Sensitivity analysis for a parametric generalized variational-like inequality problem: a P-η-proximal mapping approach

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Adv. Inequal. Appl. 2019, 2019:15 https://doi.org/10.28919/aia/4357 ISSN: 2050-7461

SENSITIVITY ANALYSIS FOR A PARAMETRIC GENERALIZED

VARIATIONAL-LIKE INEQUALITY PROBLEM: A P-η-PROXIMAL MAPPING

APPROACH

F.A. KHAN∗, R.H. ALJAHANI, A.A. ALATAWI, J.S. ALATAWI

Department of Mathematics, University of Tabuk, Tabuk-71491, KSA

Copyright c2019 the author(s). 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.

Abstract. In this paper, usingP-η-proximal mapping, we study the existence and sensitivity analysis of solution of a parametric generalized variational-like inequality problem in uniformly smooth Banach space. The approach

used in this paper may be treated as an extension and unification of approaches for studying sensitivity analysis

of solution for various important classes of variational inequalities (inclusions) given by many authors, see for

example [2-7,9-11,13-16,18,19].

Keywords: parametric generalized variational-like inequality;P-η-proximal mapping; existence and sensitivity analysis of solution.

2010 AMS Subject Classification:47J25, 49J40, 49J53, 47H05.

1.

INTRODUCTION

In recent years, much attention has been given to develop general techniques for the

sen-sitivity analysis of solution of various classes of variational inequalities (inclusions). From the

∗_{Corresponding author}

E-mail address: faizan [email protected]

Received November 27, 2019

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mathematical and engineering point of view, sensitivity properties of various classes of

varia-tional inequalities can provide new insight concerning the problem being studied and can

stim-ulate ideas for solving problems. The sensitivity analysis of solution for variational inequalities

have been studied extensively by many authors using quite different techniques. By using the

projection technique, Dafermos [3], Ding and Luo [6], Mukherjee and Verma [14], Parida et al. [17], Park and Jeong [18] and Yen [19] studied the sensitivity analysis of solution of some

classes of variational inequalities with single-valued mappings. By using proximal (resolvent)

mapping technique, Adly [1], Agarwalet al. [2], Ding [4] and Noor [15] studied the sensitivity analysis of solution of some classes of variational inclusions with single-valued mappings.

Recently, by using projection and proximal mapping techniques, Ding [5], Kazmi and

Bhat [9], Kazmi and Khan [10,11], Lim [12], Liuet al. [13] and Noor [16] studied the behavior

and sensitivity analysis of solution set for some important classes of parametric variational

inequalities (inclusions) with single and set-valued mappings. It is worth mentioning that most

of the results in this direction have been obtained in the setting of Hilber space.

Inspired by recent research works in this area, in this paper, we consider a parametric

generalized variational-like inequality problem (in short, PGVLIP) in uniformly smooth Banach

space. Further, usingP-η-proximal mapping, we study the existence and sensitivity analysis of

solution of PGVLIP. Our results extend, improve, and unify the corresponding results given by

many authors, see for example [2-7,9-11,13-16,18,19].

2.

PRELIMINARIES

We assume thatEis a real Banach space equipped with normk · k.Leth·,·idenote the dual pair between E and its dual space E∗ and let J:E →2E∗ be the normalized duality mapping

defined by

J(u) = {f ∈E∗: hu,fi = kuk2, kuk = kfkE∗}, u∈E. (2.1) We note that ifE is smooth, thenJis single-valued and ifE ≡H, a Hilbert space, thenJ is the identity map onH. In sequel, we shall denote a selection of normalized duality mapping Jby

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First, we recall the following concepts and results.

Definition 2.1[9]. LetP:E→E∗, g:E→E, andη:E×E →E be single-valued mappings.

Then

(i) Pis said to beα-stronglyη-monotone, if there exists a constantα >0 such that

hP(u)−P(v),η(u,v)i ≥αku−vk2, ∀u,v∈E;

(ii) gis said to beβ-strongly accretive, if there exists a constantβ >0 and for anyu,v∈E,

j(u−v)∈J(u−v)such that

hg(u)−g(v),j(u−v)i ≥βku−vk2;

(iii) η is said to beτ-Lipschitz continuous, if there exists a constantτ>0 such that

kη(u,v)k ≤τku−vk, ∀u,v∈E.

Definition 2.2[4].Letη:E×E→Ebe a single-valued mapping. A proper functionalφ :E→

R∪ {+∞}is said to beη-subdifferentiable at a pointu∈E, if there exists a point f∗∈E∗such

that

φ(v)−φ(u)≥ hf∗,η(v,u)i, ∀v∈E,

where f∗is calledη-subgradient ofφ atu. The set of allη-subgradients ofφ atuis denoted by

∂ηφ(u). The mapping∂ηφ :E→2E

∗

defined by

∂ηφ(u) ={f∗∈E∗:φ(v)−φ(u)≥ hf∗,η(v,u)i, ∀v∈E} (2.2)

is said to beη-subdifferential ofφ atu.

Definition 2.3[8].A functionalφ :E×E→_R∪ {+∞}is said to be 0-diagonally quasi-concave

(0-DQCV) inu, if for any finite set{u1, ...,un} ⊂E and for anyv=∑n_i=1λiui withλi≥0 and

∑n_i₌₁λi=1, min1≤i≤nφ(ui,v)≤0 holds.

Definition 2.4[9]. Letη :E×E→E be a single-valued mapping. Letφ :E →_R∪ {+∞}be

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P:E →E∗ be a nonlinear mapping. If for any given pointu∗∈E∗ andρ >0, there exists a

unique pointu∈Esatisfying

hP(u)−u∗,η(v,u)i+ρ φ(v)−ρ φ(u)≥0, ∀v∈E,

then the mappingu∗7→u, denoted byP∂ηφ

ρ (u∗), is calledP-η-proximal mapping ofφ. Clearly, u∗−P(u)∈ρ ∂ηφ(u)and then it follow that

P∂ηφ

ρ (u

∗_{) = (}_P₊

ρ ∂ηφ)

−1₍_u∗₎_. ₍

2.3)

Remark 2.1[9].

(i) Ifη(v,u) =v−u,for allu,v∈Eandφ is a lower semicontinuous and proper functional

onE, then theP-η-proximal mapping ofφ reduces to the P-η-proximal mapping ofφ

discussed by Ding and Xia [7].

(ii) If E =H, a Hilbert space, η(v,u) =v−u, for all u,v∈H and φ is a convex, lower

semicontinuous and proper functional onE, andP is the identity mapping onH, then theP-η-proximal mapping ofφ reduces to the usual proximal (resolvent) mapping ofφ

on Hilbert space.

Lemma 2.1[9]. Let E be a real reflexive Banach space; let η :E×E →E be a continuous

mapping such thatη(v,v0) +η(v0,v),for allv,v0∈E; letP:E→E∗beα-stronglyη-monotone

continuous mapping; let, for any given u∗∈E∗, the function h(v,u) =hu∗−P(u),η(v,u)ibe 0-DQCV in vand let φ :E→_R∪ {+∞} be a lower semicontinuous, η-subdifferentiable and

proper functional onE.Then for any given constantρ >0 andu∗∈E∗,there exists a unique

u∈E such that

hP(u)−u∗,η(v,u)i ≥ρ φ(u)−ρ φ(v), ∀v∈E, (2.4)

that is,u=P∂ηφ

ρ (u∗).

Remark 2.2[9]. Lemma 2.1 shows that for any strongly monotone continuous mapping P:

E →E∗ andρ >0,theP-η-proximal mappingPρ∂ηφ :E∗→E of a lower semicontinuous,η -subdifferentiable and proper functionalφ is well defined and for eachu∗∈E∗, u=P_ρ∂ηφ(u∗)is

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Lemma 2.2[9].LetEbe a real reflexive Banach space and letη:E×E→Ebeτ-Lipschitz

con-tinuous such thatη(v,v0) +η(v0,v),for allv,v0∈E; letP:E→E∗beα-stronglyη-monotone

continuous mapping; let, for any given u∗∈E∗, the function h(v,u) =hu∗−P(u),η(v,u)ibe 0-DQCV inv; letφ :E→_R∪ {+∞}be a lower semicontinuous,η-subdifferentiable and proper

functional onEand letρ>0 be any given constant. ThenP-η-proximal mappingPρ∂ηφ ofφ is τ

δ-Lipschitz continuous.

Throughout the rest of the paper unless otherwise stated, letE be a real uniformly smooth

Banach space withρE(q)≤cq2for somec>0, where the modulus of smoothness ofE is the functionρE :[0,∞)→[0,∞), defined below in the Lemma 2.3.

Lemma 2.3[4,11]. LetE be a real uniformly smooth Banach space and letJ:E →E∗be the

normalized duality mapping. Then, for allu,v∈E, we have

(a) ku+vk2_{≤ k}_u_k2₊_2h_v_,_J₍_u₊_v₎_i;

(b) hu−v,J(u)−J(v)i ≤2d2ρE(4ku−vk/d), whered= p

(kuk2₊_k_v_k2₎_/₂_,

ρE(q) =sup

n₍_k_u_k₊_k_v_k₎

2 −1 : kuk=1, kvk=q o

.

3.

FORMULATION OFPROBLEM

Let T,A,S:E →E∗, g:E →E, η :E×E →E, N :E∗×E∗×E∗→E∗ be nonlinear

mappings. Assume thatφ :E×E→R∪ {+∞}is a lower semicontinuous,η-subdifferentiable

(may not be convex) and proper functional such thatg(u)∈∂ηφ(u,z),for allu,z∈E,then we consider the following generalized variational-like inequality problem (in short, GVLIP): find

u∈E such that

hN(T(u),A(u),S(u)),η(v,g(u))i+φ(v,u)−φ(g(u),u)≥0, ∀v∈E. (3.1)

For a suitable choices of mappingsT,A,S,N,P,g,P◦g,φ,η and the spaceE, it is easy to

see that GVLIP (3.1) includes a number of known classes of variational inequalities studied by

many authors as special cases, see for example [3,6,11,12,14,17-19].

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Let Ω be a nonempty open subset of E in which the parameter λ takes the values. Let

T,A,S:E×Ω→E∗, g:E×Ω→E, η :E×E→E, N:E∗×E∗×E∗×Ω→E∗ be

single-valued mappings. Assume that φ :E×E×Ω→R∪ {+∞} is a lower semicontinuous, η

-subdifferentiable (may not be convex) and proper functional such thatg(u,λ)∈∂ηφ(u,v,λ),for allu,v∈E,λ∈Ω. We consider the following parametric generalized variational-like inequality

problem (in short, PGVLIP): findu∈E such that

hN(T(u,λ),A(u,λ),S(u,λ),λ),η(v,g(u,λ))i+φ(v,u,λ)−φ(g(u,λ),u,λ)≥0,∀v∈E. (3.2)

The aim of this paper is to study the existence and sensitivity analysis of the solution for

PGVLIP (3.2), and the conditions on these mappingsT,A,S,N,P,g,P◦g,φ,η,under which the

solution of PGVLIP (3.2) is nonempty and Lipschitz continuous with respect to the parameter

λ ∈Ω.

4.

EXISTENCE ANDSENSITIVITY ANALYSIS OFSOLUTION

First, we prove the following technical lemma.

Lemma 4.1. u∈Eis the solution of PGVLIP (3.2) if and only if it satisfies the relation

g(u,λ) =P_ρ∂ηφ(·,u,λ)[P◦g(u,λ)−ρN(T(u,λ),A(u,λ),S(u,λ),λ)], (4.1)

where P∂ηφ(·,u,λ)

ρ = (P+ρ ∂ηφ(·,u,λ))−1 is the P-η-proximal mapping of φ for each fixed u∈E, λ ∈Ω, P:E→E∗, P◦g(·,λ)denotesPcompositiong(·,λ),andρ>0 is a constant.

Proof.Assume thatu∈E satisfies (4.1), that is,

g(u,λ) =P_ρ∂ηφ(·,u,λ)[P◦g(u,λ)−ρN(T(u,λ),A(u,λ),S(u,λ),λ)]. (4.2)

SinceP∂ηφ(·,u,λ)

ρ = (P+ρ ∂ηφ(·,u,λ))−1,the above relation holds if and only if

P◦g(u,λ)−ρN(T(u,λ),A(u,λ),S(u,λ),λ)∈P◦g(u,λ) +ρ ∂ηφ(g(u,λ),u,λ). (4.3) By the definition ofη-subdifferential ofφ(g(u,λ),u,λ),the above inclusion holds if and only

if

φ(v,u,λ)−φ(g(u,λ),u,λ)≥ hN(T(u,λ),A(u,λ),S(u,λ),λ),η(v,g(u,λ))i, ∀v∈E, (4.4)

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Now, assume that for some ¯λ∈Ω,PGVLIP (3.2) has a solution ¯uandKis a closed sphere

inE centered at ¯u.We are interested in investigating those conditions under which, for eachλ

in a neighborhood of ¯λ,PGVLIP (3.2) has a unique solutionu(λ)near ¯uand the solutionu(λ)

is Lipschitz continuous.

Next, we define the following concepts.

Definition 4.1. A mappingg:K×Ω→E is said to be

(i) locallyβ-strongly accretive, if there exists a constantβ >0 such that

hg(u,λ)−g(v,λ),J(u−v)i ≥βku−vk2, ∀u,v∈K, λ ∈Ω;

(ii) locally(L_g,l_g)-mixed Lipschitz continuous, if there exist constantsL_g,l_g>0 such that

kg(u,λ)−g(v,λ˜)k ≤Lgku−vk+lgkλ−λ˜k, ∀u,v∈K, λ,λ˜ ∈Ω.

Definition 4.2.LetP:E→E∗, g:K×Ω→E, T,A,S:K×Ω→E∗, N:E∗×E∗×E∗×Ω→

E∗.ThenNis said to be

(i) locallyα-stronglyP◦g-accretive with respect toT, Aand S, if there exists a constant

α >0 such that

hN(T(u,λ),A(u,λ),S(u,λ),λ)−N(T(v,λ),A(v,λ),S(v,λ),λ),J∗(P◦g(u,λ)−P◦g(v,λ))i

≥αku−vk2, ∀u,v∈K, λ ∈Ω,

whereJ∗:E∗→E is a normalized duality mapping;

(ii) locally (L₍_N_,₁₎,L₍_N_,₂₎,L₍_N_,₃₎,l_N)-mixed Lipschitz continuous, if there exist constants L₍_N_,₁₎,L₍_N_,₂₎,L₍_N_,₃₎,lN >0 such that

kN(u1,v1,w1,λ)−N(u2,v2,w2,λ˜)k ≤L(N,1)ku1−u2k+L(N,2)kv1−v2k+L(N,3)kw1−w2k

+l_Nkλ−λ˜k|, ∀u1,u2,v1,v2,w1,w2∈K, λ,λ˜ ∈Ω.

Using the technique of Dafermos [3], we consider the mappingF(·,λ):K×Ω→Edefined

by

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Remark 4.1. It follows from Lemma 4.1 that the fixed point of the mappingFdefined by (4.5)

is the solution of PGVLIP (3.2).

Now, we show that the mapping F(u,λ) defined by (4.5) is a contraction mapping with

respect touuniformly inλ ∈Ω.

Theorem 4.1. Let E be a real uniformly smooth Banach space with ρE(q)≤cq2 for some c>0. Let the mappingg:K×Ω→Ebe locallyβ-strongly accretive and locally(Lg,lg)-mixed Lipschitz continuous. LetT,A,S:K×Ω→E∗be the mappings such thatT,AandSare locally

Lipschitz continuous in the first argument with constantsLT,LAandLS, respectively; letη:E×

E→Ebeτ-Lipschitz continuous such thatη(u,v) +η(v,u) =0,for allu,v∈Eand letP:E→

E∗ beδ-stronglyη-monotone continuous mapping; the function h(v,u) =hu∗−P(u),η(v,u)i

be 0-DQCV inv.Letφ:E×E→_R∪{+∞}be a lower semicontinuous,η-subdifferentiable and

proper functional such thatg(u,λ)∈∂ηφ(u,v,λ),for allu,v∈E,λ ∈Ω; letP◦g:K×Ω→E∗ be locally(L_P_◦_g,l_P_◦_g)-mixed Lipschitz continuous. LetN:E∗×E∗×E∗×Ω→E∗be locallyα

-stronglyP◦g-accretive with respect toT, AandS, and locally(L₍_N_,₁₎,L₍_N_,₂₎,L₍_N_,₃₎,lN)-mixed Lipschitz continuous. Suppose that there exist some real constantsk₁>0 andρ>0 such that

kP∂ηφ(·,u1,λ)

ρ (z)−P

∂ηφ(·,u2,λ)

ρ (z)k ≤k1ku1−u2k, ∀u1,u2∈E,z∈E

∗_,

λ ∈Ω. (4.6)

Then for eachu₁,u₂∈E, λ ∈Ω,

kF(u1,λ)−F(u2,λ)k ≤θku1−u2k, (4.7)

whereθ :=l+τ

δt(ρ)∈(0,1);l:=k1+ q

1−2β+64cL2_g;

t(ρ):=

q

L2_P_◦_g−2ρ α+64cρ2L2_N ; LN := (L(N,1)LT +L(N,2)LA+L(N,3)LS), that is, F isθ-contraction uniformly inλ ∈Ω.

Proof. For all u₁,u₂ ∈E, λ ∈Ω, using condition(4.6), locally (LP◦g,lP◦g)-mixed Lipschitz continuity ofP◦gand locallyL_T-Lipschitz continuity ofT, we have

kF(u₁,λ)−F(u2,λ)k

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−hu2−g(u2,λ) +P

∂ηφ(·,u2,λ)

ρ [P◦g(u2,λ)−ρN(T(u2,λ),A(u2,λ),S(u2,λ),λ)] i

≤ ku1−u2−(g(u1,λ)−g(u2,λ))k +

P

∂ηφ(·,u1,λ)

ρ [P◦g(u1,λ)−ρN(T(u1,λ),A(u1,λ),S(u1,λ),λ)] −P∂ηφ(·,u2,λ)

ρ [P◦g(u1,λ)−ρN(T(u1,λ),A(u1,λ),S(u1,λ),λ)]

+ P

∂ηφ(·,u2,λ)

ρ [P◦g(u1,λ)−ρN(T(u1,λ),A(u1,λ),S(u1,λ),λ)] −P∂ηφ(·,u2,λ)

ρ [P◦g(u2,λ)−ρN(T(u2,λ),A(u2,λ),S(u2,λ),λ)]

≤ ku₁−u₂−(g(u₁,λ)−g(u2,λ))k+k1ku1−u2k+

τ δ

h

P◦g(u1,λ)−P◦g(u2,λ) −ρ[N(T(u1,λ),A(u1,λ),S(u1,λ),λ)−N(T(u2,λ),A(u2,λ),S(u2,λ),λ)]

i

. (4.8)

Using Lemma 2.3, locallyβ-strongly accretiveness and locally(Lg,lg)-mixed Lipschitz

conti-nuity ofg, we have

ku1−u2−(g(u1,λ)−g(u2,λ))k2

≤ ku1−u2k2−2hg(u1,λ)−g(u2,λ),J(u1−u2−(g(u1,λ)−g(u2,λ)))i

≤ ku1−u2k2−2hg(u1,λ)−g(u2,λ),J(u1−u2)i

+2hg(u₁,λ)−g(u2,λ),J(u1−u2)−J(u1−u2−(g(u1,λ)−g(u2,λ)))i ≤(1−2β)ku1−u2k2+64ckg(u1,λ)−g(u2,λ)k2

≤(1−2β+64cL2_g)ku1−u2k2. (4.9) Since N is locally α-strongly P◦g-accretive with respect to T, A and S, and locally (L₍_N_,₁₎,L₍_N_,₂₎,L₍_N_,₃₎,lN)-mixed Lipschitz continuous; T,A,S are locally LT-Lipschitz contin-uous, locally L_A-Lipschitz continuous and locally L_S-Lipschitz continuous, respectively, we have

kN(T(u₁,λ),A(u1,λ),S(u1,λ),λ)−N(T(u2,λ),A(u2,λ),S(u2,λ),λ)k

≤L₍_N_,₁₎kT(u₁,λ)−T(u2,λ)k+L(N,2)kA(u1,λ)−A(u2,λ)k+L(N,3)kS(u1,λ)−S(u2,λ)k ≤L₍_N_,₁₎L_Tku₁−u₂k+L₍_N_,₂₎L_Aku₁−u₂k+L₍_N_,₃₎L_Sku₁−u₂k

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SinceP◦gis locally(L_P◦g,lP◦g)-mixed Lipschitz continuous, using Lemma 2.3, we have

kP◦g(u1,λ)−P◦g(u2,λ)−ρ[N(T(u1,λ),A(u1,λ),S(u1,λ),λ)−N(T(u2,λ),A(u2,λ),S(u2,λ),λ)]k2

≤ kP◦g(u1,λ)−P◦g(u2,λ)k2

−2ρhN(T(u1,λ),A(u1,λ),S(u1,λ),λ)−N(T(u2,λ),A(u2,λ),S(u2,λ),λ),J∗(P◦g(u1,λ)−P◦g(u2,λ))i

+2ρhN(T(u1,λ),A(u1,λ),S(u1,λ),λ)−N(T(u2,λ),A(u2,λ),S(u2,λ),λ),J∗(P◦g(u1,λ)−P◦g(u2,λ))

−J∗(P◦g(u1,λ)−P◦g(u2,λ)−ρ[N(T(u1,λ),A(u1,λ),S(u1,λ),λ)−N(T(u2,λ),A(u2,λ),S(u2,λ),λ)])i

≤(L2_P_◦_g−2ρ α)ku1−u2k2

+64cρ2kN(T(u1,λ),A(u1,λ),S(u1,λ),λ)−N(T(u2,λ),A(u2,λ),S(u2,λ),λ)k2. (4.11)

Now, from (4.8)-(4.11), we have

kF(u₁,λ)−F(u2,λ)k ≤ θku1−u2k, (4.12)

whereθ :=l+τ

δt(ρ); l:=k1+ q

1−2β+64cL2_g;

t(ρ):=

q

L2_P_◦_g−2ρ α+64cρ2L2_N; LN:= (L(N,1)LT+L(N,2)LA+L(N,3)LS).

Next, we have to show thatθ <1.It is clear thatt(ρ)assumes its minimum value for ˜ρ=

α/64cL2_N with t(ρ˜) =

q

L2_P_◦_g−(α2_/₆₄_cL2

N), where LN := (L(N,1)LT +L(N,2)LA+L(N,3)LS). Forρ =ρ˜, l+τ

δt(ρ)<1→ l<1,then it follows thatθ <1 for allρ satisfying (4.7). Hence, it follows that F defined by (4.5) is aθ-contraction mapping uniformly inλ ∈Ω. Therefore,

invoking Banach contraction principle, F admits a unique fixed point, sayu(λ), which in turn

is a solution of PGVLIP (3.2). This completes the proof.

Remark 4.2. From Theorem 4.1, it is clear that the mappingF defined by (4.5) has a unique fixed pointu(λ), that is,u(λ) =F(u,λ).

It also follows from our assumption that the function ˜uforλ =λ˜ is a solution of PGVLIP

(3.2). Again by using Theorem 4.1, we observe that forλ =λ˜, u˜is a fixed point ofF(u,λ)and

it is a fixed point ofF(u,λ˜). Consequently, we conclude that

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Next, using Theorem 4.1, we show the Lipschitz continuity of the solutionu(λ)of PGVLIP (3.2).

Theorem 4.2. Let the mappingsT,A,S,N,P,g,h,P◦gbe same as in Theorem 4.1 and let

condi-tions (4.6)-(4.7) of Theorem 4.1 hold. Suppose thatλ →Pρ∂ηφ(·,u,λ) isk2-Lipschitz continuous atλ =λ˜,then the functionu(λ)is Lipschitz continuous atλ =λ˜.

Proof.For allλ,λ˜ ∈Ω,using (4.13) and Theorem 4.1, we have

ku(λ)−u(λ˜)k=kF(u(λ),λ)−F(u(λ˜),λ˜)k

≤ kF(u(λ),λ)−F(u(λ˜),λ)k+kF(u(λ˜),λ)−F(u(λ˜),λ˜)k

≤θku(λ)−u(λ˜)k+kF(u(λ˜),λ)−F(u(λ˜),λ˜)k, (4.14)

where θ is given by (4.7). Using (4.5), Lemma 2.2 and the conditions on mappings

T,A,S,N,P,g,h,P◦gandP∂ηφ(·,u,λ)

ρ ,we have

kF(u(λ˜),λ)−F(u(λ˜),λ˜)k

=

u(

˜

λ)−g(u(λ˜),λ) +P∂ηφ(·,u(

˜ λ),λ)

ρ [P◦g(u(λ˜),λ)−ρN(T(u(λ˜)),A(u(λ˜)),S(u(λ˜)),λ)] −hu(λ˜)−g(u(λ˜),λ˜) +P∂ηφ(·,u(

˜ λ),λ˜)

ρ [P◦g(u(λ˜),λ˜)−ρN(T(u(λ˜)),A(u(λ˜)),S(u(λ˜)),λ˜)] i

≤l_gkλ−λ˜k+k2kλ−λ˜k+

τ δ

h

kP◦g(u(λ˜),λ)−P◦g(u(λ˜),λ˜)k

+ρkN(T(u(λ˜)),A(u(λ˜)),S(u(λ˜)),λ)−N(T(u(λ˜)),A(u(λ˜)),S(u(λ˜)),λ˜)

i

≤(l_g+k₂)kλ−λ˜k+τ

δ[lP◦gkλ−

˜

λk+ρlNkλ−λ˜k]

≤hl_g+k₂+(lP◦g+ρlN)τ δ

i

kλ−λ˜k. (4.15)

Combining (4.14) and (4.15), we have

ku(λ)−u(λ˜)k=θku(λ)−u(λ˜)k+

h

lg+k2+

(l_P_◦_g+ρlN)τ

δ

i

kλ−λ˜k, (4.16)

which implies

ku(λ)−u(λ˜)k ≤

h(l_g+k₂)δ+ (l_P_◦_g+_ρl_N)τ

δ(1−θ)

i

kλ−λ˜k. (4.17)

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Remark 4.3. Since the PGVLIP (3.2) includes many known classes of parametric variational

inequalities as special cases, Theorems 4.1-4.2 improve and generalize the known results given

in [3,6,11,14,17-20].

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgements

This work has been supported by Research Project (Grant Number: S-0169-1440) sanctioned by the

Deanship of Scientific Research Unit, University of Tabuk, KSA.

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