝑎_𝑛+𝑏_𝑛,𝑛=0,1,2,…. (15) Then lim_𝑛→∞𝑎_𝑛 exists.

Lemma 7 (see [15]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying 𝑎_𝑛+1≤1−𝜆_𝑛𝑎_𝑛+𝜆_𝑛𝑏_𝑛+𝑐_𝑛,∀𝑛≥0, (16) where {𝜆_𝑛},{𝑏_𝑛} and {𝑐_𝑛} satisfy the following conditions: (i){𝜆_𝑛}⊂[0,1],∑^∞_𝑛=0𝜆_𝑛=∞ or, equivalently, Π^∞_𝑛=0(1−𝜆_𝑛)=0;(ii)limsup_𝑛→∞𝑏_𝑛≤0; (iii)𝑐_𝑛≥0(𝑛≥0),∑^∞_𝑛=0𝑐_𝑛<∞. Then lim_𝑛→∞𝑎_𝑛=0.

If Banach space 𝑋 admits sequentially continuous duality mapping 𝐽 from weak topology to weak * topology, then by [16, Lemma 1] we get that duality mapping 𝐽 is single-valued. In this case, duality mapping 𝐽 is also said to be weakly sequentially continuous, that is, for each {𝑥_𝑛}⊂𝑋 with 𝑥_𝑛⇀𝑥, then 𝐽(𝑥_𝑛)⇁𝐽𝑥 [16, 17].

Recall that a Banach space 𝑋 is said to be satisfying Opial’s condition if for any sequence {𝑥_𝑛} in 𝐸, 𝑥_𝑛⇀𝑥(𝑛→∞) implies that limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝐸,with𝑦≠𝑥. (17) By [16, Lemma 1], we know that if 𝑋 admits a weakly sequentially continuous duality mapping, then 𝑋 satisfies Opial’s condition.

In the sequel, we also need the following lemmas.

Lemma 8 (see [17]). Let 𝐶 be a nonempty, closed and convex subset of a reflexive Banach space 𝑋 which satisfies Opial’s condition and 𝑇∶𝐶→𝑋 a nonexpansive mapping. Then the mapping 𝐼−𝑇 is demiclosed at zero, that is, 𝑥_𝑛𝑥⇀𝑥_𝑛−𝑇𝑥_𝑛⟶0imply𝑥=𝑇𝑥. (18)
Let 𝐶 be a nonempty and convex subset of a Banach space 𝑋. Then for 𝑥∈𝐶, one defines the inward set 𝐼_𝐶(𝑥) as follows [2, 3]: 𝐼_𝐶(𝑥)=𝑦∈𝑋∶𝑦=𝑥+𝜆(𝑧−𝑥),𝑧∈𝐶,𝜆≥0. (19) A mapping 𝑇∶𝐶→𝑋 is said to satisfy the inward condition if 𝑇𝑥∈𝐼_𝐶(𝑥) for all 𝑥∈𝐶. 𝑇 is also said to satisfy the weakly inward condition if for each 𝑥∈𝐶, 𝑇𝑥∈𝐼_𝐶(𝑥)(𝐼_𝐶(𝑥) is the closure of 𝐼_𝐶(𝑥)). Clearly 𝐶⊂𝐼_𝐶(𝑥) and it is not hard to show that 𝐼_𝐶(𝑥) is a convex set if 𝐶 does.

Lemma 9 (see [18, Theorem 2.4]). Let 𝑋 be a reflexive Banach space which admits a weakly sequentially continuous duality mapping 𝐽 from 𝑋 to 𝑋^∗. Suppose 𝐶 is a nonempty closed convex subset of 𝑋 which is also a sunny nonexpansive retract of 𝑋, and 𝑇∶𝐶→𝑋 is a nonexpansive mapping satisfying the weakly inward condition and 𝐹(𝑇)≠∅. Let {𝑢_𝑛} be defined by 𝑢₀𝑢∈𝐶,_𝑛+1=𝑄𝛼_𝑛𝑓𝑢_𝑛+1−𝛼_𝑛𝑇𝑢_𝑛, (20) where 𝑄 is a sunny nonexpansive retract of 𝑋 onto 𝐶 and 𝛼_𝑛∈(0,1) satisfy the following conditions: (i)𝛼_𝑛→0,as𝑛→∞; (ii)∑^∞_𝑛=0𝛼_𝑛=∞; (iii)either ∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞(𝛼_𝑛/𝛼_𝑛+1)=1.Then {𝑥_𝑛} converges strongly to a fixed point 𝑝 of 𝑇 such that 𝑝 is the unique solution in 𝐹(𝑇) to the following variational inequality: 𝐼−𝑓𝑝,𝑗𝑝−𝑢≤0,∀𝑢∈𝐹(𝑇). (21)

Remark 10. If a Banach space 𝑋 admits a sequentially continuous duality mapping 𝐽 from weak topology to weak star topology, from Lemma 1 of [16] it follows that 𝑋 is smooth. So for Lemma 9, if 𝑋 is a reflexive and strictly convex Banach space which admits a weakly sequentially continuous duality mapping 𝐽, by Lemma 4, the weakly inward condition of 𝑇 can be removed.

2. Main Results

Theorem 11. Let 𝑋 be a reflexive and strictly convex Banach space which admits a weakly sequentially continuous duality mapping 𝐽∶𝑋→𝑋^∗ and 𝐶 a nonempty, closed and convex subset of 𝑋 which is also a sunny nonexpansive retract of 𝑋. Let 𝑆∶𝐶→𝑋 be a nonexpansive nonself-mapping, 𝑓∶𝐶→𝐶 a contractive mapping with a contractive constant 𝜌∈(0,1) and 𝑇_𝑖∶𝐶→𝑋(𝑖={1,2,…}) an infinite family of nonexpansive nonself-mappings such that ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝑇∶𝐶→𝑋 be defined by (13) and 𝑄 a sunny nonexpansive retraction of 𝑋 onto 𝐶. Let {𝑥_𝑛} be the sequence generated by (14), and {𝛼_𝑛} and {𝛽_𝑛} the sequences in(0,1)satisfying the following conditions: (i)𝛼_𝑛→0(𝑛→∞),∑^∞_𝑛=0𝛼_𝑛=∞; (ii)lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0; (iii)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞. Then {𝑥_𝑛} converges strongly to some point 𝑥^∗∈𝐹(𝑇)=⋂^∞_𝑖=1𝐹(𝑇_𝑖), which is the unique solution to the following variational inequality: 𝐼−𝑓𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐹(𝑇). (22)

Proof. From condition (ii), without loss of generality, we can assume that 𝛽_𝑛≤𝛼_𝑛,forall𝑛≥0.
First we prove that the sequence {𝑥_𝑛} is bounded.
In fact, for any 𝑢∈𝐹(𝑇), we have 𝑥_𝑛+1=𝑄𝛼−𝑢_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛−𝑄𝑢≤𝛼_𝑛𝑓𝑥_𝑛−𝑢+1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛−𝑢≤𝛼_𝑛𝜌𝑥_𝑛+𝑓(𝑢)+−𝑢−𝑢1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛−𝑢+1−𝛽_𝑛𝑇𝑥_𝑛≤−𝑢1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑢+𝛼_𝑛𝑓(𝑢)+−𝑢1−𝛼_𝑛𝛽_𝑛‖‖≤𝑆𝑢−𝑢1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑢+𝛼_𝑛𝑓(𝑢)+‖‖𝑥−𝑢𝑆𝑢−𝑢≤max_𝑛,𝑓(𝑢)+‖‖−𝑢−𝑢𝑆𝑢−𝑢.1−𝜌 (23) By induction, 𝑥_𝑛+1−𝑢≤max𝑥₀,𝑓(𝑢)+‖‖−𝑢−𝑢𝑆𝑢−𝑢.1−𝜌 (24) Thus {𝑥_𝑛} is bounded, so {𝑆𝑥_𝑛} and {𝑇𝑥_𝑛} are also bounded.
Next we prove that ‖𝑥_𝑛−𝑢_𝑛‖→0,as𝑛→∞, where the sequence {𝑢_𝑛} is defined by 𝑢₀=𝑥₀𝑢∈𝐶,_𝑛+1=𝑄𝛼_𝑛𝑓𝑢_𝑛+1−𝛼_𝑛𝑇𝑢_𝑛. (25)
By Lemma 9 and Remark 10, {𝑢_𝑛} converges strongly to some point 𝑥^∗∈𝐹(𝑇), which is the unique solution to the following variational inequality: 𝐼−𝑓𝑥^∗,𝑗𝑥^∗−𝑥≤0,∀𝑥∈𝐹(𝑇). (26) Furthermore, we obtain 𝑥_𝑛+1−𝑢_𝑛+1≤𝑄𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛−𝑄𝛼_𝑛𝑓𝑢_𝑛+1−𝛼_𝑛𝑇𝑢_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑢_𝑛+1−𝛼_𝑛𝑦_𝑛−𝑇𝑢_𝑛≤𝛼_𝑛𝜌𝑥_𝑛−𝑢_𝑛+1−𝛼_𝑛×𝛽_𝑛𝑆𝑥_𝑛−𝑇𝑢_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑢_𝑛≤1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑢_𝑛+1−𝛼_𝑛𝛽_𝑛𝑀≤1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑢_𝑛+𝛽_𝑛𝑀, (27) where 𝑀=sup_𝑛≥0‖𝑆𝑥_𝑛−𝑇𝑢_𝑛‖. It follows from conditions (i)-(ii) and Lemma 7 we have lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0. Since as 𝑛→∞, 𝑢_𝑛→𝑥^∗∈𝐹(𝑇), we get 𝑥_𝑛→𝑥^∗(𝑛→∞), which is the unique solution to the variational inequality (22).

Remark 12. Theorem 11 extends Theorem 2.1 in [5] from the following aspects: (i) from Hilbert spaces to reflexive and strictly convex Banach spaces which admits a weakly sequentially continuous duality mapping; (ii) for the infinite family of mappings {𝑇_𝑖} from self-mappings to nonself-mappings. In addition, the existence of the sunny nonexpansive retraction has been proved in [19, Theorem 3.10].

Remark 13. If we take 𝛼_𝑛=1(1+𝑛)^𝛼,𝛽_𝑛=1(1+𝑛)^𝛽,0<𝛼<𝛽<1, (28) then since |𝛼_𝑛+1−𝛼_𝑛|≈1/𝑛^𝛼+1 and |𝛽_𝑛+1−𝛽_𝑛|≈1/𝑛^𝛽+1(as 𝑛→∞), it is not hard to find that the conditions (i)–(iii) are satisfied. For details, see [12, Remark 3.2].

In the sequel, we consider the result in the setting of Hilbert spaces.

Theorem 14. Let 𝐻 be a Hilbert space and 𝐶 a nonempty, closed and convex subset of 𝐻. Let 𝑆∶𝐶→𝐻 be a nonexpansive nonself-mapping, 𝑓∶𝐶→𝐶 a contractive mapping with a contractive constant 𝜌∈(0,1), and 𝑇_𝑖∶𝐶→𝐻(𝑖={1,2,…}) an infinite family of nonexpansive nonself-mappings such that 𝐹(𝑇)=⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let {𝑥_𝑛} be the sequence generated by (14) and {𝛼_𝑛} and {𝛽_𝑛} the sequences in(0,1)satisfying the following conditions: (i)𝛼_𝑛→0,∑^∞_𝑛=0𝛼_𝑛=∞; (ii)lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=𝜏∈(0,+∞); (iii)lim_𝑛→∞((|𝛽_𝑛−𝛽_𝑛−1|+|𝛼_𝑛−𝛼_𝑛−1|)/𝛼_𝑛𝛽_𝑛)=0; (iv)there exists a constant 𝐾>0 such that 1/𝛼_𝑛|(1/𝛽_𝑛)−(1/𝛽_𝑛−1)|≤𝐾 for all 𝑛>0.Then {𝑥_𝑛} converges strongly to some point 𝑥^∗∈𝐹(𝑇), which is the unique solution to the following variational inequality: 1𝜏𝐼−𝑓𝑥^∗+(𝐼−𝑆)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐹(𝑇). (29)

Proof. By condition (ii), without loss of generality, we can assume that 𝛽_𝑛≤(𝜏+1)𝛼_𝑛,forall𝑛≥0. Similar to the proof of (24), for any 𝑢∈𝐹(𝑇), we have 𝑥_𝑛+1𝑥−𝑢≤max₀,()𝑓(𝑢)+‖‖−𝑢𝜏+1−𝑢𝑆𝑢−𝑢.1−𝜌 (30) Thus {𝑥_𝑛} is bounded. Furthermore, {𝑓(𝑥_𝑛)},{𝑇𝑥_𝑛},{𝑦_𝑛},{𝑆𝑥_𝑛} are all bounded. Put 𝑢_𝑛=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑦_𝑛 and 𝑀=sup_𝑛≥0{‖𝑓(𝑥_𝑛)‖+‖𝑦_𝑛‖,‖𝑇𝑥_𝑛‖+‖𝑆𝑥_𝑛‖}. So {𝑢_𝑛} and {𝑃_𝐶(𝑢_𝑛)} are also bounded.
Step 1. We prove that ‖𝑥_𝑛+1−𝑥_𝑛‖→0(𝑛→∞).
From (14), we obtain 𝑥_𝑛+1−𝑥_𝑛=𝑃_𝐶𝑢_𝑛−𝑃_𝐶𝑢_𝑛−1≤𝑢_𝑛−𝑢_𝑛−1≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+1−𝛼_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝑦_𝑛−1≤𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛×𝑦_𝑛−𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑦𝑀,_𝑛−𝑦_𝑛−1≤𝛽_𝑛𝑆𝑥_𝑛−𝑆𝑥_𝑛−1+1−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝑇𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑀. (31)(32) Substituting (32) into (31), we have 𝑥_𝑛+1−𝑥_𝑛≤1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑀𝛼_𝑛. (33) By conditions (i), (iii), and Lemma 7, we have ‖𝑥_𝑛+1−𝑥_𝑛‖→0(𝑛→∞).
Step 2. We prove that 𝜔_𝑤(𝑥_𝑛)⊂𝐹(𝑇), where 𝜔_𝑤(𝑥_𝑛) is the 𝜔-limit point set of {𝑥_𝑛} in the weak topology: 𝑥_𝑛+1−𝑄𝑇𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑆𝑥_𝑛+𝛼_𝑛+𝛽_𝑛+𝛼_𝑛𝛽_𝑛𝑇𝑥_𝑛. (34) Noting that 𝛼_𝑛→0 and 𝛽_𝑛→0, we have ‖𝑥_𝑛+1−𝑄𝑇𝑥_𝑛‖→0(𝑛→∞). Then from Step 1 we have ‖𝑥_𝑛−𝑄𝑇𝑥_𝑛‖→0(𝑛→∞). Furthermore, it follows from Lemmas 4 and 8 that 𝜔_𝑤(𝑥_𝑛)⊂𝐹(𝑄𝑇)=𝐹(𝑇), where 𝑄=𝑃_𝐶.
Step 3. We show that ‖𝑥_𝑛+1−𝑥_𝑛‖/𝛽_𝑛→0(𝑛→∞).
It follows from (31) and (33) that 𝑥_𝑛+1−𝑥_𝑛𝛽_𝑛≤𝑢_𝑛−𝑢_𝑛−1𝛽_𝑛≤1−𝛼_𝑛𝑥1−𝜌^𝑛−𝑥_𝑛−1𝛽_𝑛−1+1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−11𝛽_𝑛−1𝛽_𝑛−1+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑀𝛽_𝑛≤1−𝛼_𝑛𝑥1−𝜌^𝑛−𝑥_𝑛−1𝛽_𝑛−1+𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1𝐾+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑀𝛼_𝑛𝛽_𝑛𝛼_𝑛. (35) By conditions (i) and (iii), ‖𝑥_𝑛−𝑥_𝑛−1‖→0(𝑛→∞), and Lemma 7, we have 𝑥_𝑛+1−𝑥_𝑛𝛽_𝑛⟶0(𝑛⟶∞). (36) Thus from (35), we get 𝑢_𝑛−𝑢_𝑛−1𝛽_𝑛⟶0(𝑛⟶∞). (37)
Step 4. We show that {𝑥_𝑛} converges strongly to some point 𝑥^′∈𝐹(𝑇), which is the unique solution of (29).
Setting 𝑊_𝑛=𝛽_𝑛𝑆+(1−𝛽_𝑛)𝑇, we have 𝑥_𝑛+1=𝑃_𝐶𝑢_𝑛−𝑢_𝑛+𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑊_𝑛𝑥_𝑛. (38) Then 𝑥_𝑛−𝑥_𝑛+1=𝑢_𝑛−𝑃_𝐶𝑢_𝑛+𝛼_𝑛𝐼−𝑓𝑥_𝑛+1−𝛼_𝑛𝐼−𝑊_𝑛𝑥_𝑛. (39) Letting 𝑣_𝑛=(𝑥_𝑛−𝑥_𝑛+1)/(1−𝛼_𝑛)𝛽_𝑛, from condition (i) and (36), we have 𝑣_𝑛→0(𝑛→∞). Noting that 𝐼−𝑊_𝑛 is monotone and 𝐼−𝑓 is (1−𝜌)-strongly monotone, for any 𝑥^∗∈𝐹(𝑇), from Lemma 3 we obtain 𝑣_𝑛,𝑥_𝑛−𝑥^∗=11−𝛼_𝑛𝛽_𝑛𝑢_𝑛−𝑃_𝐶𝑢_𝑛,𝑥_𝑛−𝑥^∗+𝛼_𝑛1−𝛼_𝑛𝛽_𝑛𝐼−𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗+1𝛽_𝑛⟨𝐼−𝑊_𝑛𝑥_𝑛,𝑥_𝑛−𝑥^∗⟩=11−𝛼_𝑛𝛽_𝑛⟨𝑢_𝑛−𝑃_𝐶𝑢_𝑛,𝑥_𝑛−𝑥^∗⟩+𝛼_𝑛1−𝛼_𝑛𝛽_𝑛⟨𝐼−𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗⟩+1𝛽_𝑛⟨𝐼−𝑊_𝑛𝑥_𝑛−𝐼−𝑊_𝑛𝑥^∗,𝑥_𝑛−𝑥^∗⟩+1𝛽_𝑛𝐼−𝑊_𝑛𝑥^∗,𝑥_𝑛−𝑥^∗=11−𝛼_𝑛𝛽_𝑛𝑢_𝑛−𝑃_𝐶𝑢_𝑛,𝑃_𝐶𝑢_𝑛−𝑥^∗+11−𝛼_𝑛𝛽_𝑛×𝑢_𝑛−𝑃_𝐶𝑢_𝑛,−𝑃_𝐶𝑢_𝑛−𝑥^∗+𝑃_𝐶𝑢_𝑛−1−𝑥^∗+𝛼_𝑛1−𝛼_𝑛𝛽_𝑛𝐼−𝑓𝑥_𝑛−𝐼−𝑓𝑥^∗,𝑥_𝑛−𝑥^∗+𝛼_𝑛1−𝛼_𝑛𝛽_𝑛𝐼−𝑓𝑥^∗,𝑥_𝑛−𝑥^∗+1𝛽_𝑛𝐼−𝑊_𝑛𝑥_𝑛−𝐼−𝑊_𝑛𝑥^∗,𝑥_𝑛−𝑥^∗+1𝛽_𝑛⟨𝐼−𝑊_𝑛𝑥^∗,𝑥_𝑛−𝑥^∗⟩≥11−𝛼_𝑛𝛽_𝑛𝑢_𝑛−𝑃_𝐶𝑢_𝑛,𝑃_𝐶𝑢_𝑛−1−𝑃_𝐶𝑢_𝑛+𝛼_𝑛1−𝜌1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2+𝛼_𝑛1−𝛼_𝑛𝛽_𝑛⟨𝐼−𝑓𝑥^∗,𝑥_𝑛−𝑥^∗⟩+(𝐼−𝑆)𝑥^∗,𝑥_𝑛−𝑥^∗. (40) Thus we have 𝑥_𝑛−𝑥^∗2≤1−𝛼_𝑛𝛽_𝑛𝛼_𝑛𝑣1−𝜌^𝑛,𝑥_𝑛−𝑥^∗−1−𝛼_𝑛𝛽_𝑛𝛼_𝑛)𝑥1−𝜌(𝐼−𝑆^∗,𝑥_𝑛−𝑥^∗−1𝛼_𝑛1−𝜌⟨𝑢_𝑛−𝑃_𝐶𝑢_𝑛,𝑃_𝐶𝑢_𝑛−1−𝑃_𝐶𝑢_𝑛⟩−1⟨𝑥1−𝜌𝐼−𝑓^∗,𝑥_𝑛−𝑥^∗⟩≤1−𝛼_𝑛𝛽_𝑛𝛼_𝑛𝑣1−𝜌^𝑛𝑥_𝑛−𝑥^∗−1−𝛼_𝑛𝛽_𝑛𝛼_𝑛)𝑥1−𝜌(𝐼−𝑆^∗,𝑥_𝑛−𝑥^∗+1𝑢1−𝜌^𝑛−𝑃_𝐶𝑢_𝑛𝑢_𝑛−1−𝑢_𝑛𝛼_𝑛−1𝑥1−𝜌𝐼−𝑓^∗,𝑥_𝑛−𝑥^∗. (41) Since 𝛽_𝑛≤(𝜏+1)𝛼_𝑛, by (37) we have 𝑢_𝑛−𝑢_𝑛−1𝛼_𝑛⟶0(𝑛⟶∞). (42) Combining condition (ii), 𝑣_𝑛→0(𝑛→∞), (41), and (42), every weak cluster point of {𝑥_𝑛} is also a strong cluster point. From (40), we obtain 𝐼−𝑓𝑥_𝑛,𝑥_𝑛−𝑥^∗=1−𝛼_𝑛𝛽_𝑛𝛼_𝑛⟨𝑣_𝑛,𝑥_𝑛−𝑥^∗⟩−1𝛼_𝑛𝑢_𝑛−𝑃_𝐶𝑢_𝑛,𝑥_𝑛−𝑥^∗−1−𝛼_𝑛𝛼_𝑛𝐼−𝑊_𝑛𝑥_𝑛,𝑥_𝑛−𝑥^∗=1−𝛼_𝑛𝛽_𝑛𝛼_𝑛⟨𝑣_𝑛,𝑥_𝑛−𝑥^∗⟩−1𝛼_𝑛⟨𝑢_𝑛−𝑃_𝐶𝑢_𝑛,𝑃_𝐶𝑢_𝑛−𝑥^∗⟩−1𝛼_𝑛⟨𝑢_𝑛−𝑃_𝐶𝑢_𝑛,𝑃_𝐶𝑢_𝑛−1−𝑃_𝐶𝑢_𝑛⟩−1−𝛼_𝑛𝛼_𝑛×𝐼−𝑊_𝑛𝑥_𝑛−𝐼−𝑊_𝑛𝑥^∗,𝑥_𝑛−𝑥^∗−1−𝛼_𝑛𝛼_𝑛𝐼−𝑊_𝑛𝑥^∗,𝑥_𝑛−𝑥^∗≤1−𝛼_𝑛𝛽_𝑛𝛼_𝑛𝑣_𝑛𝑥_𝑛−𝑥^∗+1𝛼_𝑛𝑢_𝑛−𝑃_𝐶𝑢_𝑛𝑃_𝐶𝑢_𝑛−1−𝑃_𝐶𝑢_𝑛−1−𝛼_𝑛𝛽_𝑛𝛼_𝑛(𝐼−𝑆)𝑥^∗,𝑥_𝑛−𝑥^∗≤1−𝛼_𝑛𝛽_𝑛𝛼_𝑛𝑣_𝑛𝑥_𝑛−𝑥^∗+𝑢_𝑛−1−𝑢_𝑛𝛼_𝑛𝑢_𝑛−𝑃_𝐶𝑢_𝑛−1−𝛼_𝑛𝛽_𝑛𝛼_𝑛⟨(𝐼−𝑆)𝑥^∗,𝑥_𝑛−𝑥^∗⟩. (43) Note that the sequence {𝑥_𝑛} is bounded; thus there exists a subsequence {𝑥_𝑛𝑗} converging to a point 𝑥^′∈𝐻. From Step 2, we have 𝑥^′∈𝐹(𝑇). Then it follows from the above inequality, (42), and 𝑣_𝑛→0(𝑛→∞) that 𝐼−𝑓𝑥^′,𝑥^′−𝑥^∗≤−𝜏(𝐼−𝑆)𝑥^∗,𝑥^′−𝑥^∗,∀𝑥^∗∈𝐹(𝑇). (44) Replacing 𝑥^∗ with 𝑥^′+𝜇(𝑥^∗−𝑥^′), where 𝜇∈(0,1) and 𝑥^∗∈𝐹(𝑇), we have ⟨𝐼−𝑓𝑥^′,𝑥^′−𝑥^∗⟩≤−𝜏⟨(𝐼−𝑆)𝑥^′+𝜇𝑥^∗−𝑥^′,𝑥^′−𝑥^∗⟩,∀𝑥^∗∈𝐹(𝑇). (45) Letting 𝜇→0, we have 𝐼−𝑓𝑥^′,𝑥^′−𝑥^∗≤−𝜏(𝐼−𝑆)𝑥^′,𝑥^′−𝑥^∗,∀𝑥^∗∈𝐹(𝑇). (46) If there exists another subsequence {𝑥^′_𝑛𝑗} of {𝑥_𝑛} converging to a point 𝑥^′′∈𝐻. From Step 2, we also have 𝑥^′′∈𝐹(𝑇). Then from (46) we obtain ⟨𝐼−𝑓𝑥^′,𝑥^′−𝑥^′′⟩≤−𝜏(𝐼−𝑆)𝑥^′,𝑥^′−𝑥^′′ (47) and, via interchanging 𝑥^′ and 𝑥^′′, 𝐼−𝑓𝑥^′′,𝑥^′′−𝑥^′≤−𝜏(𝐼−𝑆)𝑥^′′,𝑥^′′−𝑥^′. (48) Adding up these two inequalities yields 1−𝜌𝑥^′−𝑥^′′2≤𝐼−𝑓𝑥^′−𝐼−𝑓𝑥^′′,𝑥^′−𝑥^′′≤0, (49) which implies 𝑥^′=𝑥^′′. Then {𝑥_𝑛} converges strongly to 𝑥^′∈𝐹(𝑇), which is the solution to the following variational inequality: 1𝜏𝐼−𝑓𝑥^′+(𝐼−𝑆)𝑥^′,𝑥−𝑥^′≥0,∀𝑥∈𝐹(𝑇). (50) Since 𝐼−𝑓 is (1−𝜌)-strongly monotone and 𝐼−𝑆 is monotone, it is easy to see that the above variational inequality has a unique solution.

Remark 15. Theorem 14 extends Theorem 3.2 in [12] from the following aspects: (i) from a nonexpansive mapping 𝑇 to an infinite family of nonexpansive mappings {𝑇_𝑖}; (ii) from self-mappings to nonself-mappings.

Acknowledgments

The author is extremely grateful to the referees for their useful suggestions that improved the content of the paper. Supported by the China Postdoctoral Science Foundation Funded Project (no. 2012M511928).