Algorithms for a System of General Variational Inequalities in Banach Spaces

Abstract

The purpose of this paper is using Korpelevich's extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results.

1. Introduction

Throughout this paper, we denote by ℕ and ℝ the sets of positive integers and real numbers, respectively. We also assume that 𝐸 is a real Banach space, 𝐸^∗ is the dual space of 𝐸,𝐶 is a nonempty closed convex subset of 𝐸, and ⟨⋅,⋅⟩ is the pairing between 𝐸 and 𝐸^∗.

In this paper, we are concerned a finite family of a general system of nonlinear variational inequalities in Banach spaces, which involves finding (𝑥^∗1,𝑥^∗2,…,𝑥^∗𝑛)∈𝐶×𝐶×⋯×𝐶 such that 𝜆𝜆₁𝐴₁𝑥^∗2+𝑥^∗1−𝑥^∗2,𝑗𝑥−𝑥^∗1≥0,∀𝑥∈𝐶,₂𝜆𝐴₂𝑥^∗3+𝑥^∗2−𝑥^∗3,𝑗𝑥−𝑥^∗2≥0,∀𝑥∈𝐶,₃𝜆𝐴₃𝑥^∗4+𝑥^∗3−𝑥^∗4,𝑗𝑥⋮−𝑥^∗3≥0,∀𝑥∈𝐶,_𝑁−1𝜆𝐴_𝑁−1𝑥^∗𝑁+𝑥^∗𝑁−1−𝑥^∗𝑁,𝑗𝑥−𝑥^∗𝑁−1≥0,∀𝑥∈𝐶,_𝑁𝐴_𝑁𝑥^∗1+𝑥^∗𝑁−𝑥^∗1,𝑗𝑥−𝑥^∗𝑁≥0,∀𝑥∈𝐶,(1.1) where {𝐴_𝑖∶𝐶→𝐸,𝑖=1,2,…,𝑁} is a finite family of nonlinear mappings and 𝜆_𝑖(𝑖=1,2,…,𝑁) are positive real numbers.

As special cases of the problem (1.1), we have the following.

(I) If 𝐸 is a real Hilbert space and 𝑁=2, then (1.1) reduces to𝜆𝜆₁𝐴₁𝑥^∗2+𝑥^∗1−𝑥^∗2,𝑥−𝑥^∗1≥0,∀𝑥∈𝐶,₂𝐴₂𝑥^∗1+𝑥^∗2−𝑥^∗1,𝑥−𝑥^∗2≥0,∀𝑥∈𝐶,(1.2) which was considered by Ceng et al. [1]. In particular, if 𝐴₁=𝐴₂=𝐴, then the problem (1.2) reduces to finding (𝑥^∗1,𝑥^∗2)∈𝐶×𝐶 such that𝜆𝜆₁𝐴𝑥^∗2+𝑥^∗1−𝑥^∗2,𝑥−𝑥^∗1≥0,∀𝑥∈𝐶,₂𝐴𝑥^∗1+𝑥^∗2−𝑥^∗1,𝑥−𝑥^∗2≥0,∀𝑥∈𝐶,(1.3) which is defined by Verma [2]. Furthermore, if 𝑥^∗1=𝑥^∗2, then (1.3) reduces to the following variational inequality (VI) of finding 𝑥^∗∈𝐶 such that⟨𝐴𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶.(1.4)

This problem is a fundamental problem in variational analysis and, in particular, in optimization theory. Many algorithms for solving this problem are projection algorithms that employ projections onto the feasible set 𝐶 of the VI or onto some related set, in order to iteratively reach a solution. In particular, Korpelevich’s extragradient method which was introduced by Korpelevich [3] in 1976 generates a sequence {𝑥_𝑛} via the recursion𝑥_𝑛+1𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜆𝐴𝑥_𝑛,=𝑃_𝐶𝑥_𝑛−𝜆𝐴𝑦_𝑛,𝑛≥0,(1.5) where 𝑃_𝐶 is the metric projection from ℝ^𝑛 onto 𝐶, 𝐴∶𝐶→𝐻 is a monotone operator, and 𝜆 is a constant. Korpelevich [3] proved that the sequence {𝑥_𝑛} converges strongly to a solution of 𝑉𝐼(𝐶,A). Note that the setting of the space is Euclid space ℝ^𝑛.

The literature on the VI is vast, and Korpelevich’s extragradient method has received great attention by many authors, who improved it in various ways. See, for example, [4–16] and references therein.

(II) If 𝐸 is still a real Banach space and 𝑁=1, then the problem (1.1) reduces to finding 𝑥^∗∈𝐶 such that𝐴𝑥^∗,𝑗𝑥−𝑥^∗≥0,∀𝑥∈C,(1.6) which was considered by Aoyama et al. [17]. Note that this problem is connected with the fixed point problem for nonlinear mapping, the problem of finding a zero point of a nonlinear operator, and so on. It is clear that problem (1.6) extends problem (1.4) from Hilbert spaces to Banach spaces.

In order to find a solution for problem (1.6), Aoyama et al. [17] introduced the following iterative scheme for an accretive operator 𝐴 in a Banach space 𝐸:𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_nΠ_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑛≥1,(1.7) where Π_𝐶 is a sunny nonexpansive retraction from 𝐸 to 𝐶. Then they proved a weak convergence theorem in a Banach space. For related works, please see [18] and the references therein.

It is an interesting problem of constructing some algorithms with strong convergence for solving problem (1.1) which contains problem (1.6) as a special case.

Our aim in this paper is to construct two algorithms for solving problem (1.1). For this purpose, we first prove that the system of variational inequalities (1.1) is equivalent to a fixed point problem of some nonexpansive mapping. Finally, we prove the strong convergence of the proposed methods which solve problem (1.1).

2. Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence {𝑥_𝑛} by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively.

For 𝑞>1, the generalized duality mapping 𝐽_𝑞∶𝐸→2^𝐸∗ is defined by𝐽_𝑞(𝑥)=𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖^𝑞,‖𝑓‖=‖𝑥‖^𝑞−1(2.1) for all 𝑥∈𝐸. In particular, 𝐽=𝐽₂ is called the normalized duality mapping. It is known that 𝐽_𝑞(𝑥)=||𝑥||^𝑞−2 for all 𝑥∈𝐸. If 𝐸 is a Hilbert space, then 𝐽=𝐼, the identity mapping. Let 𝑈={𝑥∈𝐸∶||𝑥||=1}. A Banach space 𝐸 is said to be uniformly convex if, for any 𝜀∈(0,2], there exists 𝛿>0 such that, for any 𝑥,𝑦∈𝑈,‖𝑥−𝑦‖≥𝜀implies‖‖‖𝑥+𝑦2‖‖‖≤1−𝛿.(2.2)

It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space 𝐸 is said to be smooth if the limitlim_𝑛→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.3) exists for all 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the previous limit is attained uniformly for 𝑥,𝑦∈𝑈. The norm of 𝐸 is said to be Fréchet differentiable if, for each 𝑥∈𝑈, the previous limit is attained uniformly for all 𝑦∈𝑈. The modulus of smoothness of 𝐸 is defined by𝜌(𝜏)=sup12(‖𝑥+𝑦‖+‖𝑥−𝑦‖)−1∶𝑥,𝑦∈𝐸,‖𝑥‖=1,‖𝑦‖=𝜏,(2.4) where 𝜌∶[0,∞)→[0,∞) is function. It is known that 𝐸 is uniformly smooth if and only if lim_𝜏→0(𝜌(𝜏)/𝜏)=0. Let 𝑞 be a fixed real number with 1<𝑞≤2. Then a Banach space 𝐸 is said to be 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌(𝜏)≤𝑐𝜏^𝑞 for all 𝜏>0. Note the following.(1)𝐸 is a uniformly smooth Banach space if and only if 𝐽 is single valued and uniformly continuous on any bounded subset of 𝐸.(2)All Hilbert spaces, 𝐿_𝑝 (or 𝑙_𝑝) spaces (𝑝≥2) and the Sobolev spaces 𝑊^𝑝𝑚(𝑝≥2) are 2-uniformly smooth, while 𝐿_𝑝 (or 𝑙_𝑝) and 𝑊^𝑃𝑚 spaces (1<𝑝≤2) are 𝑝-uniformly smooth.(3)Typical examples of both uniformly convex and uniformly smooth Banach spaces are 𝐿^𝑝, where 𝑝>1. More precisely, 𝐿^𝑝 is min{𝑝,2}-uniformly smooth for every 𝑝>1.

In our paper, we focus on a 2-uniformly smooth Banach space with the smooth constant 𝐾.

Let 𝐸 be a real Banach space, 𝐶 a nonempty closed convex subset of 𝐸, 𝑇∶𝐶→𝐶 a mapping, and 𝐹(𝑇) the set of fixed points of 𝑇.

Recall that a mapping 𝑇∶𝐶→𝐶 is called nonexpansive if‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(2.5) A bounded linear operator 𝐹∶𝐶∈𝐸 is called strongly positive if there exists a constant 𝛾>0 with the property⟨𝐹(𝑥),𝑗(𝑥)⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐶.(2.6) A mapping 𝐴∶𝐶→𝐸 is said to be accretive if there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that⟨𝐴𝑥−𝐴𝑦,𝑗(𝑥−𝑦)⟩≥0,(2.7) for all 𝑥,𝑦∈𝐶, where 𝐽 is the duality mapping.

A mapping 𝐴 of 𝐶 into 𝐸 is said to be 𝛼-strongly accretive if, for 𝛼>0,⟨𝐴𝑥−𝐴𝑦,𝑗(𝑥−𝑦)⟩≥𝛼‖𝑥−𝑦‖²,(2.8) for all 𝑥,𝑦∈𝐶.

A mapping 𝐴 of 𝐶 into 𝐸 is said to be 𝛼-inverse-strongly accretive if, for 𝛼>0,⟨𝐴𝑥−𝐴𝑦,𝑗(𝑥−𝑦)⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖²,(2.9) for all 𝑥,𝑦∈𝐶.

Remark 2.1. Evidently, the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for instance, [6, 19, 20].
Let 𝐷 be a subset of 𝐶, and let Π be a mapping of 𝐶 into 𝐷. Then Π is said to be sunny if Π[Π(𝑥)+𝑡(𝑥−Π(𝑥))]=Π(𝑥)(2.10) whenever Π(𝑥)+𝑡(𝑥−Π(𝑥))∈𝐶 for 𝑥∈𝐶 and 𝑡≥0. A mapping Π of 𝐶 into itself is called a retraction if Π²=Π. If a mapping Π of 𝐶 into itself is a retraction, then Π(𝑧)=𝑧 for every 𝑧∈𝑅(Π), where 𝑅(Π) is the range of Π. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷. Then following lemma concerns the sunny nonexpansive retraction.

Lemma 2.2 (see [21]). Let 𝐶 be a closed convex subset of a smooth Banach space 𝐸, let 𝐷 be a nonempty subset of 𝐶, and let Π be a retraction from 𝐶 onto 𝐷. Then Π is sunny and nonexpansive if and only if ⟨𝑢−Π(𝑢),𝑗(𝑦−Π(𝑢))⟩≤0,(2.11) for all 𝑢∈𝐶 and 𝑦∈𝐷.

Remark 2.3. (1) It is well known that if 𝐸 is a Hilbert space, then a sunny nonexpansive retraction Π_𝐶 is coincident with the metric projection from 𝐸 onto 𝐶.
(2) Let 𝐶 be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space 𝐸, and let 𝑇 be a nonexpansive mapping of 𝐶 into itself with the set 𝐹(𝑇)≠∅. Then the set 𝐹(𝑇) is a sunny nonexpansive retract of 𝐶.

In what follows, we need the following lemmas for proof of our main results.

Lemma 2.4 (see [22]). Assume that {𝛼_𝑛} is a sequence of nonnegative real numbers such that 𝛼_𝑛+1≤1−𝛾_𝑛𝛼_𝑛+𝛿_𝑛,(2.12) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that(a)Σ^∞𝑛=1𝛾_𝑛=∞,(b)limsup_𝑛→∞(𝛿_𝑛/𝛾_𝑛)≤0 or Σ^∞𝑛=1|𝛿_𝑛|<∞.Then lim_𝑛→∞𝛼_𝑛=0.

Lemma 2.5 (see [23]). Let 𝑋 be a Banach space, {𝑥_𝑛},{𝑦_𝑛} be two bounded sequences in 𝑋 and {𝛽_𝑛} be a sequence in [0,1] satisfying 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.(2.13) Suppose that 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑦_𝑛, for all 𝑛≥1 and limsup_𝑛→∞‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0,(2.14) then lim_n→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.6 (see [24]). Let 𝐸 be a real 2-uniformly smooth Banach space with the best smooth constant 𝐾. Then the following inequality holds: ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝐽𝑥⟩+2‖𝐾𝑦‖²,∀𝑥,𝑦∈𝐸.(2.15)

Lemma 2.7 (see [25]). Let 𝐶 be a nonempty bounded closed convex subset of a uniformly convex Banach space 𝐸, and let 𝐺 be a nonexpansive mapping of 𝐶 into itself. If {𝑥_𝑛} is a sequence of 𝐶 such that 𝑥_𝑛⇀𝑥 and 𝑥_𝑛−𝐺𝑥_𝑛→0, then 𝑥 is a fixed point of 𝐺.

Lemma 2.8 (see [26]). Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸. Assume that the mapping 𝐹∶𝐶→𝐸 is accretive and weakly continuous along segments (i.e., 𝐹(𝑥+𝑡𝑦)⇀𝐹(𝑥) as 𝑡→0). Then the variational inequality 𝑥^∗∈𝐶,𝐹𝑥^∗,𝑗𝑥−𝑥^∗≥0,𝑥∈𝐶,(2.16) is equivalent to the dual variational inequality 𝑥^∗∈𝐶,𝐹𝑥,𝑗𝑥−𝑥^∗≥0,𝑥∈𝐶.(2.17)

Lemma 2.9. Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝐸. Let Π_𝐶 be a sunny nonexpansive retraction from 𝐸 onto 𝐶. Let {𝐴_𝑖∶𝐶→𝐸,𝑖=1,2,…,𝑁} be a finite family of 𝛾_𝑖-inverse-strongly accretive. For given (𝑥^∗1,𝑥^∗2,…,𝑥^∗𝑛)∈𝐶×𝐶×⋯×𝐶, where 𝑥^∗=𝑥^∗1,𝑥^∗𝑖=Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖)𝑥^∗𝑖+1,𝑖∈{𝑖,2,…,𝑁−1},𝑥^∗𝑁=Π_𝐶(𝐼−𝜆_𝑁𝐴_𝑁)𝑥^∗1, then (𝑥^∗1,𝑥^∗2,…,𝑥^∗𝑛) is a solution of the problem (1.1) if and only if 𝑥^∗ is a fixed point of the mapping 𝑄 defined by 𝑄(𝑥)=Π_𝐶𝐼−𝜆₁𝐴₁Π_𝐶𝐼−𝜆₂𝐴₂⋯Π_𝐶𝐼−𝜆_𝑁𝐴_𝑁(𝑥),(2.18) where 𝜆_𝑖(𝑖=1,2,…,𝑁) are real numbers.

Proof. We can rewrite (1.1) as ⟨𝑥𝑥^∗1_∗2−𝑥^∗2−𝜆₁𝐴₁𝑥^∗2,𝑗𝑥−𝑥^∗1≥0,∀𝑥∈𝐶,𝑥−𝑥^∗3−𝜆₂𝐴₂𝑥^∗3,𝑗𝑥−𝑥^∗2⟩≥0,∀𝑥∈𝐶,_∗3𝑥−𝑥^∗4−𝜆₃𝐴₃𝑥^∗4,𝑗⋮𝑥−𝑥^∗3≥0,∀𝑥∈𝐶,_∗𝑁−1𝑥−𝑥^∗𝑁−𝜆_𝑁−1𝐴_𝑁−1𝑥^∗𝑁,𝑗𝑥−𝑥^∗𝑁−1≥0,∀𝑥∈C,_∗𝑁−𝑥^∗1−𝜆_𝑁𝐴_𝑁𝑥^∗1,𝑗𝑥−𝑥^∗𝑁≥0,∀𝑥∈𝐶.(2.19) By Lemma 2.2, we can check (2.19) is equivalent to 𝑥𝑥^∗1_∗2=Π_𝐶𝐼−𝜆₁𝐴₁𝑥^∗2,𝑥_∗𝑁−1=Π_𝐶⋮𝐼−𝜆₂𝐴₂𝑥^∗3,𝑥_∗𝑁=Π_𝐶𝐼−𝜆_𝑁−1𝐴_𝑁−1𝑥^∗𝑁,𝑄𝑥_∗=⟺Π_𝐶𝐼−𝜆_𝑁𝐴_𝑁𝑥^∗1.=Π_𝐶𝐼−𝜆₁𝐴₁Π_𝐶𝐼−𝜆₂𝐴₂⋯Π_𝐶𝐼−𝜆_𝑁𝐴_𝑁𝑥^∗=𝑥^∗.(2.20) This completes the proof.

Throughout this paper, the set of fixed points of the mapping 𝑄 is denoted by Ω.

Lemma 2.10. Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝐸. Let Π_𝐶 be a sunny nonexpansive retraction from 𝐸 onto 𝐶. Let {𝐴_𝑖∶𝐶→𝐸,𝑖=1,2,…,𝑁} be a finite family of 𝛾_𝑖-inverse-strongly accretive. Let 𝑄 be defined as Lemma 2.9. If 0≤𝜆_𝑖≤𝛾_𝑖/𝐾², then 𝑄∶𝐶→𝐶 is nonexpansive.

Proof. First, we show that for all 𝑖∈{𝑖,2,…,𝑁}, the mapping Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖) is nonexpansive. Indeed, for all 𝑥,𝑦∈𝐶, from the condition 𝜆_𝑖∈[0,𝛾_𝑖/𝐾²] and Lemma 2.6, we have ‖‖Π_𝐶𝐼−𝜆_𝑖𝐴_𝑖𝑥−Π_𝐶𝐼−𝜆_𝑖𝐴_𝑖𝑦‖‖²≤‖‖𝐼−𝜆_𝑖𝐴_𝑖𝑥−𝐼−𝜆_𝑖𝐴_𝑖𝑦‖‖²=‖‖(𝑥−𝑦)−𝜆_𝑖𝐴_𝑖𝑥−𝐴_𝑖𝑦‖‖²≤+‖𝑥2𝐾−₂𝑦‖²−2𝜆_𝑖⟨𝐴_𝑖𝑥−𝐴_𝑖𝑦,𝑗(𝑥−𝑦)⟩𝜆^2𝑖‖‖𝐴_𝑖𝑥−𝐴_𝑖𝑦‖‖²≤‖𝑥−𝑦‖²−2𝜆_𝑖𝛾_𝑖‖‖𝐴_𝑖𝑥−𝐴_𝑖𝑦‖‖²+2𝐾²𝜆^2𝑖‖‖𝐴_𝑖𝑥−𝐴_𝑖𝑦‖‖²=‖𝑥−𝑦‖²+2𝜆_𝑖𝐾²𝜆_𝑖−𝛾_𝑖‖‖𝐴_𝑖𝑥−𝐴_𝑖𝑦‖‖²≤‖𝑥−𝑦‖²,(2.21) which implies for all 𝑖∈{1,2,…,𝑁}, the mapping Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖) is nonexpansive, so is the mapping 𝑄.

3. Main Results

In this section, we introduce our algorithms and show the strong convergence theorems.

Algorithm 3.1. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝐸. Let Π_𝐶 be a sunny nonexpansive retraction from 𝐸 to 𝐶. Let {𝐴_𝑖∶𝐶→𝐸,𝑖=1,2,…,𝑁} be a finite family of 𝛾_𝑖-inverse-strongly accretive. Let 𝐵∶𝐶→𝐸 be a strongly positive bounded linear operator with coefficient 𝛼>0 and 𝐹∶𝐶→𝐸 be a strongly positive bounded linear operator with coefficient 𝜌∈(0,𝛼). For any 𝑡∈(0,1), define a net {𝑥_𝑡} as follows: 𝑥𝑦_𝑡𝑡=Π_𝐶(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡,=Π_𝐶𝐼−𝜆₁𝐴₁Π_𝐶𝐼−𝜆₂𝐴₂⋯Π_𝐶𝐼−𝜆_𝑁𝐴_𝑁𝑥_𝑡,(3.1) where, for any 𝑖,𝜆_𝑖∈(0,𝛾_𝑖/𝐾²) is a real number.

Remark 3.2. We notice that the net {𝑥_𝑡} defined by (3.1) is well defined. In fact, we can define a self-mapping 𝑊_𝑡∶𝐶→𝐶 as follows: 𝑊_𝑡𝑥∶=Π_𝐶(𝑡𝐹+(𝐼−𝑡𝐵))Π_𝐶𝐼−𝜆₁𝐴₁Π_𝐶𝐼−𝜆₂𝐴₂⋯Π_𝐶𝐼−𝜆_𝑁𝐴_𝑁𝑥,∀𝑥∈𝐶.(3.2)
From Lemma 2.10, we know that if, for any 𝑖,𝜆_𝑖∈(0,𝛾_𝑖/𝐾²), the mapping Π_𝐶(𝐼−𝜆₁𝐴₁)Π_𝐶(𝐼−𝜆₂𝐴₂)⋯Π_𝐶(𝐼−𝜆_𝑁𝐴_𝑁)=𝑄 is nonexpansive and ||𝐼−𝑡𝐵||≤1−𝑡𝛼. Then, for any 𝑥,𝑦∈𝐶, we have ‖‖𝑊_𝑡𝑥−𝑊_𝑡𝑦‖‖≤=≤≤==‖𝑡𝑡(‖‖‖𝜌𝜌1𝑡Π(‖‖(−(_𝐶𝐹𝑥𝑥𝑡𝐹𝑥((−−𝛼𝑡𝐹+−−𝑦𝑦+(𝐹‖‖𝐼𝜌𝑦()++−𝐼)𝑡+)−(‖𝑡‖1𝐵𝐼(𝑡𝑥𝐼−)𝐵−)−−)𝑄𝑡𝑡)𝛼𝐵𝑦𝑡)𝑄)𝐵‖‖𝑥‖(.)𝑥−𝑥‖()𝑄𝑄−(−(𝑥𝑥𝑡𝑦𝐹Π‖−−+_𝐶𝑄𝑄((𝑦𝑡𝑦𝐼𝐹)‖‖−+𝑡(𝐵𝐼)−)𝑄𝑡𝐵)𝑦))‖𝑄(𝑦)‖‖(3.3) This shows that the mapping 𝑊_𝑡 is contraction. By Banach contractive mapping principle, we immediately deduce that the net (3.1) is well defined.

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝐸. Let Π_𝐶 be a sunny nonexpansive retraction from E to 𝐶. Let {𝐴_𝑖∶𝐶→𝐸,𝑖=1,2,…,𝑁} be a finite family of 𝛾_𝑖-inverse-strongly accretive. Let 𝐵∶𝐶→𝐸 be a strongly positive bounded linear operator with coefficient 𝛼>0, and let 𝐹∶𝐶→𝐸 be a strongly positive bounded linear operator with coefficient 𝜌∈(0,𝛼). Assume that Ω≠∅ and 𝜆_𝑖∈(0,𝛾_𝑖/𝐾²). Then the net {𝑥_𝑡} generated by the implicit method (3.1) converges in norm, as 𝑡→0⁺ to the unique solution 𝑥̃ of VI 𝑥̃∈Ω,⟨(𝐵−𝐹)𝑥̃,𝑗(𝑧−𝑥̃)⟩≥0,𝑧∈Ω.(3.4)

Proof. We divide the proof of Theorem 3.3 into four steps.(I) Next we prove that the net {𝑥_𝑡} is bounded.
Take that 𝑥^∗∈Ω, we have ‖‖𝑥_𝑡−𝑥^∗‖‖==‖‖‖‖ΠΠ_𝐶𝐶(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡−𝑥^∗‖‖≤‖‖(𝑡𝐹(𝑡+𝐹(+𝐼−(𝐼𝑡−𝐵)𝑡)𝐵𝑦)_𝑡)𝑦_𝑡−Π_𝐶𝑥^∗‖‖=‖‖𝑡𝐹𝑦_𝑡−𝑥^∗‖‖≤𝑡‖‖𝐹𝑦_𝑡−𝐹𝑥^∗+(𝐼−𝑡𝐵)𝑦_𝑡−𝑥^∗+𝑡𝐹𝑥^∗−𝑡𝐵𝑥^∗‖‖≤𝑡𝜌‖‖𝑦_𝑡−𝐹𝑥^∗‖‖+‖𝐼−𝑡𝐵‖‖‖𝑦_𝑡−𝑥^∗‖‖+𝑡‖‖𝐹𝑥^∗−𝐵𝑥^∗‖‖=(1−(𝛼−−𝑥^∗𝜌‖‖)𝑡+)‖‖(𝑦1_𝑡−𝑡𝛼)‖‖𝑦_𝑡−𝑥^∗‖‖+𝑡‖‖𝐹𝑥^∗−𝐵𝑥^∗‖‖=(1−(𝛼−𝜌)𝑡)‖‖𝑄−𝑥𝑥^∗_𝑡‖‖+𝑡‖‖𝐹𝑥^∗−𝐵𝑥^∗‖‖≤(1−(𝛼−𝜌)𝑡)‖‖𝑥_𝑡−𝑄𝑥^∗‖‖+𝑡‖‖𝐹𝑥^∗−𝐵𝑥^∗‖‖−𝑥^∗‖‖+𝑡‖‖𝐹𝑥^∗−𝐵𝑥^∗‖‖.(3.5) It follows that ‖‖𝑥_𝑡−𝑥^∗‖‖≤‖‖𝐹𝑥^∗−𝐵𝑥^∗‖‖𝛼−𝜌.(3.6) Therefore, {𝑥_𝑡} is bounded. Hence, {𝑦_𝑡},{𝐵𝑦_𝑡},{𝐴_𝑖𝑥_𝑡}, and {𝐹(𝑦_𝑡)} are also bounded. We observe that ‖‖𝑥_𝑡−𝑦_𝑡‖‖≤=‖‖‖‖Π(𝑡_𝐶𝐹(𝑡+𝐹(+𝐼−(𝐼𝑡−𝐵)𝑡)𝐵𝑦)_𝑡)𝑦_𝑡−Π_𝐶𝑦_𝑡‖‖=𝑡‖‖𝐹𝑦_𝑡−𝑦_𝑡‖‖⟶0.−𝐵𝑦_𝑡‖‖(3.7)
From Lemma 2.10, we know that 𝑄∶𝐶→𝐶 is nonexpansive. Thus, we have ‖‖𝑦_𝑡−𝑄𝑦_𝑡‖‖=‖‖𝑄𝑥_𝑡−𝑄𝑦_𝑡‖‖≤‖‖𝑥_𝑡−𝑦_𝑡‖‖⟶0.(3.8) Therefore, lim_𝑡→0‖‖𝑥_𝑡−𝑄𝑥_𝑡‖‖=0.(3.9)
(II) {𝑥_𝑡} is relatively norm-compact as 𝑡→0⁺.
Let {𝑡_𝑛}⊂(0,1) be any subsequence such that 𝑡_𝑛→0⁺ as 𝑛→∞. Then, there exists a positive integer 𝑛₀ such that 0<𝑡_𝑛<1/2, for all 𝑛≥𝑛₀. Let 𝑥_𝑛∶=𝑥_𝑡𝑛. It follows from (3.9) that ‖‖𝑥_𝑛−𝑄𝑥_𝑛‖‖⟶0.(3.10) We can rewrite (3.1) as 𝑥_𝑡=Π_𝐶(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡−(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡+(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡.(3.11)
For any 𝑥^∗∈Ω⊂𝐶, by Lemma 2.2, we have (𝑡𝐹=+((𝐼𝑡𝐹−+𝑡𝐵()𝐼)−𝑦_𝑡𝑡−𝐵)𝑥)_𝑡𝑦,𝑗_𝑡𝑥^∗−𝑥_𝑡−Π_𝐶(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡,𝑗𝑥^∗−Π_𝐶(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡≤0.(3.12) With this fact, we derive that ‖‖𝑥_𝑡−𝑥^∗‖‖²==𝑥𝑥^∗−𝑥_𝑡,𝑗𝑥^∗−𝑥_𝑡_∗+(−𝑡𝐹(+𝑡𝐹(+𝐼−(𝐼𝑡𝐵−)𝑡)𝐵𝑦)_𝑡)𝑦_𝑡,𝑗𝑥^∗−𝑥_𝑡≤(𝑡𝐹+(𝐼−𝑡𝐵))𝑥_∗−Π_C(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡,𝑗𝑥^∗−𝑥_𝑡≤(1−𝑡(𝛼−𝜌))‖‖𝑥_∗−𝑦_𝑡,𝑗𝑥^∗−𝑥_𝑡+𝑡𝐵𝑥^∗−𝐹𝑥^∗,𝑗𝑥^∗−𝑥_𝑡=(1−𝑡(𝛼−𝜌))‖‖𝑄−𝑥_∗𝑦_𝑡‖‖‖‖𝑥^∗−𝑥_𝑡‖‖+𝑡𝐵𝑥^∗−𝐹𝑥^∗,𝑗𝑥^∗−𝑥_𝑡≤(1−𝑡(𝛼−𝜌))‖‖𝑥_∗−𝑄𝑥_𝑡‖‖‖‖𝑥^∗−𝑥_𝑡‖‖+𝑡𝐵𝑥^∗−𝐹𝑥^∗,𝑗𝑥^∗−𝑥_𝑡≤(1−𝑡(𝛼−𝜌))‖‖𝑥_∗−𝑥_𝑡‖‖‖‖𝑥^∗−𝑥_𝑡‖‖+𝑡𝐵𝑥^∗−𝐹𝑥^∗,𝑗𝑥^∗−𝑥_𝑡−𝑥_𝑡‖‖²+𝑡𝐵𝑥^∗−𝐹𝑥^∗,𝑗𝑥^∗−𝑥_𝑡.(3.13)
It turns out that ‖‖𝑥_𝑡−𝑥^∗‖‖²≤1𝛼−𝜌𝐵𝑥^∗−𝐹𝑥^∗,𝑗𝑥^∗−𝑥_𝑡,𝑥^∗∈Ω.(3.14)
In particular, ‖‖𝑥_𝑛−𝑥^∗‖‖²≤1𝛼−𝜌𝐵𝑥^∗−𝐹𝑥^∗,𝑗𝑥^∗−𝑥_𝑛,𝑥^∗∈Ω.(∗∗)
Since {𝑥_𝑛} is bounded, without loss of generality, 𝑥_𝑛⇀𝑥̃∈𝐶 can be assumed. Noticing (3.10), we can use Lemma 2.7 to get 𝑥̃∈Ω=𝐹(𝑄). Therefore, we can substitute 𝑥̃ for 𝑥^∗ in (**) to get ‖‖𝑥_𝑛−𝑥̃‖‖²≤1𝛼−𝜌𝐵(𝑥̃)−𝐹(𝑥̃),𝑗𝑥̃−𝑥^𝑛.(3.15) Consequently, the weak convergence of {𝑥_𝑛} to 𝑥̃ actually implies that 𝑥_𝑛→𝑥̃ strongly. This has proved the relative norm compactness of the net {𝑥_𝑡} as 𝑡→0⁺.
(III) Now, we prove that 𝑥̃ solves the variational inequality (3.4). From (3.1), we have 𝑥_𝑡=Π_𝐶(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡−(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡+(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡⟹𝑥_𝑡=+Π𝑡_𝐶𝐹(𝑡𝐹𝑥+_𝑡(𝐼−𝑡𝐵))𝑦_𝑡−(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡−(𝑡𝐹+(𝐼−𝑡𝐵))𝑥_𝑡−𝑦_𝑡⟹𝐹𝑥_𝑡+(𝐼−𝑡𝐵)𝑥_𝑡−𝐵𝑥_𝑡=1𝑡(𝑡𝐹+(𝐼−𝑡𝐵))𝑦^𝑡−Π_𝐶(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡−(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡−𝑥_𝑡.(3.16) For any 𝑧∈Ω, we obtain 𝐹𝑥_𝑡−𝐵𝑥_𝑡,𝑗𝑧−𝑥_𝑡=−1𝑡1(𝑡𝐹+(𝐼−𝑡𝐵))𝑦^𝑡−Π_𝐶(𝑡𝐹+(𝐼−𝑡𝐵))𝑦_𝑡,𝑗𝑧−𝑥_𝑡𝑡(𝑡𝐹+(𝐼−𝑡𝐵))𝑦^𝑡≤−1−𝑥_𝑡,𝑗𝑧−𝑥_𝑡𝑡(𝑡𝐹+(𝐼−𝑡𝐵))𝑦^𝑡=−1−𝑥_𝑡,𝑗𝑧−𝑥_𝑡𝑡𝑦^𝑡−𝑥_𝑡,𝑗𝑧−𝑥_𝑡+(𝐵−𝐹)𝑦_𝑡−𝑥_𝑡,𝑗𝑧−𝑥_𝑡.(3.17)
Now we prove that ⟨𝑦_𝑡−𝑥_𝑡,𝑗(𝑧−𝑥_𝑡)⟩≥0. In fact, we can write 𝑦_𝑡=𝑄(𝑥_𝑡). At the same time, we note that 𝑧=𝑄(𝑧), so 𝑦_𝑡−𝑥_𝑡,𝑗𝑧−𝑥_𝑡=𝑧−𝑄(𝑧)−𝑥_𝑡−𝑄𝑥_𝑡,𝑗𝑧−𝑥_𝑡.(3.18) Since 𝐼−𝑄 is accretive (this is due to the nonexpansivity of 𝑄), we can deduce immediately that 𝑦_𝑡−𝑥_𝑡,𝑗𝑧−𝑥_𝑡=⟨𝑧−𝑄(𝑧)−𝑥_𝑡−𝑄𝑥_𝑡,𝑗𝑧−𝑥_𝑡≥0.(3.19) Therefore, 𝐹𝑥_𝑡−𝐵𝑥_𝑡,𝑗𝑧−𝑥_𝑡≤(𝐵−𝐹)𝑦_𝑡−𝑥_𝑡,𝑗𝑧−𝑥_𝑡.(3.20) Since 𝐵,𝐹 is strongly positive, we have 0=≤(𝛼𝐹−𝜌𝑥)_𝑡‖‖𝑧−𝑥_𝑡‖‖²≤(𝐵−𝐹)𝑧−𝑥_𝑡,𝑗𝑧−𝑥_𝑡−𝐵𝑥_𝑡−(𝐹(𝑧)−𝐵(𝑧)),𝑗𝑧−𝑥_𝑡.(3.21) It follows that 𝐹(𝑧)−𝐵(𝑧),𝑗𝑧−𝑥_𝑡≤𝐹𝑥_𝑡−𝐵𝑥_𝑡,𝑗𝑧−𝑥_𝑡.(3.22) Combining (3.20) and (3.22), we get 𝐹(𝑧)−𝐵(𝑧),𝑗𝑧−𝑥_𝑡≤(𝐵−𝐹)𝑦_𝑡−𝑥_𝑡,𝑗𝑧−𝑥_𝑡.(3.23) Now replacing 𝑡 in (3.23) with 𝑡_𝑛 and letting 𝑛→∞, noticing that 𝑥_𝑡𝑛−𝑦_𝑡𝑛→0, we obtain ⟨𝐹(𝑧)−𝐵(𝑧),𝑗(𝑧−𝑥̃)⟩≤0,𝑧∈Ω,(3.24) which is equivalent to its dual variational inequality (see Lemma 2.8) ⟨(𝐵−𝐹)𝑥̃,𝑗(𝑧−𝑥̃)⟩≥0,𝑧∈Ω,(3.25) that is, 𝑥̃∈Ω is a solution of (3.4).
(IV) Now we show that the solution set of (3.4) is singleton.
As a matter of fact, we assume that 𝑥^∗∈Ω is also a solution of (3.4) Then, we have (𝐵−𝐹)𝑥^∗,𝑗𝑥̃−𝑥^∗≥0.(3.26) From (3.25), we have (𝐵−𝐹)𝑥̃,𝑗𝑥^∗−𝑥̃≥0.(3.27) So, ⟹(𝐵−(𝐹𝐵)−𝑥^∗𝐹,𝑗)𝑥𝑥̃̃−−𝑥𝑥^∗_∗+(𝐵−𝐹)𝑥̃,𝑗𝑥^∗−𝑥̃≥0⟹(𝐵−𝐹)𝑥_∗,𝑗𝑥^∗−𝑥̃≥0⟹(𝛼−𝜌)‖𝑥_∗−𝑥̃,𝑗𝑥^∗−𝑥̃≤0−𝑥̃‖²≤0.(3.28) Therefore, 𝑥^∗=𝑥̃. In summary, we have shown that each cluster point of {𝑥_𝑡} (as 𝑡→0) equals 𝑥̃. Therefore, 𝑥_𝑡→𝑥̃ as 𝑡→0. This completes the proof.

Next, we introduce our explicit method which is the discretization of the implicit method (3.1).

Algorithm 3.4. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝐸. Let Π_𝐶 be a sunny nonexpansive retraction from 𝐸 to 𝐶. Let {𝐴_𝑖∶𝐶→𝐸,𝑖=1,2,…,𝑁} be a finite family of 𝛾_𝑖-inverse-strongly accretive. Let 𝐵∶𝐶→𝐸 be a strongly positive bounded linear operator with coefficient 𝛼>0, and let 𝐹∶𝐶→𝐸 be a strongly positive bounded linear operator with coefficient 𝜌∈(0,𝛼). For arbitrarily given 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated iteratively by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛Π_𝐶𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵Π_𝐶𝐼−𝑛𝜆≥₁𝐴0₁,Π_𝐶𝐼−𝜆₂𝐴₂⋯Π_𝐶𝐼−𝜆_𝑁𝐴_𝑁𝑥_𝑛,(3.29) where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in [0,1] and, for any 𝑖,𝜆_𝑖∈(0,𝛾_𝑖/𝐾²) is a real number.

Theorem 3.5. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝐸, and let Π_𝐶 be a sunny nonexpansive retraction from 𝐸 to 𝐶. Let {𝐴_𝑖∶𝐶→𝐸,𝑖=1,2,…,𝑁} be a finite family of 𝛾_𝑖-inverse-strongly accretive.Let 𝐵∶𝐶→𝐸 be a strongly positive bounded linear operator with coefficient 𝛼>0, and let 𝐹∶𝐶→𝐸 be a strongly positive bounded linear operator with coefficient 𝜌∈(0,𝛼). Assume that Ω≠∅. For given 𝑥₀∈𝐶, let {𝑥_𝑛} be generated iteratively by (3.29). Suppose the sequences {𝛼_𝑛} and {𝛽_𝑛} satisfy the following conditions:(1)lim_𝑛→∞𝛼_𝑛=0 and Σ^∞𝑛=1𝛼_𝑛=∞,(2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛≤1.Then {𝑥_𝑛} converges strongly to 𝑥̃∈Ω which solves the variational inequality (3.4).

Proof. Set 𝑦_𝑛=Π_𝐶(𝐼−𝜆₁𝐴₁)Π_𝐶(𝐼−𝜆₂𝐴₂)⋯Π_𝐶(𝐼−𝜆_𝑁𝐴_𝑁)𝑥_𝑛 for all 𝑛≥0. Then 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)Π_𝐶(𝛼_𝑛𝐹+(𝐼−𝛼_𝑛𝐵))𝑦_𝑛 for all 𝑛≥0. Pick up 𝑥^∗∈Ω.
From Lemma 2.10, we have ‖‖𝑦_𝑛−𝑥^∗‖‖=‖‖𝑄𝑥_𝑛−𝑄𝑥^∗‖‖≤‖‖𝑥_𝑛−𝑥^∗‖‖.(3.30) Hence, it follows that ‖‖𝑥_𝑛+1−𝑥^∗‖‖==‖‖‖‖𝛽𝛽_𝑛𝑛𝑥_𝑛+1−𝛽_𝑛Π_𝐶𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝑥^∗‖‖≤𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛Π_𝐶𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝑥^∗‖‖≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛽_𝑛‖‖Π_𝐶𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−Π_𝐶𝑥^∗‖‖=𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛽_𝑛‖‖𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝑥^∗‖‖≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛽_𝑛‖‖𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝑥^∗+𝛼_𝑛𝐹𝑥^∗−𝐵𝑥^∗‖‖≤1‖‖−𝑥_𝑛𝛼−_𝑛𝑥^∗‖‖+1−𝛽_𝑛𝛼_𝑛𝜌+1−𝛼_𝑛𝛼‖‖𝑦_𝑛−𝑥^∗‖‖+1−𝛽_𝑛𝛼_𝑛‖‖𝐹𝑥^∗−𝐵𝑥^∗‖‖1−𝛽_𝑛(𝛼−𝜌)‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛1−𝛽_𝑛(𝛼−𝜌)‖‖𝐹𝑥^∗−𝐵𝑥^∗‖‖𝛼−𝜌.(3.31) By induction, we deduce that ‖‖𝑥_𝑛+1−𝑥^∗‖‖≤max‖‖𝑥₀−𝑥^∗‖‖,‖‖𝐹𝑥^∗−𝐵𝑥^∗‖‖𝛼−𝜌.(3.32) Therefore, {𝑥_𝑛} is bounded. Hence, {𝐴_𝑖𝑥_𝑖}(𝑖=1,2,…,𝑁),{𝑦_𝑛},{𝐵𝑦_𝑛}, and {𝐹(𝑦_𝑛)} are also bounded. We observe that ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖=‖‖𝑄𝑥_𝑛+1−𝑄𝑥_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.33) Set 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑧_𝑛 for all 𝑛≥0. Then 𝑧_𝑛=Π_𝐶(𝛼_𝑛𝐹+(𝐼−𝛼_𝑛𝐵))𝑦_𝑛. It follows that ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤=‖‖‖‖Π_𝐶𝛼𝛼_𝑛+1𝑛+1𝐹+𝐼−𝛼_n+1𝐵𝑦_𝑛+1−Π_𝐶𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛‖‖=‖‖𝑦_𝑛+1𝐹+𝐼−𝛼_𝑛+1𝐵𝑦_𝑛+1−𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛‖‖≤‖‖𝑦−𝑦_𝑛+𝛼_𝑛+1𝐹𝑦_𝑛+1−𝐵𝑦_𝑛+1−𝛼_𝑛𝐹𝑦_𝑛−𝐵𝑦_𝑛_𝑛+1‖‖≤‖‖𝑥_𝑛+1−𝑦_𝑛‖‖+𝛼_𝑛+1‖‖𝐹𝑦_𝑛+1−𝐵𝑦_𝑛+1‖‖−𝛼_𝑛‖‖𝐹𝑦_𝑛−𝐵𝑦_𝑛‖‖−𝑥_𝑛‖‖+𝛼_𝑛+1‖‖𝐹𝑦_𝑛+1−𝐵𝑦_𝑛+1‖‖−𝛼_𝑛‖‖𝐹𝑦_𝑛−𝐵𝑦_𝑛‖‖.(3.34) This implies that limsup_𝑛→∞‖‖𝑧_𝑛+1−𝑧_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.35) Hence, by Lemma 2.5, we obtain lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0. Consequently, lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞1−𝛽_𝑛‖‖𝑧_𝑛−𝑥_𝑛‖‖=0.(3.36) At the same time, we note that ‖‖𝑧_𝑛−𝑦_𝑛‖‖==‖‖‖‖ΠΠ_𝐶𝐶𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝑦_𝑛‖‖≤‖‖𝛼_𝑛𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−Π_𝐶𝑦_𝑛‖‖=𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝑦_𝑛‖‖⟶0‖‖.𝐹𝑦_𝑛−𝐵𝑦_𝑛‖‖(3.37) It follows that lim_𝑛→∞‖‖𝑥_𝑛−𝑦_𝑛‖‖=0.(3.38) From Lemma 2.10, we know that 𝑄∶𝐶→𝐶 is nonexpansive. Thus, we have ‖‖𝑦_𝑛−𝑄𝑦_𝑛‖‖=‖‖𝑄𝑥_𝑛−𝑄𝑦_𝑛‖‖≤‖‖𝑥_𝑛−𝑦_𝑛‖‖⟶0.(3.39) Thus, lim_𝑛→∞‖𝑥_𝑛−𝑄(𝑥_𝑛)‖=0. We note that ‖‖𝑧_𝑛−𝑄𝑧_𝑛‖‖≤≤2‖‖‖‖𝑧_𝑛𝑧_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑄𝑥_𝑛‖‖+‖‖𝑄𝑥_𝑛−𝑄𝑧_𝑛‖‖=2‖‖Π_𝐶−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑄𝑥_𝑛‖‖≤2‖‖𝑦_𝑛𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−Π_𝐶𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑄𝑥_𝑛‖‖⟶0.−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝐹𝑦_𝑛−𝐵𝑦_𝑛‖‖+‖‖𝑥_𝑛−𝑄𝑥_𝑛‖‖(3.40) Next, we show that limsup_𝑛→∞𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧_𝑛−𝑥̃≤0,(3.41) where 𝑥̃∈Ω is the unique solution of VI(3.4).
To see this, we take a subsequence {𝑧_𝑛𝑗} of {𝑧_𝑛} such that lim_𝑛→∞𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧_𝑛−𝑥̃=lim_{𝑛𝑗→∞}𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧^𝑛_𝑗−𝑥̃.(3.42) We may also assume that 𝑧_𝑛𝑗⇀𝑧. Note that 𝑧∈Ω in virtue of Lemma 2.7 and (3.40). It follows from the variational inequality (3.4) that lim_𝑛→∞𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧_𝑛−𝑥̃==⟨l𝐹im(_{𝑛𝑗→∞}𝑥̃)−𝐹𝐵(𝑥(̃)𝑥−̃),𝐵𝑗((𝑧𝑥̃)−,𝑗𝑥̃)⟩𝑧≤_𝑛𝑗−0.𝑥̃(3.43) Since 𝑧_𝑛=Π_𝐶(𝛼_𝑛𝐹+(𝐼−𝛼_𝑛𝐵))𝑦_𝑛, according to Lemma 2.2, we have 𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−Π_𝐶𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛,𝑗𝑥̃−𝑧_𝑛≤0.(3.44) From (3.44), we have ‖‖𝑧_𝑛−𝑥̃‖‖²==ΠΠ_𝐶𝛼_𝑛𝐹+I−𝛼_𝑛𝐵𝑦_𝑛−𝑥̃,𝑗𝑧_𝑛−𝑥̃_𝐶+𝛼_𝑛𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛,𝑗𝑧_𝑛−𝑥̃≤𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝑥̃,𝑗𝑧_𝑛−𝑥̃=𝛼_𝑛𝐹+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝑥̃,𝑗𝑧_𝑛−𝑥̃≤1−𝐹𝛼_𝑛+𝐼−𝛼_𝑛𝐵𝑦_𝑛−𝑥̃,𝑗𝑧_𝑛−𝑥̃+𝛼_𝑛𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧_𝑛−𝑥̃≤1−𝛼_𝑛(𝛼−𝜌)‖‖𝑦_𝑛−𝑥̃‖‖‖‖𝑧_𝑛−𝑥̃‖‖+𝛼_𝑛𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧_𝑛−𝑥̃(𝛼−𝜌)²2‖‖𝑦^𝑛−𝑥̃‖‖²+12‖‖𝑧^𝑛−𝑥̃‖‖²+𝛼_𝑛𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧_𝑛−𝑥̃.(3.45) It follows that ‖‖𝑧_𝑛−𝑥̃‖‖²≤≤11−−𝛼𝛼_𝑛𝑛(𝛼−𝜌)‖‖𝑦_𝑛−𝑥̃‖‖²+2𝛼_𝑛𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧_𝑛−𝑥̃,(𝛼−𝜌)‖‖𝑥_𝑛−𝑥̃‖‖²+2𝛼_𝑛𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧_𝑛−𝑥̃.(3.46) Finally, we prove 𝑥_𝑛→𝑥̃. From 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑧_𝑛 and (3.46), we have ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+1−𝛽_𝑛‖‖𝑧_𝑛−𝑥̃‖‖²≤=𝛽1_𝑛‖‖−𝑥_𝑛𝛼−_𝑛𝑥̃‖‖²+1−𝛽_𝑛1−𝛼_𝑛(𝛼−𝜌)‖‖𝑥_𝑛−𝑥̃‖‖²+2𝛼_𝑛𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧_𝑛−𝑥̃×21−𝛽_𝑛(𝛼−𝜌)‖‖𝑥_𝑛−𝑥̃‖‖²+𝛼_𝑛1−𝛽_𝑛(𝛼−𝜌)𝛼−𝜌𝐹(𝑥̃)−𝐵(𝑥̃),𝑗𝑧^𝑛−𝑥̃.(3.47) We can apply Lemma 2.4 to the relation (3.47) and conclude that 𝑥_𝑛→𝑥̃. This completes the proof.

Acknowledgments

The authors would like to express their thanks to the referees and the editor for their helpful suggestion and comments. This work was supported by the Scientific Reserch Fund of Sichuan Provincial Education Department (09ZB102,11ZB146) and Yunnan University of Finance and Economics.