PARAMETRIC PROBLEM OF COMPLETELY GENERALIZED QUASI-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
Let H be a real Hilbert space with (cid:2)x(cid:2)2 = (cid:3)x,x(cid:4), 2H the family of all nonempty bounded subsets of H and C(H) the family of all nonempty compact subsets of H. Let δ : 2H → [0, ∞) be defined by
(cid:2) (cid:3)
(2.1)
(cid:2)a − b(cid:2) : a ∈ A, b ∈ B
, ∀A,B ∈ 2H ,
δ(A,B) = sup
and let (cid:4)H : C(H) → [0, ∞) be defined by
(cid:5) (cid:6)
(2.2)
(cid:4)H(A,B) = max
d(A, y)
, ∀A,B ∈ C(H),
sup x∈A
d(x,B), sup y∈B
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 86869, Pages 1–12 DOI 10.1155/JIA/2006/86869
2
Parametric problem of quasi-variational inequalities
where
(2.3)
(cid:2)x − y(cid:2).
d(x,B) = inf y∈B
Then, (2H ,δ) and (C(H), (cid:4)H) are complete metric spaces, (cid:4)H is the Hausdorff metric on C(H).
We now consider the parametric problem of completely generalized quasi-variational inequalities. Let Ω be a nonempty open subset of H in which the parameter λ takes val- ues and K : 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 fixed λ ∈ Ω, we write Gλ(x) = G(x,λ), uλ(x) = u(x,λ) unless otherwise specified. The parametric problem of completely generalized quasi-variational inequality (PPCGQVI) consists in finding x ∈ H, uλ(x) ∈ Aλ(x), wλ(x) ∈ Rλ(x), zλ(x) ∈ Tλ(x) such that Gλ(x) ∈ Kλ(x) and (cid:9)
(cid:9)(cid:9) (cid:10) (cid:7) (cid:8) (cid:9) −
(2.4)
(cid:8) uλ(x)
pλ
(cid:8) wλ(x)
fλ
(cid:8) zλ(x) − gλ
, y − Gλ(x)
≥ 0, ∀y ∈ Kλ(x).
In many important applications, Kλ(x) has the form
(2.5)
Kλ(x) = m(x) + Kλ, ∀(x,λ) ∈ H × Ω,
where m : H → H and {Kλ : λ ∈ Ω} is a family of nonempty closed and convex subsets of H, see, for example, [13] and the references therein.
For each λ ∈ 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) (cid:10)= ∅ near S(λ) and the set- valued mappings S(λ) is continuous or Lipschitz continuous under the metric δ or (cid:4)H.
We need the following concepts and results.
Lemma 2.1 [5]. For each x,v ∈ H,
(2.6)
x = PK (v)
if and only if
(2.7)
(cid:3)x − v, y − v(cid:4) ≥ 0, ∀y ∈ K,
where PK (v) is the projection of v ∈ H onto K. Lemma 2.2 [9]. Let m : H → H be a single-valued mapping and
(2.8)
K(x) = m(x) + K, ∀x ∈ H.
Then
(cid:8)
(2.9)
y − m(x)
(cid:9) , ∀x, y ∈ H.
PK(x)(y) = m(x) + PK
Definition 2.3 [12]. A single-valued mapping G : H × Ω → H is called: (i) α-strongly monotone if there exists a constant α > 0 such that (cid:10)
(2.10)
≥ α(cid:2)x − y(cid:2)2, ∀(x, y,λ) ∈ H × H × Ω; (cid:7) Gλ(x) − Gλ(y),x − y
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3
(ii) β- Lipschitz continuous if there exists a constant β > 0 such that
(2.11)
(cid:11) (cid:11) ≤ β(cid:2)x − y(cid:2), ∀(x, y,λ) ∈ H × H × Ω. (cid:11) (cid:11)Gλ(x) − Gλ(y)
Definition 2.4 [1]. A set-valued mapping R : H × Ω → 2H is said to be
(i) relaxed Lipschitz with respect to a mapping f : H × Ω → H if there exists a constant
r ≥ 0 such that (cid:7)
(cid:9) (cid:10) (cid:9) ,x − y ≤ −r(cid:2)x − y(cid:2)2, (cid:8) wλ(x)
fλ
(cid:8) wλ(y) − fλ
(2.12)
∀(x, y,λ) ∈ H × H × Ω, wλ(x) ∈ Rλ(x), wλ(y) ∈ Rλ(y);
(ii) relaxed monotone with respect to a mapping g : H × Ω → H if there exists a constant
s > 0 such that (cid:7)
(cid:9) (cid:10) (cid:9) ,x − y ≥ −s(cid:2)x − y(cid:2)2, (cid:8) wλ(x)
gλ
(cid:8) wλ(y) − gλ
(2.13)
∀(x, y,λ) ∈ H × H × Ω, wλ(x) ∈ Rλ(x), wλ(y) ∈ Rλ(y).
Definition 2.5 [2]. A set-valued mapping A : H × Ω → 2H [A : H × Ω → C(H)] is said to be η-δ-Lipschitz [η- (cid:4)H-Lipschitz ] continuous if there exists a constant η ≥ 0 such that
(cid:9)
(cid:9)
(2.14)
δ (cid:4)H
≤ η(cid:2)x − y(cid:2), ∀(x, y,λ) ∈ H × H × Ω, ≤ η(cid:2)x − y(cid:2), ∀(x, y,λ) ∈ H × H × Ω. (cid:8) Aλ(x),Aλ(y) (cid:8) Aλ(x),Aλ(y)
Lemma 2.6. Let Kλ(x) be defined as (2.5). Then for each fixed λ ∈ Ω, problem (2.4) has a solution (x(λ),uλ(x(λ)),wλ(x(λ)),zλ(x(λ))) if and only if x = x(λ) is a fixed point of the set-valued mapping φ : H × Ω → 2H defined by
(cid:12)
φλ(x) =
(2.15)
uλ(x)∈Aλ(x),wλ(x)∈Rλ(x),zλ(x)∈Tλ(x) (cid:13) x − Gλ(x) + m(x) (cid:8)
(cid:2) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) − (cid:3)(cid:14) , − m(x)
Gλ(x) − ρ
pλ
(cid:8) uλ(x) (cid:8) wλ(x)
fλ
(cid:8) zλ(x) − gλ
+ PKλ
for each x ∈ H, where λ = λ, ρ > 0 is some constant and PKλ(v) is the projection of v ∈ H onto Kλ. Proof. For any fixed λ ∈ Ω , let (x,uλ(x),wλ(x),zλ(x)) be a solution of problem (2.4). Then x ∈ H, uλ(x) ∈ Aλ(x), wλ(x) ∈ Rλ(x) and zλ(x) ∈ Tλ(x) such that Gλ(x) ∈ Kλ(x) and
(cid:15) (cid:16) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9) −
(2.16)
(cid:8) uλ(x)
p λ
(cid:8) wλ(x)
f λ
(cid:8) zλ(x) − g λ
, y − Gλ(x)
≥ 0, ∀y ∈ Kλ(x).
Hence for any ρ > 0,
(cid:8) (cid:9) (cid:8) (cid:9) − (cid:15) Gλ(x) −
(2.17)
p λ (cid:9)(cid:9)(cid:9)(cid:18)
(cid:17) Gλ(x) − ρ (cid:8) − g λ zλ(x) (cid:8) uλ(x) f λ (cid:16) , y − Gλ(x) (cid:8) wλ(x) ≥ 0, ∀y ∈ Kλ(x).
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Parametric problem of quasi-variational inequalities
From Lemmas 2.1 and 2.2, we have
(cid:17) (cid:8) (cid:8) (cid:9) (cid:9) (cid:9)(cid:9)(cid:9)(cid:18) −
Gλ(x) = PK λ(x)
f λ (cid:9)
− g λ (cid:9) (cid:9)(cid:9)(cid:9)
(2.18)
− m(x) (cid:18) .
Gλ(x) − ρ p λ (cid:17) Gλ(x) − ρ
(cid:8) uλ(x) (cid:8) (cid:8) uλ(x)
p λ
(cid:8) wλ(x) (cid:8) (cid:8) − wλ(x)
f λ
(cid:8) zλ(x) (cid:8) − g λ zλ(x) = m(x) + PK λ
We can also write
(cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) (cid:18) − − m(x) (cid:8) uλ(x)
p λ
(cid:8) wλ(x)
f λ
(cid:8) zλ(x) − g λ
+ PK λ
x = x − Gλ(x) + m(x) (cid:8) (cid:17) Gλ(x) − ρ (cid:12)
∈
uλ(x)∈Aλ(x),w λ(x)∈R λ(x),z λ(x)∈T λ(x) (cid:13) x − Gλ(x) + m(x) Gλ(x) − ρ
(cid:2) (cid:8) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) (cid:3)(cid:14) − − m(x) (cid:8) uλ(x)
p λ
(cid:8) wλ(x)
f λ
(cid:8) zλ(x) − g λ = φλ(x),
+ PK λ
(2.19)
that is, x = x(λ) is a fixed point of φλ(x).
Now, for any fixed λ ∈ Ω, let x(λ) be a fixed point of φλ(x). By Lemma 2.1 there exist
uλ(x) ∈ Aλ(x), wλ(x) ∈ Rλ(x) and zλ(x) ∈ Tλ(x) such that
(cid:8) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) (cid:18) − m(x)
p λ
f λ
− g λ (cid:8) zλ(x) (cid:17) − (cid:8) (cid:8) wλ (cid:9) (cid:8) uλ(x) (cid:8) (cid:9) − (cid:9)(cid:9)(cid:9)(cid:18) . (cid:17) Gλ(x) − ρ (cid:8) (cid:8) uλ(x)
p λ
Gλ(x) − ρ
f λ
wλ(x)
− g λ
(x) (cid:8) zλ(x)
Gλ(x) = m(x) + PK λ = PK λ(x)
(2.20)
Hence, we have Gλ(x) ∈ Kλ(x) and
(cid:16) (cid:8) (cid:8) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9)(cid:18) − ≥ 0,
(2.21)
(cid:15) Gλ(x) − (cid:17) Gλ(x) − ρ
p λ
uλ(x)
(cid:8) wλ(x)
f λ
(cid:8) zλ(x) − g λ
, y − Gλ(x)
for all y ∈ Kλ(x).
Noting that ρ > 0, we have
(cid:15) (cid:16) (cid:9) (cid:8) (cid:8) (cid:9) (cid:9)(cid:9) −
(2.22)
(cid:8) uλ(x)
p λ
f λ
wλ(x)
− g λ (cid:8) zλ(x)
, y − Gλ(x)
≥ 0, ∀y ∈ Kλ(x),
(cid:2)
that is, (x,uλ(x),wλ(x),zλ(x)) is a solution of the problem (2.4). Lemma 2.7. Let Kλ(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
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5
relaxed 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
(cid:20)(cid:8) (cid:9)
(r − s) + ξη(q − 1)
(cid:19) (cid:19) (cid:19) (cid:19) (cid:19) (cid:19) (cid:19) < (cid:8) (γχ + σν)2 − (ξη)2 (cid:9)2 (cid:9)2 − (cid:19)ρ − (r − s) + ξη(q − 1) (cid:8) (cid:9)2 (cid:8) γχ + σν ξη (cid:20) (cid:9)
(r − s) > (1 − q)ξη +
q(q − 1)
(cid:9)2 − q(q − 1) (cid:8) (cid:9)2 − (cid:8) γχ + σν ξη (cid:8) (γχ + σν)2 − (ξη)2
ρξη < γχ + σν,
(cid:20) (cid:9)
q = 2
(cid:8) 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, (cid:20)
(2.24)
t(ρ) =
1 − 2ρ(r − s) + ρ2(γχ + σν)2.
Proof. By the definition of φ, for any x, y ∈ H, λ ∈ Ω, a ∈ φλ(x) and b ∈ φλ(y), there exist uλ(x) ∈ Aλ(x), uλ(y) ∈ Aλ(y), wλ(x) ∈ Rλ(x), wλ(y) ∈ Rλ(y), zλ(x) ∈ Tλ(x) and zλ(y) ∈ Tλ(y) such that
(cid:17) (cid:8) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) − − m(x)
Gλ(x) − ρ
pλ
fλ
− gλ
a = x − Gλ(x) + m(x) + PKλ
(cid:8) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) − − m(y) (cid:17) Gλ(y) − ρ (cid:8) uλ(x) (cid:8) uλ(y)
pλ
(cid:8) wλ(x) (cid:8) wλ(y)
fλ
(cid:8) zλ(x) (cid:8) zλ(y) − gλ
b = y − Gλ(y) + m(y) + PKλ
(cid:18) , (cid:18) . (2.25)
Since projection operator is nonexpansive, we have
(cid:2)a − b(cid:2) ≤ 2 (cid:8) Gλ(x) − Gλ(y) (cid:8) (cid:9) (cid:9)(cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:11) (cid:11)
+
− ρ (cid:11) (cid:11) (cid:11) (cid:11)m(x) − m(y) (cid:8) zλ(x)
gλ
(cid:8) zλ(y) − gλ
(2.26)
fλ (cid:9)
(cid:9)(cid:11) (cid:11) + 2 (cid:8) − fλ wλ(y) (cid:9)(cid:11) (cid:11).
+ ρ
(cid:11) (cid:11)x − y − (cid:11) (cid:11)x − y + ρ (cid:11) (cid:8) (cid:11)pλ uλ(x) (cid:8) wλ(y) (cid:8) uλ(y) − pλ
Since G is strongly monotone and Lipschitz continuous, we have
(cid:11) (cid:11)x − y − (cid:8) 1 − 2α + β2 (cid:9) (cid:2)x − y(cid:2)2, (cid:8) Gλ(x) − Gλ(y)
(2.27)
(cid:9) (cid:9) ≤ ξη(cid:2)x − y(cid:2). (cid:9)(cid:11) (cid:11)2 ≤ (cid:11) (cid:11) (cid:11) ≤ μ(cid:2)x − y(cid:2), (cid:11)m(x) − m(y) (cid:11) (cid:9)(cid:11) (cid:8) (cid:11) ≤ ξδ (cid:11) ≤ ξ Aλ(x),Aλ(y) (cid:11) (cid:11)uλ(x) − uλ(y) (cid:11) (cid:11)pλ (cid:8) μλ(x) (cid:8) uλ(y) − pλ
6
Parametric problem of quasi-variational inequalities
(cid:9) (cid:9) (cid:8) (cid:9)(cid:9) (cid:9)(cid:9)(cid:11) (cid:11)2
Again (cid:11) (cid:11)x − y + ρ
(cid:8) gλ (cid:10) (cid:8) wλ(y) (cid:9) − gλ (cid:10) (cid:8) zλ(y) (cid:7) (cid:9) (cid:9) ,x − y (cid:8) zλ(y) − gλ
gλ
(cid:8) zλ(x) (cid:9) ,x − y (cid:9) (cid:8) zλ(x) (cid:9)(cid:9)(cid:11) (cid:11)2 (cid:8) wλ(x) fλ = (cid:2)x − y(cid:2)2 + 2ρ (cid:11) (cid:11) fλ (cid:8) wλ(x) − gλ − 2ρ (cid:8) zλ(y) − fλ (cid:9) − (cid:18) ≤ − fλ (cid:8) (cid:7) wλ(x) fλ (cid:8) (cid:9) + ρ2 − fλ wλ(y) (cid:17) 1 − 2ρ(r − s) + ρ2(γχ + σν)2 − ρ (cid:8) wλ(y) (cid:8) (cid:8) zλ(x) gλ (cid:2)x − y(cid:2)2.
(2.28)
From (2.26)–(2.28), we have
(cid:17)
(2.29)
(cid:2)a − b(cid:2) ≤
q + t(ρ) + ρξη
(cid:18) (cid:2)x − y(cid:2) ≤ θ(cid:2)x − y(cid:2),
where
θ = q + t(ρ) + ρξη,
(cid:20)
t(ρ) =
1 − 2ρ(r − s) + ρ2(γχ + ρν)2,
(2.30)
(cid:20) (cid:22) (cid:21) μ +
1 − 2α + β2
.
q = 2
By the arbitrariness of a and b, we have
(cid:9)
(2.31)
δ
≤ θd(x, y). (cid:8) φλ(x),φλ(y)
By conditions (2.23) and (2.24), we have θ < 1. This proves that θ is a uniform θ-δ-set- valued contraction with respect to λ ∈ Ω. (cid:2) Lemma 2.8 [6]. Let X be a complete metric space and T1,T2 : X → C(X) be θ- (cid:4)H-contraction mapping. Then
(cid:24) (cid:23) (cid:9)(cid:9) ≤ (cid:9) ,
(2.32)
(cid:8) F (cid:9) ,F (cid:4)H (cid:4)H (cid:8) T1 (cid:8) T2 (cid:8) T1(x),T2(x)
1 1 − θ
sup x∈X
where F(T1) and F(T2) are the sets of fixed points of T1 and T2, respectively.
3. Sensitivity analysis
(cid:8)
Theorem 3.1. Assume that Aλ(x), Rλ(x) and Tλ(x) are δ-Lipschitz continuous at λ. Let Rλ(x) be the relaxed Lipschitz continuous with fλ(·) at λ, and Tλ(x) the relaxed monotone with gλ(·) at λ. Suppose that Gλ(x), pλ(·), fλ(·), gλ(·) and PKλ(v) are Lipschitz continuous at λ, where x = x(λ) ∈ S(λ), uλ(x) ∈ Aλ(x), w λ(x) ∈ Rλ(x), z λ(x) ∈ Tλ(x) and (cid:9)
(cid:9)(cid:9)(cid:9) (cid:8) (cid:9) −
(3.1)
− m(x).
v = Gλ(x) − ρ
(cid:8) w λ(x) (cid:8) uλ(x) (cid:8) z λ(x) − g λ
p λ
f λ 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 that x(λ) ∈ φλ(x(λ)). From Lemma 2.6, we have x(λ) ∈ S(λ), hence S(λ) (cid:10)= ∅ and S(λ) coincides with the set of fixed point of φλ(x). In particular, S(λ) coincides with the set of
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7
fixed point of φλ(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 uλ(x(λ)) ∈ Aλ(x(λ)), wλ(x(λ)) ∈ Rλ(x(λ)), zλ(x(λ)) ∈ Tλ(x(λ)), uλ(x(λ)) ∈ Aλ(x(λ)), w λ(x(λ)) ∈ Rλ(x(λ)) and z λ(x(λ)) ∈ Tλ(x(λ)) such that
(cid:9) (cid:9)
(cid:8) x(λ) (cid:9) (cid:9)(cid:9) (cid:8) (cid:9)(cid:9) (cid:9)(cid:9)(cid:9)(cid:9) (cid:9)(cid:18) −
,
(cid:8) x(λ) (cid:8) x(λ) (cid:8) x(λ) − m (cid:8) x(λ) (cid:8) uλ (cid:8) wλ
fλ
(cid:8) zλ − gλ
+ m (cid:8) − ρ (cid:9)
(cid:8) x(λ) (cid:9)
(cid:9)(cid:9) (cid:8) (cid:9)(cid:9)(cid:9)(cid:9) −
x(λ)
+ m (cid:8) −ρ
(cid:8) x (cid:9)(cid:9)(cid:9) (cid:8) λ (cid:8) x(λ) (cid:8) x −m
x(λ)
pλ (cid:8) x(λ) (cid:8) (cid:8) uλ
p λ
(cid:8) w λ
f λ
(cid:8) z λ −g λ
x(λ) = x(λ) − Gλ (cid:8) (cid:17) x(λ) + PKλ Gλ (cid:8) x(λ) = x(λ) − Gλ x(λ) (cid:9) (cid:8) (cid:17) +PK λ Gλ
(cid:8) (cid:9)(cid:9)(cid:18) . λ (3.2)
Write x = x(λ) and x = x(λ). Taking any uλ(x) ∈ Aλ(x), wλ(x) ∈ Rλ(x) and zλ(x) ∈
(cid:2)x − x(cid:2) ≤ (cid:17) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) (cid:18) − − m(x) (cid:8) uλ(x) (cid:8) wλ(x)
fλ
(cid:8) zλ(x) − gλ
Gλ(x) − ρ (cid:9) (cid:8) x
Tλ(x), we have (cid:11) (cid:11)x − Gλ(x) + m(x) (cid:8) + PKλ pλ (cid:17) − x − Gλ (cid:2)
+ m(x) (cid:8)
(cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) (cid:3)(cid:18)(cid:11) (cid:11) − − m(x) (cid:8) uλ(x) (cid:8) wλ(x)
fλ
(cid:8) zλ(x) − gλ
+
(cid:17) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) (cid:18) − − m(x) (cid:8) uλ(x) (cid:8) wλ(x)
fλ
(cid:8) zλ(x) − gλ
(cid:8) (cid:2) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) (cid:3)(cid:18)(cid:11) (cid:11) − − m(x) (cid:8) w λ(x)
f λ
(cid:8) z λ(x) − g λ
(cid:17)
uλ(x) (cid:11) (cid:11) (cid:9)
(cid:8) (cid:9) (cid:9)(cid:9)(cid:9) (cid:18)
+
− (cid:9) − gλ (cid:9) − m(x) (cid:9)(cid:9)(cid:9) (cid:18)(cid:11) (cid:11)
fλ (cid:8) − (cid:8)
(cid:8) wλ(x) (cid:8) w λ(x) (cid:9) (cid:8) zλ(x) (cid:8) z λ(x) (cid:9)(cid:9)(cid:9) − m(x) (cid:18)
+
− m(x)
uλ(x) (cid:9) − (cid:9)
f λ (cid:8)
− g λ (cid:9) (cid:9)(cid:9)(cid:9) (cid:18)(cid:11) (cid:11) − m(x) (cid:9)(cid:11) (cid:11)
f λ (cid:8) w λ(x) (cid:8) w λ(x) f λ (cid:11) (cid:9) (cid:8) (cid:11)pλ uλ(x)
− g λ (cid:8) zλ(x) (cid:8) z λ(x) (cid:8) uλ(x) − g λ − p λ
+ ρ
(cid:8) wλ(x) (cid:9) (cid:11) (cid:11),
+ ρ
Gλ(x) − ρ + PKλ pλ (cid:11) (cid:11)x − Gλ(x) + m(x) (cid:8) Gλ(x) − ρ + PKλ pλ (cid:17) − x − Gλ(x) + m(x) (cid:8) Gλ(x) − ρ + PKλ p λ (cid:11) (cid:11)Gλ(x) − Gλ(x) ≤ θ(cid:2)x − x(cid:2) + (cid:11) (cid:8) (cid:8) (cid:11)PKλ Gλ(x) − ρ uλ(x) pλ (cid:8) (cid:8) (cid:17) − PKλ Gλ(x) − ρ (cid:11) (cid:8) (cid:17) (cid:11)PKλ pλ (cid:8) − PK λ ≤ θ(cid:2)x − x(cid:2) + 2 (cid:11) (cid:11) fλ (cid:11) (cid:11)gλ
(cid:8) zλ(x)
pλ (cid:8) Gλ(x) − ρ uλ(x) (cid:8) (cid:17) Gλ(x) − ρ − uλ(x) pλ (cid:11) (cid:11) (cid:11) + ρ (cid:11)Gλ(x) − Gλ(x) (cid:9)(cid:11) (cid:8) (cid:9) (cid:11) − f λ w λ(x) (cid:9)(cid:11) (cid:8) (cid:11) + − g λ z λ(x)
(cid:11) (cid:11)PKλ(v) − Pkλ(v)
(3.3)
(cid:23) (cid:9) (cid:9) (cid:9) ≤
where, v =Gλ(x) − ρ(p λ(uλ(x)) − ( f λ(w λ(x)) − g λ(z λ(x)))) − m(x). Since, x =x(λ) ∈ S(λ) and x = x(λ) ∈ S(λ) are arbitrary, it follows that (cid:24)(cid:13) 2
(cid:8) S(λ),S(λ) (cid:9)(cid:11) (cid:11) + ρ
δ
(cid:11) (cid:11)pλ
1 1 − θ
(cid:8) uλ(x) (cid:9) (cid:11) (cid:11)Gλ(x) − Gλ(x) (cid:9)(cid:11) (cid:11) + ρ (cid:11) (cid:11) + ρ (cid:11) (cid:8) (cid:11)gλ
zλ(x)
(cid:8) w λ(x) − f λ − p λ (cid:8) z λ(x) − g λ (cid:11) (cid:8) (cid:8) (cid:11) fλ wλ(x) uλ(x) (cid:11) (cid:11) (cid:9)(cid:11) (cid:11) (cid:11)PKλ(v) − PK λ(v) (cid:11) + (cid:14) . (3.4)
8
Parametric problem of quasi-variational inequalities
From the δ-Lipschitz continuity of A, R, T at λ; Lipschitz continuity of G and PKλ(v) at (cid:2) λ, it follows that S(λ) is δ-Lipschitz continuous.
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- (cid:4)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), Rλ(x), Tλ(x) ∈ C(H) and PKλ are continu- ous, follows from (2.15) of φλ(x) ∈ C(H). Now, we show that φλ(x) is a uniform θ- (cid:4)H-contraction mapping with respect to λ ∈ Ω. For any a ∈ φλ(x), there exist uλ(x) ∈ Aλ(x) ∈ C(H), wλ(x) ∈ Rλ(x) ∈ C(H) and zλ(x) ∈ Tλ(x) ∈ C(H) such that
(cid:8) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) − − m(x) (cid:17) Gλ(x) − ρ (cid:8) uλ(x)
pλ
(cid:8) wλ(x)
fλ
(cid:8) zλ(x) − gλ
a = x − Gλ(x) + m(x) + PKλ
(cid:18) . (3.5)
(cid:8) uλ(y) (cid:8) uλ(x) (cid:9) (cid:9) , (cid:9) ,
(3.6)
− fλ (cid:9) (cid:8) Aλ(x),Aλ(y) (cid:8) Rλ(x),Rλ(y) (cid:9) .
Note that (y,λ) ∈ H × Ω; Aλ(y), Rλ(y), Tλ(y) ∈ C(H), then there exist uλ(y) ∈ Aλ(y), wλ(y) ∈ Rλ(y) and zλ(y) ∈ Tλ(y) such that (cid:11) (cid:11) (cid:9)(cid:11) (cid:9) (cid:11) ≤ ξ (cid:4)H (cid:11)uλ(x) − uλ(y) (cid:11) ≤ ξ (cid:11) (cid:11) (cid:9)(cid:11) (cid:11) ≤ χ (cid:4)H (cid:11)wλ(x) − wλ(y) (cid:11) ≤ χ (cid:11) (cid:11) (cid:9)(cid:11) (cid:8) (cid:11) ≤ σ (cid:4)H (cid:11)zλ(x) − zλ(y) (cid:11) ≤ σ Tλ(x),Tλ(y)
(cid:11) (cid:11)pλ (cid:11) (cid:8) (cid:11) fλ wλ(x) (cid:11) (cid:8) (cid:11)gλ zλ(x) − pλ (cid:8) wλ(y) (cid:8) zλ(y) − gλ
Let
(cid:17) (cid:8) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) − − m(y)
Gλ(y) − ρ
pλ
(cid:8) uλ(y) (cid:8) wλ(y)
fλ
(cid:8) zλ(y) − gλ
b = y − Gλ(y) + m(y) + PKλ
(cid:18) , (3.7)
then
(3.8)
b ∈ φλ(y).
(cid:20)
By using the similar argument as in the proof of Lemma 2.7, we can obtain (cid:20)
(cid:9) (cid:13) 2
+
1 − 2ρ(r − s) + ρ2(γχ + σν)2 + ρξη
(cid:14) (cid:2)x − y(cid:2) (cid:2)a − b(cid:2) ≤ (cid:8) μ +
(3.9)
(cid:17) ≤
1 − 2α + β2 (cid:18) (cid:2)x − y(cid:2) ≤ θ(cid:2)x − y(cid:2),
q + t(ρ) + ρξη
where
θ = q + t(ρ) + ρξη,
(cid:20)
1 − 2ρ(r − s) + ρ2(γχ + σν)2,
t(ρ) =
(3.10)
(cid:20) (cid:22) (cid:21) μ +
.
1 − 2α + β2
q = 2
Salahuddin et al.
9
By conditions (2.23) and (2.24), θ < 1, and hence we have
(cid:8) (cid:9)
(3.11)
d
≤ θ(cid:2)x − y(cid:2).
a,φλ(y)
sup a∈φλ(x)
By the similar arguments, we have
(cid:8) (cid:9)
(3.12)
d
≤ θ(cid:2)x − y(cid:2).
φλ(x),b
sup b∈φλ(y)
Hence, by the Hausdorff metric (cid:4)H, we obtain (cid:9)
(3.13)
(cid:4)H ≤ θ(cid:2)x − y(cid:2). (cid:8) φλ(x),φλ(y)
Therefore φλ(x) is a uniform θ- (cid:4)H-contraction mapping with respect to λ ∈ Ω.
(ii) Since φλ(x) is a uniform θ- (cid:4)H-contraction with respect to λ ∈ Ω, hence by Nadler theorem [8], φλ(x) has a fixed point x(λ). Since S(λ) (cid:10)= ∅, then let {xn} ⊂ S(λ) and xn → x0 as n → ∞. Therefore,
(3.14)
n = 1,2,....
xn ∈ φλ(xn),
From (i), we have
(cid:9)
(3.15)
(cid:4)H (cid:8) φλ(xn),φλ(x0) ≤ θ(cid:2)xn − x0(cid:2).
If follows that
(cid:8) (cid:9) (cid:9) (cid:8) (cid:9) ≤
d
x0,φλ(x0)
φλ(xn),φλ(x0)
(3.16)
(cid:8) xn,φλ(xn) (cid:11) (cid:11) −→ 0,
+ (cid:4)H as n −→ ∞,
(cid:11) (cid:11) (cid:11) + d (cid:11)x0 − xn (cid:11) (cid:11)xn − x0 ≤ (1 + θ)
(cid:2)
hence x0 ∈ φλ(x0) and x0 ∈ 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 (cid:4)H-Lipschitz continuous at λ.
Proof. From Theorem 3.2(ii), the solution set S(λ) of (2.4) is a nonempty closed set in H. Now, we show that S(λ) is (cid:4)H-Lipschitz continuous at λ. By Theorem 3.2(i), φλ(x) and φλ(x) are both θ- (cid:4)H-contraction mappings. From Lemma 2.8, we have
(cid:24) (cid:23) (cid:8) (cid:9) ≤
(3.17)
(cid:4)H
S(λ),S(λ)
(cid:9) . (cid:4)H (cid:8) φλ(x),φλ(x)
1 1 − θ
sup x∈H
Taking any a ∈ φλ(x), ∃uλ(x) ∈ Aλ(x), wλ(x) ∈ Rλ(x) and zλ(x) ∈ Tλ(x) such that (cid:8)
(cid:9)(cid:9)(cid:9) (cid:8) (cid:9) (cid:9) − − m(x) (cid:17) Gλ(x) − ρ (cid:8) uλ(x)
pλ
(cid:8) wλ(x)
fλ
(cid:8) zλ(x) − gλ
a = x − Gλ(x) + m(x) + PKλ
(cid:18) . (3.18)
10
Parametric problem of quasi-variational inequalities
For uλ(x) ∈ Aλ(x) ∈ C(H), wλ(x) ∈ Rλ(x) ∈ C(H), zλ(x) ∈ Tλ(x) ∈ C(H), there exist uλ(x) ∈ Aλ(x), wλ(x) ∈ Rλ(x) and zλ(x) ∈ Tλ(x) such that
(cid:9) , (cid:9) ,
(3.19)
(cid:8) Aλ(x),Aλ(x) (cid:8) Rλ(x),Rλ(x) (cid:9) . (cid:11) (cid:11) (cid:11) ≤ (cid:4)H (cid:11)uλ(x) − uλ(x) (cid:11) (cid:11) (cid:11) ≤ (cid:4)H (cid:11)wλ(x) − wλ(x) (cid:11) (cid:11) (cid:8) (cid:11) ≤ (cid:4)H (cid:11)zλ(x) − zλ(x) Tλ(x),Tλ(x)
Let
(cid:8) (cid:9) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) − − m(x) (cid:17) Gλ(x) − ρ (cid:8) uλ(x)
p λ
(cid:8) wλ(x)
f λ
(cid:8) zλ(x) − g λ
b = x − Gλ(x) + m(x) + PK λ
(cid:18) , (3.20)
then
(3.21)
b ∈ φλ(x).
(cid:11) (cid:11) (cid:2)a − b(cid:2) ≤ (cid:9) (cid:8) (cid:8) (cid:9) (cid:9)(cid:9)(cid:9) (cid:3)
fλ (cid:8)
− (cid:9) − gλ (cid:9) − m(x) (cid:9)(cid:9)(cid:9) (cid:11) (cid:11) − m(x)} − (cid:8) (cid:8) wλ(x) (cid:8) wλ(x) (cid:9) (cid:8) zλ(x) (cid:8) zλ(x) (cid:9)(cid:9)(cid:9) (cid:3)
+
− m(x)
f λ (cid:8)
(cid:8) uλ(x) (cid:8) uλ(x) (cid:9) − (cid:9) (cid:9)(cid:9)(cid:9) (cid:3)(cid:11) (cid:11) − m(x) − g λ (cid:8) zλ(x) (cid:8) zλ(x) − (cid:9) − g λ (cid:9) − g λ (cid:9)(cid:11) (cid:11) ≤ 2
f λ (cid:8) wλ(x) (cid:8) wλ(x) (cid:8) uλ(x) (cid:9)
(cid:9)
(3.22)
(cid:9)(cid:11) (cid:11)
p λ (cid:8) uλ(x) (cid:8) uλ(x) (cid:8) uλ(x) (cid:9)(cid:11) (cid:11) + ρ
(cid:8) zλ(x) − g λ
+
(cid:9) (cid:9) (cid:8) uλ(x) (cid:8) uλ(x) (cid:9) (cid:9) (cid:9)(cid:11) (cid:11) (cid:9)(cid:11) (cid:11)
+ ρ
− p λ (cid:8) − f λ wλ(x) (cid:9)(cid:11) (cid:8) (cid:11) zλ(x) − g λ
+
It follows that (cid:11) (cid:11)Gλ(x) − Gλ(x) (cid:11) (cid:11)PKλ {Gλ(x) − ρ + pλ (cid:8) {Gλ(x) − ρ − PKλ (cid:11) (cid:8) (cid:2) (cid:11)PKλ Gλ(x) − ρ p λ (cid:8) (cid:2) Gλ(x) − ρ − PK λ p λ f λ (cid:11) (cid:11) (cid:11) (cid:11)pλ (cid:11) + ρ (cid:11)Gλ(x) − Gλ(x) − p λ (cid:11) (cid:11) (cid:8) (cid:8) (cid:8) (cid:11) fλ (cid:11)gλ − f λ wλ(x) + ρ wλ(x) zλ(x) (cid:11) (cid:11) (cid:11) (cid:11) (cid:11)Gλ(x) − Gλ(x) (cid:11)PKλ(v) − PK λ(v) (cid:11) (cid:11) ≤ 2 (cid:11) (cid:11) (cid:9)(cid:11) (cid:8) (cid:8) (cid:11)p λ (cid:11)pλ (cid:11) + ρ − p λ uλ(x) uλ(x) + ρ (cid:11) (cid:11) (cid:9)(cid:11) (cid:8) (cid:8) (cid:8) (cid:11) f λ (cid:11) fλ (cid:11) + ρ − f λ wλ(x) wλ(x) wλ(x) (cid:11) (cid:11) (cid:9)(cid:11) (cid:9) (cid:8) (cid:8) (cid:9) (cid:8) (cid:11)g λ (cid:11)gλ (cid:11) + ρ − g λ zλ(x) + ρ zλ(x) zλ(x) (cid:11) (cid:11) (cid:11), (cid:11)PKλ(v) − PK λ(v)
where v = Gλ(x) − ρ(p λ(uλ(x)) − ( f λ(wλ(x)) − g λ(zλ(x)))) − m(x).
Write
(cid:9) (cid:9)(cid:11) (cid:11)
M = 2
(cid:9) (cid:8) uλ(x) (cid:9) (cid:8) (cid:9)(cid:11) (cid:11) − f λ − p λ (cid:11) (cid:8) (cid:11)gλ zλ(x) (cid:9) (cid:9)
(3.23)
(cid:11) (cid:11) (cid:11)pλ (cid:11) + ρ (cid:8) wλ(x) (cid:9) + ρχ (cid:4)H − g λ + ρσ (cid:4)H (cid:8) uλ(x) (cid:9)(cid:11) (cid:11) + ρ (cid:8) Rλ(x),Rλ(x)
zλ(x) (cid:8) Tλ(x),Tλ(x)
+
(cid:11) (cid:11)Gλ(x) − Gλ(x) (cid:11) (cid:8) (cid:11) fλ + ρ wλ(x) (cid:8) + ρξ (cid:4)H Aλ(x),Aλ(x) (cid:11) (cid:11) (cid:11). (cid:11)PKλ(v) − PK λ(v)
Salahuddin et al.
11
Then we have
(cid:8) (cid:9)
(3.24)
d
≤ M.
a,φλ(x)
sup a∈φλ(x)
By the similar arguments, we obtain
(cid:8) (cid:9)
(3.25)
d
≤ M.
φλ(x),b
sup b∈φ λ(x)
It follows that
(cid:9)
(3.26)
(cid:4)H ≤ M. (cid:8) φλ(x),φλ(x)
If Aλ(x), Rλ(x) and Tλ(x) are uniformly (cid:4)H-Lipschitz continuous at λ with respect to x ∈ H, and Gλ(x), PKλ(v) are uniformly Lipschitz continuous at λ with respect to x ∈ H, then it follows that: for any (cid:2) > 0, there exists a δ > 0 such that for all λ ∈ Ω with (cid:2)λ − λ(cid:2) < δ,
(cid:9)
(3.27)
(cid:4)H ≤ M < (cid:2), ∀x ∈ H. (cid:8) φλ(x),φλ(x)
From (3.17), we obtain
(cid:9)
(3.28)
,
(cid:8) S(λ),S(λ)
<
H
(cid:2) 1 − θ
hence S(λ) is (cid:4)H-continuous at λ. If Aλ(x), Rλ(x) and Tλ(x) are uniformly (cid:4)H-Lipschitz continuous at λ with respect to x ∈ H, and Gλ(x), PKλ(v) are also uniformly Lipschitz continuous at λ with respect to x ∈ H, then by the above arguments, we can prove that S(λ) is (cid:4)H-Lipschitz continuous. (cid:2)
Acknowledgment
A. H. Siddiqi would like to thank King Fahd University of Petroleum Minerals, Dhahran, Saudi Arabia, for providing excellent research environment.
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Salahuddin: Department of Mathematics, Aligarh Muslim University, Aligarh 202002 (UP), India E-mail address: salahuddin12@mailcity.com
M. K. Ahmad: Department of Mathematics, Aligarh Muslim University, Aligarh 202002 (UP), India E-mail address: ahmad kalimuddin@yahoo.co.in
A. H. Siddiqi: Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals, P.O. Box 1745. Dhahran 31261, Saudi Arabia E-mail address: ahasan@kfupm.edu.sa
