# Sensitivity Analysis for a System of Generalized Nonlinear Mixed Quasi Variational Inclusions withH-Monotone Operators

Authors: Han-Wen Cao
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

The existence of the solution for a new system of  generalized nonlinear mixed
quasi variational inclusions withH-monotone operators is proved by using implicit resolvent technique, and the sensitivity
analysis of solution in Hilbert spaces is given. Our results improve and generalize some results of the recent ones.


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## Body

## 1. Introduction

Sensitivity analysis of solution for variational inequalities and variational inclusions has been studied by many authors via quite different technique. (See [1–7] and the reference therein).In 2004, Agarwal et al. [1] introduced and studied the following problem which is called the system of parametric generalized nonlinear mixed quasi variational inclusions.Letℋ be a real Hilbert space endowed with the product 〈·,·〉 and norm ∥·∥, respectively. Let Ω and Λ be two nonempty open subsets of ℋ in which the parametric ω and λ take values. Let M:ℋ×Ω→2ℋ and N:ℋ×Λ→2ℋ be two maximal monotone mappings with respect to the first argument. H1,S:ℋ×Ω→ℋ and H2,T:ℋ×Λ→ℋ be nonlinear single-valued mappings. The system of parametric generalized nonlinear mixed quasi variational inclusions problem [1] is to find (x,y)∈ℋ×ℋ such that
(1.1)0∈x-y+ρ(H1(y,ω),S(y,ω))+ρM(x,ω),0∈y-x+γ(H2(x,λ)+T(x,λ))+γN(y,λ),
where ρ>0 and γ>0 are two constants.In this paper, we introduce a new system of parametric generalized nonlinear mixed quasi variational inclusions problem.For eachω∈Ω, λ∈Λ, find x=x(ω,λ), y=y(ω,λ) such that
(1.2)0∈f(x,y,ω)+ρ1(F(x,y,ω)+M(x,ω)),0∈g(x,y,λ)+ρ2(G(x,y,λ)+N(y,λ)),
where f,F:ℋ×ℋ×Ω→ℋ, g,G:ℋ×ℋ×Λ→ℋ are nonlinear single-valued mappings, M:ℋ×Ω→2ℋ, N:ℋ×Λ→2ℋ are multivalued mappings, ρ1>0, ρ2>0 are constants. By using implicit resolvent equations technique of H-monotone operator, the existence of solution is proved and the sensitivity analysis of solution for the problem (1.2) is given. Our results improve and generalize the known results of [1, 2, 8, 9].


## 2. Preliminaries

Letℋ be a real Hilbert space, Ω, Λ be two nonempty open subsets of ℋ.Definition 2.1 (see [1]).
(i)
A mappingT:ℋ×Ω→ℋ is said to be monotone with respect to the first argument if
(2.1)〈T(x,ω)-T(y,ω),x-y〉≥0,∀(x,ω),(y,ω)∈H×Ω.(ii)
T is said to be κ-strongly monotone with respect to the first argument if there exists a constant κ>0 such that
(2.2)〈T(x,ω)-T(y,ω),x-y〉≥κ‖x-y‖2,∀(x,ω),(y,ω)∈H×Ω.(iii)
T is said to be (ξ,η)-Lipschitz continuous if there exist ξ>0,η>0 such that
(2.3)‖T(x,ω1)-T(y,ω2)‖≤ξ‖x-y‖+η‖ω1-ω2‖,∀(x,ω1),(y,ω2)∈H×Ω.Definition 2.2 (see [1]).
A mappingF:ℋ×ℋ×Ω→ℋ is said to be (ξ,η,ζ)-Lipschitz continuous if there exist ξ>0,η>0,ζ>0 such that
(2.4)‖F(x1,y1,ω1)-F(x2,y2,ω2)‖≤ξ‖x1-x2‖+η‖y1-y2‖+ζ‖ω1-ω2‖,∀(x1,y1,ω1),(x2,y2,ω2)∈H×H×Ω.Definition 2.3 (see [1]).
A mappingF:ℋ×ℋ×Ω→ℋ is said to be α-strongly monotone with respect to H in the first argument if there exists α>0 such that
(2.5)〈F(x1,y,ω)-F(x2,y,ω),Hx1-Hx2〉≥α‖x1-x2‖2,∀(x1,y,ω),(x2,y,ω)∈H×H×Ω.
In a similar way, we can define the strong monotonicity of F with respect to H in the second argument.Definition 2.4 (see [9]).
LetH:ℋ×Ω→ℋ be a single-valued mapping and M:ℋ×Ω→2ℋ be a multi-valued mapping. M is said to be H-monotone if M is monotone with respect to the first argument and (H(·,ω)+ρM(·,ω))(ℋ)=ℋholds for all ρ>0 and ω∈Ω.Definition 2.5 (see [9]).
LetH:ℋ×Ω→ℋ be a strictly monotone mapping and M:ℋ×Ω→2ℋ be an H-monotone mapping. The resolvent operator JM(·,ω),ρH:ℋ→ℋ is defined by
(2.6)JM(⋅,ω),ρH(u)=(H(⋅,ω)+ρM(⋅,ω))-1(u),∀u∈H.Lemma 2.6 (see [9]).
LetH1:ℋ×Ω→ℋ be γ1-strongly monotone with respect to the first argument, H2:ℋ×Λ→ℋ be γ2-strongly monotone with respect to the first argument, and M:ℋ×Ω→2ℋ be H1-monotone, andN:ℋ×Λ→2ℋ be H2-monotone. Then for any fixed ω∈Ω,λ∈Λ, the resolvent operator JM(·,ω),ρ1H1 and JN(·,λ),ρ2H2 are Lipschitz continuous:
(2.7)‖JM(⋅,ω),ρ1H1(u)-JM(⋅,ω),ρ1H1(v)‖≤1γ1‖u-v‖,∀u,v∈H,‖JN(⋅,λ),ρ2H2(u)-JN(⋅,λ),ρ2H2(v)‖≤1γ2‖u-v‖,∀u,v∈H,
where γ1>0, γ2>0 are constants.Lemma 2.7.
LetM:ℋ×Ω→2ℋ be H1-monotone and N:ℋ×Λ→2ℋ be H2-monotone. For any fixed (ω,λ)∈Ω×Λ. (x(ω,λ),y(ω,λ)) is a solution of (1.2) if and only if(2.81)x(ω,λ)=JM(⋅,ω),ρ1H1(H1(x(ω,λ),ω)-f(x(ω,λ),y(ω,λ),ω)-ρ1F(x(ω,λ),y(ω,λ),ω))≜T(x,y,ω,λ),(2.82)y(ω,λ)=JN(⋅,λ),ρ2H2(H2(y(ω,λ),λ)-g(x(ω,λ),y(ω,λ),λ)-ρ2G(x(ω,λ),y(ω,λ),λ))≜S(x,y,ω,λ), here x=x(ω,λ), y=y(ω,λ).Proof.
Assume that(x(ω,λ),y(ω,λ)) satisfies relations 2.81 and 2.82, since JM(·,ω),ρ1H1=(H1(·,ω)+ρ1M(·,ω))-1, JM(·,ω),ρ2H2=(H2(·,ω)+ρ2N(·,ω))-1, then 2.81 and 2.82 holds if and only if
(2.9)0∈f(x(ω,λ),y(ω,λ),ω)+ρ1F(x(ω,λ),y(ω,λ),ω)+ρ1M(x(ω,λ),ω),0∈g(x(ω,λ),y(ω,λ),λ)+ρ2G(x(ω,λ),y(ω,λ),λ)+ρ2N(y(ω,λ),λ),
and hence (x(ω,λ)),y(ω,λ) is a solution of 2.81 and 2.82.


## 3. Main Results

Lemma 3.1.
LetT:ℋ×ℋ→ℋ, S:ℋ×ℋ→ℋ be two continuous mappings. If there exist Θ1, Θ2, 0<Θ1,Θ2<1, such that
(3.1)‖T(x1,y1)-T(x2,y2)‖+‖S(x1,y1)-S(x2,y2)‖≤Θ1‖x1-x2‖+Θ2‖y1-y2‖,∀x1,x2,y1,y2∈H.
then there exist x*,y*∈ℋ such that
(3.2)x*=T(x*,y*),y*=S(x*,y*).Proof.
For anyx0,y0∈ℋ, let xn+1=T(xx,yn), yn+1=S(xn,yn),n=0,1,2,…, then by (3.1),forallx1,x2,…,xn,xn+1,y1,y2,…,yn,yn+1∈ℋ,we have
(3.3)‖xn+1-xn‖+‖yn+1-yn‖≤Θ1‖xn-xn-1‖+Θ2‖yn-yn-1‖≤Θ(‖xn-xn-1‖+‖yn-yn-1‖),
where Θ=max{Θ1,Θ2}. Let a=∥x1-x0∥+∥y1-y0∥, then
(3.4)‖xn+1-xn‖+‖yn+1-yn‖≤Θ(‖xn-xn-1‖+‖yn-yn-1‖)≤Θ2(‖xn-1-xn-2‖+‖yn-1-yn-2‖)≤⋯≤Θn(‖x1-x0‖+‖y1-y0‖)=Θna,
and hence,
(3.5)0≤‖xn+1-xn‖≤Θna,0≤‖yn+1-yn‖≤Θna.
Since 0<Θ=max{Θ1,Θ2}<1, (3.5) implies that {xn} and {yn} are both cauchy sequences. Therefore, there exist x*,y*∈ℋ such that xn→x*,yn→y*(n→∞). By continuity of T and S, x*=T(x*,y*),y*=S(x*,y*).Theorem 3.2.
LetH1:ℋ×Ω→ℋ be (ξ1,η1)-Lipschitz continuous and H2:ℋ×Λ→ℋ be (ξ2,η2)-Lipschitz continuous. Let f:ℋ×ℋ×Ω→ℋ be (ξf,ηf,ζf)-Lipschitz continuous and γf-strongly monotone with respect to H in the first argument, F:ℋ×ℋ×Ω→ℋ be (ξF,ηF,ζF)-Lipschitz continuous,g:ℋ×ℋ×Λ→ℋ be (ξg,ηg,ζg)-Lipschitz continuous and γg-strongly monotone with respect to H2 in the second argument,G:ℋ×ℋ×Λ→ℋ be (ξG,ηG,ζG)-Lipschitz continuous. Suppose that M:ℋ×Ω→2ℋ is H1-monotone and N:ℋ×Λ→2ℋ is H2-monotone If
(3.6)Θ1=1γ1(ξ12-2γf+ξf2+ρ1ξF)+ξg+ρ2ξGγ2<1,Θ2=1γ2(ξ22-2γg+ηg2+ρ2ηG)+ηf+ρ1ηFγ1<1,
then, for each ω,λ∈Ω×Λ, the problem (1.2) has an unique solution (x*(ω,λ),y*(ω,λ)).Proof.
Letx1=x1(ω,λ),x2=x2(ω,λ),y1=y1(ω,λ),y2=y2(ω,λ). By defining 2.81 of T and Lemma 2.6, we have
(3.7)‖T(x1,y1,ω,λ)-T(x2,y2,ω,λ)‖≤1γ1‖H1(x1,ω)-f(x1,y1,ω)-ρ1F(x1,y1,ω)-H1(x2,ω)+f(x2,y2,ω)+ρ1F(x2,y2,ω)‖≤1γ1‖H1(x1,ω)-H1(x2,ω)-f(x1,y1,ω)+f(x2,y2,ω)‖+1γ1ρ1‖F(x1,y1,ω)-F(x2,y2,ω)‖.
Since f is (ξf,ηf,ζf)-Lipschitz continuous and γf-strongly monotone with respect to H1 in the first argument, F is (ξF,ηF,ζF)-Lipschitz continuous, we have
(3.8)‖H1(x1,ω)-H1(x2,ω)-f(x1,y1,ω)+f(x2,y2,ω)‖≤‖H1(x1,ω)-H1(x2,ω)-f(x1,y1,ω)+f(x2,y1,ω)‖+‖f(x2,y1,ω)-f(x2,y2,ω)‖≤ξ12-2γf+ξf2‖x1-x2‖+ηf‖y1-y2‖,(3.9)‖F(x1,y1,ω)-F(x2,y2,ω)‖≤ξF‖x1-x2‖+ηF‖y1-y2‖.
Combining (3.7)–(3.9), we have
(3.10)‖T(x1,y1,ω,λ)-T(x2,y2,ω,λ)‖≤1γ1(ξ12-2γf+ξf2+ρ1ξF)‖x1-x2‖+ηf+ρ1ηFγ1‖y1-y2‖.
By defining 2.82 of S and Lemma 2.6, we have
(3.11)‖S(x1,y1,ω,λ)-S(x2,y2,ω,λ)‖≤1γ2‖H2(y1,λ)-g(x1,y1,λ)-ρ2G(x1,y1,λ)-H2(y2,λ)+g(x2,y2,λ)+ρ2G(x2,y2,λ)‖≤1γ2‖H2(y1,λ)-H2(y2,λ)-g(x1,y1,λ)+g(x2,g2,λ)‖+ρ2γ2‖G(x1,y1,λ)-G(x2,y2,λ)‖.
Since g is (ξg,ηg,ζg)-Lipschitz continuous and γg-strongly monotone with respect to H2 in the second argument, G is (ξG,ηG,ζG)-Lipschitz continuous. We have
(3.12)‖H2(y1,λ)-H2(y2,λ)-g(x1,y1,λ)+g(x2,y2,λ)‖≤‖H2(y1,λ)-H2(y2,λ)-g(x1,y1,λ)+g(x1,y2,λ)‖+‖g(x1,y2,λ)-g(x2,y2,λ)‖≤ξ22-2γg+ηg2‖y1-y2‖+ξg‖x1-x2‖,(3.13)‖G(x1,y1,λ)-G(x2,y2,λ)‖≤ξG‖x1-x2‖+ηG‖y1-y2‖.
Combining (3.11)–(3.13), we have
(3.14)‖S(x1,y1,ω,λ)-S(x2,y2,ω,λ)‖≤1γ2(ξ22-2γg+ηg2+ρ2ηG)‖y1-y2‖+ξg+ρ2ξGγ2‖x1-x2‖.
By (3.10) and (3.14), we have
(3.15)‖T(x1,y1,ω,λ)-T(x2,y2,ω,λ)‖+‖S(x1,y1,ω,λ)-S(x2,y2,ω,λ)‖≤[1γ1(ξ12-2γf+ξf2+ρ1ξF)+ξg+ρ2ξGγ2]‖x1-x2‖+[1γ2(ξ22-2γg+ηg2+ρ2ηG)+ηf+ρ1ηFγ1]‖y1-y2‖=Θ1‖x1-x2‖+Θ2‖y1-y2‖.
By (3.6) and Lemma 3.1, there exist x*=x*(ω,λ),y*=y*(ω,λ).(x*,y*) such that x*=T(x*,y*,ω,λ),y*=S(x*,y*,ω,λ). By Lemma 2.7, (x*,y*) is a solution of (1.2). From (3.15), we easily see that the solution of (1.2) is unique.Assumption 1.
For implicit resolvent operatorJM(·,ω),ρ1H1,JN(·,λ),ρ2H2, there are two constants ξ>0 and η>0 such that
(3.16)‖JM(⋅,ω),ρ1H1(u)-JM(⋅,ω¯),ρ1H1(u)‖≤ξ‖ω-ω¯‖,‖JN(⋅,λ),ρ2H2(v)-JN(⋅,λ¯),ρ2H2(v)‖≤η‖λ-λ¯‖,∀u,v∈H.Theorem 3.3.
Suppose that the mappingsH1,f,F,M,H2,g,G, and N are the same as in Theorem 3.2, and for any fixed x,y∈ℋ, the mappings ω→H1(x,ω),ω→f(x,y,ω),ω→F(x,y,ω),λ→H2(y,λ),λ→g(x,y,λ),λ→G(x,y,λ) are continuous (or Lipschitz continuous.) If Assumption 1 and condition (3.6) hold, then the solution (x(ω,λ),y(ω,λ)) for the problem (1.2) is continuous (or Lipschitz continuous).Proof.
Suppose thatω,ω¯∈Ω, λ,λ¯∈Λ such that ω→ω¯,λ→λ¯. From Theorem 3.2, we know that the problem (1.2) has solution (x(ω,λ),y(ω,λ)),x(ω¯,λ)¯,y(ω¯,λ¯)), (x(ω¯,λ),y(ω¯,λ)) and (x(ω,λ¯),y(ω,λ¯)).
(A) Estimate ∥x(ω,λ)-x(ω¯,λ)∥ and ∥y(ω,λ)-y(ω¯,λ)∥.  By Lemma 2.7,we have
(3.17)x(ω,λ)=JM(⋅,ω),ρ1H1(H1(x(ω,λ),ω)-f(x(ω,λ),y(ω,λ),ω)-ρ1F(x(ω,λ),y(ω,λ),ω))≜JM(⋅,ω),ρ1H1(p),x(ω¯,λ)=JM(⋅,ω¯),ρ1H1(H1(x(ω¯,λ),ω¯)-f(x(ω¯,λ),y(ω¯,λ),ω¯)-ρ1F(x(ω¯,λ),y(ω¯,λ),ω¯))≜JM(⋅,ω¯),ρ1H1(q).
It follows from Assumption 1, Lemmas 2.6 and 2.7 that
(3.18)‖x(ω,λ)-x(ω¯,λ)‖=‖JM(⋅,ω),ρ1H1(p)-JM(⋅,ω¯),ρ1H1(q)‖≤‖JM(⋅,ω),ρ1H1(p)-JM(⋅,ω),ρ1H1(q)‖+‖JM(⋅,ω),ρ1H1(q)-JM(⋅,ω¯),ρ1H1(q)‖≤1γ1‖p-q‖+ξ‖ω-ω¯‖.
Since f is (ξf,ηf,ζf)-Lipschitz continuous, γf -strongly monotone with respect to H1 in the first argument and F is (ξF,ηF,ζF)-Lipschitz continuous, we conclude
(3.19)‖p-q‖=‖H1(x(ω,λ),ω)-f(x(ω,λ),y(ω,λ),ω)-ρ1F(x(ω,λ),y(ω,λ),ω)-H1(x(ω¯,λ),ω¯)+f(x(ω¯,λ),y(ω¯,λ),ω¯)+ρ1F(x(ω¯,λ),y(ω¯,λ),ω¯)‖≤‖H1(x(ω,λ),ω)-H1(x(ω¯,λ),ω)-f(x(ω,λ),y(ω,λ),ω)+f(x(ω¯,λ),y(ω,λ),ω)‖+‖H1(x(ω¯,λ),ω)-H1(x(ω¯,λ),ω¯)‖+‖f(x(ω¯,λ),y(ω,λ),ω)-f(x(ω¯,λ),y(ω¯,λ),ω)‖+‖f(x(ω¯,λ),y(ω¯,λ),ω)-f(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ρ1‖F(x(ω,λ),y(ω,λ),ω)-F(x(ω¯,λ),y(ω¯,λ),ω)‖+ρ1‖F(x(ω¯,λ),y(ω¯,λ),ω)-F(x(ω¯,λ),y(ω¯,λ),ω¯)‖≤ξ12-2γf+ξf2‖x(ω,λ)-x(ω¯,λ)‖+‖H1(x(ω¯,λ),ω)-H1(x(ω¯,λ),ω¯)‖+ηf‖y(ω,λ)-y(ω¯,λ)‖+‖f(x(ω¯,λ),y(ω¯,λ),ω)-f(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ρ1ξF‖x(ω,λ)-x(ω¯,λ)‖+ρ1ηF‖y(ω,λ)-y(ω¯,λ)‖+ρ1‖F(x(ω¯,λ),y(ω¯,λ),ω)-F(x(ω¯,λ),y(ω¯,λ),ω¯)‖.
It follows from (3.18) and (3.19) that
(3.20)‖x(ω,λ)-x(ω¯,λ)‖≤θ1‖x(ω,λ)-x(ω¯,λ)‖+1γ1(ηf+ρ1ηF)‖y(ω,λ)-y(ω¯,λ)‖+1γ1‖H1(x(ω¯,λ),ω)-H1(x(ω¯,λ),ω¯)‖+1γ1‖f(x(ω¯,λ),y(ω¯,λ),ω)-f(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ρ1γ1‖F(x(ω¯,λ),y(ω¯,λ),ω)-F(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ξ‖ω-ω¯‖,
that is,
(3.21)‖x(ω,λ)-x(ω¯,λ)‖≤ηf+ρ1ηF(1-θ1)γ1‖y(ω,λ)-y(ω¯,λ)‖+11-θ1{1γ1‖H1(x(ω¯,λ),ω)-H1(x(ω¯,λ),ω¯)‖+1γ1‖f(x(ω¯,λ),y(ω¯,λ),ω)-f(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ρ1γ1‖F(x(ω¯,λ),y(ω¯,λ),ω)-F(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ξ‖ω-ω¯‖1γ1},
where
(*)θ1=1γ1(ξ12-2γf+ξf2+ρ1ξF),
By Lemma 2.7, we have
(3.22)y(ω,λ)=JN(⋅,λ)ρ2H2(H2(y(ω,λ),λ)-g(x(ω,λ),y(ω,λ),λ)-ρ2G(x(ω,λ),y(ω,λ),λ))≜JN(⋅,λ),ρ2H2(s),y(ω¯,λ)=JN(⋅,λ),ρ2H2(H2(y(ω¯,λ),λ)-g(x(ω¯,λ),y(ω¯,λ),λ)-ρ2G(x(ω¯,λ),y(ω¯,λ),λ))≜JN(⋅,λ),ρ2H2(t).
It follows from Assumption 1, Lemmas 2.6 and 2.7 that
(3.23)‖y(ω,λ)-y(ω¯,λ)‖=‖JN(⋅,λ),ρ2H2(s)-JN(⋅,λ),ρ2H2(t)‖≤1γ2‖s-t‖.
Since g is (ξg,ηg,ζg)-Lipschitz continuous and γg-strongly monotone with respect to H2 in the second argument and G is (ξG,ηG,ζG)-Lipschitz continuous,we conclude
(3.24)‖s-t‖=‖H2(y(ω,λ),λ)-g(x(ω,λ),y(ω,λ),λ)-ρ2G(x(ω,λ),y(ω,λ),λ)-H2(y(ω¯,λ),λ)+g(x(ω¯,λ),y(ω¯,λ),λ)+ρ2G(x(ω¯,λ),y(ω¯,λ),λ)‖≤‖H2(y(ω,λ),λ)-H2(y(ω¯,λ),λ)-g(x(ω,λ),y(ω,λ),λ)+g(x(ω,λ),y(ω¯,λ),λ)‖+‖g(x(ω,λ),y(ω¯,λ),λ)-g(x(ω¯,λ),y(ω¯,λ),λ)‖+ρ2‖G(x(ω,λ),y(ω,λ),λ)-G(x(ω¯,λ),y(ω¯,λ),λ)‖≤ξ22-2γg+ηg2‖y(ω,λ)-y(ω¯,λ)‖+ξg‖x(ω,λ)-x(ω¯,λ)‖+ρ2ξG‖x(ω,λ)-x(ω¯,λ)‖+ρ2ηG‖y(ω,λ)-y(ω¯,λ)‖.
It follows from (3.23) and (3.24) that
(3.25)‖y(ω,λ)-y(ω¯,λ)‖≤1γ2(ξ22-2γg+ηg2+ρ2ηG)‖y(ω,λ)-y(ω¯,λ)‖+1γ2(ξg+ρ2ξG)‖x(ω,λ)-x(ω¯,λ)‖.
That is,
(3.26)‖y(ω,λ)-y(ω¯,λ)‖≤11-θ21γ2(ξg+ρ2ξG)‖x(ω,λ)-x(ω¯,λ)‖,
where
(**)θ2=1γ2(ξ22-2γg+ηg2+ρ2ηG).
By combining (3.21) and (3.26), we derive
(3.27)‖x(ω,λ)-x(ω¯,λ)‖≤(ηf+ρ1ηF)(ξg+ρ2ξG)(1-θ1)(1-θ2)γ1γ2‖x(ω,λ)-x(ω¯,λ)‖+11-θ1{1γ1‖H1(x(ω¯,λ),ω)-H1(x(ω¯,λ),ω¯)‖+1γ1‖f(x(ω¯,λ),y(ω¯,λ),ω)-f(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ρ1γ1‖F(x(ω¯,λ),y(ω¯,λ),ω)-F(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ξ‖ω-ω¯‖1γ1s}.
That is,
(3.28)‖x(ω,λ)-x(ω¯,λ)‖≤11-κ11-θ1{1γ1‖H1(x(ω¯,λ),ω)-H1(x(ω¯,λ),ω¯)‖+1γ1‖f(x(ω¯,λ),y(ω¯,λ),ω)-f(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ρ1γ1‖F(x(ω¯,λ),y(ω¯,λ),ω)-F(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ξ‖ω-ω¯‖1γ1},(3.29)‖y(ω,λ)-y(ω¯,λ)‖≤ξg+ρ2ξG(1-θ2)γ211-κ11-θ1{1γ1‖H1(x(ω¯,λ),ω)-H1(x(ω¯,λ),ω¯)‖+1γ1‖f(x(ω¯,λ),y(ω¯,λ),ω)-f(x(ω¯,λ),y(ω¯,λ),ω¯)‖+ρ1γ1‖F(x(ω¯,λ),y(ω¯,λ),ω)-F(x(ω¯,λ),y(ω¯,λ),ω¯)‖+‖ξω-ω¯‖1γ1},
where
(***)κ=(ηf+ρ1ηF)(ξg+ρ2ξG)(1-θ1)(1-θ2)γ1γ2.
By (3.6),
(3.30)1-θ1>1γ2(ξg+ρ2ξG),1-θ2>1γ1(ηf+ρ1ηF),
and hence
(3.31)0<(ξg+ρ2ξG)(ηf+ρ1ηF)(1-θ1)(1-θ2)γ1γ2<1,thatis,0<κ<1.
(B) Estimate ∥y(ω¯,λ¯)-y(ω¯,λ)∥ and ∥x(ω¯,λ¯)-x(ω¯,λ)∥.
By Lemma2.7, we have
(3.32)y(ω¯,λ¯)=JN(⋅,λ¯),ρ2H2(H2(y(ω¯,λ¯),λ¯)-g(x(ω¯,λ¯),y(ω¯,λ¯),λ¯)-ρ2G(x(ω¯,λ¯),y(ω¯,λ¯),λ¯))=JN(⋅,λ¯),ρ2H2(m),y(ω¯,λ¯)=JN(⋅,λ),ρ2H2(H2(y(ω¯,λ),λ)-g(x(ω¯,λ),y(ω¯,λ),λ)-ρ2G(x(ω¯,λ),y(ω¯,λ),λ))=JN(⋅,λ),ρ2H2(n).
It follows from Assumption 1, Lemmas 2.6 and 2.7 that
(3.33)‖y(ω¯,λ¯)-y(ω¯,λ)‖=‖JN(⋅,λ¯),ρ2H2(m)-JN(⋅,λ),ρ2H2(n)‖≤‖JN(⋅,λ¯),ρ2H2(m)-JN(⋅,λ¯),ρ2H2(n)‖+‖JN(⋅,λ¯),ρ2H2(n)-JN(⋅,λ),ρ2H2(n)‖≤1γ2‖m-n‖+η‖λ-λ¯‖.
Since g is (ξg,ηg,ζg)-Lipschitz continuous and γg-strongly monotone with respect to H2 in the second argument and G is (ξG,ηG,ζG)-Lipschitz continuous,we conclude
(3.34)‖m-n‖=‖H2(y(ω¯,λ¯),λ¯)-g(x(ω¯,λ¯),y(ω¯,λ¯),λ¯)-ρ2G(x(ω¯,λ¯),y(ω¯,λ¯),λ¯)-H2(y(ω¯,λ),λ)+g(x(ω¯,λ),y(ω¯,λ),λ)+ρ2G(x(ω¯,λ),y(ω¯,λ),λ)‖≤‖H2(y(ω¯,λ¯),λ¯)-H2(y(ω¯,λ),λ¯)-g(x(ω¯,λ¯),y(ω¯,λ¯),λ¯)-g(x(ω¯,λ¯),y(ω¯,λ),λ¯)‖+‖H2(y(ω¯,λ),λ¯)-H2(y(ω¯,λ),λ)‖+‖g(x(ω¯,λ¯),y(ω¯,λ),λ¯)-g(x(ω¯,λ),y(ω¯,λ),λ¯)‖+‖g(x(ω¯,λ),y(ω¯,λ),λ¯)-g(x(ω¯,λ),y(ω¯,λ),λ)‖+ρ2‖G(x(ω¯,λ¯),y(ω¯,λ¯),λ¯)-G(x(ω¯,λ),y(ω¯,λ),λ¯)‖+ρ2‖G(x(ω¯,λ),y(ω¯,λ),λ¯)-G(x(ω¯,λ),y(ω¯,λ),λ)‖≤ξ22-2γg+ηg2‖y(ω¯,λ¯)-y(ω¯,λ)‖+ξg‖x(ω¯,λ¯)-x(ω¯,λ)‖+ρ2(ξG‖x(ω¯,λ¯)-x(ω¯,λ)‖+ηG‖y(ω¯,λ¯)-y(ω¯,λ)‖)+‖g(x(ω¯,λ),y(ω¯,λ),λ¯)-g(x(ω¯,λ),y(ω¯,λ),λ)‖+‖H2(y(ω¯,λ),λ¯)-H2(y(ω¯,λ),λ)‖+ρ2‖G(x(ω¯,λ),y(ω¯,λ),λ¯)-G(x(ω¯,λ),y(ω¯,λ),λ)‖.
It follows from (3.33) and (3.34) that
(3.35)‖y(ω¯,λ¯)-y(ω¯,λ)‖≤ξg+ρ2ξG(1-θ2)γ2‖x(ω¯,λ¯)-x(ω¯,λ)‖+11-θ2{1γ2‖H2(y(ω¯,λ),λ¯)-H2(y(ω¯,λ),λ)‖+1γ2‖g(x(ω¯,λ),y(ω¯,λ),λ¯)-g(x(ω¯,λ),y(ω¯,λ),λ)g(x(ω¯,λ),y(ω¯,λ),λ¯)‖+ρ2γ2‖G(x(ω¯,λ),y(ω¯,λ),λ¯)-G(x(ω¯,λ),y(ω¯,λ),λ)G(x(ω¯,λ),y(ω¯,λ),λ¯)‖+η‖λ-λ¯‖1γ2},
where θ2 defined by **.
By Lemma2.7, we have
(3.36)x(ω¯,λ¯)=JM(⋅,ω¯),ρ1H1(H1(x(ω¯,λ¯),ω¯)-f(x(ω¯,λ¯),y(ω¯,λ¯),ω¯)-ρ1F(x(ω¯,λ¯),y(ω¯,λ¯),ω¯)),x(ω¯,λ)=JM(⋅,ω¯),ρ1H1(H1(x(ω¯,λ),ω¯)-f(x(ω¯,λ),y(ω¯,λ),ω¯)-ρ1F(x(ω¯,λ),y(ω¯,λ),ω¯)).
As the proof of (3.25),we have
(3.37)‖x(ω¯,λ¯)-x(ω¯,λ)‖≤(1γ1ξ12-2γf+ξf2+ρ1ξF)‖x(ω¯,λ¯)-x(ω¯,λ)‖+1γ1(ηf+ρ1ηF)‖y(ω¯,λ¯)-y(ω¯,λ)‖,
that is,
(3.38)‖x(ω¯,λ¯)-x(ω¯,λ)‖≤ηf+ρ1ηF(1-θ1)γ1‖y(ω¯,λ¯)-y(ω¯,λ)‖,
where θ1 is defined by *.
Therefore(3.39)‖y(ω¯,λ¯)-y(ω¯,λ)‖≤11-κ11-θ2{1γ2‖H2(y(ω¯,λ),λ¯)-H2(y(ω¯,λ),λ)‖+1γ2‖g(x(ω¯,λ),y(ω¯,λ),λ¯)-g(x(ω¯,λ),y(ω¯,λ),λ)‖+ρ2γ2‖G(x(ω¯,λ),y(ω¯,λ),λ¯)-G(x(ω¯,λ),y(ω¯,λ),λ)‖+η‖λ-λ¯‖1γ2},(3.40)‖x(ω¯,λ¯)-x(ω¯,λ)‖≤ηf+ρ1ηF(1-θ1)γ111-κ11-θ2{1γ2‖H2(y(ω¯,λ),λ¯)-H2(y(ω¯,λ),λ)‖+1γ2‖g(x(ω¯,λ),y(ω¯,λ),λ¯)-g(x(ω¯,λ),y(ω¯,λ),λ)‖+ρ2γ2‖G(x(ω¯,λ),y(ω¯,λ),λ¯)-G(x(ω¯,λ),y(ω¯,λ),λ)‖+η‖λ-λ¯‖1γ2},
where κ is defined by ***.
(C) Prove that the conclusion of Theorem3.3.
From (3.28) and (3.40), by the assumptions for H1,f,F,H2,g, and G in Theorem 3.3 and relation
(3.41)‖x(ω,λ)-x(ω¯,λ¯)‖≤‖x(ω,λ)-x(ω¯,λ)‖+‖x(ω¯,λ)-x(ω¯,λ¯)‖,
we know that x(ω,λ) is continuous (or Lipschitz continuous).
From (3.29) and (3.39), by the assumptions for H1,f,F,H2,g and G in Theorem 3.3 and relation
(3.42)‖y(ω,λ)-y(ω¯,λ¯)‖≤‖y(ω,λ)-y(ω¯,λ)‖+‖y(ω¯,λ)-y(ω¯,λ¯)‖,
we know that y(ω,λ) is continuous (or Lipschitz continuous).



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*Source: 921835-2011-08-04.xml*