Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points of Nonexpansive Semigroups

Abstract

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point 𝑥^∗ such that 𝑥^∗∈Ω,⟨(𝐴−𝛾𝑓)𝑥^∗−(𝐼−𝐵)𝑆𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈Ω, where Ω is the set of the solutions of the following variational inequality: 𝑥^∗∈ϝ,⟨(𝐴−𝑆)𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈ϝ, where 𝐴,𝐵 are two strongly positive bounded linear operators, 𝑓 is a 𝜌-contraction, 𝑆 is a nonexpansive mapping, and ϝ is the fixed points set of a nonexpansive semigroup {𝑇(𝑠)}_𝑠≥0. We present a double-net convergence hierarchical to some elements in ϝ which solves the above hierarchical constrained variational inequalities.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻. Recall that a self-mapping 𝑓 of 𝐶 is said to be contractive if there exists a constant 𝜌∈[0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜌‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1)

We denote by Fix(𝑇) the set of fixed points of 𝑇; that is, Fix(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. A bounded linear operator 𝐵 is called strongly positive on 𝐻 if there exists a constant 𝛾^∼>0 such that ⟨𝐵𝑥,𝑥⟩≥𝛾^∼‖𝑥‖²,∀𝑥∈𝐻. (2)

It is well known that the variational inequality for an operator, 𝜑∶𝐻→𝐻, over a nonempty, closed, and convex set, 𝐶⊂𝐻, is to find a point 𝑥^∗∈𝐶 with the property 𝜑𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶. (3)

The set of the solutions of the variational inequality (3) is denoted by VI(𝐶,𝜑). If the mapping 𝜑 is a monotone operator, then we say that VI(3) is monotone. It is well known that if 𝜑 is Lipschitzian and strongly monotone, then for small enough 𝛿>0, the mapping 𝑃_𝐶(𝐼−𝛿𝜑) is a contraction on 𝐶 and so the sequence {𝑥_𝑛} of Picard iterates, given by 𝑥_𝑛=𝑃_𝐶(𝐼−𝛿𝜑)𝑥_𝑛−1(𝑛≥1), converges strongly to the unique solution of the VI(3). This sort of VI(3) where 𝜑 is strongly monotone and Lipschitzian is originated from Yamada [1]. However, if 𝜑 is only monotone (not strongly monotone), then their iterative methods do not apply to VI.

Many practical problems such as signal processing and network resource allocation are formulated as the variational inequality over the set of the solutions of some nonlinear mappings (e.g., the fixed point set of nonexpansive mappings), and algorithms to solve these problems have been proposed. Iterative algorithms have been presented for the convex optimization problem with a fixed point constraint along with proof that these algorithms strongly converge to the unique solution of problems with a strongly monotone operator. The strong monotonicity condition guarantees the uniqueness of the solution. For some related works on the variational inequalities, please see [2–23] and the references therein. Particulary, the variational inequality problems over the fixed points of nonexpansive mappings have been considered. The reader can consult [16, 24]. On the other hand, we note that in the literature, nonlinear ergodic theorems for nonexpansive semigroups have been considered by many authors; see, for example, [25–32]. In this paper, we will consider a general variational inequality problem with the variational inequality constraint is the fixed points of nonexpansive semigroups.

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point 𝑥^∗ such that 𝑥^∗∈Ω,𝐴−𝛾𝑓𝑥^∗−(𝐼−𝐵)𝑆𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ω, (4) where Ω is the set of the solutions of the following variational inequality: 𝑥^∗∈ϝ,(𝐴−𝑆)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈ϝ, (5) where 𝐴 and 𝐵 are two strongly positive bounded linear operators, 𝑓 is a 𝜌-contraction, 𝑆 is a nonexpansive mapping, and ϝ is the fixed points set of a nonexpansive semigroup {𝑇(𝑠)}_𝑠≥0. We present a double-net convergence hierarchical to some elements in ϝ which solves the above hierarchical constrained variational inequalities.

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. The metric (or the nearest point) projection from 𝐻 onto 𝐶 is the mapping 𝑃_𝐶∶𝐻→𝐶 which assigns to each point 𝑥∈𝐶 the unique point 𝑃_𝐶𝑥∈𝐶 satisfying the property 𝑥−𝑃_𝐶𝑥=inf_𝑦∈𝐶𝑥−𝑦=∶𝑑(𝑥,𝐶). (6)

It is well known that 𝑃_𝐶 is a nonexpansive mapping and satisfies 𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦≥𝑃_𝐶𝑥−𝑃_𝐶𝑦²,∀𝑥,𝑦∈𝐻. (7)

Moreover, 𝑃_𝐶 is characterized by the following property: 𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥≤0,∀𝑥∈𝐻,𝑦∈𝐶. (8)

Recall that a family {𝑇(𝑠)}_𝑠≥0 of mappings of 𝐻 into itself is called a nonexpansive semigroup if it satisfies the following conditions:(S1)𝑇(0)𝑥=𝑥 for all 𝑥∈𝐻;(S2)𝑇(𝑠+𝑡)=𝑇(𝑠)𝑇(𝑡) for all 𝑠,𝑡≥0;(S3)‖𝑇(𝑠)𝑥−𝑇(𝑠)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻 and 𝑠≥0;(S4) for all 𝑥∈𝐻, 𝑠→𝑇(𝑠)𝑥 is continuous.

We denote by Fix(𝑇(𝑠)) the set of fixed points of 𝑇(𝑠) and by ϝ the set of all common fixed points of {𝑇(𝑠)}_𝑠≥0; that is, ϝ=⋂_𝑠≥0Fix(𝑇(𝑠)). It is known that ϝ is closed and convex.

We need the following lemmas for proving our main results.

Lemma 1 (see [33]). Let 𝐶 be a nonempty bounded closed convex subset of a Hilbert space 𝐻 and let {𝑇(𝑠)}_𝑠≥0 be a nonexpansive semigroup on 𝐶. Then, for every ℎ≥0, lim_𝑡→∞sup_𝑥∈𝐶1𝑡^𝑡₀𝑇(𝑠)(ℎ)1𝑥𝑑𝑠−𝑇𝑡^𝑡₀𝑇(𝑠)𝑥𝑑𝑠=0. (9)

Lemma 2 (see [34]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻 and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping. Then, the mapping 𝐼−𝑆 is demiclosed. That is, if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛→𝑥^∗ weakly and (𝐼−𝑆)𝑥_𝑛→𝑦 strongly, then (𝐼−𝑆)𝑥^∗=𝑦.

Lemma 3 (see [35]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Assume that a mapping 𝐹∶𝐶→𝐻 is monotone and weakly continuous along segments (i.e., 𝐹(𝑥+𝑡𝑦)→𝐹(𝑥) weakly, as 𝑡→0, whenever 𝑥+𝑡𝑦∈𝐶 for 𝑥,𝑦∈𝐶). Then the variational inequality 𝑥^∗∈𝐶,𝐹𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (10) is equivalent to the dual variational inequality 𝑥^∗∈𝐶,𝐹𝑥,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (11)

3. Main Results

Now we consider the following hierarchical variational inequality with the variational inequality constraint over the fixed points set of nonexpansive semigroups {𝑇(𝑠)}_𝑠≥0.

Problem 1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝐻 be a 𝜌-contraction with coefficient 𝜌∈[0,1) and let𝑆∶𝐶→𝐶 be a nonexpansive mapping. Let {𝑇(𝑠)}_𝑠≥0 be a nonexpansive semigroup on 𝐶 and let 𝐴,𝐵∶𝐻→𝐻 be two strongly positive bounded linear operators with coefficients 𝜆^∼(1≤𝜆^∼<2) and 𝛾^∼(0<𝛾^∼<1), respectively. Let 𝛾 be a constant satisfying 0<𝛾𝜌<𝛾^∼.
Now, our objective is to find 𝑥^∗ such that 𝑥^∗∈Ω,𝐴−𝛾𝑓𝑥^∗−(𝐼−𝐵)𝑆𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ω, (12) where Ω∶=VI(ϝ,𝐴−𝑆) is the set of the solutions of the following variational inequality: 𝑥^∗∈ϝ,⟨(𝐴−𝑆)𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈ϝ. (13)

We observe that (𝐴−𝛾𝑓)−(𝐼−𝐵)𝑆 is strongly monotone and Lipschitz continuous. In fact, we have ()()=𝑓(𝑥)𝑦−(≥𝐴−𝛾𝑓𝑥−𝐼−𝐵𝑆𝑥−𝐴−𝛾𝑓𝑦−𝐼−𝐵𝑆𝑦,𝑥−𝑦𝐴𝑥−𝐴𝑦,𝑥−𝑦−𝛾−𝑓,𝑥−𝑦𝐼−𝐵)𝑆𝑥−𝑆𝑦,𝑥−𝑦𝜆^∼𝑥−𝑦²−𝛾𝜌𝑥−𝑦²−1−𝛾^∼𝑥−𝑦²=𝜆^∼−1+𝛾^∼−𝛾𝜌𝑥−𝑦²,()()≤𝑓(𝑥)𝑦+‖≤‖𝐴+𝐴−𝛾𝑓𝑥−𝐼−𝐵𝑆𝑥−𝐴−𝛾𝑓𝑦−𝐼−𝐵𝑆𝑦𝐴𝑥−𝐴𝑦+𝛾−𝑓𝐼−𝐵‖𝑆𝑥−𝑆𝑦‖𝑥−𝑦+𝛾𝜌𝑥−𝑦1−𝛾^∼=‖𝐴‖𝑥−𝑦1++𝛾𝜌−𝛾^∼𝑥−𝑦. (14)

Hence, the existence and the uniqueness of the solution to Problem 1 are guaranteed.

In order to solve the above hierarchical constrainted variational inequality, we present the following double net.

Algorithm 4. Set 𝜅=1/(𝜆^∼+𝛾^∼−𝛾𝜌). Then, 0<𝜅≤1. For each (𝑠,𝑡)∈(0,𝜅)×(0,1), we define a double net {𝑥_𝑠,𝑡} implicitly by 𝑥_𝑠,𝑡=𝑃_𝐶𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡+(𝐼−𝑡𝐵)𝑆𝑥_𝑠,𝑡+()1𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈. (15)

Note that this implicit manner algorithm is well defined. In fact, we define the mapping 𝑥↦𝑊_𝑠,𝑡(𝑥)∶=𝑃_𝐶𝑠(𝑥)+()+()1𝑡𝛾𝑓𝐼−𝑡𝐵𝑆𝑥𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈),()∈()×().𝑥𝑑𝜈𝑠,𝑡0,𝜅0,1 (16)

Note that this self-mapping is a contraction. As a matter of fact, we have 𝑊_𝑠,𝑡(𝑥)−𝑊_𝑠,𝑡𝑦=𝑃_𝐶𝑠(𝑥)+()+()1𝑡𝛾𝑓𝐼−𝑡𝐵𝑆𝑥𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥𝑑𝜈−𝑃_𝐶𝑠𝑦+()+()1𝑡𝛾𝑓𝐼−𝑡𝐵𝑆𝑦𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑓(𝑥)𝑦‖+‖1𝑦𝑑𝜈≤𝑠𝑡𝛾−𝑓+𝑠𝐼−𝑡𝐵‖𝑆𝑥−𝑆𝑦𝐼−𝑠𝐴‖𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)−1𝑥𝑑𝜈𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)≤𝑦𝑑𝜈1−𝜆^∼−1𝑠−𝛾^∼−𝛾𝜌𝑠𝑡𝑥−𝑦. (17) Since (𝑠,𝑡)∈(0,𝜅)×(0,1), 0<1−(𝜆^∼−1)𝑠−(𝛾^∼−𝛾𝜌)𝑠𝑡<1. Hence, 𝑊_𝑠,𝑡 is a contraction. Therefore, by Banach’s Contraction Principle, 𝑊_𝑠,𝑡 has a unique fixed point which is denoted by 𝑥_𝑠,𝑡∈𝐶.

Next we show the behavior of the net {𝑥_𝑠,𝑡} as 𝑠→0 and 𝑡→0 successively.

Theorem 5. Assume that 𝑉𝐼(ϝ,𝐴−𝑆)≠∅. Then, for each fixed 𝑡∈(0,1), the net {𝑥_𝑠,𝑡} defined by (15) converges in norm, as 𝑠→0+, to a solution 𝑥_𝑡∈ϝ. Moreover, as 𝑡→0+, the net {𝑥_𝑡} converges in norm to the unique solution 𝑥^∗ of Problem 1.

Proof. We first show that the sequence {𝑥_𝑠,𝑡} is bounded. Take 𝑦^∗∈ϝ. From (15), we have 𝑥_𝑠,𝑡−𝑦^∗=𝑃_𝑐𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡+(𝐼−𝑡𝐵)𝑆𝑥_𝑠,𝑡+()1𝐼−𝑠A𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥𝑑𝜈−𝑦^∗≤𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡+(𝐼−𝑡𝐵)𝑆𝑥_𝑠,𝑡+()1𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥𝑑𝜈−𝑦^∗=𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡−𝑓𝑦^∗+𝑠(𝐼−𝑡𝐵)𝑆𝑥_𝑠,𝑡−𝑆𝑦^∗+𝑠𝑡𝛾𝑓𝑦^∗+𝑠(𝐼−𝑡𝐵)𝑆𝑦^∗−𝑠𝐴𝑦^∗+()1𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥𝑑𝜈−𝑦^∗≤𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡−𝑓𝑦^∗+𝑠‖𝐼−𝑡𝐵‖𝑆𝑥_𝑠,𝑡−𝑆𝑦^∗+‖‖×1𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥𝑑𝜈−𝑦^∗+𝑠𝑡𝛾𝑓𝑦^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗≤𝑠𝑡𝛾𝜌𝑥_𝑠,𝑡−𝑦^∗+𝑠1−𝑡𝛾^∼𝑥_𝑠,𝑡−𝑦^∗+1−𝑠𝜆^∼𝑥_𝑠,𝑡−𝑦^∗+𝑠𝑡𝛾𝑓𝑦^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗=1−𝜆^∼−1𝑠−𝛾^∼−𝛾𝜌𝑠𝑡𝑥_𝑠,𝑡−𝑦^∗+𝑠𝑡𝛾𝑓𝑦^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗. (18)
Hence 𝑥_𝑠,𝑡−𝑦^∗≤_∼1𝛾−𝛾𝜌𝑡+^∼×𝑦𝜆−1𝑡𝛾𝑓^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗. (19) It follows that for each fixed 𝑡∈(0,1), {𝑥_𝑠,𝑡} is bounded.
Next, we show that lim_𝑠→0‖𝑇(𝜏)𝑥_𝑠,𝑡−𝑥_𝑠,𝑡‖=0 for all 0≤𝜏<∞ and consequently, as 𝑠→0+, the entire net {𝑥_𝑠,𝑡} converges in norm to 𝑥_𝑡∈ϝ.
For each fixed 𝑡∈(0,1), we set 𝑅_𝑡∶=(1/((𝛾^∼−𝛾𝜌)𝑡+𝜆^∼−1))‖𝑡𝛾𝑓(𝑦^∗)+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗‖. It is clear that for each fixed 𝑡∈(0,1), {𝑥_𝑠,𝑡}⊂𝐵(𝑦^∗,𝑅_𝑡). Notice that 1𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈−𝑦^∗≤𝑥_𝑠,𝑡−𝑦^∗≤𝑅_𝑡. (20)
Moreover, we observe that if 𝑥∈𝐵(𝑦^∗,𝑅_𝑡), then 𝑇(𝑠)𝑥−𝑦^∗≤𝑇(𝑠)𝑥−𝑇(𝑠)𝑦^∗≤𝑥−𝑦^∗≤𝑅_𝑡, (21)
that is, 𝐵(𝑦^∗,𝑅_𝑡) is 𝑇(𝑠)-invariant for all 𝑠.
From (15), we deduce 𝑇(𝜏)𝑥_𝑠,𝑡−𝑥_𝑠,𝑡≤𝑇(𝜏)𝑥_𝑠,𝑡(𝜏)1−𝑇𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡+𝑇(𝜏)1𝑑𝜈𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡1𝑑𝜈−𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡+1𝑑𝜈𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈−𝑥_𝑠,𝑡≤𝑇(𝜏)1𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡1𝑑𝜈−𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑥𝑑𝜈+2_𝑠,𝑡−1𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑥𝑑𝜈≤2𝑠𝑡𝛾𝑓_𝑠,𝑡+(𝐼−𝑡𝐵)𝑆𝑥_𝑠,𝑡−𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡+𝑇(𝜏)1𝑑𝜈𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡1𝑑𝜈−𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈. (22)
Since {𝑥_𝑠,𝑡} is bounded, {𝑓(𝑥_𝑠,𝑡)} and {𝑆𝑥_𝑠,𝑡} are also bounded. Then, from Lemma 1, we deduce for all 0≤𝜏<∞ and fixed 𝑡∈(0,1) that lim_𝑠→0𝑇(𝜏)𝑥_𝑠,𝑡−𝑥_𝑠,𝑡=0. (23)
Set 𝑦_𝑠,𝑡=𝑠(𝑡𝛾𝑓(𝑥_𝑠,𝑡)+(𝐼−𝑡𝐵)𝑆𝑥_𝑠,𝑡)+(𝐼−𝑠𝐴)(1/𝜆_𝑠)∫^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈 for all (𝑠,𝑡)∈(0,𝜅)×(0,1). We then have 𝑥_𝑠,𝑡=𝑃_𝐶𝑦_𝑠,𝑡, and for any 𝑦^∗∈ϝ, 𝑥_𝑠,𝑡−𝑦^∗=𝑥_𝑠,𝑡−𝑦_𝑠,𝑡+𝑦_𝑠,𝑡−𝑦^∗=𝑥_𝑠,𝑡−𝑦_𝑠,𝑡+𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡+(𝐼−𝑡𝐵)𝑆𝑥_𝑠,𝑡+()1𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈−𝑦^∗=𝑥_𝑠,𝑡−𝑦_𝑠,𝑡+𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡−𝑓𝑦^∗+𝑠(𝐼−𝑡𝐵)𝑆𝑥_𝑠,𝑡−𝑆𝑦^∗+𝑠𝑡𝛾𝑓𝑦^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗+()1𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈−𝑦^∗. (24)
Notice that 𝑥_𝑠,𝑡−𝑦_𝑠,𝑡,𝑥_𝑠,𝑡−𝑦^∗≤0. (25)
Thus, we have 𝑥_s,𝑡−𝑦^∗2=𝑥_𝑠,𝑡−𝑦_𝑠,𝑡,𝑥_𝑠,𝑡−𝑦^∗+𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡−𝑓𝑦^∗,𝑥_𝑠,𝑡−𝑦^∗+𝑠(𝐼−𝑡𝐵)𝑆𝑥_𝑠,𝑡−𝑆𝑦^∗,𝑥_𝑠,𝑡−𝑦^∗+()×1𝐼−𝑠𝐴𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈−𝑦^∗,𝑥_𝑠,𝑡−𝑦^∗+𝑠𝑡𝛾𝑓𝑦^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑠,𝑡−𝑦^∗≤𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡−𝑓𝑦^∗𝑥_𝑠,𝑡−𝑦^∗+𝑠‖𝐼−𝑡𝐵‖𝑆𝑥_𝑠,𝑡−𝑆𝑦^∗𝑥_𝑠,𝑡−𝑦^∗+𝑠𝑡𝛾𝑓𝑦^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑠,𝑡−𝑦^∗+‖1𝐼−𝑠𝐴‖𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈−𝑦^∗𝑥_𝑠,𝑡−𝑦^∗≤𝑠𝑡𝛾𝜌𝑥_𝑠,𝑡−𝑦^∗2+𝑠1−𝑡𝛾^∼𝑥_𝑠,𝑡−𝑦^∗2+𝑠𝑡𝛾𝑓𝑦^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑠,𝑡−𝑦^∗+1−𝑠𝜆^∼𝑥_𝑠,𝑡−𝑦^∗2=1−𝜆^∼−1𝑠−𝛾^∼−𝛾𝜌𝑠𝑡𝑥_𝑠,𝑡−𝑦^∗2+𝑠𝑡𝛾𝑓𝑦^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑠,𝑡−𝑦^∗. (26)
So 𝑥_𝑠,𝑡−𝑦^∗2≤_∼1𝛾−𝛾𝜌𝑡+^∼×𝑦𝜆−1𝑡𝛾𝑓^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑠,𝑡−𝑦^∗,𝑦^∗∈ϝ. (27)
Assume {𝑠_𝑛}⊂(0,1) such that 𝑠_𝑛→0 as 𝑛→∞. By (27), we obtain immediately that 𝑥_{𝑠𝑛,𝑡}−𝑦^∗2≤_∼1𝛾−𝛾𝜌𝑡+^∼×𝑦𝜆−1𝑡𝛾𝑓^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_{𝑠𝑛,𝑡}−𝑦^∗,𝑦^∗∈ϝ. (28)
Since {𝑥_{𝑠𝑛,𝑡}} is bounded, there exists a subsequence {𝑠_𝑛𝑖} of {𝑠_𝑛} such that {𝑥_{𝑠𝑛𝑖,𝑡}} converges weakly to a point 𝑥_𝑡. From (23) and Lemma 2, we get 𝑥_𝑡∈ϝ. We can substitute 𝑥_𝑡 for 𝑦^∗ in (28) to get 𝑥_{𝑠𝑛,𝑡}−𝑥_𝑡²≤_∼1𝛾−𝛾𝜌𝑡+^∼×𝑥𝜆−1𝑡𝛾𝑓_𝑡+(𝐼−𝑡𝐵)𝑆𝑥_𝑡−𝐴𝑥_𝑡,𝑥_{𝑠𝑛,𝑡}−𝑥_𝑡. (29)
The weak convergence of {𝑥_{𝑠𝑛,𝑡}} to 𝑥_𝑡 actually implies that 𝑥_{𝑠𝑛,𝑡}→𝑥_𝑡 strongly. This has proved the relative norm-compactness of the net {𝑥_𝑠,𝑡} as 𝑠→0+ for each fixed 𝑡∈(0,1).
In (28), we take the limit as 𝑛→∞ to get 𝑥_𝑡−𝑦^∗2≤_∼1𝛾−𝛾𝜌𝑡+^∼×𝑦𝜆−1𝑡𝛾𝑓^∗+(𝐼−𝑡𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑡−𝑦^∗,∀𝑦^∗∈ϝ. (30)
In particular, 𝑥_𝑡 solves the following variational inequality: 𝑥_𝑡∈ϝ,𝐴𝑦^∗−𝑡𝛾𝑓𝑦^∗−(𝐼−𝑡𝐵)𝑆𝑦^∗,𝑦^∗−𝑥_𝑡≥0,∀𝑦^∗∈ϝ. (31)
Note that the mapping 𝐴−𝑡𝛾𝑓−(𝐼−𝑡𝐵)𝑆 is monotone for all 𝑡∈(0,1), since (𝑥)−()−𝑦−()=𝑓(𝑥)𝑦−(≥𝐴𝑥−𝑡𝛾𝑓𝐼−𝑡𝐵𝑆𝑥𝐴𝑦−𝑡𝛾𝑓𝐼−𝑡𝐵𝑆𝑦,𝑥−𝑦𝐴𝑥−𝐴𝑦,𝑥−𝑦−𝑡𝛾−𝑓,𝑥−𝑦𝐼−𝑡𝐵)𝑆𝑥−𝑆𝑦,𝑥−𝑦𝜆^∼𝑥−𝑦²−𝑡𝛾𝜌𝑥−𝑦²−1−𝑡𝛾^∼𝑥−𝑦²=𝜆^∼−1+𝛾^∼−𝛾𝜌𝑡𝑥−𝑦²≥0. (32)
By Lemma 3, (31) is equivalent to its dual VI: 𝑥_𝑡∈ϝ,𝐴𝑥_𝑡−𝑡𝛾𝑓𝑥_𝑡−(𝐼−𝑡𝐵)𝑆𝑥_𝑡,𝑦^∗−𝑥_𝑡≥0,∀𝑦^∗∈ϝ. (33)
Next we show that as 𝑠→0+, the entire net {𝑥_𝑠,𝑡} converges in norm to 𝑥_𝑡∈ϝ. We assume 𝑥_{𝑠′𝑛,𝑡}→𝑥^′_𝑡, where 𝑠^′_𝑛→0. Similarly, by the above proof, we deduce 𝑥^′_𝑡∈ϝ which solves the following variational inequality: 𝑥^′_𝑡∈ϝ,𝐴𝑥^′_𝑡−𝑡𝛾𝑓𝑥^′_𝑡−(𝐼−𝑡𝐵)𝑆𝑥^′_𝑡,𝑦^∗−𝑥^′_𝑡≥0,∀𝑦^∗∈ϝ. (34)
In (33), we take 𝑦^∗=𝑥^′_𝑡 to get 𝐴𝑥_𝑡−𝑡𝛾𝑓𝑥_𝑡−(𝐼−𝑡𝐵)𝑆𝑥_𝑡,𝑥^′_𝑡−𝑥_𝑡≥0. (35)
In (34), we take 𝑦^∗=𝑥_𝑡 to get 𝐴𝑥^′_𝑡−𝑡𝛾𝑓𝑥^′_𝑡−(𝐼−𝑡𝐵)𝑆𝑥^′_𝑡,𝑥_𝑡−𝑥^′_𝑡≥0. (36)
Adding up (35) and (36) yields 𝐴𝑥_𝑡−𝐴𝑥^′_𝑡,𝑥_𝑡−𝑥^′_𝑡−𝑡𝛾𝑓𝑥_𝑡−𝑓𝑥^′_𝑡,𝑥_𝑡−𝑥^′_𝑡−(𝐼−𝑡𝐵)𝑆𝑥_𝑡−𝑆𝑥^′_𝑡,𝑥_𝑡−𝑥^′_𝑡≤0. (37)
At the same time we note that 𝐴𝑥_𝑡−𝐴𝑥^′_𝑡,𝑥_𝑡−𝑥^′_𝑡−𝑡𝛾𝑓𝑥_𝑡−𝑓𝑥^′_𝑡,𝑥_𝑡−𝑥^′_𝑡−(𝐼−𝑡𝐵)𝑆𝑥_𝑡−𝑆𝑥^′_𝑡,𝑥_𝑡−𝑥^′_𝑡≥𝜆^∼−1+𝛾^∼−𝛾𝜌𝑡𝑥_𝑡−𝑥^′_𝑡²≥0. (38)
Therefore, by (37) and (38), we deduce 𝑥^′_𝑡=𝑥_𝑡. (39)Hence the entire net {𝑥_𝑠,𝑡} converges in norm to 𝑥_𝑡∈ϝ as 𝑠→0+.
As 𝑡→0+, the net {𝑥_𝑡} converges to the unique solution 𝑥^∗ of Problem 1.
In (33), we take any 𝑦^∗∈Ω to deduce 𝐴𝑥_𝑡−𝑡𝛾𝑓𝑥_𝑡−(𝐼−𝑡𝐵)𝑆𝑥_𝑡,𝑦^∗−𝑥_𝑡≥0. (40)
By virtue of the monotonicity of 𝐴−𝑆 and the fact that 𝑦^∗∈Ω, we have 𝐴𝑥_𝑡−𝑆𝑥_𝑡,𝑦^∗−𝑥_𝑡≤𝐴𝑦^∗−𝑆𝑦^∗,𝑦^∗−𝑥_𝑡≤0. (41)
We can rewrite (40) as 𝑡𝐴𝑥_𝑡−𝛾𝑓𝑥_𝑡−(𝐼−𝐵)𝑆𝑥_𝑡+(1−𝑡)𝐴𝑥_𝑡−𝑆𝑥_𝑡,𝑦^∗−𝑥_𝑡≥0. (42)
It follows from (41) and (42) that ⟨𝐴𝑥_𝑡−𝛾𝑓𝑥_𝑡−(𝐼−𝐵)𝑆𝑥_𝑡,𝑦^∗−𝑥_𝑡⟩≥0,∀𝑦^∗∈Ω. (43)
Hence 𝑥_𝑡−𝑦^∗2≤𝑥_𝑡−𝑦^∗,𝑥_𝑡−𝑦^∗+𝛾𝑓𝑥_𝑡+(𝐼−𝐵)𝑆𝑥_𝑡−𝐴𝑥_𝑡,𝑥_𝑡−𝑦^∗=𝛾𝑓𝑥_𝑡−𝑓𝑦^∗,𝑥_𝑡−𝑦^∗+(𝐼−𝐵)𝑆𝑥_𝑡−𝑆𝑦^∗,𝑥_𝑡−𝑦^∗+𝐴𝑦^∗−𝐴𝑥_𝑡,𝑥_𝑡−𝑦^∗+𝛾𝑓𝑦^∗+(𝐼−𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑡−𝑦^∗≤𝛾𝜌𝑥_𝑡−𝑦^∗2+1−𝛾^∼𝑥_𝑡−𝑦^∗2−𝜆^∼𝑥_𝑡−𝑦^∗2+𝛾𝑓𝑦^∗+(𝐼−𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑡−𝑦^∗=1−𝜆^∼+𝛾^∼−𝛾𝜌𝑥_𝑡−𝑦^∗2+𝛾𝑓𝑦^∗+(𝐼−𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑡−𝑦^∗. (44)
Therefore, 𝑥_𝑡−𝑦^∗2≤_∼1𝜆+^∼×𝑦𝛾−𝛾𝜌𝛾𝑓^∗+(𝐼−𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑥_𝑡−𝑦^∗,𝑦^∗∈Ω. (45)
In particular, 𝑥_𝑡−𝑦^∗≤_∼1𝜆+^∼×𝑦𝛾−𝛾𝜌𝛾𝑓^∗+(𝐼−𝐵)𝑆𝑦^∗−𝐴𝑦^∗,∀𝑡∈(0,1), (46)
which implies that {𝑥_𝑡} is bounded.
We next prove that 𝜔_𝑤(𝑥_𝑡)⊂Ω; namely, if (𝑡_𝑛) is a null sequence in (0,1) such that 𝑥_𝑡𝑛→𝑥^′ weakly as 𝑛→∞, then 𝑥^′∈Ω. To see this, we use (33) to get (𝐴−𝑆)𝑥_𝑡,𝑦^∗−𝑥_𝑡≥𝑡𝑥1−𝑡𝛾𝑓_𝑡+(𝐼−𝐵)𝑆𝑥_𝑡−𝐴𝑥_𝑡,𝑦^∗−𝑥_𝑡,𝑦^∗∈ϝ. (47)
However, since 𝐴−𝑆 is monotone, (𝐴−𝑆)𝑦^∗,𝑦^∗−𝑥_𝑡≥(𝐴−𝑆)𝑥_𝑡,𝑦^∗−𝑥_𝑡. (48)
Combining the last two relations yields (𝐴−𝑆)𝑦^∗,𝑦^∗−𝑥_𝑡≥𝑡𝑥1−𝑡𝛾𝑓_𝑡+(𝐼−𝐵)𝑆𝑥_𝑡−𝐴𝑥_𝑡,𝑦^∗−𝑥_𝑡,𝑦^∗∈ϝ. (49)
Letting 𝑡=𝑡_𝑛→0+ as 𝑛→∞ in (49), we get (𝐴−𝑆)𝑦^∗,𝑦^∗−𝑥^′≥0,𝑦^∗∈ϝ. (50)
The equivalent dual VI of (50) is (𝐴−𝑆)𝑥^′,𝑦^∗−𝑥^′≥0,𝑦^∗∈ϝ. (51)
Namely, 𝑥^′ is a solution of VI(13); hence 𝑥^′∈Ω.
We further prove that 𝑥^′=𝑥^∗, the unique solution of VI(12). As a matter of fact, we have by (45) 𝑥_𝑡𝑛−𝑥^′2≤_∼1𝜆+^∼×𝑥𝛾−𝛾𝜌𝛾𝑓^′+(𝐼−𝐵)𝑆𝑥^′−𝐴𝑥^′,𝑥_𝑡𝑛−𝑥^′,𝑥^′∈Ω. (52)
Therefore, the weak convergence to 𝑥^′ of {𝑥_𝑡𝑛} right implies that 𝑥_𝑡𝑛→𝑥^′ in norm. Now we can let 𝑡=𝑡_𝑛→0 in (45) to get 𝛾𝑓𝑦^∗+(𝐼−𝐵)𝑆𝑦^∗−𝐴𝑦^∗,𝑦^∗−𝑥^′≤0,∀𝑦^∗∈Ω, (53)
which is equivalent to its dual VI 𝛾𝑓𝑥^′+(𝐼−𝐵)𝑆𝑥^′−𝐴𝑥^′,𝑦^∗−𝑥^′≤0,∀𝑦^∗∈Ω. (54)
It turns out that 𝑥^′∈Ω solves VI(12). By uniqueness, we have 𝑥^′=𝑥^∗. This is sufficient to guarantee that 𝑥_𝑡→𝑥^∗ in norm, as 𝑡→0+. This completes the proof.

Corollary 6. For each (𝑠,𝑡)∈(0,𝜅)×(0,1), let {𝑥_𝑠,𝑡} be a double net defined by 𝑥_𝑠,𝑡=𝑃_𝐶𝑠𝑡𝛾𝑓𝑥_𝑠,𝑡+(1−𝑡𝐵)𝑆𝑥_𝑠,𝑡+()11−𝑠𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈, (55)
for all (𝑠,𝑡)∈(0,𝜅)×(0,1). Then, for each fixed 𝑡∈(0,1), the net {𝑥_𝑠,𝑡} defined by (55) converges in norm, as 𝑠→0+, to a solution 𝑥_𝑡∈ϝ. Moreover, as 𝑡→0+, the net {𝑥_𝑡} converges in norm to 𝑥^∗ which solves the following variational inequality: 𝑥^∗∈Ω,𝐼−𝛾𝑓𝑥^∗−(𝐼−𝐵)𝑆𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ω, (56) where Ω is the set of the solutions of the following variational inequality: 𝑥^∗∈ϝ,(𝐼−𝑆)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈ϝ. (57)

Corollary 7. For each (𝑠,𝑡)∈(0,𝜅)×(0,1), let {𝑥_𝑠,𝑡} be a double net defined by 𝑥_𝑠,𝑡=𝑃_𝐶𝑠(1−𝑡)𝑆𝑥_𝑠,𝑡+()11−𝑠𝜆_𝑠^𝜆𝑠₀𝑇(𝜈)𝑥_𝑠,𝑡𝑑𝜈, (58)
for all (𝑠,𝑡)∈(0,𝜅)×(0,1). Then, for each fixed 𝑡∈(0,1), the net {𝑥_𝑠,𝑡} defined by (58) converges in norm, as 𝑠→0+, to a solution 𝑥_𝑡∈ϝ. Moreover, as 𝑡→0+, the net {𝑥_𝑡} converges to the minimum norm solution 𝑥^∗ of the following variational inequality: 𝑥^∗∈ϝ,(𝐼−𝑆)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈ϝ. (59)

Proof. In (55), we take 𝑓=0 and 𝐵=𝐼. Then (55) reduces to (58). Hence, the net {𝑥_𝑡} defined by (58) converges in norm to 𝑥^∗∈Ω which satisfies 𝑥^∗∈Ω,𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ω. (60)
This indicates that 𝑥^∗2≤𝑥^∗,𝑥≤𝑥^∗‖𝑥‖,∀𝑥∈Ω. (61)
Therefore, 𝑥^∗ is the minimum norm solution of the VI(59). This completes the proof.

Acknowledgment

The author was supported in part by NSFC 71161001-G0105, NNSF of China (10671157 and 61261044), NGY2012097 and Beifang University of Nationalities scientific research project.