SIDDIQI Received 29 August 2004; Revised 27 January 2005; Accepted 29 June 2005 This paper is devoted to the study of behaviour and sensitivity analysis of the solution for a class of pa
Trang 1QUASI-VARIATIONAL INEQUALITIES
SALAHUDDIN, M K AHMAD, AND A H SIDDIQI
Received 29 August 2004; Revised 27 January 2005; Accepted 29 June 2005
This paper is devoted to the study of behaviour and sensitivity analysis of the solution for
a class of parametric problem of completely generalized quasi-variational inequalities Copyright © 2006 Salahuddin et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
Sensitivity analysis of solutions for variational inequalities with single-valued mappings has been studied by many authors with different techniques in finite dimensional spaces and Hilbert spaces [3,4,7,11,14] Robinson [10] has dealt with the sensitivity analysis
of solutions for the classical variational inequalities over polyhedral convex sets in finite dimensional spaces
In this paper, we study the behaviour and sensitivity analysis of solutions for a class
of parametric problem of completely generalized quasi-variational inequalities with set-valued mappings without the differentiability assumptions
2 Preliminaries
LetH be a real Hilbert space with x 2= x, x , 2 Hthe family of all nonempty bounded subsets ofH and C(H) the family of all nonempty compact subsets of H Let δ : 2 H →
[0,∞) be defined by
δ(A, B) =sup
a − b :a ∈ A, b ∈ B
, ∀ A, B ∈2H, (2.1) and letH : C(H) →[0,∞) be defined by
H(A, B) =max
sup
x ∈ A
d(x, B), sup
y ∈ B d(A, y)
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006, Article ID 86869, Pages 1 12
DOI 10.1155/JIA/2006/86869
Trang 2d(x, B) =inf
Then, (2H,δ) and (C(H), H) are complete metric spaces, H is the Hausdor ff metric on
C(H).
We now consider the parametric problem of completely generalized quasi-variational inequalities LetΩ be a nonempty open subset of H in which the parameter λ takes
val-ues andK : H ×Ω→2H set-valued mapping with nonempty closed convex valued Let
A, R, T : H ×Ω→2H be the set-valued mappings and p, f , g, G : H ×Ω→ H the
single-valued mappings For each fixedλ ∈ Ω, we write Gλ( x) = G(x, λ), u λ( x) = u(x, λ) unless
otherwise specified The parametric problem of completely generalized quasi-variational inequality (PPCGQVI) consists in findingx ∈ H, u λ( x) ∈ A λ( x), w λ( x) ∈ R λ( x), z λ( x) ∈
T λ( x) such that G λ( x) ∈ K λ( x) and
p λ
u λ( x)
−f λ
w λ( x)
− g λ
z λ( x)
,y − G λ( x) ≥0, ∀ y ∈ K λ( x). (2.4)
In many important applications,K λ( x) has the form
K λ( x) = m(x) + K λ, ∀( x, λ) ∈ H ×Ω, (2.5) wherem : H → H and { K λ:λ ∈Ω}is a family of nonempty closed and convex subsets of
H, see, for example, [13] and the references therein
For eachλ ∈ Ω, let S(λ) denote the set of solutions to the problem (2.4) For some
λ ∈ Ω, we fix those conditions under which for each λ in a neighborhood (say N(λ)) of
λ, problem (2.4) has a nonempty solution set, that is,S(λ) nearS( λ) and the
set-valued mappingsS(λ) is continuous or Lipschitz continuous under the metric δ or H.
We need the following concepts and results
Lemma 2.1 [5] For each x, v ∈ H,
if and only if
where P K(v) is the projection of v ∈ H onto K.
Lemma 2.2 [9] Let m : H → H be a single-valued mapping and
Then
P K(x)( y) = m(x) + P K
y − m(x)
Definition 2.3 [12] A single-valued mappingG : H ×Ω→ H is called:
(i)α-strongly monotone if there exists a constant α > 0 such that
G λ(x) − G λ(y), x − y ≥ α x − y 2, ∀( x, y, λ) ∈ H × H ×Ω; (2.10)
Trang 3(ii)β- Lipschitz continuous if there exists a constant β > 0 such that
G λ( x) − G λ( y) ≤ β x − y , ∀( x, y, λ) ∈ H × H × Ω. (2.11)
Definition 2.4 [1] A set-valued mappingR : H ×Ω→2His said to be
(i) relaxed Lipschitz with respect to a mapping f : H ×Ω→ H if there exists a constant
r ≥0 such that
f λ
w λ( x)
− f λ
w λ( y)
,x − y ≤ − r x − y 2,
∀( x, y, λ) ∈ H × H × Ω, wλ( x) ∈ R λ( x), w λ( y) ∈ R λ( y); (2.12) (ii) relaxed monotone with respect to a mapping g : H ×Ω→ H if there exists a constant
s > 0 such that
g λ
w λ( x)
− g λ
w λ( y)
,x − y ≥ − s x − y 2,
∀( x, y, λ) ∈ H × H × Ω, wλ( x) ∈ R λ( x), w λ( y) ∈ R λ( y). (2.13) Definition 2.5 [2] A set-valued mappingA : H ×Ω→2H [A : H ×Ω→ C(H)] is said to
beη-δ-Lipschitz [η- H-Lipschitz ] continuous if there exists a constant η ≥0 such that
δ
A λ(x), A λ(y)
≤ η x − y , ∀( x, y, λ) ∈ H × H ×Ω,
H
A λ( x), A λ( y)
≤ η x − y , ∀( x, y, λ) ∈ H × H × Ω. (2.14) Lemma 2.6 Let K λ( x) be defined as ( 2.5 ) Then for each fixed λ ∈ Ω, problem (2.4 ) has a solution (x( λ), u λ(x( λ)), w λ(x( λ)), z λ(x( λ))) if and only if x = x( λ) is a fixed point of the set-valued mapping φ : H ×Ω→2H defined by
u λ(x) ∈ A λ(x), w λ(x) ∈ R λ(x), z λ(x) ∈ T λ(x)
x − G λ( x) + m(x)
+P K λ
G λ( x) − ρ
p λ
u λ( x)
−f λ
w λ( x)
− g λ
z λ( x)
− m(x)
, (2.15)
for each x ∈ H, where λ = λ, ρ > 0 is some constant and P K λ(v) is the projection of v ∈ H onto K λ
Proof For any fixed λ ∈ Ω , let (x,u λ(x), w λ(x), z λ(x)) be a solution of problem (2.4) Thenx ∈ H, u λ(x) ∈ A λ(x), w λ(x) ∈ R λ(x) and z λ(x) ∈ T λ(x) such that G λ(x) ∈ K λ(x)
and
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
,y − G λ(x)
≥0, ∀ y ∈ K λ(x). (2.16) Hence for anyρ > 0,
G λ(x) −G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
,y − G λ(x)
≥0, ∀ y ∈ K λ(x). (2.17)
Trang 4From Lemmas2.1and2.2, we have
G λ(x) = P K λ(x)
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
= m(x) + P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
. (2.18)
We can also write
x = x − G λ(x) + m(x)
+P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
u λ(x) ∈ A λ(x), w λ(x) ∈ R λ(x), z λ(x) ∈ T λ(x)
x − G λ(x) + m(x)
+P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
= φ λ(x),
(2.19)
that is,x = x( λ) is a fixed point of φ λ(x).
Now, for any fixedλ ∈ Ω, let x(λ) be a fixed point of φ λ(x) ByLemma 2.1there exist
u λ(x) ∈ A λ(x), w λ(x) ∈ R λ(x) and z λ(x) ∈ T λ(x) such that
G λ(x) = m(x) + P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ (x)
− g λ
z λ(x)
− m(x)
= P K λ(x)
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
.
(2.20)
Hence, we haveG λ(x) ∈ K λ(x) and
G λ(x) −G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
,y − G λ(x)
≥0, (2.21)
for ally ∈ K λ(x).
Noting thatρ > 0, we have
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
,y − G λ(x)
≥0, ∀ y ∈ K λ(x), (2.22)
that is, (x, u λ(x), w λ(x), z λ(x)) is a solution of the problem (2.4)
Lemma 2.7 Let K λ( x) be defined as ( 2.5 ), A, R, T : H ×Ω→2H the δ-Lipschitz continuous with respect to constants η, γ, ν, respectively, and p, f ,g,G : H ×Ω→ H the Lipschitz con-tinuous with respect to the constants ξ, χ, σ and β, respectively Let G be strongly monotone with constant α > 0, R relaxed Lipschitz continuous with respect to f with constant r ≥0,T
Trang 5relaxed monotone with respect to g with constant s > 0, and m : H → H is μ-Lipschitz con-tinuous If there exists a constant ρ > 0 such that
ρ −( − s) + ξη(q −1)
γχ + σ ν 2
−ξη 2
<
( − s) + ξη(q −1) 2
− q(q −1)
(γχ + σ ν)2−(ξη)2
γχ + σ ν 2
−ξη 2
( − s) > (1 − q)ξη +
q(q −1)
(γχ + σ ν)2−(ξη)2
ρξη < γχ + σ ν,
q =2
μ +
1−2α + β2
< 1,
(2.23)
then the set-valued mapping φ : H ×Ω→2H defined by ( 2.15 ) is a uniform θ-δ-set-valued contraction with respect to λ ∈ Ω, where
θ = q + t(ρ) + ρξη < 1, t(ρ) =1−2ρ(r − s) + ρ2(γχ + σ ν)2. (2.24)
Proof By the definition of φ, for any x, y ∈ H, λ ∈ Ω, a ∈ φ λ( x) and b ∈ φ λ( y), there
existu λ( x) ∈ A λ( x), u λ( y) ∈ A λ( y), w λ( x) ∈ R λ( x), w λ( y) ∈ R λ( y), z λ( x) ∈ T λ( x) and
z λ(y) ∈ T λ(y) such that
a = x − G λ(x) + m(x) + P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
,
b = y − G λ( y) + m(y) + P K λ
G λ( y) − ρ
p λ
u λ( y)
−f λ
w λ( y)
− g λ
z λ( y)
− m(y)
.
(2.25) Since projection operator is nonexpansive, we have
a − b ≤2 x − y −
G λ( x) − G λ( y) + 2 m(x) − m(y)
+ x − y + ρ
f λ
w λ(y)
− f λ
w λ(y)
− ρ
g λ
z λ(x)
− g λ
z λ(y)
+ρ p λ
u λ( x)
− p λ
SinceG is strongly monotone and Lipschitz continuous, we have
x − y −
G λ(x) − G λ(y) 2
≤1−2α + β2
x − y 2,
m(x) − m(y) ≤ μ x − y ,
p λ
μ λ( x)
− p λ
u λ( y) ≤ ξ u λ( x) − u λ( y) ≤ ξδ
A λ( x), A λ( y)
≤ ξη x − y
(2.27)
Trang 6x − y + ρ
f λ
w λ( x)
− f λ
w λ( y)
− ρ
g λ
z λ( x)
− g λ
z λ( y) 2
= x − y 2+ 2ρ
f λ
w λ( x)
− f λ
w λ( y)
,x − y −2
g λ
z λ( x)
− g λ
z λ( y)
,x − y
+ρ2 f λ
w λ( x)
− f λ
w λ( y)
−g λ
z λ( x)
− g λ
z λ( y) 2
≤1−2ρ(r − s) + ρ2(γχ + σν)2
x − y 2.
(2.28) From (2.26)–(2.28), we have
a − b ≤q + t(ρ) + ρξη
x − y ≤ θ x − y , (2.29) where
θ = q + t(ρ) + ρξη, t(ρ) =1−2ρ(r − s) + ρ2(γχ + ρν)2,
q =2
μ +
1−2α + β2
.
(2.30)
By the arbitrariness ofa and b, we have
δ
φ λ(x), φ λ(y)
By conditions (2.23) and (2.24), we haveθ < 1 This proves that θ is a uniform
Lemma 2.8 [6] Let X be a complete metric space and T1,T2:X → C(X) be θ- H-contraction
mapping Then
H
F
T1
,F
T2
≤
1
1− θ
sup
x ∈ X
H
T1(x), T2(x)
where F(T1) and F(T2) are the sets of fixed points of T1and T2, respectively.
3 Sensitivity analysis
Theorem 3.1 Assume that A λ( x), R λ( x) and T λ( x) are δ-Lipschitz continuous at λ Let
R λ( x) be the relaxed Lipschitz continuous with f λ(·) at λ, and T λ( x) the relaxed monotone with g λ(·) at λ Suppose that G λ( x), p λ(·), f λ(·), g λ(·) and P K λ(v) are Lipschitz continuous
at λ, where x = x(λ) ∈ S( λ), u λ(x) ∈ A λ(x), w λ(x) ∈ R λ(x), z λ(x) ∈ T λ(x) and
v = G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
Then for all λ ∈ Ω, the solution set S(λ) of the problem (2.4 ) is nonempty and S(λ) is δ-Lipschitz continuous at λ.
Proof For each fixed λ ∈ Ω, φλ( x) has a fixed point, that is, there exists a x( λ) ∈ H such
thatx(λ) ∈ φ λ( x(λ)) FromLemma 2.6, we havex(λ) ∈ S(λ), hence S(λ) andS(λ)
coincides with the set of fixed point ofφ λ( x) In particular, S(λ) coincides with the set of
Trang 7fixed point ofφ λ(x) Now we show that S(λ) is δ-Lipschitz continuous at λ For all x(λ) ∈ S(λ) and x(λ) ∈ S(λ) there exist u λ(x(λ)) ∈ A λ(x(λ)), w λ(x(λ)) ∈ R λ(x(λ)), z λ(x(λ)) ∈
T λ( x(λ)), u λ(x(λ)) ∈ A λ(x(λ)), w λ(x(λ)) ∈ R λ(x(λ)) and z λ(x(λ)) ∈ T λ(x(λ)) such that x(λ) = x(λ) − G λ
x(λ)
+m
x(λ)
+P K λ
G λ
x(λ)
− ρ
p λ
u λ
x(λ)
−f λ
w λ
x(λ)
− g λ
z λ
x(λ)
− m
x(λ)
,
x( λ) = x( λ) − G λ
x( λ)
+m
x( λ)
+P K λ
G λ
x( λ)
− ρ
p λ
u λ
x( λ)
−f λ
w λ
x
λ
− g λ
z λ
x(λ)
− m
x
λ
.
(3.2) Writex = x(λ) and x = x(λ) Taking any u λ( x) ∈ A λ( x), w λ( x) ∈ R λ( x) and z λ( x) ∈
T λ( x), we have
x − x ≤ x − G λ( x) + m(x)
+P K λ
G λ( x) − ρ
p λ
u λ( x)
−f λ
w λ( x)
− g λ
z λ( x)
− m(x)
−x − G λ
x
+m(x)
+P K λ
G λ( x) − ρ
p λ
u λ( x)
−f λ
w λ( x)
− g λ
z λ( x)
− m(x)
+ x − G λ( x) + m(x)
+P K λ
G λ( x) − ρ
p λ
u λ( x)
−f λ
w λ( x)
− g λ
z λ( x)
− m(x)
−x − G λ(x) + m(x)
+P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
≤ θ x − x + G λ( x) − G
λ(x)
+ P K
λ
G λ( x) − ρ
p λ
u λ( x)
−f λ
w λ( x)
− g λ
z λ( x)
− m(x)
− P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
+ P K
λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
− P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
≤ θ x − x + 2 G λ( x) − G
λ(x) +ρ p λ
u λ( x)
− p λ
u λ(x)
+ρ f λ
w λ( x)
− f λ
w λ(x)
+ρ g λ
z λ(x)
− g λ
z λ(x) + P K
λ(v) − P k λ(v) ,
(3.3) where,v = G λ(x) − ρ(p λ(u λ(x)) −(f λ(w λ(x)) − g λ( λ(x)))) − m(x) Since, x = x(λ) ∈ S(λ)
andx = x(λ) ∈ S(λ) are arbitrary, it follows that
δ
S(λ), S(λ)
≤
1
1− θ 2 G λ( x) − G
λ(x) +ρ p λ
u λ( x)
− p λ
u λ(x) +ρ f λ
w λ( x)
− f λ
w λ(x) +ρ g λ
z λ( x)
− g λ
z λ(x) + P K
λ(v) − P K λ(v) .
(3.4)
Trang 8From theδ-Lipschitz continuity of A, R, T at λ; Lipschitz continuity of G and P K λ(v) at
Theorem 3.2 If we assume the hypothesis of Lemma 2.7 , then
(i)φ : H ×Ω→ C(H) defined by ( 2.15 ) is a compact valued uniform θ- H-contraction
mapping with respect to λ ∈ Ω;
(ii) for each λ ∈ Ω, (2.4 ) has nonempty solution set S(λ), closed in H.
Proof (i) For each (x, λ) ∈ H × Ω; Aλ( x), R λ( x), T λ( x) ∈ C(H) and P K λ are continu-ous, follows from (2.15) of φ λ( x) ∈ C(H) Now, we show that φ λ( x) is a uniform
θ-
H-contraction mapping with respect to λ ∈ Ω For any a ∈ φ λ(x), there exist u λ(x) ∈
A λ( x) ∈ C(H), w λ( x) ∈ R λ( x) ∈ C(H) and z λ( x) ∈ T λ( x) ∈ C(H) such that
a = x − G λ( x) + m(x) + P K λ
G λ( x) − ρ
p λ
u λ( x)
−f λ
w λ( x)
− g λ
z λ( x)
− m(x)
.
(3.5) Note that (y, λ) ∈ H × Ω; Aλ( y), R λ( y), T λ( y) ∈ C(H), then there exist u λ( y) ∈ A λ( y),
w λ(y) ∈ R λ(y) and z λ(y) ∈ T λ(y) such that
p λ
u λ( x)
− p λ
u λ( y) ≤ ξ u λ( x) − u λ( y) ≤ ξ H
A λ( x), A λ( y)
,
f λ
w λ( x)
− f λ
w λ( y) ≤ χ w λ( x) − w λ( y) ≤ χ HR λ( x), R λ( y) ,
g λ
z λ( x)
− g λ
z λ( y) ≤ σ z λ( x) − z λ( y) ≤ σ HT λ( x), T λ( y) .
(3.6)
Let
b = y − G λ(y) + m(y) + P K λ
G λ(y) − ρ
p λ
u λ(y)
−f λ
w λ(y)
− g λ
z λ(y)
− m(y)
, (3.7) then
By using the similar argument as in the proof ofLemma 2.7, we can obtain
a − b ≤ 2
μ +
1−2α + β2
+
1−2ρ(r − s) + ρ2(γχ + σ ν)2+ρξη
x − y
≤q + t(ρ) + ρξη
where
θ = q + t(ρ) + ρξη, t(ρ) =1−2ρ(r − s) + ρ2(γχ + σ ν)2,
q =2
μ +
1−2α + β2
.
(3.10)
Trang 9By conditions (2.23) and (2.24),θ < 1, and hence we have
sup
a ∈ φ λ(x)
d
a, φ λ( y)
By the similar arguments, we have
sup
b ∈ φ λ(y)
d
φ λ( x), b
Hence, by the Hausdorff metricH, we obtain
H
φ λ( x), φ λ( y)
Thereforeφ λ( x) is a uniform θ- H-contraction mapping with respect to λ ∈Ω
(ii) Sinceφ λ(x) is a uniform θ- H-contraction with respect to λ ∈Ω, hence by Nadler theorem [8],φ λ( x) has a fixed point x(λ) Since S(λ) , then let{ x n} ⊂ S(λ) and x n →
x0asn → ∞ Therefore,
x n ∈ φ λ( x n), n =1, 2, . (3.14) From (i), we have
H
φ λ( x n), φ λ( x0)
If follows that
d
x0,φ λ( x0)
≤ x0− x n +d
x n, φ λ( x n)
+Hφ λ( x n), φ λ( x0)
≤(1 +θ) x n − x0 −→0, asn −→ ∞, (3.16)
hencex0∈ φ λ( x0) andx0∈ S(λ) Therefore S(λ) is closed in H.
Theorem 3.3 Assume the hypothesis as in Theorem 3.1 Then for all λ ∈ Ω, the solution set S(λ) of ( 2.4 ) is nonempty and S(λ) is H-Lipschitz continuous at λ.
Proof FromTheorem 3.2(ii), the solution setS(λ) of (2.4) is a nonempty closed set in
H Now, we show that S(λ) is H-Lipschitz continuous at λ By Theorem 3.2(i),φ λ( x) and
φ λ(x) are both θ- H-contraction mappings From Lemma 2.8, we have
H
S(λ), S( λ)
≤
1
1− θ
sup
x ∈ H
H
φ λ( x), φ λ(x)
Taking anya ∈ φ λ( x), ∃ u λ( x) ∈ A λ( x), w λ( x) ∈ R λ( x) and z λ( x) ∈ T λ( x) such that
a = x − G λ( x) + m(x) + P K λ
G λ( x) − ρ
p λ
u λ( x)
−f λ
w λ( x)
− g λ
z λ( x)
− m(x)
.
(3.18)
Trang 10For u λ( x) ∈ A λ( x) ∈ C(H), w λ( x) ∈ R λ( x) ∈ C(H), z λ( x) ∈ T λ( x) ∈ C(H), there exist
u λ(x) ∈ A λ(x), w λ(x) ∈ R λ(x) and z λ(x) ∈ T λ(x) such that
u λ( x) − u
λ(x) ≤ H
A λ( x), A λ(x)
,
w λ(x) − w
λ(x) ≤ H
R λ(x), R λ(x)
,
z λ( x) − z
λ(x) ≤ H
T λ( x), T λ(x)
.
(3.19)
Let
b = x − G λ(x) + m(x) + P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
, (3.20) then
It follows that
a − b ≤ G λ( x) − G
λ(x)
+ P K
λ { G λ( x) − ρ
p λ
u λ( x)
−f λ
w λ( x)
− g λ
z λ( x)
− m(x)
− P K λ { G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x) }
+ P K
λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
− P K λ
G λ(x) − ρ
p λ
u λ(x)
−f λ
w λ(x)
− g λ
z λ(x)
− m(x)
≤2 G λ( x) − G
λ(x) +ρ p λ
u λ( x)
− p λ
u λ(x)
+ρ f λ
w λ( x)
− f λ
w λ(x) +ρ g λ
z λ( x)
− g λ
z λ(x)
+ P K
λ(v) − P K λ(v) ≤2 G λ( x) − G
λ(x)
+ρ p λ
u λ( x)
− p λ
u λ( x) +ρ p
λ
u λ( x)
− p λ
u λ(x)
+ρ f λ
w λ( x)
− f λ
w λ( x) +ρ f
λ
w λ( x)
− f λ
w λ(x)
+ρ g λ
z λ( x)
− g λ
z λ( x) +ρ g
λ
z λ( x)
− g λ
z λ(x)
+ P K
λ(v) − P K λ(v) ,
(3.22)
wherev = G λ(x) − ρ(p λ(u λ(x)) −(f λ(w λ(x)) − g λ( λ(x)))) − m(x).
Write
M =2 G λ( x) − G
λ(x) +ρ p λ
u λ( x)
− p λ
u λ( x)
+ρ f λ
w λ( x)
− f λ
w λ( x) +ρ g λ
z λ( x)
− g λ
z λ( x)
+ρξ H
A λ( x), A λ(x)
+ρχ H
R λ( x), R λ(x)
+ρσ H
T λ( x), T λ(x)
+ P K (v) − P K (v) .
(3.23)