# Split General Strong Nonlinear Quasi-Variational Inequality Problem

Authors: Yali Zhao; Dongxue Han
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

We introduce a split general strong nonlinear quasi-variational inequality problem which is a natural extension of a split general quasi-variational inequality problem, split variational inequality problem, and quasi-variational and variational inequality problems in Hilbert spaces. Using the projection method, we propose an iterative algorithm for the split general strongly nonlinear quasi-variational inequality problem and discuss the convergence criteria of the iterative algorithm. The results presented here generalized, unify, and improve many previously known results for quasi-variational and variational inequality problems.


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## Body

## 1. Introduction

Variational inequalities are a very powerful tool of the current mathematical technology and have become a rich source of inspiration for scientists and engineers. These have been extended and generalized to study a wide class of problems arising in mechanics, optimization and control problem, operations research and engineering sciences, and so forth. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On one hand, it reveals the fundamental facts on the qualitative behavior of solutions to important classes of problems. On the other hand, it enables us to develop highly efficient and powerful numerical methods to solve, for example, obstacle, unilateral, free, and moving boundary value problems. In the last five decades, considerable interest has been shown in developing various classes of variational inequality problems, both for their own sake and for their applications.An important generalization of the variational inequality problem is the quasi-variational inequality problem introduced and studied by Bensoussan et al. [1] in connection with impulse control problem. Recently, Kazmi [2] introduced and studied the following split general quasi-variational inequality problem (in short, SpGQVIP). For each i∈1,2, let Hi be a Hilbert space with inner product ·,· and induced norm ·, let Ci:Hi→2Hi be a nonempty, closed, and convex set-valued mapping, let fi:Hi→Hi and gi:Hi→Hi be nonlinear mappings, and let A:H1→H2 be a bounded linear operator with its adjoint operator A∗. Then, the SpGQVIP is to find x1∗∈H1 such that g1x1∗∈C1x1∗ and(1a)f1x1∗,x1-g1x1∗≥0∀x1∈C1x1∗and such that, x2∗=Ax1∗∈H2, g2(x2∗)∈C2(x2∗) solves (1b)f2x2∗,x2-g2x2∗≥0∀x2∈C2x2∗.SpGQVIP (1a)-(1b) amounts to saying the following: find a solution of general quasi-variational inequality GQVI (1a) whose image under a given bounded linear operator is a solution of GQVIP (1b). If gi=Ii, where Ii is an identity mapping on Hi, Ci(xi)=Ci for all xi∈Hi, and then SpGQVIP (1a)-(1b) is reduced to the following SpVIP. Find x1∗∈C1 such that (2a)f1x1∗,x1-x1∗≥0∀x1∈C1and such that x2∗=Ax1∗∈C2 solves(2b)f2x2∗,x2-x2∗≥0∀x2∈C2.SpVIP (2a)-(2b) has been introduced and studied by Censor et al. [3]. It is worth mentioning that the SpVIP (2a)-(2b) is quite general and permits split minimization between two spaces so that the imagine of a minimizer of a given function, under a bounded linear operator, is a minimizer of another function and it includes as a special case the split zero problem and the split feasibility problem which have already been studied and used in practice as a model in the intensity-modulated radiation therapy planning; see [4–6] and the references therein.In this paper, we introduced the following split general strongly nonlinear quasi-variational inequality problem: for eachi∈1,2, let Ci:Hi→2Hi be a nonempty, closed, and convex set-valued mapping, let fi:Hi→Hi, hi:Hi→Hi, and gi:Hi→Hi be three nonlinear mappings, and let A:H1→H2 be a bounded linear operator with its adjoint operator A∗. Then, we consider the problem: find x1∗∈H1 such that g1x1∗∈C1x1∗ and (3a)f1x1∗-h1x1∗,x1-g1x1∗≥0∀x1∈C1x1∗and such that x2∗=Ax1∗∈H2, g2(x2∗)∈C2(x2∗) solves(3b)f2x2∗-h2x2∗,x2-g2x2∗≥0∀x2∈C2x2∗.We call problem (3a)-(3b) the split general strongly nonlinear quasi-variational inequality problem (in short, SpGSNQVIP).Remark 1.
Ifhi=0, then SpGSNQVIP (3a)-(3b) is reduced to SpGQVIP (1a)-(1b). So SpGSNQVIP (3a)-(3b) is the generalization of SpGQVIP (1a)-(1b).Remark 2.
Note that general strongly nonlinear variational inequality problem〈f1(x1∗)-h1(x1∗),x1-x1∗〉≥0, ∀x1∈C1, is an important class of variational inequalities, which is the optimal condition of the following minimization problem:(4)minx∈C12f1x,x-T1x,where T1′=h1(x). We denote the solution set of SpGSNQVIP (3a)-(3b) and the solution set of SpGQVIP (1a)-(1b) by Γ1 and Γ2, respectively.Example 3.
LettingH1=H2=R, C1(x1)=[2,5](∀x1∈H1), and C2(x2)=[1,4](∀x2∈H2),(5)f1x1=2x1,h1x1=3x1,∀x1∈H1;f2x2=-x2,h2x2=-2x2,∀x2∈H2;g1x=g2x=x,∀x∈H1=H2;Ax=15x,∀x∈H1.It is easy to see that x1∗=5 satisfies (3a) and A(5)=1 satisfies (3b). So 5∈Γ1 and thus Γ1≠⌀.Example 4.
LetH1=R with the norm · and H2=R2 with the norm(6)y=y12+y221/2,∀y=y1,y2∈R2.Set(7)C1x=2,+∞∀x∈H1,C2y=y=y1,y2∈R2∣y2≥y1+1∀y∈H2,f1x=3x,h1x=2x,f2y=y12,y22,h2y=y22,y12,g1x=x∀x∈H1,g2y=y∀y∈H2.It is easy to see that the solution sets of (3a) and (3b) are S1=2 and S2=y∗=y1∗,y2∗∣y2∗-y1∗=1, respectively. Letting Ax=(x,(3/2)x)(∀x∈H1), then A:H1→H2 is a bounded linear operator. It is obvious that if x∗=2∈S1, then A(x∗)=(2,3)∈S2, and so 2∈Γ1≠⌀.In this paper, using the projection method, we propose an iterative algorithm for SpGSNQVIP (3a)-(3b) and discuss the convergence of the iterative algorithm. The results presented here generalized, unify, and improve the many previously known results for quasi-variational and variational inequality problems.


## 2. Iterative Algorithms and Convergence Results

For eachi∈1,2, a mapping PCi is said to be the metric projection of Hi on Ci if, for every point xi∈Hi, there exists a unique nearest point in Ci denoted by PCi(xi) such that(8)xi-PCixi≤xi-yi∀yi∈Ci.It is well known that PCi is nonexpansive and satisfies(9)xi-yi,PCixi-PCiyi≥PCixi-PCiyi2∀xi,yi∈Hi.Moreover, PCi(xi) is characterized by(10)xi-PCixi,yi-PCixi≤0∀yi∈Ci.Further, it is easy to see the following fact: x1∗ satisfied QVIP ⇔ we find x1∗∈C1(x1∗) such that (11)f1x1∗,x1-x1∗≥0∀x1∈C1x1∗⟺x1∗=PC1x1∗x1∗-ρ1f1x1∗,ρ1>0.Hence, SpGSNQVIP (3a)-(3b) can be reformulated as follows: find x1∗∈H1 with x2∗=Ax1∗ such that gixi∗∈Cixi∗ and(12)gixi∗=PCixi∗gixi∗-ρifixi∗-hixi∗for ρi>0.Based on the above-mentioned arguments, we propose the following iterative algorithm for approximating a solution to SpGSNQVIP (3a)-(3b).Letαn⊂(0,1) be a sequence such that ∑n=1∞αn=∞, and let ρ1, ρ2, and γ be the parameters with positive values.Algorithm 5.
Givenx10∈H1, compute the iterative sequence x1n by the iterative schemes:(13a)g1yn=PC1x1ng1x1n-ρ1f1x1n-h1x1n,(13b)g2zn=PC2Ayng2Ayn-ρ2f2Ayn-h2Ayn,(13c)x1n+1=1-αnx1n+αnyn+γA∗zn-Aynfor all n=0,1,2,…, ρ1,ρ2,γ>0.Ifhi=0, then Algorithm 5 is reduced to the following iterative algorithm for SpGQVIP (1a)-(1b).Algorithm 6.
Givenx10∈H1, compute the iterative sequence x1n by the iterative schemes:(14a)g1yn=PC1x1ng1x1n-ρ1f1x1n,(14b)g2zn=PC2Ayng2Ayn-ρ2f2Ayn,(14c)x1n+1=1-αnx1n+αnyn+γA∗zn-Aynfor all n=0,1,2,…, ρ1,ρ2,γ>0.Ifgi=Ii, Ci(xi)=Ci(∀xi∈Hi), where Ci is a nonempty closed convex subset of Hi, and then Algorithm 5 is reduced to the following iterative algorithm for SpVIP (2a)-(2b).Algorithm 7.
Givenx10∈H1, compute the iterative sequence x1n by the iterative schemes:(15a)yn=PC1x1n-ρ1f1x1n,(15b)zn=PC2Ayn-ρ2f2Ayn,(15c)x1n+1=1-αnx1n+αnyn+γA∗zn-Aynfor all n=0,1,2,…, ρ1,ρ2,γ>0.Remark 8.
Algorithms6 and 7 are proposed by Kazmi in [2] and [7], respectively. Note that Algorithm 5 concludes them as special cases.In order to obtain our main results, we need the following assumption, definition, and lemmas.Assumption 9.
For allxi,yi,zi∈Hi, the operator PCi(xi) satisfies the following condition:(16)PCixizi-PCiyizi≤vixi-yifor some constant vi>0.Definition 10.
A nonlinear mappingf1:H1→H1 is said to be (i)
α1-strongly monotone with respect to g1:H1→H1 if there exists a constant α1>0 such that(17)f1x-f1y,g1x-g1y≥α1x-y2,∀x,y∈H1,(ii)
β1-Lipschitz continuous if there exists a constant β1>0 such that (18)f1x-f1y≤β1x-y,∀x,y∈H.Remark 11.
Ifg1=I1, where I1 is an identity mapping on H1, then Definition 10(i) is reduced to the definition of α1-strong monotonicity of f.Lemma 12.
LetH be a real Hilbert space. Then, the following inequalities hold:(1)
x+y2≤x2+2〈y,x+y〉,∀x,y∈H,(2)
〈x,y〉=(1/2)(x2+y2-x-y2),∀x,y∈H.Lemma 13 (see [8]).
Assume thatan is a sequence of nonnegative numbers such that an+1≤(1-γn)an+δn, where γn is a sequence in (0,1) and δn is a sequence such that(i)
∑n=1∞γn=∞;(ii)
lim⁡supn→∞⁡(δn/γn)≤0or ∑n=1∞δn<∞; then limn→∞⁡an=0.Now we study the convergence of Algorithm5 for SpGSNQVIP (3a)-(3b).Theorem 14.
For eachi∈1,2, let Ci:Hi→2Hi be a nonempty, closed, and convex set-valued mapping, and let gi:Hi→Hi be δi-Lipschitz continuous such that (gi-Ii) is σi-strongly monotone, where Ii is the identity mapping on Hi. Let fi:Hi→Hi be αi-strongly monotone with respect to gi and βi-Lipschitz continuous. Let hi:Hi→Hi be ξi-Lipschitz continuous and let A:H1→H2 be a bounded linear operator and A∗ be its adjoint mapping. Suppose x1∗∈H1 is a solution to SpGSNQVIP (3a)-(3b) and Assumption 9 holds. Then, the sequence x1n generated by Algorithm 5 convergences strongly to x1∗ provided that the constants ρi and γ satisfy the following conditions:(19)ρ1-α1-k1ξ1β12-ξ12≤α1-k1ξ12-δ12-k12β12-ξ12β12-ξ12,k1=1+2σ11+2θ2-v1,α1-k1ξ1>δ12-k12β12-ξ12,δ1>k1,β1>ξ1,0<θ2=11+2σ2v2+δ2-2ρ2α2+ρ22β22+ρ2ξ2,ρ2>0,γ∈0,2A2.Proof.
Sincex1∗∈H1 is a solution of SpGSNQVIP (3a)-(3b), x1∗∈H1 such that gi(xi∗)∈Ci(xi∗) and (20)g1x1∗=PC1x1∗g1x1∗-ρ1f1x1∗-h1x1∗,(21)g2Ax1∗=PC2Ax1∗g2Ax1∗-ρ2f2Ax1∗-h2Ax1∗for ρi>0. It follows from Algorithm 5(13a), Assumption 9, and (20) that(22)g1yn-g1x1∗=PC1x1ng1x1n-ρ1f1x1n-h1x1n-PC1x1∗g1x1∗-ρ1f1x1∗-h1x1∗≤PC1x1ng1x1n-ρ1f1x1n-h1x1n-PC1x1ng1x1∗-ρ1f1x1∗-h1x1∗+PC1x1ng1x1∗-ρ1f1x1∗-h1x1∗-PC1x1∗g1x1∗-ρ1f1x1∗-h1x1∗≤g1x1n-g1x1∗-ρ1f1x1n-f1x1∗+ρ1h1x1n-h1x1∗+v1x1n-x1∗.Noting that f1 is α1-strongly monotone with respect to g1 and β1-Lipschitz continuous and g1 is δ1-Lipschitz continuous, we have (23)g1x1n-g1x1∗-ρ1f1x1n-f1x1∗2=g1x1n-g1x1∗2-2ρ1f1x1n-f1x1∗,g1x1n-g1x1∗+ρ12f1x1n-f1x1∗2≤δ12-2ρ1α1+ρ12β12x1n-x1∗2.
Combining (22) and (23), we get(24)g1yn-g1x1∗≤δ12-2ρ1α1+ρ12β12+ρ1ξ1+v1x1n-x1∗.Since (g1-I1) is σ1-strongly monotone, by virtue of Lemma 12(1), we have(25)yn-x1∗2≤g1yn-g1x1∗2-2g1-I1yn-g1-I1x1∗,yn-x1∗≤g1yn-g1x1∗2-2σ1yn-x1∗2,which implies that(26)yn-x1∗2≤11+2σ1g1yn-g1x1∗.It follows from (24) and (26) that we have(27)yn-x1∗≤θ1x1n-x1∗,where θ1=(1/1+2σ1)δ12-2ρ1α1+ρ12β12+ρ1ξ1+v1.
Similarly, we obtain(28)g2zn-g2Ax1∗≤δ22-2ρ2α2+ρ22β22+ρ2ξ2+v2Ayn-Ax1∗,(29)zn-Ax1∗≤θ2Ayn-Ax1∗,where θ2=(1/1+2σ2)δ22-2ρ2α2+ρ22β22+ρ2ξ2+v2.
Furthermore, in view of Algorithm5(13c), we have(30)x1n+1-x1∗≤1-αnx1n-x1∗+αnyn-x1∗-γA∗Ayn-Ax1∗+γA∗zn-Ax1∗.
Note thatA∗ is a bounded linear operator with A=A∗ and the given condition on γ, we get(31)yn-x1∗-γA∗Ayn-Ax1∗2=yn-x1∗2-2γyn-x1∗,A∗Ayn-Ax1∗+γ2A∗Ayn-Ax1∗2≤yn-x1∗2-γ2-γA2Ayn-Ax1∗≤yn-x1∗2.
And using (29), we have(32)A∗zn-Ax1∗≤Azn-Ax1∗≤θ2AAyn-Ax1∗≤θ2A2yn-x1∗.From (30)–(32),(33)x1n+1-x1∗≤1-1-θαnx1n-x1∗,where θ=θ11+γA2θ2. It follows from the conditions on ρ1, ρ2, and γ that θ∈(0,1). Thus, (1-θ)αn⊂(0,1) and ∑n=1∞(1-θ)αn=∞ for ∑n=1∞αn=∞. So it follows from Lemma 13 that x1n converges strongly to x1∗ as n→∞. Since A is continuous, it follows from (24), (27), (28), and (29) that(34)g1yn⟶g1x1∗,yn⟶x1∗,Ayn⟶Ax1∗,g2zn⟶g2Ax1∗,zn⟶Ax1∗asn⟶∞.This completes the proof.Ifhi=0, then Theorem 14 reduced to the following result of the convergence of Algorithm 6 for SpGQVIP (1a)-(1b).Corollary 15.
For eachi∈1,2, let Ci:Hi→2Hi be a nonempty, closed, and convex set-valued mapping, and let gi:Hi→Hi be δi-Lipschitz continuous such that (gi-Ii) is σi-strongly monotone, where Ii is the identity mapping on Hi. Let fi:Hi→Hi be αi-strongly monotone with respect to gi and βi-Lipschitz continuous. Let A:H1→H2 be a bounded linear operator and A∗ be its adjoint mapping. Suppose x1∗∈H1 is a solution to SpGQVIP (1a)-(1b) and Assumption 9 holds. Then, the sequence x1n generated by Algorithm 6 convergences strongly to x1∗ provided that the constants ρi and γ satisfy the following conditions:(35)ρ1-α1β12≤α12-β12δ12-k12β12,α1>β1δ12-k12,k1=1+2σ11+2θ2-v1,δ1>k1,0<θ2=11+2σ2v2+δ22-2ρ2α2+ρ22β22,ρ2>0,γ∈0,2A2.
IfCi(xi)=Ci(∀xi∈Hi), where Ci is a nonempty closed and convex subset of Hi, gi=Ii, and hi=0, then Theorem 14 reduces to the following convergence result of Algorithm 7 for SpVIP (2a)-(2b).Corollary 16.
For eachi∈1,2, let Ci be a nonempty, closed, and convex subset of Hi. Let fi:Hi→Hi be αi-strongly monotone and βi-Lipschitz continuous. Let A:H1→H2 be a bounded linear operator and A∗ be its adjoint operators. Suppose x1∗∈H1 is a solution to SpVIP (2a)-(2b). Then, the sequence x1n generated by Algorithm 7 converges strongly to x1∗ provided that the constants ρi and γ satisfy the following conditions:(36)ρ1-α1β12≤α12-β121-k12β12,α1>β11-k12,k1=11+2θ2,k1<1,θ2=1-2ρ2α2+ρ22β22.



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