# Existence and Iterative Algorithm of Solutions for a System of Generalized Nonlinear Mixed Variational-Like Inequalities

Authors: Yan-Mei Du
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce and study a system of generalized nonlinear mixed variational-like inequality problems (SGNMVLIPs) in Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the SGNMVLIP. First, the existence of solutions of the auxiliary problems for the SGNMVLIP is shown. Second, an iterative algorithm for solving the SGNMVLIP is constructed by using this existence result. Finally, not only the existence of solutions of the SGNMVLIP is shown but also the convergence of iterative sequences generated by the algorithm is also proven. The technique and results presented in this paper generalize and unify the corresponding techniques and results given in the literature.


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## Body

## 1. Introduction and Preliminaries

Variational inequality theory, which was introduced by Stampacchia [1] in 1964, is an important part of nonlinear analysis. Various kinds of iterative algorithms to solve the variational inequalities have been developed by many authors, see [2–8] and the references therein. Variational-like inequality introduced by Parida et al. [9] is an important generalization of the variational inequalities and has significant applications in nonconvex optimization. It is worth mentioning that the projection method cannot be extended for constructing iterative algorithms for variational-like inequalities. To overcome this drawback, one uses usually the auxiliary principle technique which deals with finding a suitble auxiliary problems for the original problem. Further, this auxiliary problem is used to construct an algorithm for solving the original problems. Glowinski et al. [10] introduced this technique and used it to study the existence of a solution of variational-like inequality. Later, many authors extended this technique to suggest and analyze a number of algorithms for solving various classes of variational inequalities (see [11–19]).Recently, the auxiliary principle technique was extended by Ding et al. [15] to study the existence and iterative algorithm of solutions of generalized strongly nonlinear mixed variational-like inequalities in Banach spaces. On the other hand, the auxiliary principle technique was also extended by Kazmi and Khan [16] who studied a system of generalized variational-like inequality problems in Hilbert spaces.In this paper, we still extend the auxiliary principle technique to study a system of generalized nonlinear mixed variational-like inequality problems (SGNMVLIPs) in Banach spaces. At first, the existence of solutions of the auxiliary problems for the SGNMVLIP is shown. Next, an iterative algorithm for solving SGNMVLIP is constructed by using this existence and uniqueness result. Finally, we prove the existence of solutions of the SGNMVLIP and the convergence of the algorithm. These results improve and generalize many corresponding results given in [12, 13, 16–18].Throughout the paper unless otherwise stated, letI={1,2} be an index set. For each i∈I, let Bi be a real Banach spaces with norm ∥·∥i, let Bi* be the topological dual space of Bi, and let 〈·,·〉i be the generalized duality pairing between Bi and Bi*. Let Fi:B1×B2→Bi*, ηi:Bi×Bi→Bi be nonlinear mappings, then we consider the following system of generalized nonlinear mixed variational-like inequality problems (SGNMVLIPs): for given ωi*∈Bi*, find (x,y)∈B1×B2 such that
(1.1)〈F1(x,y)-ω1*,η1(v1,x)〉1+b1(x,v1)-b1(x,x)≥0∀v1∈B1,(1.2)〈F2(x,y)-ω2*,η2(v2,x)〉2+b2(y,v2)-b2(y,y)≥0∀v2∈B2,
where for each i∈I, the bifunction bi:Bi×Bi→ℝ is a real-valued nondifferential function with the following properties:(i)
bi is linear in the first argument;(ii)
bi is convex in the second argument;(iii)
bi is bounded, that is, there exists a constant γi>0 such that
(1.3)bi(ui,vi)≤γi‖ui‖i‖vi‖i,∀ui,vi∈Bi;(iv)
bi(ui,vi)-bi(ui,wi)≤bi(ui,vi-wi), for all ui,vi,wi∈Bi;Remark 1.1 (see [15]).
(1) It follows from property (i), for anyui,vi∈Bi, bi(-ui,vi)=-bi(ui,vi). By property (iii), we have bi(-ui,vi)≤γi∥ui∥i∥vi∥i, and hence
(1.4)|bi(ui,vi)|≤γi‖ui‖i‖vi‖i,∀ui,vi∈Bi.
This shows that for any ui,vi∈Bi, bi(ui,0)=bi(0,vi)=0.(2) It follows from properties (iii) and (iv), for anyui,vi,wi∈Bi,
(1.5)bi(ui,vi)-bi(ui,wi)≤γi‖ui‖i‖vi-wi‖i,bi(ui,wi)-bi(ui,vi)≤γi‖ui‖i‖wi-vi‖i.
Therefore, we have
(1.6)|bi(ui,vi)-bi(ui,wi)|≤γi‖ui‖i‖vi-wi‖i.
This implies that bi is continuous with respect to the second argument.Some Special Cases
(1) If for eachi∈I, Bi is a real Hilbert space and ωi*=0, then the SGNMVLIP (1.1) and (1.2) reduce to the problems: find (x,y)∈B1×B2 such that
(1.7)〈F1(x,y),η1(v1,x)〉1+b1(x,v1)-b1(x,x)≥0∀v1∈B1,〈F2(x,y),η2(v2,x)〉2+b2(y,v2)-b2(y,y)≥0∀v2∈B2.
The problem (1.7) has been studied by Kazmi and Khan [16].
(2) If index setI={1}, B1=B2=B and F1(x,x)=F2(x,x)=Tx-Ax for each x∈B, where A,T:B→B* are two single-valued mappings, then the SGNMVLIP (1.1) and (1.2) reduce to the problem: find x∈B such that
(1.8)〈Tx-Ax-ω*,η(v,x)〉+b(x,v1)-b(x,x)≥0∀v∈B.
The problem (1.8) with ω*=0 was introduced and studied by Ding [17] in reflexive Banach spaces.
In brief, for appropriate and suitable choice of the mappingsF1,F2,η1,η2,b1,b2, and the linear continuous functionals ω1* and ω2*, one can obtain a number of the known classes of variational inequalities as special cases from SGNMVLIP (1.1) and (1.2) (see [6–8]).
We need the following basic concepts, basic assumptions and basic results which will be used in the sequel.Definition 1.2.
LetD be a nonempty subset of a Banach space E with the dual space E*. Let g:D→E* and η:D×D→E be two mappings, then(i)
g is said to be η-strongly monotone if there exists a constant σ>0 such that
(1.9)〈g(u)-g(v),η(u,v)〉≥σ‖u-v‖2,∀u,v∈D,(ii)
g is said to be Lipschitz continuous if there exists a constant μ>0 such that
(1.10)‖g(u)-g(v)‖≤μ‖u-v‖,∀u,v∈D,(iii)
η is said to be Lipschitz continuous if there exists a constant δ>0 such that
(1.11)‖η(u,v)‖≤δ‖u-v‖,∀u,v∈D.Definition 1.3.
LetD be a nonempty subset of a Banach space E with the dual space E*. Let F:D×D→E*, η:D×D→E be two mappings, then F is said to be(i)
(λ,ξ)-Lipschitz continuous if there exist constants λ,ξ>0 such that
(1.12)‖F(u1,v1)-F(u2,v2)‖≤λ‖u1-u2‖+ξ‖v1-v2‖,∀u1,v1,u2,v2∈D,(ii)
η-strongly monotone in the first argument if there exists a constant ε1>0 such that
(1.13)〈F(u1,v)-F(u2,v),η(u1,u2)〉≥ε1‖u1-u2‖2,∀u1,u2,v∈D,(iii)
η-strongly monotone in the second argument if there exists a constant ε2>0 such that
(1.14)〈F(u,v1)-F(u,v2),η(v1,v2)〉≥ε2‖v1-v2‖2,∀v1,v2,u∈D.Assumption 1.4.
For eachi∈I, the mapping ηi:Bi×Bi→Bi satisfies the following conditions:(1)
ηi(u,v)=ηi(u,z)+ηi(z,v), for all u,v,z∈Bi;(2)
ηi is affine in the second argument;(3)
for each fixedu∈Bi, the function v↦ηi(u,v) is continuous from the weak topology to the weak topology.Remark 1.5.
It follows from Assumption1.4 (1) that ηi(u,v)=-ηi(v,u) and ηi(u,u)=0 for any u,v∈Bi. Moreover, we can prove that ηi is also affine in the first argument by Assumption 1.4 (1) and (2).Lemma 1.6 (see [20]).
LetX be a nonempty close convex subset of a Hausdorff linear topological space E, and let ϕ,ψ:X×X→ℝ be mappings satisfying the following conditions:(i)
for each(x,y)∈X×X, ψ(x,y)≤ϕ(x,y) and ψ(x,x)≥0 for each x∈X;(ii)
for eachx∈X, ϕ(x,y) is upper semicontinuous with respect to y;(iii)
for eachy∈X, the set {x∈X:ψ(x,y)<0} is convex;(iv)
there exists a nonempty compact setΩ⊂X and x0∈Ω such that ψ(x0,y)<0 for any y∈X∖Ω.
Then, there exists any̅∈Ω such that ϕ(x,y̅)≥0 for any x∈X.


## 2. Auxiliary Problems and Algorithm

In this section, we introduce the auxiliary problems to study the SGNMVLIP (1.1) and (1.2), and we give an existence theorem for a solution of the auxiliary problems. By using the existence theorem, we construct the iterative algorithm for solving the SGNMVLIP (1.1) and (1.2).For eachi∈I, let gi:Bi→Bi* be a single-valued mappings Given (x1,x2)∈B1×B2, we consider the following problems P(x1,x2): find (z1,z2)∈B1×B2 such that
(2.1)〈g1(z1)-g1(x1)+ρ[F1(x1,x2)-ω1*],η1(v1,z1)〉1+ρ[b1(x1,v1)-b1(x1,z1)]≥0∀v1∈B1,〈g2(z2)-g2(x2)+ρ[F2(x1,x2)-ω2*],η2(v2,z2)〉2+ρ[b2(x2,v2)-b2(x2,z2)]≥0∀v2∈B2,
where ρ>0 is a constant. The problems are called the auxiliary problems for SGNMVLIP (1.1) and (1.2).Remark 2.1.
If for eachi∈I, Bi is a real Hilbert spaces, ωi*=0 and gi=I are the identity mappings on Bi, then the auxiliary problems reduce to Kazmi and Khan’s auxiliary problems in [16].Theorem 2.2.
For eachi∈I, let ηi:Bi×Bi→Bi be Lipschtiz continuous with constant δi>0, and let gi:Bi→Bi* be ηi-strongly monotone and Lipschtiz continuous with constant σi>0 and μi>0, respectively. Let bi:Bi×Bi→ℝ satisfy the properties (i)–(iv). If Assumption 1.4 holds, then the auxiliary problems P(x1,x2) have a unique solution.Proof.
For eachi∈I, define the mappings ϕi,ψi:Bi×Bi→ℝ by
(2.2)ϕi(vi,zi)=〈gi(vi)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(vi,zi)〉i+ρ[bi(xi,vi)-bi(xi,zi)],ψi(vi,zi)=〈gi(zi)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(vi,zi)〉i+ρ[bi(xi,vi)-bi(xi,zi)],
respectively.
We claim that the mappingsϕi,ψi satisfy all conditions of Lemma 1.6 in the weak topology. Indeed, since gi is ηi-strongly monotone with constant σi>0 and Remark 1.5, it is clear that ϕi and ψi satisfy condition (i) of Lemma 1.6. Since the bifunction bi is convex in the second argument and ηi is affine in the second argument, it follows from Assumption 1.4 (3) and Remark 1.1 (2) that zi↦ϕi(vi,zi) is weakly upper semicontinuous. By Assumption 1.4 (1) and (2), and the property (ii) of bi, it is easy to prove that the set {vi∈Bi:ψi(vi,zi)<0} is convex, hence the conditions (ii) and (iii) of Lemma 1.6 hold.
Letwi=σi-1[δi∥gi(xi)-gi(0)∥i+δiρ∥Fi(x1,x2)-ωi*∥i+ργi∥xi∥i] and Ki={zi∈Bi:∥zi∥i≤wi}, then Ki is a weakly compact subset of Bi. For any zi∈Bi∖Ki, take vi0=0∈Ki. From Assumption 1.4 (1), Remark 1.1 (1), Lipschitz continuity of ηi, and the ηi-strongly monotonicity of gi, we have
(2.3)ψi(vi0,zi)=ψi(0,zi)=〈gi(zi)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(0,zi)〉i+ρ[bi(xi,0)-bi(xi,zi)]=-〈gi(0)-gi(zi),ηi(0,zi)〉i-〈gi(xi)-gi(0),ηi(0,zi)〉i+ρ〈[Fi(x1,x2)-ωi*],ηi(0,zi)〉i+ρ[bi(xi,0)-bi(xi,zi)]≤-σi‖zi‖i2+δi‖gi(xi)-gi(0)‖i‖zi‖i+δiρ‖Fi(x1,x2)-ωi*‖i‖zi‖i+ργi‖xi‖i‖zi‖i=-σi‖zi‖i{‖zi‖i-σi-1[δi‖gi(xi)-gi(0)‖i+δiρ‖Fi(x1,x2)-ωi*‖i+ργi‖xi‖i]}<0.
Therefore, the condition (iv) of Lemma 1.6 holds. By Lemma 1.6 there exists an zi*∈Bi such that ϕi(vi,zi*)≥0 for all vi∈Bi, that is
(2.4)〈gi(vi)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(vi,zi*)〉i+ρ[bi(xi,vi)-bi(xi,zi*)]≥0∀vi∈Bi.
For arbitraryt∈(0,1] and vi∈Bi, let yi,t=tvi+(1-t)zi*. Replacing vi by yi,t in (2.4) and utilizing Assumption 1.4 (1) and (2), Remark 1.5, and the property (ii) of bi, we obtain
(2.5)0≤〈gi(yi,t)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(yi,t,zi*)〉i+ρ[bi(xi,yi,t)-bi(xi,zi*)]=-〈gi(yi,t)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(zi*,yi,t)〉i+ρ[bi(xi,yi,t)-bi(xi,zi*)]≤-t〈gi(yi,t)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(zi*,vi)〉i+ρt[bi(xi,vi)-bi(xi,zi*)].
Hence, we derive
(2.6)〈gi(yi,t)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(vi,zi*)〉i+ρ[bi(xi,vi)-bi(xi,zi*)]≥0.
Let t→0+, by the Lipschitz continuity of gi, we have
(2.7)〈gi(zi*)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(vi,zi*)〉i+ρ[bi(xi,vi)-bi(xi,zi*)]≥0.
Therefore, (z1*,z2*) is a solution of the auxiliary problems P(x1,x2).
Now, let(z1′,z2′) be another solution of the auxiliary problems P(x1,x2) which is different from (z1*,z2*), then we have
(2.8)〈gi(zi′)-gi(xi)+ρ[Fi(x1,x2)-ωi*],ηi(vi,zi′)〉i+ρ[bi(xi,vi)-bi(xi,zi′)]≥0.
Taking vi=zi′ in (2.7) and vi=zi* in (2.8) and adding these two inequalities, we obtain
(2.9)〈gi(zi′)-gi(zi*),ηi(zi*,zi′)〉i≥0.
Since gi is ηi-strongly monotone, we obtain
(2.10)σi‖zi*-zi′‖i2≤〈gi(zi*)-gi(zi′),ηi(zi*,zi′)〉i≤0,
and so (z1′,z2′)=(z1*,z2*). This completes the proof.By virtue of Theorem2.2, we now construct an iterative algorithm for solving the SGNMVLIP (1.1) and (1.2).For given(x0,y0)∈B1×B2, from Theorem 2.2, we know that the auxiliary problems P(x0,y0) have a solution (x1,y1)∈B1×B2, that is,
(2.11)〈g1(x1)-g1(x0)+ρ[F1(x0,y0)-ω1*],η1(v1,x1)〉1+ρ[b1(x0,v1)-b1(x0,x1)]≥0∀v1∈B1,〈g2(y1)-g2(y0)+ρ[F2(x0,y0)-ω2*],η2(v2,y1)〉2+ρ[b2(y0,v2)-b2(y0,y1)]≥0∀v2∈B2.
Again by Theorem 2.2, the auxiliary problems P(x1,y1) have a solution (x2,y2)∈B1×B2, that is,
(2.12)〈g1(x2)-g1(x1)+ρ[F1(x1,y1)-ω1*],η1(v1,x2)〉1+ρ[b1(x1,v1)-b1(x1,x2)]≥0∀v1∈B1,〈g2(y2)-g2(y1)+ρ[F2(x1,y1)-ω2*],η2(v2,y2)〉2+ρ[b2(y1,v2)-b2(y1,y2)]≥0∀v2∈B2.
By induction, we can get the iterative algorithm for solving the SGNMVLIP (1.1) and (1.2) as follows.Algorithm 2.3.
For given(x0,y0)∈B1×B2, there exists a sequence {(xn,yn)} such that
(2.13)〈g1(xn+1)-g1(xn)+ρ[F1(xn,yn)-ω1*],η1(v1,xn+1)〉1+ρ[b1(xn,v1)-b1(xn,xn+1)]≥0∀v1∈B1;(2.14)〈g2(yn+1)-g2(yn)+ρ[F2(xn,yn)-ω2*],η2(v2,yn+1)〉2+ρ[b2(yn,v2)-b2(yn,yn+1)]≥0∀v2∈B2,n=0,1,2,…,
where ρ>0 is a constant.


## 3. Existence of Convergence Theorem

In this section, we will prove not only that the sequence{(xn,yn)} generated by Algorithm 2.3 converges strongly to (x*,y*), and also that (x*,y*) is a solution of SGNMVLIP (1.1) and (1.2).Theorem 3.1.
For eachi∈I, assume that the following conditions are satisfied:(1)
gi:Bi→Bi* is ηi-strongly monotone and Lipschtiz continuous with constant σi>0 and μi>0, respectively;(2)
ηi:Bi×Bi→Bi is Lipschtiz continuous with constant δi>0;(3)
Fi:B1×B2→Bi* is (λi,ξi)-Lipschtiz continuous;(4)
Fi:B1×B2→Bi* is ηi-strongly monotone in the ith argument with constant εi>0;(5)
bi:Bi×Bi→ℝ satisfies the properties (i)–(iv).
If Assumption1.4 holds and there exists a constant ρ>0 such that
(3.1)P1=2σ1-14ρε1[δ12μ12+ρδ12μ1λ1+2ρ2ε1(γ1+δ1ξ1+2δ1λ1)]>0,P2=2σ2-14ρε2[δ22μ22+ρδ22μ2ξ2+2ρ2ε2(γ2+δ2λ2+2δ2ξ2)]>0,(3.2)14ε1[δ12μ1λ1+ρ(δ12λ12+2ε1γ1)]+12ρλ2δ2<min⁡{P1,P2},14ε2[δ22μ2ξ2+ρ(δ22ξ22+2ε2γ2)]+12ρξ1δ1<min⁡{P1,P2},
then the sequence {(xn,yn)} generated by Algorithm 2.3 converges strongly to (x*,y*), and (x*,y*) is a solution of SGNMVLIP (1.1) and (1.2).Proof.
For any(v1,v2)∈B1×B2, it follows from Algorithm 2.3 that
(3.3)〈g1(xn)-g1(xn-1)+ρ[F1(xn-1,yn-1)-ω1*],η1(v1,xn)〉1+ρ[b1(xn-1,v1)-b1(xn-1,xn)]≥0,(3.4)〈g2(yn)-g2(yn-1)+ρ[F2(xn-1,yn-1)-ω2*],η2(v2,yn)〉2+ρ[b2(yn-1,v2)-b2(yn-1,yn)]≥0,(3.5)〈g1(xn+1)-g1(xn)+ρ[F1(xn,yn)-ω1*],η1(v1,xn+1)〉1+ρ[b1(xn,v1)-b1(xn,xn+1)]≥0,(3.6)〈g2(yn+1)-g2(yn)+ρ[F2(xn,yn)-ω2*],η2(v2,yn+1)〉2+ρ[b2(yn,v2)-b2(yn,yn+1)]≥0.
Taking v1=xn+1 in (3.3) and v1=xn in (3.5), respectively, we get
(3.7)〈g1(xn)-g1(xn-1)+ρ[F1(xn-1,yn-1)-ω1*],η1(xn+1,xn)〉1+ρ[b1(xn-1,xn+1)-b1(xn-1,xn)]≥0,(3.8)〈g1(xn+1)-g1(xn)+ρ[F1(xn,yn)-ω1*],η1(xn,xn+1)〉1+ρ[b1(xn,xn)-b1(xn,xn+1)]≥0.
Adding (3.7) and (3.8), we obtain
(3.9)〈g1(xn)-g1(xn+1),η1(xn,xn+1)〉1≤〈g1(xn-1)-g1(xn)-ρ[F1(xn-1,yn-1)-F1(xn,yn)],η1(xn,xn+1)〉1+ρ[b1(xn-xn-1,xn)-b1(xn-xn-1,xn+1)].
From conditions (1) and (2), we have
(3.10)〈g1(xn)-g1(xn+1),η1(xn,xn+1)〉1≥σ1‖xn-xn+1‖12,〈g1(xn-1)-g1(xn),η1(xn,xn+1)〉1=〈g1(xn-1)-g1(xn+1),η1(xn,xn+1)〉1+〈g1(xn+1)-g1(xn),η1(xn,xn+1)〉1≤δ1μ1‖xn+1-xn-1‖1‖xn+1-xn‖1-σ1‖xn+1-xn‖12.
From conditions (3) and (4), we obtain
(3.11)〈F1(xn,yn)-F1(xn-1,yn-1),η1(xn,xn+1)〉1=〈F1(xn,yn)-F1(xn+1,yn-1),η1(xn,xn+1)〉1+〈F1(xn+1,yn-1)-F1(xn-1,yn-1),η1(xn,xn-1)〉1+〈F1(xn+1,yn-1)-F1(xn-1,yn-1),η1(xn-1,xn+1)〉1≤δ1λ1‖xn+1-xn‖12+δ1ξ1‖xn+1-xn‖1‖yn-yn-1‖2+δ1λ1‖xn+1-xn-1‖1‖xn-xn-1‖1-ε1‖xn+1-xn-1‖12.
From condition (5) and Remark 1.1 (2), we have
(3.12)b1(xn-xn-1,xn)-b1(xn-xn-1,xn+1)≤γ1‖xn+1-xn‖1‖xn-xn-1‖1.
Therefore, from (3.9)–(3.12), we derive
(3.13)(2σ1-ρδ1λ1)∥xn-xn+1∥12≤-ρε1‖xn+1-xn-1‖12+[δ1μ1‖xn+1-xn‖1+ρδ1λ1‖xn-xn-1‖1]‖xn+1-xn-1‖1+ρδ1ξ1‖xn+1-xn‖1‖yn-yn-1‖2+ργ1‖xn+1-xn‖1‖xn-xn-1‖1≤14ρε1[δ1μ1‖xn+1-xn‖1+ρδ1λ1‖xn-xn-1‖1]2+ρδ1ξ1‖xn+1-xn‖1‖yn-yn-1‖2+ργ1‖xn+1-xn‖1‖xn-xn-1‖1=δ12μ124ρε1‖xn+1-xn‖12+ρδ12λ124ε1‖xn-xn-1‖12+[δ12μ1λ12ε1+ργ1]‖xn+1-xn‖1‖xn-xn-1‖1+ρδ1ξ1‖xn+1-xn‖1‖yn-yn-1‖2≤14ρε1[δ12μ12+ρδ12μ1λ1+2ρ2ε1(γ1+δ1ξ1)]‖xn+1-xn‖12+14ε1[δ12μ1λ1+ρ(δ12λ12+2ε1γ1)]‖xn-xn-1‖12+12ρδ1ξ1‖yn-yn-1‖22,
which implies
(3.14)P1‖xn-xn+1‖12≤Q1‖xn-xn-1‖12+R1‖yn-yn-1‖22,
where Q1=(1/4ε1)[δ12μ1λ1+ρ(δ12λ12+2ε1γ1)], R1=(1/2)ρδ1ξ1.
Takingv2=yn+1 in (3.4) and v2=yn in (3.6), respectively, we get
(3.15)〈g2(yn)-g2(yn-1)+ρ[F2(xn-1,yn-1)-ω2*],η2(yn+1,yn)〉2+ρ[b2(yn-1,yn+1)-b2(yn-1,yn)]≥0,〈g2(yn+1)-g2(yn)+ρ[F2(xn,yn)-ω2*],η2(yn,yn+1)〉2+ρ[b2(yn,yn)-b2(yn,yn+1)]≥0.
Adding (3.15), we obtain
(3.16)〈g2(yn)-g2(yn+1),η2(yn,yn+1)〉2≤〈g2(yn-1)-g2(yn)-ρ[F2(xn-1,yn-1)-F2(xn,yn)],η2(yn,yn+1)〉2+ρ[b2(yn-yn-1,yn)-b2(yn-yn-1,yn+1)].
From conditions (1) and (2), we have
(3.17)〈g2(yn)-g2(yn+1),η2(yn,yn+1)〉2≥σ2‖yn-yn+1‖22,〈g2(yn-1)-g2(yn),η2(yn,yn+1)〉2=〈g2(yn-1)-g2(yn+1),η2(yn,yn+1)〉2+〈g2(yn+1)-g2(yn),η2(yn,yn+1)〉2≤δ2μ2‖yn+1-yn-1‖2‖yn+1-yn‖2-σ2‖yn+1-yn‖22.
From conditions (3) and (4), we obtain
(3.18)〈F2(xn,yn)-F2(xn-1,yn-1),η2(yn,yn+1)〉2=〈F2(xn,yn)-F2(xn-1,yn+1),η2(yn,yn+1)〉2+〈F2(xn-1,yn+1)-F2(xn-1,yn-1),η2(yn,yn-1)〉2+〈F2(xn-1,yn+1)-F2(xn-1,yn-1),η2(yn-1,yn+1)〉2≤δ2ξ2‖yn+1-yn‖22+δ2λ2‖yn+1-yn‖2‖xn-xn-1‖1+δ2ξ2‖yn+1-yn-1‖2‖yn-yn-1‖2-ε2‖yn+1-yn-1‖22.
From condition (5) and Remark 1.1 (2), we have
(3.19)b2(yn-yn-1,yn)-b2(yn-yn-1,yn+1)≤γ2‖yn+1-yn‖2‖yn-yn-1‖2.
Therefore, from (3.16)–(3.19), we derive
(3.20)(2σ2-ρδ2ξ2)‖yn-yn+1‖22≤-ρε2‖yn+1-yn-1‖22+[δ2μ2‖yn+1-yn‖2+ρδ2ξ2‖yn-yn-1‖2]‖yn+1-yn-1‖2+ρδ2λ2‖yn+1-yn‖2‖xn-xn-1‖1+ργ2‖yn+1-yn‖2‖yn-yn-1‖2≤14ρε2[δ2μ2‖yn+1-yn‖2+ρδ2ξ2‖yn-yn-1‖2]2+ρδ2λ2‖yn+1-yn‖2‖xn-xn-1‖1+ργ2‖yn+1-yn‖2‖yn-yn-1‖2=δ22μ224ρε2‖yn+1-yn‖22+ρδ22ξ224ε2‖yn-yn-1‖22+[δ22μ2ξ22ε2+ργ2]‖yn+1-yn‖2‖yn-yn-1‖2+ρδ2λ2‖yn+1-yn‖2‖xn-xn-1‖1≤14ρε2[δ22μ22+ρδ22μ2ξ2+2ρ2ε2(γ2+δ2λ2)]‖yn+1-yn‖22+14ε2[δ22μ2ξ2+ρ(δ22ξ22+2ε2γ2)]‖yn-yn-1‖22+12ρδ2λ2‖xn-xn-1‖12,
which implies
(3.21)P2‖yn-yn+1‖22≤Q2‖yn-yn-1‖22+R2‖xn-xn-1‖12,
where Q2=(1/4ε2)[δ22μ2ξ2+ρ(δ22ξ22+2ε2γ2)], R2=(1/2)ρδ2λ2.
Adding (3.14) and (3.21), we have
(3.22)P1‖xn-xn+1‖12+P2‖yn-yn+1‖22≤(Q1+R2)‖xn-xn-1‖12+(Q2+R1)‖yn-yn-1‖22.
Define the norm∥·∥*on B1×B2 by
(3.23)‖(u,v)‖*=‖u‖12+‖v‖22∀(u,v)∈B1×B2,
it is easy to prove that (B1×B2,∥·∥*) is a Banach space.
From (3.22), by conditions (3.1), we have
(3.24)‖(xn+1,yn+1)-(xn,yn)‖*2≤max⁡{θ1,θ2}‖(xn,yn)-(xn-1,yn-1)‖*2,
where θ1=(Q1+R2)/min{P1,P2},θ2=(Q2+R1)/min{P1,P2}. From condition (3.2), which implies θ1,θ2∈(0,1), hence {(xn,yn)} is a cauchy sequence, let (xn,yn)→(x*,y*)(n→∞).
By the Lipschitz continuities ofη1 and g, and xn→x*(n→∞), we have
(3.25)|〈g1(xn+1)-g1(xn),η1(v1,xn+1)〉1|≤δ1μ1‖xn+1-xn‖1‖v1-xn+1‖1⟶0asn⟶∞.
Since F1 is (λ1,ξ1)-Lipschitz, xn→x*(n→∞) and yn→y*(n→∞), then we obtain
(3.26)|〈F1(xn,yn)-ω1*,η1(v1,xn+1)〉1-〈F1(x*,y*)-ω1*,η1(v1,x*)〉1|≤|〈F1(xn,yn)-F1(x*,y*),η1(v1,xn+1)〉1|+|〈F1(x*,y*)-ω1*,η1(v1,xn+1)-η1(v1,x*)〉1|≤δ1(λ1‖xn-x*‖1+ξ1‖yn-y*‖2)‖v1-xn+1‖1+δ1‖F1(x*,y*)-ω1*‖1‖xn+1-x*‖1⟶0asn⟶∞.
From condition (5) and Remark 1.1 (2), we have
(3.27)|b1(xn,xn+1)-b1(x*,x*)|≤|b1(xn,xn+1)-b1(xn,x*)|+|b1(xn,x*)-b1(x*,x*)|≤γ1‖xn‖1‖xn+1-x*‖1+|b1(xn-x*,x*)|≤γ1‖xn‖1‖xn+1-x*‖1+γ1‖xn-x*‖1‖x*‖1⟶0asn⟶∞.
Hence, as n→∞ in (2.13), we obtain
(3.28)〈F1(x*,y*)-ω1*,η1(v1,x*)〉1+b1(x*,v1)-b1(x*,x*)≥0∀v1∈B1.
It is similar as above, we can obtain
(3.29)〈F2(x*,y*)-ω1*,η2(v2,y*)〉2+b2(y*,v2)-b2(y*,y*)≥0∀v2∈B2.
Therefore, (x*,y*) is a solution of SGNMLVIP (1.1) and (1.2).Example 3.2.
LetB1=B2=L2[0,1]={x:[0,1]→ℝ|x be Lebesgue measurable and ∫01x2(t)dt<+∞}, then the dual space B1*=B2*=L2[0,1]. For each x∈Bi, (x,y)∈B1×B2, let the norm ∥x∥=(∫01x2(t)dt)1/2, and the inner product 〈x,y〉=∫01x(t)y(t)dt.
For eachi∈I, let the mappings Fi:B1×B2→Bi*, ηi:Bi×Bi→Bi, bi:Bi×Bi→ℝ, gi:Bi→Bi*, be defined as for any (x,y)∈B1×B2, (x′,y′)∈Bi×Bi, z∈Bi,
(3.30)F1(x,y)=19x+arctanx-y,F2(x,y)=19y+arctany-x;ηi(x′,y′)=910(x′-y′);bi(x′,y′)=15〈x′,y′〉,gi(z)=120(19z+arctanz),
respectively, then we have(1)
F1 is Lipschitz continuous with constant (λ1,ξ1)=(20,1) and η1-strongly monotone in the first argument with constant ε1=18;(2)
F2 is Lipschitz continuous with constant (λ2,ξ2)=(1,20) and η2-strongly monotone in the second argument with constant ε2=18;(3)
for eachi∈I, ηi is Lipschitz continuous with constant δi=9/10;(4)
for eachi∈I, bi satisfy properties (i)–(iv) with constant γi=1/5;(5)
for eachi∈I, gi is Lipschitz continuous with constant μi=1 and ηi-strongly monotone with constant σi=9/10.After simple calculations, conditions (3.1) and (3.2) imply that ρ∈(0.01,0.047).Remark 3.3.
Example3.2 shows that the constant ρ which satisfies the conditions (3.1) and (3.2) can be obtained.



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*Source: 176283-2012-03-27.xml*