Strong Convergence Theorems for Variational Inequalities and Split Equality Problem

Abstract

Let 𝐻₁,𝐻₂, and 𝐻₃ be real Hilbert spaces, let 𝐶⊆𝐻₁,𝑄⊆𝐻₂ be two nonempty closed convex sets, and let 𝐴∶𝐻₁→𝐻₃,𝐵∶𝐻₂→𝐻₃ be two bounded linear operators. The split equality problem (SEP) is to find 𝑥∈𝐶,𝑦∈𝑄 such that 𝐴𝑥=𝐵𝑦. Let 𝐻=𝐻₁×𝐻₂; consider 𝑓∶𝐻→𝐻 a contraction with coefficient 0<𝛼<1, a strongly positive linear bounded operator 𝑇∶𝐻→𝐻 with coefficient 𝛾>0, and 𝑀∶𝐻→𝐻 is a 𝛽-inverse strongly monotone mapping. Let 0<𝛾<𝛾/𝛼, 𝑆=𝐶×𝑄 and 𝐺∶𝐻→𝐻₃ be defined by restricting to 𝐻₁ is 𝐴 and restricting to 𝐻₂ is −𝐵, that is, 𝐺 has the matrix form 𝐺=[𝐴,−𝐵]. It is proved that the sequence {𝑤_𝑛}={(𝑥_𝑛,𝑦_𝑛)}⊆𝐻 generated by the iterative method 𝑤_𝑛+1=𝑃_𝑆[𝛼_𝑛𝛾𝑓(𝑤_𝑛)+(𝐼−𝛼_𝑛𝑇)𝑃_𝑆(𝐼−𝛾_𝑛𝐺^∗𝐺)𝑃_𝑆(𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛)] converges strongly to 𝑤 which solves the SEP and the following variational inequality: ⟨(𝑇−𝜆𝑓)𝑤,𝑤−𝑤⟩≥0 and ⟨𝑀𝑤,𝑤−𝑤⟩≥0 for all 𝑤∈𝑆. Moreover, if we take 𝑀=𝐺^∗𝐺∶𝐻→𝐻,𝛾_𝑛=0, then 𝑀 is a 𝛽-inverse strongly monotone mapping, and the sequence {𝑤_𝑛} generated by the iterative method 𝑤_𝑛+1=𝛼_𝑛𝛾𝑓(𝑤_𝑛)+(𝐼−𝛼_𝑛𝑇)𝑃_𝑆(𝑤_𝑛−𝜆_𝑛𝐺^∗𝐺𝑤_𝑛) converges strongly to 𝑤 which solves the SEP and the following variational inequality: ⟨(𝑇−𝜆𝑓)𝑤,𝑤−𝑤⟩≥0 for all 𝑤∈𝑆.

1. Introduction and Preliminaries

Throughout this paper, let 𝐻 be a real Hilbert space and 𝑆 a nonempty closed convex subset of 𝐻, and let 𝐼 denote the identity operator on Hilbert space 𝐻, Fix(𝑇) the set of the fixed points of an operator 𝑇. An operator 𝑇 on a Hilbert space 𝐻 is nonexpansive if, for each 𝑥 and 𝑦 in 𝐻, ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖. 𝑇 is said to be strongly positive, if there exists 𝛾>0 such that ⟨𝑇𝑥,𝑥⟩≥𝛾‖𝑥‖² for all 𝑥∈𝐻. 𝑇 is said to be inverse strongly monotone if there exists 𝛽>0 such that ⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩≥𝛽‖𝑇𝑥−𝑇𝑦‖² for all 𝑥,𝑦∈𝐻. A bounded linear operator 𝑇 on a Hilbert space 𝐻 is positive operator if 𝑇^∗=𝑇 and 𝜆≥0 for all 𝜆 in the spectrum of 𝑇 or equivalent to ⟨𝑇𝑥,𝑥⟩≥0 for all 𝑥∈𝐻.

Let 𝑃_𝑆 denote the projection from 𝐻 onto a nonempty closed convex subset 𝑆 of 𝐻; that is, 𝑃_𝑆(𝑤)=min_𝑥∈𝑆‖𝑥−𝑤‖. (1) It is well known that 𝑃_𝑆(𝑤) is characterized by the inequality 𝑤−𝑃_𝑆(𝑤),𝑥−𝑃_𝑆(𝑤)≤0,∀𝑥∈𝑆, (2) and 𝑃_𝑆 is nonexpansive.

Let 𝐶 and 𝑄 be nonempty closed convex subsets of real Hilbert spaces 𝐻₁ and 𝐻₂, respectively, and let 𝐴∶𝐻₁→𝐻₂ be a bounded linear operator. The split feasibility problem (SFP) is to find a point 𝑥 satisfying the property 𝑥∈𝐶,𝐴𝑥∈𝑄. (3) The SFP was first introduced by Censor and Elfving [1], which attracts many authors’ attention due to its application in signal processing [1]. Various algorithms have been invented to solve it (see [2–7]).

Recently, Moudafi [8] proposes a new split equality problem (SEP): let 𝐻₁, 𝐻₂, and 𝐻₃ be real Hilbert spaces, let 𝐶⊆𝐻₁, 𝑄⊆𝐻₂ be two nonempty closed convex sets, and let 𝐴∶𝐻₁→𝐻₃, 𝐵∶𝐻₂→𝐻₃ be two bounded linear operators. Find 𝑥∈𝐶, 𝑦∈𝑄 satisfying 𝐴𝑥=𝐵𝑦. (4) When 𝐵=𝐼, SEP reduces to the well-known SFP. In the paper [8], Moudafi gives the alternating CQ-algorithm and relaxed alternating CQ-algorithm iterative algorithm to solve the split equality problem. We must point out that the above algorithms converge weakly to a solution of the SEP.

We use Γ={(𝑥,𝑦)∈𝐻₁×𝐻₂,𝐴𝑥=𝐵𝑦,𝑥∈𝐶,𝑦∈𝑄} to denote the solution set of SEP, and letting 𝑆=𝐶×𝑄 in 𝐻=𝐻₁×𝐻₂ define 𝐺∶𝐻→𝐻₃ by 𝐺=[𝐴,−𝐵], then 𝐺^∗𝐺∶𝐻→𝐻 has the matrix form 𝐺^∗𝐴𝐺=^∗𝐴−𝐴^∗𝐵−𝐵^∗𝐴𝐵^∗𝐵. (5) The original problem can now be reformulated as finding 𝑤=(𝑥,𝑦)∈𝑆 with 𝐺𝑤=0 or, more generally, minimizing the function ‖𝐺𝑤‖ over 𝑤∈𝑆. Therefore solving SEP (4) is equivalent to solving the following minimization problem: min_𝑤∈𝑆𝑓(𝑤)=12‖𝐺𝑤‖². (6) In the paper [9], we use the well-known Tychonov regularization to get some algorithms converge strongly to the minimum-norm solution of the SEP. Moreover, in the paper [9], we obtain that, for 0<𝛾<2/‖𝐺‖², Fix(𝑃_𝑆(𝐼−𝛾𝐺^∗𝐺))=Fix(𝑃_𝑆)∩Fix(𝐼−𝛾𝐺^∗𝐺)=Γ. That is to say, the set of the solution of SEP is equal to the fixed points set of the nonexpansive mapping 𝑃_𝑆(𝐼−𝛾𝐺^∗𝐺).

Recall that the variational inequality problem is to find 𝑝∈𝐻 such that ⟨𝑀𝑝,𝑥−𝑝⟩≥0, for all 𝑥∈𝑆, where 𝑆 is closed, convex, and nonempty, and 𝑀 is a mapping on 𝐻. The set of solutions of the variational inequality is denoted by VI(𝑆,𝑀). In the paper [9], we also get that 𝑤∈Fix(𝑃_𝑆(𝐼−𝛾𝐺^∗𝐺)) if and only if 𝑤 is a solution of the variational inequality ⟨𝐺^∗𝐺𝑤,𝑣−𝑤⟩≥0, for all 𝑣∈𝑆.

In this paper, we consider a contraction 𝑓 on 𝐻=𝐻₁×𝐻₂ with coefficient 0<𝛼<1, a strongly positive linear bounded operator 𝑇 with coefficient 𝛾>0, and 𝑀 is a 𝛽-inverse strongly monotone mapping. Let 0<𝛾<𝛾/𝛼, 𝑆=𝐶×𝑄. We prove that the sequence {𝑤_𝑛}={(𝑥_𝑛,𝑦_𝑛)}⊆𝐻 generated by the iterative method 𝑤_𝑛+1=𝑃_𝑆[𝛼_𝑛𝛾𝑓(𝑤_𝑛)+(𝐼−𝛼_𝑛𝑇)𝑃_𝑆(𝐼−𝛾_𝑛𝐺^∗𝐺)𝑃_𝑆(𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛)] converges strongly to 𝑤 which solves the SEP and the following variational inequality: ⟨(𝐴−𝜆𝑓)𝑤,𝑤−𝑤⟩≥0 and ⟨𝑀𝑤,𝑤−𝑤⟩≥0 for all 𝑤∈𝑆 under some mild conditions for 𝛼_𝑛, 𝛾_𝑛, and 𝜆_𝑛.

We now collect some elementary facts which will be used in the proofs of our main results.

Lemma 1. Assuming 𝑇 is a positive operator, then 𝑇 is 1/‖𝑇‖-inverse strongly monotone.

Proof. Since 𝑇 is a positive operator on 𝐻, by the functional calculus of positive operator (for more details see [10, Chapter 4]), there exists positive operator 𝑇₁ such that 𝑇=𝑇₁𝑇₁ and ‖𝑇₁‖=‖𝑇‖^1/2.
Hence, for any 𝑥,𝑦∈𝐻, ‖𝑇𝑥−𝑇𝑦‖²=‖𝑇₁𝑇₁(𝑥−𝑦)‖²≤‖𝑇₁‖²‖𝑇₁(𝑥−𝑦)‖²=‖𝑇‖⟨𝑇₁(𝑥−𝑦),𝑇₁(𝑥−𝑦)⟩=‖𝑇‖⟨𝑥−𝑦,𝑇₁𝑇₁(𝑥−𝑦)⟩=‖𝑇‖⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩. It follows that ⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩≥(1/‖𝑇‖)‖𝑇𝑥−𝑇𝑦‖². That is to say, 𝑇 is 1/‖𝑇‖-inverse strongly monotone.

Lemma 2 (see [11, 12]). Let 𝑋 be a Banach space, 𝐶 a closed convex subset of 𝑋, and 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦.

Lemma 3 (see [13]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers, {𝛼_𝑛} a sequence of real numbers in [0,1] with ∑^∞_𝑛=1𝛼_𝑛=∞, {𝑢_𝑛} a sequence of nonnegative real numbers with ∑^∞_𝑛=1𝑢_𝑛<∞, and {𝑡_𝑛} a sequence of real numbers with limsup_𝑛𝑡_𝑛≤0. Suppose that 𝑠_𝑛+1=1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝑡_𝑛+𝑢_𝑛,∀𝑛∈ℕ. (7) Then lim_𝑛→∞𝑠_𝑛=0.

Lemma 4 (see [14]). Assuming 𝑇 is a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾>0, then ‖𝐼−𝜌𝑇‖≤1−𝜌𝛾 for 0<𝜌≤‖𝑇‖⁻¹.

Lemma 5 (see [13]). Let 𝑆 be a nonempty and closed subset of a Banach space 𝑋, and let {𝑅_𝑛} be a family of mappings of 𝑆 into itself which satisfies the condition: for each bounded subset 𝐸 of 𝑆, sup𝑅^∞_𝑛=1𝑛+1𝑤−𝑅_𝑛𝑤,𝑤∈𝐸<∞. (8) Then, for each 𝑤∈𝑆, {𝑅_𝑛𝑤} converges strongly to a point in 𝑆. Moreover, 𝑅 is defined by 𝑅𝑤=lim_𝑛→∞𝑅_𝑛𝑤,𝑤∈𝐸. (9) Then, for each bounded subset 𝐸 of 𝑆, limsup_𝑛→∞𝑅𝑤−𝑅_𝑛𝑤,𝑤∈𝐸=0, (10) and Fix(𝑅)=∩Fix(𝑅_𝑛), if ∩Fix(𝑅_𝑛)≠∅.

2. Main Results

Theorem 6. Let Γ denote the solutions set of the SEP, 𝑆=𝐶×𝑄, 𝑓 a contraction on 𝐻=𝐻₁×𝐻₂ with coefficient 0<𝛼<1, a strongly positive linear bounded operator 𝑇 with coefficient 𝛾>0, and 𝑀 a 𝛽-inverse strongly monotone mapping. Assume that 0<𝛾<𝛾/𝛼, Γ∩𝑉𝐼(𝑆,𝑀)≠∅, and {𝑤_𝑛}={(𝑥_𝑛,𝑦_𝑛)}⊆𝐻 is the sequence generated by the following algorithm: 𝑤₀∈𝐻, 𝑤_𝑛+1=𝑃_𝑆𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑃_𝑆𝐼−𝛾_𝑛𝐺^∗𝐺𝑃_𝑆𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛 (11) for all 𝑛=0,1,2,…, where 𝛼_𝑛∈(0,1),𝜆_𝑛∈(0,2𝛽), and 𝛾_𝑛∈(0,2/‖𝐺‖²). If 𝛼_𝑛, 𝜆_𝑛, and 𝛾_𝑛 are chosen such that 𝜆_𝑛∈[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<2𝛽 and(i)lim_𝑛𝛼_𝑛=0,(ii)∑^∞_𝑛=0𝛼_𝑛=∞, (iii)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞,(v)∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞,(iv)∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞,then the sequence {𝑤_𝑛} converges strongly to 𝑤 which solves the SEP and the following variational inequality: ⟨(𝑇−𝜆𝑓)𝑤,𝑤−𝑤⟩≥0 and ⟨𝑀𝑤,𝑤−𝑤⟩≥0 for all 𝑤∈𝑆.

Proof. First, we show that the sequence {𝑤_𝑛} is bounded. Consider the mapping 𝐼−𝜆_𝑛𝑀. Since 𝑀 is a 𝛽-inverse strongly monotone mapping, we have that, for all 𝑤,𝑣∈𝑆, 𝐼−𝜆_𝑛𝑀𝑤−𝐼−𝜆_𝑛𝑀𝑣²=(𝑤−𝑣)−𝜆_𝑛(𝑀𝑤−𝑀𝑣)²=‖𝑤−𝑣‖²−2𝜆_𝑛⟨𝑤−𝑣,𝑀𝑤−𝑀𝑣⟩+𝜆²_𝑛‖𝑀𝑤−𝑀𝑣‖²≤‖𝑤−𝑣‖²+𝜆_𝑛𝜆_𝑛−2𝛽‖𝑀𝑤−𝑀𝑣‖². (12) Since 0<𝜆_𝑛<2𝛽, we deduce that ‖(𝐼−𝜆_𝑛𝑀)𝑤−(𝐼−𝜆_𝑛𝑀)𝑣‖²≤‖𝑤−𝑣‖². That is to say, 𝐼−𝜆_𝑛𝑀 is nonexpansive.
Letting 𝑣_𝑛=𝑃_𝑆(𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛), 𝑅_𝑛=𝑃_𝑆(𝐼−𝛾_𝑛𝐺^∗𝐺), choosing 𝑢∈Γ∩VI(𝑆,𝑀), then 𝑢=𝑃_𝑆(𝐼−𝛾𝐺^∗𝐺)𝑢 and 𝑢=𝑃_𝑆(𝐼−𝜆_𝑛𝑀)𝑢. 𝑃_𝑆 is nonexpansive, which implies that 𝑣_𝑛−𝑢=𝑃_𝑆𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛−𝑃_𝑆𝑢−𝜆_𝑛≤𝑤𝑀𝑢_𝑛−𝜆_𝑛𝑀𝑤_𝑛−𝑢−𝜆_𝑛=𝑀𝑢𝐼−𝜆_𝑛𝑀𝑤_𝑛−𝐼−𝜆_𝑛𝑀𝑢≤𝑤_𝑛,𝑤−𝑢_𝑛+1−𝑢=𝑃_𝑆𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛≤𝛼−𝑢_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛=𝛼−𝑢_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑢. (13) Since 𝑇 is strongly positive linear bounded operator, 𝑅_𝑛 is nonexpansive, and 𝑓 is contraction, we have 𝑤_𝑛+1−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢+1−𝛼_𝑛𝛾𝑅_𝑛𝑣_𝑛−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝛾𝑓(𝑢)+𝛼_𝑛(𝑢)+𝛾𝑓−𝑇𝑢1−𝛼_𝑛𝛾𝑅_𝑛𝑣_𝑛−𝑢≤𝛼𝛾𝛼_𝑛𝑤_𝑛−𝑢+𝛼_𝑛(𝑢)+𝛾𝑓−𝑇𝑢1−𝛼_𝑛𝛾𝑅_𝑛𝑣_𝑛−𝑢=𝛼𝛾𝛼_𝑛𝑤_𝑛−𝑢+𝛼_𝑛(𝑢)+𝛾𝑓−𝑇𝑢1−𝛼_𝑛𝛾𝑅_𝑛𝑣_𝑛−𝑅_𝑛𝑢≤𝛼𝛾𝛼_𝑛𝑤_𝑛−𝑢+𝛼_𝑛(𝑢)+𝛾𝑓−𝑇𝑢1−𝛼_𝑛𝛾𝑣_𝑛−𝑢≤𝛼𝛾𝛼_𝑛𝑤_𝑛−𝑢+𝛼_𝑛(𝑢)+𝛾𝑓−𝑇𝑢1−𝛼_𝑛𝛾𝑤_𝑛=−𝑢1−𝛼𝛾−𝛾𝛼_𝑛𝑤_𝑛−𝑢+𝛼_𝑛(𝑢)𝑤𝛾𝑓−𝑇𝑢≤max_𝑛,1−𝑢𝛾−𝛾𝛼_𝑛𝛾𝑓(𝑢)−𝑇𝑢. (14) It follows from induction that 𝑤_𝑛−𝑢≤max𝑤₀,1−𝑢𝛾−𝛾𝛼_𝑛𝛾𝑓(𝑢)−𝑇𝑢,𝑛>0. (15) Hence, {𝑤_𝑛} is bounded, so are {𝑣_𝑛}, {𝑅_𝑛𝑣_𝑛}, {𝑀𝑤_𝑛}, and {𝑓(𝑤_𝑛)}.
Next, we show that ‖𝑤_𝑛+1−𝑤_𝑛‖→0 and ‖𝑣_𝑛−𝑅_𝑛𝑣_𝑛‖→0. In fact, in view that 𝑃_𝑆 and 𝐼−𝜆_𝑛𝑀 are nonexpansive, we can get that 𝑣_𝑛+1−𝑣_𝑛=𝑃_𝑆𝑤_𝑛+1−𝜆_𝑛+1𝑀𝑤_𝑛+1−𝑃_𝑆𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛≤𝑤_𝑛+1−𝜆_𝑛+1𝑀𝑤_𝑛+1−𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛≤𝑤_𝑛+1−𝜆_𝑛+1𝑀𝑤_𝑛+1−𝑤_𝑛−𝜆_𝑛+1𝑀𝑤_𝑛+𝜆_𝑛+1−𝜆_𝑛𝑀𝑤_𝑛≤𝑤_𝑛+1−𝑤_𝑛+𝜆_𝑛+1−𝜆_𝑛𝑀𝑤_𝑛. (16) Hence, 𝑤_𝑛+1−𝑤_𝑛=𝑃_𝑆𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑃_𝑆𝛼_𝑛−1𝛾𝑓𝑤_𝑛−1+𝐼−𝛼_𝑛−1𝑇𝑅_𝑛−1𝑣_𝑛−1≤𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝛼_𝑛−1𝛾𝑓𝑤_𝑛−1+𝐼−𝛼_𝑛−1𝑇𝑅_𝑛−1𝑣_𝑛−1=𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑓𝑤_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1×𝑓𝑤_𝑛−1+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑅_𝑛−1𝑣_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑇𝑅_𝑛−1𝑣_𝑛−1≤𝛼_𝑛𝛼𝛾𝑤_𝑛−𝑤_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑤_𝑛−1+1−𝛼_𝑛𝛾𝑅_𝑛𝑣_𝑛−𝑅_𝑛−1𝑣_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑇𝑅_𝑛−1𝑣_𝑛−1≤𝛼_𝑛𝛼𝛾𝑤_𝑛−𝑤_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑤_𝑛−1+1−𝛼_𝑛𝛾𝑅_𝑛𝑣_𝑛−𝑅_𝑛𝑣_𝑛−1+1−𝛼_𝑛𝛾𝑅_𝑛𝑣_𝑛−1−𝑅_𝑛−1𝑣_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑇𝑅_𝑛−1𝑣_𝑛−1≤𝛼_𝑛𝛼𝛾𝑤_𝑛−𝑤_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑤_𝑛−1+1−𝛼_𝑛𝛾𝑣_𝑛−𝑣_𝑛−1+1−𝛼_𝑛𝛾×𝑅_𝑛𝑣_𝑛−1−𝑅_𝑛−1𝑣_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑇𝑅_𝑛−1𝑣_𝑛−1≤𝛼_𝑛𝛼𝛾𝑤_𝑛−𝑤_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑤_𝑛−1+1−𝛼_𝑛𝛾𝑤_𝑛−𝑤_𝑛−1+1−𝛼_𝑛𝛾𝜆_𝑛−𝜆_𝑛−1×𝑀𝑤_𝑛−1+1−𝛼_𝑛𝛾𝑅_𝑛𝑣_𝑛−1−𝑅_𝑛−1𝑣_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑇𝑅_𝑛−1𝑣_𝑛−1=1−𝛼_𝑛𝛾−𝛼𝛾𝑤_𝑛−𝑤_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑤_𝑛−1+1−𝛼_𝑛𝛾×𝜆_𝑛−𝜆_𝑛−1𝑀𝑤_𝑛−1+1−𝛼_𝑛𝛾×𝑅_𝑛𝑣_𝑛−1−𝑅_𝑛−1𝑣_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑇𝑅_𝑛−1𝑣_𝑛−1. (17) Based on 𝑅_𝑛𝑣_𝑛−1−𝑅_𝑛−1𝑣_𝑛−1=𝑃_𝑆𝐼−𝛾_𝑛𝐺^∗𝐺𝑣_𝑛−1−𝑃_𝑆𝐼−𝛾_𝑛−1𝐺^∗𝐺𝑣_𝑛−1≤𝐼−𝛾_𝑛𝐺^∗𝐺𝑣_𝑛−1−𝐼−𝛾_𝑛−1𝐺^∗𝐺𝑣_𝑛−1≤𝛾_𝑛−𝛾_𝑛−1‖𝐺‖²𝑣_𝑛−1, (18) we deduce that 𝑤_𝑛+1−𝑤_𝑛≤1−𝛼_𝑛𝛾−𝛼𝛾𝑤_𝑛−𝑤_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑤_𝑛−1+1−𝛼_𝑛𝛾𝜆_𝑛−𝜆_𝑛−1𝑀𝑤_𝑛−1+1−𝛼_𝑛𝛾𝛾_𝑛−𝛾_𝑛−1‖𝐺‖²𝑣_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑇𝑅_𝑛−1𝑣_𝑛−1. (19) By Lemma 3, we have 𝑤_𝑛+1−𝑤_𝑛⟶0. (20) On the other hand, since 𝑢=𝑃_𝑆(𝑢−𝜆_𝑛𝑀𝑢) and 𝑢=𝑅_𝑛𝑢 we have 𝑤_𝑛+1−𝑢²=𝑃_𝑆𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑃_𝑆(𝑢)²≤𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑢²=𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑢²≤𝛼_𝑛𝛾𝑓𝑤_𝑛+−𝑇𝑢𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑢²≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+𝐼−𝛼_𝑛𝛾𝑅_𝑛𝑣_𝑛−𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑅_𝑛𝑣_𝑛−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+𝐼−𝛼_𝑛𝛾𝑣_𝑛−𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+𝐼−𝛼_𝑛𝛾×𝐼−𝜆_𝑛𝑀𝑤_𝑛−𝐼−𝜆_𝑛𝑀𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+𝐼−𝛼_𝑛𝛾×𝑤_𝑛−𝑢²−2𝜆_𝑛𝑤_𝑛−𝑢,𝑀𝑤_𝑛−𝑀𝑢+𝜆²_𝑛𝑀𝑤_𝑛−𝑀𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾×𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+𝐼−𝛼_𝑛𝛾𝑤_𝑛−𝑢²−2𝜆_𝑛𝛽𝑀𝑤_𝑛−𝑀𝑢²+𝜆²_𝑛𝑀𝑤_𝑛−𝑀𝑢+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+𝐼−𝛼_𝑛𝛾×𝑤_𝑛−𝑢²+𝜆_𝑛𝜆_𝑛−2𝛽𝑀𝑤_𝑛−𝑀𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+𝑤_𝑛−𝑢²+𝐼−𝛼_𝑛𝛾𝑏𝑏−2𝛽𝑀𝑤_𝑛−𝑀𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢. (21) Therefore, we have 𝐼−𝛼_𝑛𝛾𝑏2𝛽−𝑏𝑀𝑤_𝑛−𝑀𝑢²≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+𝑤_𝑛−𝑢²−𝑤_𝑛+1−𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢=𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾×𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢+𝑤_𝑛−𝑢−𝑤_𝑛+1×𝑤−𝑢_𝑛−𝑢−𝑤_𝑛+1−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛+𝑤−𝑢_𝑛−𝑤_𝑛+1𝑤_𝑛−𝑢−𝑤_𝑛+1−𝑢. (22) Since 𝛼_𝑛→0, ‖𝑤_𝑛−𝑤_𝑛+1‖→0, we can get that 𝑀𝑤_𝑛−𝑀𝑢⟶0. (23) Furthermore, by the property of the projection 𝑃_𝑆, we have 𝑣_𝑛−𝑢=𝑃_𝑆𝐼−𝜆_𝑛𝑀𝑤_𝑛−𝑃_𝑆𝐼−𝜆_𝑛𝑀𝑢≤𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛−𝑢−𝜆_𝑛𝑀𝑢,𝑣_𝑛=1−𝑢2𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛−𝑢−𝜆_𝑛𝑀𝑢²+𝑣_𝑛−𝑢²−𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛−𝑢−𝜆_𝑛𝑀𝑢−𝑣_𝑛−𝑢²≤12𝑤_𝑛−𝑢²+𝑣_𝑛−𝑢²−𝑤_𝑛−𝑣_𝑛−𝜆_𝑛𝑀𝑤_𝑛−𝑀𝑢²≤𝑤_𝑛−𝑢²−𝑤_𝑛−𝑣_𝑛²+2𝜆_𝑛𝑤_𝑛−𝑣_𝑛,𝑀𝑤_𝑛−𝑀𝑢−𝜆²_𝑛𝑀𝑤_𝑛−𝑀𝑢². (24) Then, we have 𝑤_𝑛+1−𝑢²=𝑃_𝑆𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑃_𝑆(𝑢)²≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+1−𝛼_𝑛𝛾𝑣_𝑛−𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+1−𝛼_𝑛𝛾𝑤_𝑛−𝑢²−1−𝛼_𝑛𝛾𝑤_𝑛−𝑣_𝑛²+21−𝛼_𝑛𝛾×𝜆_𝑛𝑤_𝑛−𝑣_𝑛,𝑀𝑤_𝑛−𝑀𝑢−1−𝛼_𝑛𝛾×𝜆²_𝑛𝑀𝑤_𝑛−𝑀𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾×𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢, (25) which yields that 1−𝛼_𝑛𝛾𝑤_𝑛−𝑣_𝑛²≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑢²+𝑤_𝑛−𝑢+𝑤_𝑛+1−𝑢𝑤_𝑛−𝑤_𝑛+1+21−𝛼_𝑛𝛾𝜆_𝑛𝑤_𝑛−𝑣_𝑛,𝑀𝑤_𝑛−−𝑀𝑢1−𝛼_𝑛𝛾𝜆²_𝑛𝑀𝑤_𝑛−𝑀𝑢²+2𝛼_𝑛𝐼−𝛼_𝑛𝛾𝛾𝑓𝑤_𝑛−𝑇𝑢𝑣_𝑛−𝑢. (26) Since 𝛼_𝑛→0, ‖𝑤_𝑛−𝑤_𝑛+1‖→0, and ‖𝑀𝑤_𝑛−𝑀𝑢‖→0, we can get that 𝑤_𝑛−𝑣_𝑛⟶0. (27) Next, since 𝛼_𝑛→0 we can get that 𝑤_𝑛+1−𝑅_𝑛𝑣_𝑛=𝑃_𝑆𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑃_𝑆𝑅_𝑛𝑣_𝑛≤𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑅_𝑛𝑣_𝑛=𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑅_𝑛𝑣_𝑛⟶0. (28) Hence, 𝑤_𝑛−𝑅_𝑛𝑣_𝑛≤𝑤_𝑛−𝑤_𝑛+1+𝑤_𝑛+1−𝑅_𝑛𝑣_𝑛𝑤⟶0,_𝑛−𝑅_𝑛𝑤_𝑛≤𝑤_𝑛−𝑅_𝑛𝑣_𝑛+𝑅_𝑛𝑣_𝑛−𝑅_𝑛𝑤_𝑛≤𝑤_𝑛−𝑅_𝑛𝑣_𝑛+𝑣_𝑛−𝑤_𝑛𝑣⟶0,_𝑛−𝑅_𝑛𝑣_𝑛≤𝑣_𝑛−𝑤_𝑛+𝑤_𝑛−𝑅_𝑛𝑣_𝑛⟶0. (29)
Let the mapping 𝑅 be defined by 𝑅𝑤=lim_𝑛→∞𝑅_𝑛𝑤,∀𝑤∈𝐻. (30) By Lemma 5 we can get that Fix(𝑅)=∩Fix(𝑅_𝑛) and 𝑅𝑤_𝑛−𝑤_𝑛≤𝑅𝑤_𝑛−𝑅𝑣_𝑛+𝑅𝑣_𝑛−𝑅_𝑛𝑣_𝑛+𝑅_𝑛𝑣_𝑛−𝑤_𝑛≤𝑤_𝑛−𝑣_𝑛+sup𝑅_𝑛𝑧−𝑅𝑧,𝑧∈𝑣_𝑛+𝑅_𝑛𝑣_𝑛−𝑤_𝑛⟶0. (31) Note that it is easy to check that 𝑃_{Γ∩VI(𝑆,𝑀)}(𝛾𝑓+𝐼−𝑇) is a contraction; by Banach contraction mapping principle, 𝑃_{Γ∩VI(𝑆,𝑀)}(𝛾𝑓+𝐼−𝑇) has a unique fixed point 𝑝∈𝐻. By the property of the projection 𝑃_{Γ∩VI(𝑆,𝑀)}, we can get that 𝛾𝑓−𝑇𝑝,𝑤−𝑝≤0,∀𝑤∈Γ∩VI(𝑆,𝑀). (32) Choose a subsequence 𝑣_𝑛𝑘 of 𝑣_𝑛 such that limsup_𝑛→∞𝛾𝑓−𝑇𝑝,𝑅_𝑛𝑣_𝑛−𝑝=lim_𝑛→∞𝛾𝑓−𝑇𝑝,𝑅_𝑛𝑘𝑣_𝑛𝑘−𝑝. (33) Since {𝑣_𝑛} is bounded, there exists a subsequence {𝑣_𝑛𝑘𝑖} of {𝑣_𝑛𝑘} which converges weakly 𝑞. Without loss of generality, we may assume that 𝑣_𝑛𝑘 converges weakly 𝑞. Note that, by ‖𝑣_𝑛−𝑅_𝑛𝑣_𝑛‖→0, we can get 𝑅_𝑛𝑘𝑣_𝑛𝑘 converges weakly 𝑞. Furthermore, ‖𝑤_𝑛−𝑅𝑤_𝑛‖→0,‖𝑣_𝑛−𝑤_𝑛‖→0; by Lemmas 2 and 5, we obtain that 𝑞∈Fix(𝑅)=∩Fix(𝑅_𝑛).
Next we will prove that 𝑞∈VI(𝑆,𝑀).
Choose 𝑣∈𝑆, and let 𝑤_𝑣∈{𝑤∈𝐻,⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝑆}. Since 𝑣_𝑛∈𝑆, we can get ⟨𝑣−𝑣_𝑛,𝑤_𝑣⟩≥0. On the other hand, by the property of the projection 𝑃_𝑆 and 𝑣_𝑛=𝑃_𝑆(𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛), we have 𝑣−𝑣_𝑛,𝑣_𝑛−𝑤_𝑛−𝜆_𝑛𝑀𝑤_𝑛≥0,𝑣−𝑣_𝑛,𝑣_𝑛−𝑤_𝑛𝜆_𝑛+𝑀𝑤_𝑛≥0. (34) It follows that 𝑣−𝑣_𝑛𝑘,𝑀𝑣+𝑤_𝑣≥𝑣−𝑣_𝑛𝑘≥,𝑀𝑣𝑣−𝑣_𝑛𝑘−,𝑀𝑣𝑣−𝑣_𝑛𝑘,𝑣_𝑛𝑘−𝑤_𝑛𝑘𝜆_𝑛𝑘+𝑀𝑤_𝑛𝑘=𝑣−𝑣_𝑛𝑘,𝑀𝑣−𝑀𝑤_𝑛𝑘−𝑣_𝑛𝑘−𝑤_𝑛𝑘𝜆_𝑛𝑘=𝑣−𝑣_𝑛𝑘,𝑀𝑣−𝑀𝑣_𝑛𝑘+𝑣−𝑣_𝑛𝑘,𝑀𝑣_𝑛𝑘−𝑀𝑤_𝑛𝑘−𝑣−𝑣_𝑛𝑘,𝑣_𝑛𝑘−𝑤_𝑛𝑘𝜆_𝑛𝑘≥𝑣−𝑣_𝑛𝑘,𝑀𝑣_𝑛𝑘−𝑀𝑤_𝑛𝑘−𝑣−𝑣_𝑛𝑘,𝑣_𝑛𝑘−𝑤_𝑛𝑘𝜆_𝑛𝑘. (35) Since ‖𝑀𝑣_𝑛−𝑀𝑤_𝑛‖≤‖𝑣_𝑛−𝑤_𝑛‖→0 and 𝑣_𝑛𝑘 converges weakly 𝑞, we can get that ⟨𝑣−𝑞,𝑀𝑣+𝑤_𝑣⟩≥0.
Note that 𝑀 is a 𝛽-inverse strongly monotone mapping, and if 𝑤_𝑣∈{𝑤∈𝐻,⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝑆}, then 𝛼𝑤_𝑣∈{𝑤∈𝐻,⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝑆} for all 𝛼>0. Hence, we can choose 𝑤_𝑣∈{𝑤∈𝐻,⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝑆} such that ⟨𝑣−𝑞,𝑤_𝑣⟩≥⟨𝑣−𝑞,𝑀𝑣−𝑀𝑞⟩. By 𝑣−𝑞,𝑀𝑣+𝑤_𝑣≥0, (36) we can get that 𝑣−𝑞,𝑀𝑣−𝑀𝑞+𝑣−𝑞,𝑀𝑞−𝑤_𝑣≥0. (37) Hence 𝑣−𝑞,𝑀𝑞≥𝑣−𝑞,𝑤_𝑣−𝑣−𝑞,𝑀𝑣−𝑀𝑞≥0. (38) It follows that 𝑞∈VI(𝑆,𝑀). We obtain that 𝑞∈𝑃_{Γ∩VI(𝑆,𝑀)} and ⟨(𝛾𝑓−𝑇)𝑝,𝑞−𝑝⟩≤0, for all 𝑤∈Γ∩VI(𝑆,𝑀).
Finally, we prove that lim_𝑛→∞‖𝑤_𝑛−𝑞‖=0. Since ‖𝑣_𝑛−𝑢‖≤‖𝑤_𝑛−𝑢‖, we have 𝑤_𝑛+1−𝑝²=𝑃_𝑆𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑃_𝑆𝑝²≤𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑝+𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑝²≤𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑝²+𝛼²_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑞²+2𝛼_𝑛𝐼−𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑤_𝑛≤−𝑇𝑞1−𝛼_𝑛𝛾²𝑣_𝑛−𝑝²+𝛼²_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑞²+2𝛼_𝑛𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑤_𝑛−𝑇𝑝−2𝛼²_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑤_𝑛≤−𝑇𝑝1−𝛼_𝑛𝛾²𝑣_𝑛−𝑝²+𝛼²_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑞²+2𝛼_𝑛𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑤_𝑛−𝛾𝑓𝑝+2𝛼_𝑛𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑝−𝑇𝑝−2𝛼²_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑤_𝑛≤−𝑇𝑝1−𝛼_𝑛𝛾²𝑣_𝑛−𝑝²+𝛼²_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑞²+2𝛼_𝑛𝑅_𝑛𝑣_𝑛−𝑝𝛾𝑓𝑤_𝑛−𝛾𝑓𝑝+2𝛼_𝑛𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑝−𝑇𝑝−2𝛼²_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑤_𝑛≤−𝑇𝑝1−𝛼_𝑛𝛾²𝑣_𝑛−𝑝²+𝛼²_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑞²+2𝛾𝛼𝛼_𝑛𝑣_𝑛−𝑝𝑤_𝑛−𝑝+2𝛼_𝑛𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑝−𝑇𝑝−2𝛼²_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑤_𝑛≤−𝑇𝑝1−𝛼_𝑛𝛾²𝑣_𝑛−𝑝²+𝛼²_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑞²+2𝛾𝛼𝛼_𝑛𝑤_𝑛−𝑝²+2𝛼_𝑛𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑝−𝑇𝑝−2𝛼²_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑤_𝑛≤−𝑇𝑝1−𝛼_𝑛𝛾²+2𝛾𝛼𝛼_𝑛𝑤_𝑛−𝑝²+𝛼_𝑛2𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑝−𝑇𝑝+𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑞²+2𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛−𝑝𝛾𝑓𝑤_𝑛=−𝑇𝑝1−2𝛾−𝛾𝛼𝛼_𝑛𝑤_𝑛−𝑝²+𝛼_𝑛2𝑅_𝑛𝑣_𝑛−𝑝,𝛾𝑓𝑝−𝑇𝑝+𝛼_𝑛𝛾𝑓𝑤_𝑛−𝑇𝑞²+2𝛼_𝑛𝑇𝑅_𝑛𝑣_𝑛×𝑤−𝑝𝛾𝑓_𝑛−𝑇𝑝+𝛼_𝑛𝛾²𝑤_𝑛−𝑝². (39) Since {𝑤_𝑛}, {𝑓(𝑤_𝑛)}, {𝑅_𝑛𝑣_𝑛}, and limsup_𝑛→∞⟨(𝛾𝑓−𝑇)𝑝,𝑅_𝑛𝑣_𝑛−𝑝⟩≤0, using Lemma 3, we can conclude that lim_𝑛→∞𝑤_𝑛−𝑞=0, (40) which completes the proof.

By Lemma 1, we know that 𝐺^∗𝐺 is 1/‖𝐺‖²-inverse strongly monotone. Hence, letting 𝑀=𝐺^∗𝐺, we can get the following Corollary.

Corollary 7. Let Γ denote the solutions set of the SEP, 𝑆=𝐶×𝑄, 𝑓 a contraction on 𝐻=𝐻₁×𝐻₂ with coefficient 0<𝛼<1, a strongly positive linear bounded operator 𝑇 with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼, Γ≠∅, and {𝑤_𝑛}={(𝑥_𝑛,𝑦_𝑛)}⊆𝐻 is the sequence generated by the following algorithm: 𝑤₀∈𝐻, 𝑤_𝑛+1=𝑃_𝑆𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑃_𝑆×𝐼−𝛾_𝑛𝐺^∗𝐺𝑃_𝑆𝑤_𝑛−𝜆_𝑛𝐺^∗𝐺𝑤_𝑛 (41) for all 𝑛=0,1,2,…, where 𝛼_𝑛∈(0,1), 𝜆_𝑛,𝛾_𝑛∈(0,2/‖𝐺‖²). If 𝛼_𝑛, 𝜆_𝑛, and 𝛾_𝑛 are chosen so that 𝜆_𝑛∈[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<2/‖𝐺‖² and(i)lim_𝑛𝛼_𝑛=0,(ii)∑^∞_𝑛=0𝛼_𝑛=∞,(iii)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞,(v)∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞,(iv)∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞,then the sequence {𝑤_𝑛} converges strongly to 𝑤 which solves the SEP and the following variational inequality: ⟨(𝑇−𝜆𝑓)𝑤,𝑤−𝑤⟩≥0 for all 𝑤∈𝑆.

Furthermore, if 𝑀=𝐺^∗𝐺 and 𝛾_𝑛=0, changed slightly for Theorem 6, we get the following result.

Corollary 8. Let Γ denote the solutions set of the SEP, 𝑆=𝐶×𝑄, 𝑓 a contraction on 𝐻=𝐻₁×𝐻₂ with coefficient 0<𝛼<1, a strongly positive linear bounded operator 𝑇 with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼, Γ≠∅, and {𝑤_𝑛}={(𝑥_𝑛,𝑦_𝑛)}⊆𝐻 is the sequence generated by the following algorithm: 𝑤₀∈𝐻, 𝑤_𝑛+1=𝛼_𝑛𝛾𝑓𝑤_𝑛+𝐼−𝛼_𝑛𝑇𝑃_𝑆𝑤_𝑛−𝜆_𝑛𝐺^∗𝐺𝑤_𝑛 (42) for all 𝑛=0,1,2,…, where 𝛼_𝑛∈(0,1), 𝜆_𝑛∈(0,2/‖𝐺‖²). If 𝛼_𝑛, 𝜆_𝑛 are chosen so that 𝜆_𝑛∈[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<2/‖𝐺‖² and(i)lim_𝑛𝛼_𝑛=0,(ii)∑^∞_𝑛=0𝛼_𝑛=∞,(iii)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞,(v)∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞,then the sequence {𝑤_𝑛} converges strongly to 𝑤 which solves the SEP and the following variational inequality: ⟨(𝑇−𝜆𝑓)𝑤,𝑤−𝑤⟩≥0 for all 𝑤∈𝑆.

Acknowledgments

This research was supported by NSFC Grants nos. 11071279, 11226125, and 11301379.