Báo cáo hóa học: "PARAMETRIC PROBLEM OF COMPLETELY GENERALIZED QUASI-VARIATIONAL INEQUALITIES" pptx

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, dist ribution, 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 x 2 =x, x,2 H the family of all nonempty bounded subsets of H and C(H) the family of al l nonempty compact subsets of H.Letδ :2 H → [0,∞)bedefinedby δ(A, B) = sup   a − b : a ∈ A, b ∈ B  , ∀A,B ∈ 2 H , (2.1) and let  H : C(H) → [0,∞)bedefinedby  H(A,B) = max  sup x∈A d(x,B), sup y∈B d(A, y)  , ∀A,B ∈ C(H), (2.2) 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 d(x,B) = inf y∈B x − y. (2.3) Then, (2 H ,δ)and(C(H),  H) are complete metric spaces,  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 → 2 H set-valued mapping with nonempty closed convex valued. Let A,R,T : H × Ω → 2 H be the set-valued mappings and p, f ,g,G : H × Ω → H the single- valued mappings. For each fixed λ ∈ Ω,wewriteG λ (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 finding x ∈ H, u λ (x) ∈ A λ (x), w λ (x) ∈ R λ (x), z λ (x) ∈ T λ (x)suchthatG 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)hastheform K λ (x) = m(x)+K λ , ∀(x,λ) ∈ H × Ω, (2.5) where m : H → H and {K λ : λ ∈ Ω} is a family of n onempty closed and convex subsets of H,see,forexample,[13] and the references therein. For each λ ∈ Ω,letS(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) =∅near S( λ) and the set- valued mappings S(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, x = P K (v) (2.6) if and only if x − v, y − v≥0, ∀y ∈ K, (2.7) 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 K(x) = m(x)+K, ∀x ∈ H. (2.8) Then P K(x) (y) = m(x)+P K  y − m(x)  , ∀x, y ∈ H. (2.9) Definit ion 2.3 [12]. A single-valued mapping G : H × Ω → H is called: (i) α-strongly monotone if there exists a constant α>0suchthat  G λ (x) − G λ (y),x − y  ≥ αx − y 2 , ∀(x, y,λ) ∈ H × H × Ω; (2.10) Salahuddin et al. 3 (ii) β- Lipschitz continuous if there exists a constant β>0suchthat   G λ (x) − G λ (y)   ≤ βx − y, ∀(x, y,λ) ∈ H × H × Ω. (2.11) Definit ion 2.4 [1]. A set-valued mapping R : H × Ω → 2 H is said to be (i) relaxed Lipschitz with respect to a mapping f : H × Ω → H if there exists a constant r ≥ 0suchthat  f λ  w λ (x)  − f λ  w λ (y)  ,x − y  ≤− rx − y 2 , ∀(x, y,λ) ∈ H × H × Ω, w λ (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>0suchthat  g λ  w λ (x)  − g λ  w λ (y)  ,x − y  ≥− sx − y 2 , ∀(x, y,λ) ∈ H × H × Ω, w λ (x) ∈ R λ (x), w λ (y) ∈ R λ (y). (2.13) Definit ion 2.5 [2]. A set-valued mapping A : H × Ω → 2 H [A : H × Ω → C(H)] is said to be η-δ-Lipschitz [η-  H-Lipschitz ] continuous if there exists a constant η ≥ 0suchthat 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)hasa 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 → 2 H defined by φ λ (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)   , (2.15) for each x ∈ H,whereλ = 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). Then x ∈ H, u λ (x) ∈ A λ (x), w λ (x) ∈ R λ (x)andz λ (x) ∈ T λ (x)suchthatG 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 any ρ>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) 4 Parametric problem of quasi-variational inequalities From Lemmas 2.1 and 2.2,wehave 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 ca n al so w rite 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)isafixedpointofφ λ (x). Now, for any fixed λ ∈ Ω,letx(λ)beafixedpointofφ λ (x). By Lemma 2.1 there exist u λ (x) ∈ A λ (x), w λ (x) ∈ R λ (x)andz λ (x) ∈ T λ (x)suchthat 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 have G 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 all y ∈ K λ (x). Noting that ρ>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 → 2 H the δ-Lipschitz continuous with respect to constants η,γ,ν, re spectively, 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 w ith constant α>0, R relaxed Lipschitz continuous with respect to f with constant r ≥ 0, T Salahuddin et al. 5 relaxed monotone with respect to g with constant s>0,andm : H → H is μ-Lipschitz con- tinuous. If there exists a constant ρ>0 such that     ρ − (r − s)+ξη(q − 1)  γχ + σν  2 −  ξη  2     <   (r − s)+ξη(q − 1)  2 − q(q − 1)  (γχ + σν) 2 − (ξη) 2   γχ + σν  2 −  ξη  2 (r − s) > (1 − q)ξη+  q(q − 1)  (γχ + σν) 2 − (ξη) 2  ρξη < γχ + σν, q = 2  μ +  1 − 2α + β 2  < 1, (2.23) then the set-valued mapping φ : H × Ω → 2 H defined by (2.15)isauniformθ-δ-set-valued contraction with respect to λ ∈ Ω,where θ = q + t(ρ)+ρξη < 1, t(ρ) =  1 − 2ρ(r − s)+ρ 2 (γχ + σν) 2 . (2.24) Proof. By the definition of φ,foranyx, y ∈ H, λ ∈ Ω, a ∈ φ λ (x)andb ∈ φ λ (y), there exist u 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)suchthat 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  u λ (y)    . (2.26) Since G is strongly monotone and Lipschitz continuous, we have   x − y −  G λ (x) − G λ (y)    2 ≤  1 − 2α + β 2   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) 6 Parametric problem of quasi-variational inequalities Again   x − 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  − 2ρ  g λ  z λ (x)  − g λ  z λ (y)  ,x − y  + ρ 2   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)+ρ 2 (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)+ρ 2 (γχ + ρν) 2 , q = 2  μ +  1 − 2α + β 2  . (2.30) By the arbitrariness of a and b,wehave δ  φ λ (x), φ λ (y)  ≤ θd(x, y). (2.31) 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.  Lemma 2.8 [6]. Let X be a complete metric space and T 1 ,T 2 : X → C(X) be θ-  H-contraction mapping. Then  H  F  T 1  ,F  T 2  ≤  1 1 − θ  sup x∈X  H  T 1 (x), T 2 (x)  , (2.32) where F(T 1 ) and F(T 2 ) are the sets of fixed points of T 1 and T 2 ,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,andT λ (x) the relaxed monotone with g λ (·) at λ.SupposethatG λ (x), p λ (·), f λ (·), g λ (·) and P K λ (v) are Lipschitz continuous at λ,wherex = 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)  − m(x). (3.1) Then for all λ ∈ Ω, the solution set S(λ) of the problem (2.4)isnonemptyandS(λ) is δ- Lipschitz continuous at λ. Proof. For each fixed λ ∈ Ω, φ λ (x) has a fixed point, that is, there exists a x(λ) ∈ H such that x(λ) ∈ φ λ (x( λ)). From Lemma 2.6,wehavex(λ) ∈ S(λ), hence S(λ) =∅and S(λ) coincides with the set of fixed point of φ λ (x). In particular, S(λ) coincides with the set of Salahuddin et al. 7 fixed point of φ λ (x). Now we show that S(λ)isδ-Lipschitz continuous at λ.Forallx(λ) ∈ S(λ)andx(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) Write x = x(λ)andx = x(λ). Taking any u λ (x) ∈ A λ (x), w λ (x) ∈ R λ (x)andz 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 (z λ (x)))) − m(x). Since, x = x(λ) ∈ S(λ) and x = 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) 8 Parametric problem of quasi-variational inequalities From the δ-Lipschitz continuity of A, R, T at λ; Lipschitz continuity of G and P K λ (v)at λ, it follows that S(λ)isδ-Lipschitz continuous.  Theorem 3.2. If we assume the hy pothesis 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),closedinH. Proof. (i) For each (x,λ) ∈ H × Ω; A λ (x), R λ (x), T λ (x) ∈ C(H)andP K λ are continu- ous, follows from (2.15)ofφ λ (x) ∈ C(H). Now, we show that φ λ (x)isauniformθ-  H-contrac tion mapping with respect to λ ∈ Ω.Foranya ∈ φ λ (x), there exist u λ (x) ∈ A λ (x) ∈ C(H), w λ (x) ∈ R λ (x) ∈ C(H)andz λ (x) ∈ T λ (x) ∈ C(H)suchthat 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; A λ (y), R λ (y), T λ (y) ∈ C(H), then there exist u λ (y) ∈ A λ (y), w λ (y) ∈ R λ (y)andz λ (y) ∈ T λ (y)suchthat   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  H  R λ (x), R λ (y)  ,   g λ  z λ (x)  − g λ  z λ (y)    ≤ σ   z λ (x) − z λ (y)   ≤ σ  H  T 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 b ∈ φ λ (y). (3.8) By using the similar argument as in the proof of Lemma 2.7,wecanobtain a − b≤  2  μ +  1 − 2α + β 2  +  1 − 2ρ(r − s)+ρ 2 (γχ + σν) 2 + ρξη   x − y ≤  q + t(ρ)+ρξη   x − y≤θx − y, (3.9) where θ = q + t(ρ)+ρξη, t(ρ) =  1 − 2ρ(r − s)+ρ 2 (γχ + σν) 2 , q = 2  μ +  1 − 2α + β 2  . (3.10) Salahuddin et al. 9 By conditions (2.23)and(2.24), θ<1, and hence we have sup a∈φ λ (x) d  a,φ λ (y)  ≤ θx − y. (3.11) By the similar arguments, we have sup b∈φ λ (y) d  φ λ (x), b  ≤ θx − y. (3.12) Hence, by the Hausdorff metric  H,weobtain  H  φ λ (x), φ λ (y)  ≤ θx − y. (3.13) Therefore φ λ (x)isauniformθ-  H-contraction mapping with respect to λ ∈ Ω. (ii) Since φ λ (x)isauniformθ-  H-contraction with respect to λ ∈ Ω, hence by Nadler theorem [8], φ λ (x)hasafixedpointx(λ). Since S(λ) =∅,thenlet{x n }⊂S(λ)andx n → x 0 as n →∞. Therefore, x n ∈ φ λ (x n ), n = 1,2, (3.14) From (i), we have  H  φ λ (x n ),φ λ (x 0 )  ≤ θx n − x 0 . (3.15) If follows that d  x 0 ,φ λ (x 0 )  ≤   x 0 − x n   + d  x n ,φ λ (x n )  +  H  φ λ (x n ),φ λ (x 0 )  ≤ (1 + θ)   x n − x 0   −→ 0, as n −→ ∞ , (3.16) hence x 0 ∈ φ λ (x 0 )andx 0 ∈ S(λ). Therefore S(λ)isclosedinH.  Theorem 3.3. AssumethehypothesisasinTheorem 3.1. Then for all λ ∈ Ω, the solution set S(λ) of (2.4)isnonemptyandS(λ) is  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  H-Lipschitz continuous at λ.ByTheorem 3.2(i), φ λ (x)and φ λ (x) are both θ-  H-contraction mappings. From Lemma 2.8,wehave  H  S(λ),S(λ)  ≤  1 1 − θ  sup x∈H  H  φ λ (x), φ λ (x)  . (3.17) Taking any a ∈ φ λ (x), ∃u λ (x) ∈ A λ (x), w λ (x) ∈ R λ (x)andz λ (x) ∈ T λ (x)suchthat 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) 10 Parametric problem of quasi-variational inequalities For 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)andz λ (x) ∈ T λ (x)suchthat   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 b ∈ φ λ (x) . (3.21) 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) where v = G λ (x) − ρ(p λ (u λ (x)) − ( f λ (w λ (x)) − g λ (z 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 λ (v) − P K λ (v)   . 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University of Petroleum Minerals, Dhahran, Saudi Arabia, for providing excellent research environment References [1] R Ahmad, K R Kazmi, and Salahuddin, Completely generalized non-linear variational inclusions involving relaxed Lipschitz and relaxed monotone mappings, Nonlinear Analysis Forum 5 (2000), 61–69 [2] R W Cottle, F Giannessi, and J L Lions, Variational Inequalities and Complementarity Problems,... 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 . 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. 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. 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