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Journal of Mathematical Analysis ISSN: 2217-3412, URL: www.ilirias.com/jma Volume 11 Issue 2 (2020), Pages 72-85. PARAMETRIC GENERALIZED SET-VALUED IMPLICIT QUASI-VARIATIONAL-LIKE INCLUSION PROBLEM IN UNIFORMLY SMOOTH BANACH SPACE FAIZAN A. KHAN * , ABDULAZIZ M. ALANAZI, JAVID ALI * Abstract. In this paper, using proximal-point mapping technique of P -η- accretive mapping and the property of the fixed-point set of set-valued contrac- tive mappings, we study the behavior and sensitivity analysis of the solution set of a parametric generalized set-valued implicit quasi-variational-like inclu- sion problem in real uniformly smooth Banach space. Further, under some suitable conditions, we discuss Lipschitz continuity of the solution set with respect to the parameter. The approach used in this paper may be treated as the extension and unification of approaches for studying sensitivity analysis for various important classes of parametric variational inclusions given by many authors in the literature. 1. Introduction In recent years, much attention has been given to develop general techniques for the sensitivity analysis of solution set of various classes of variational inequali- ties (inclusions). From the mathematical and engineering point of view, sensitivity properties of various classes of variational inequalities can provide new insight con- cerning the problem being studied and can stimulate ideas for solving problems. The sensitivity analysis of solution set for variational inequalities have been studied extensively by many authors using quite different techniques. By using projection technique, Dafermos [4], Ding and Luo [7], Kazmi and Khan [12], Mukherjee and Verma [16], Park and Jeong [20] and Yen [23] studied the sensitivity analysis of solution set of some classes of variational inequalities with single-valued mappings. By using resolvent operator technique, Adly [1], Agarwal et al. [2], Ding [5], Fang and Huang [8] and Noor [18] studied the sensitivity analysis of solution set of some classes of quasi-variational inclusions involving single-valued mappings. 2000 Mathematics Subject Classification. 47H25, 49J40, 49J53, 47H05. Key words and phrases. Parametric generalized set-valued implicit quasi-variational-like inclu- sion, P -η-proximal-point mapping technique, uniformly smooth Banach space, sensitivity analysis. c 2020 Ilirias Research Institute, Prishtin¨ e, Kosov¨ e. Submitted November 17, 2019. Published January 29, 2020. * Corresponding authors. First author is supported by Deanship of Scientific Research Unit (Project Grant Number: S-0169-1440), University of Tabuk, Tabuk, KSA. Communicated by Mehdi Asadi. 72

Transcript of PARAMETRIC GENERALIZED SET-VALUED IMPLICIT QUASI...

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Journal of Mathematical Analysis

ISSN: 2217-3412, URL: www.ilirias.com/jma

Volume 11 Issue 2 (2020), Pages 72-85.

PARAMETRIC GENERALIZED SET-VALUED IMPLICIT

QUASI-VARIATIONAL-LIKE INCLUSION PROBLEM IN

UNIFORMLY SMOOTH BANACH SPACE

FAIZAN A. KHAN∗, ABDULAZIZ M. ALANAZI, JAVID ALI∗

Abstract. In this paper, using proximal-point mapping technique of P -η-accretive mapping and the property of the fixed-point set of set-valued contrac-

tive mappings, we study the behavior and sensitivity analysis of the solution

set of a parametric generalized set-valued implicit quasi-variational-like inclu-sion problem in real uniformly smooth Banach space. Further, under some

suitable conditions, we discuss Lipschitz continuity of the solution set with

respect to the parameter. The approach used in this paper may be treated asthe extension and unification of approaches for studying sensitivity analysis for

various important classes of parametric variational inclusions given by manyauthors in the literature.

1. Introduction

In recent years, much attention has been given to develop general techniquesfor the sensitivity analysis of solution set of various classes of variational inequali-ties (inclusions). From the mathematical and engineering point of view, sensitivityproperties of various classes of variational inequalities can provide new insight con-cerning the problem being studied and can stimulate ideas for solving problems.The sensitivity analysis of solution set for variational inequalities have been studiedextensively by many authors using quite different techniques. By using projectiontechnique, Dafermos [4], Ding and Luo [7], Kazmi and Khan [12], Mukherjee andVerma [16], Park and Jeong [20] and Yen [23] studied the sensitivity analysis ofsolution set of some classes of variational inequalities with single-valued mappings.By using resolvent operator technique, Adly [1], Agarwal et al. [2], Ding [5], Fangand Huang [8] and Noor [18] studied the sensitivity analysis of solution set of someclasses of quasi-variational inclusions involving single-valued mappings.

2000 Mathematics Subject Classification. 47H25, 49J40, 49J53, 47H05.Key words and phrases. Parametric generalized set-valued implicit quasi-variational-like inclu-

sion, P -η-proximal-point mapping technique, uniformly smooth Banach space, sensitivity analysis.c©2020 Ilirias Research Institute, Prishtine, Kosove.

Submitted November 17, 2019. Published January 29, 2020.∗Corresponding authors.First author is supported by Deanship of Scientific Research Unit (Project Grant Number:

S-0169-1440), University of Tabuk, Tabuk, KSA.

Communicated by Mehdi Asadi.

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GENERALIZED SET-VALUED IMPLICIT 73

Recently, by using projection and resolvent techniques, Agarwal et al. [3], Ding[6], Huang [10], Kazmi and Khan [11], Khan and Ali [13], Liu et al. [15], Noor[19], Peng and Long [21] and Ram [22] studied the behavior and sensitivity analysisof solution set for some important classes of parametric generalized variationalinclusions involving single and set-valued mappings in the setting of Hilbert andBanach spaces.

Inspired by recent research works in this area, in this paper, we consider aparametric generalized set-valued implicit quasi-variational-like inclusion problem(in short, PGSIQVLIP) in real uniformly smooth Banach space. Further, usingP -η-proximal-point mapping technique of P -η-accretive mapping and the propertyof the fixed-point set of set-valued contractive mappings, we study the behavior andsensitivity analysis of the solution set for PGSIQVLIP. Furthermore, under somesuitable conditions, we prove that the solution set of PGSIQVLIP is Lipschitzcontinuous with respect to the parameter. The technique presented in this papercan be used to generalize and improve the results given by many authors, see forexample [2-6,8-11,13-15,19,21,22].

2. Preliminaries

Let E be a real Banach space equipped with norm ‖ · ‖; E∗ is the topologicaldual space of E; C(E) is the family of all nonempty compact subsets of E; 2E isthe power set of E; H(·, ·) is the Hausdorff metric on C(E) defined by

H(A,B) = max

supx∈A

infy∈B

d(x, y), supy∈B

infx∈A

d(x, y), A,B ∈ C(E);

〈·, ·〉 is the dual pair between E and E∗, and J : E → 2E∗

is the normalized dualitymapping defined by

J(x) = f ∈ E∗ : 〈x, f〉 = ‖x‖2, ‖x‖ = ‖f‖, x ∈ E.

We note that if E is smooth then J is single-valued and if E ≡ H, a Hilbert space,then J is the identity map on H. In the sequel, we shall denote a selection ofnormalized duality mapping by j.

First, we review and define the following concepts and results.

Definition 2.1[8,11]. Let η : E × E → E be a mapping. Then a mappingP : E → E is said to be

(i) η-accretive, if there exists jη(x, y) ∈ Jη(x, y) such that

〈Px− Py, jη(x, y)〉 ≥ 0, ∀x, y ∈ E;

(ii) strictly η-accretive, if there exists jη(x, y) ∈ Jη(x, y) such that

〈Px− Py, jη(x, y)〉 ≥ 0, ∀x, y ∈ E

and equality holds if and only if x = y;

(iii) δ-strongly η-accretive, if there exist jη(x, y) ∈ Jη(x, y) and δ > 0 such that

〈Px− Py, jη(x, y)〉 ≥ δ‖x− y‖2, ∀x, y ∈ E.

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74 F.A. KHAN, A.M. ALANAZI, J. ALI

Definition 2.2[8,11,12]. A mapping η : E × E → E is said to be τ -Lipschitzcontinuous, if there exists a constant τ > 0 such that

‖η(x, y)‖ ≤ τ‖x− y‖, ∀x, y ∈ E.

Definition 2.3[8,11,12]. Let η : E ×E → E be a single-valued mapping. Then aset-valued mapping M : E → 2E is said to be

(i) η-accretive, if there exists jη(x, y) ∈ Jη(x, y) such that

〈u− v, jη(x, y)〉 ≥ 0, ∀x, y ∈ E and ∀u ∈M(x), v ∈M(y);

(ii) η-m-accretive, if M is η-accretive and (I + ρM)(E) = E for any ρ > 0, whereI stands for identity mapping.

Definition 2.4[8,11,12]. Let η : E × E → E and P : E → E be nonlinearmappings. Then a set-valued mapping M : E → 2E is said to be P -η-accretive, ifM is η-accretive and (P + ρM)(E) = E for any ρ > 0.

The following theorem give some properties of P -η-accretive mappings.

Theorem 2.1[11,12]. Let η : E × E → E be a mapping and P : E → E bea strictly η-accretive mapping. Let M : E → 2E be a P -η-accretive set-valuedmapping. Then

(a) 〈u − v, jη(x, y)〉 ≥ 0, ∀(v, y) ∈ Graph(M) implies (u, x) ∈ Graph(M), whereGraph(M) := (u, x) ∈ E × E : u ∈M(x);

(b) the mapping (P + ρM)−1 is single valued for all ρ > 0.

By Theorem 2.1, we can define P -η-proximal-point mapping of M as follows:

JMρ (z) = (P + ρM)−1(z), ∀z ∈ E, (2.1)

where η : E × E → E is a nonlinear mapping, P : E → E is a strictly η-accretivemapping, and ρ > 0 is a constant.

Next, the following theorem shows that P -η-proximal-point mapping is Lipschitzcontinuous.

Theorem 2.2[11,12]. Let η : E × E → E be a τ -Lipschitz continuous mappingand let P : E → E be a δ-strongly η-accretive mapping. Let M : E → 2E bea P -η-accretive mapping. Then P -η-proximal-point mapping JMρ is τ

δ -Lipschitzcontinuous, that is,

‖JMρ (x)− JMρ (y)‖ ≤ τ

δ‖x− y‖, ∀x, y ∈ E.

Throughout the rest of the paper unless otherwise stated, let E be a real uni-formly smooth Banach space with ρE(q) ≤ cq2 for some c > 0, where modulus ofsmoothness of E is the function ρE : [0,∞)→ [0,∞), defined below in Lemma 2.1.

Lemma 2.1[11,12]. Let E be a real uniformly smooth Banach space and letJ : E → E∗ be the normalized duality mapping. Then for all x, y ∈ E, we have

(a) ‖x+ y‖2 ≤ ‖x‖2 + 2〈y, J(x+ y)〉;

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GENERALIZED SET-VALUED IMPLICIT 75

(b) 〈x− y, J(x)− J(y)〉 ≤ 2d2ρE(4‖x− y‖/d), where d =√

(‖x‖2 + ‖y‖2)/2,

ρE(q) = sup

(‖x‖+‖y‖)2 − 1 : ‖x‖ = 1, ‖y‖ = q

.

Lemma 2.2[17]. Let (X, d) be a complete metric space. Suppose that T : X →C(X) satisfies

H(T (x), T (y)) ≤ ν d(x, y), ∀x, y ∈ X,where ν ∈ (0, 1) is a constant. Then the mapping T has fixed point in X.

Lemma 2.3[14]. Let X be a complete metric space and let T1, T2 : X → C(X) beθ-H-contraction mappings, then

H(F (T1), F (T2)) ≤ (1− θ)−1 supx∈X

H(T1(x), T2(x)),

where F (T1) and F (T2) are the sets of fixed points of T1 and T2, respectively.

3. Formulation of Problem

Let Ω be a nonempty open subset of E in which the parameter λ takes values.Let P : E → E; N,M : E × E × E × Ω → E; g,m : E × Ω → E be single-valued mappings such that g 6≡ 0 and let A,B,C,R, S, T,D : E × Ω → C(E)be set-valued mappings. Suppose that W : E × E × Ω → 2E is a set-valuedmapping such that for each (y, λ) ∈ E × Ω, W (., y, λ) : E → 2E is P -η-accretiveand range (g − m)(E × λ) ∩ domain W (., y, λ) 6= ∅, where (g − m)(x, λ) =g(x, λ)−m(x, λ), for any (x, λ) ∈ E ×Ω. For each (f, λ) ∈ E ×Ω, we consider thefollowing parametric generalized set-valued implicit quasi-variational-like inclusionproblem (PGSIQVLIP):

Find x = x(λ) ∈ E, u = u(x, λ) ∈ A(x, λ), v = v(x, λ) ∈ B(x, λ), w = w(x, λ) ∈C(x, λ), r = r(x, λ) ∈ R(x, λ), s = s(x, λ) ∈ S(x, λ), t = t(x, λ) ∈ T (x, λ) andz = z(x, λ) ∈ D(x, λ) such that (g −m)(x, λ) ∈ domain W (., z, λ) and

f ∈ N(u, v, w, λ)−M(r, s, t, λ) +W((g −m)(x, λ), z, λ

). (3.1)

For a suitable choices of the mappings A,B,C,D,R, S, T,M,N, P,W, g,m, η andthe space E, it is easy to see that PGSIQVLIP (3.1) includes a number of knownclasses of parametric generalized variational inclusions (inequalities) studied bymany authors as special cases, see for example [2-6,8-15,18,20-23] and the referencestherein.

Now, for each fixed λ ∈ Ω, the solution set S(λ) of PGSIQVLIP (3.1) is denotedasS(λ) :=

x = x(λ) ∈ E : u = u(x, λ) ∈ A(x, λ), v = v(x, λ) ∈ B(x, λ), w = w(x, λ) ∈ C(x, λ),

r = r(x, λ) ∈ R(x, λ), s = s(x, λ) ∈ S(x, λ), t = t(x, λ) ∈ T (x, λ) and z = z(x, λ) ∈ D(x, λ)

such that f ∈ N(u, v, w, λ)−M(r, s, t, λ) +W((g −m)(x, λ), z, λ

). (3.2)

In this paper, our main aim is to study the behaviour and sensitivity analysis ofthe solution set S(λ), and the conditions on mappings A,B,C,D,R, S, T,M,N, P,W, g,m, η, under which the solution set S(λ) of PGSIQVLIP (3.1) is nonempty andLipschitz continuous with respect to the parameter λ ∈ Ω.

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76 F.A. KHAN, A.M. ALANAZI, J. ALI

4. Sensitivity Analysis of Solution Set S(λ)

First, we define the following concepts.

Definition 4.1[11-13]. A mapping g : E × Ω→ E is said to be

(i) (Lg, lg)-mixed Lipschitz continuous, if there exist constants Lg, lg > 0 such that

‖g(x1, λ1)− g(x2, λ2)‖ ≤ Lg‖x1 − x2‖+ lg‖λ1 − λ2‖, ∀(x1, λ1), (x2, λ2) ∈ E × Ω;

(ii) s-strongly monotone, if there exists a constant s > 0 such that

〈g(x1, λ)− g(x2, λ), j(x1 − x2)〉 ≥ s‖x1 − x2‖2, ∀(x1, λ), (x2, λ) ∈ E × Ω.

Definition 4.2[11-13]. A set-valued mapping A : E × Ω → C(E) is said to be(LA, lA)-H-mixed Lipschitz continuous, if there exist constants LA, lA > 0 suchthat

H(A(x1, λ1), A(x2, λ2)) ≤ LA‖x1−x2‖+ lA‖λ1−λ2‖, ∀ (x1, λ1), (x2, λ2) ∈ E×Ω.

Definition 4.3. Let P : E → E, g,m : E × Ω→ E be mappings and let A,B,C :E × Ω → C(E) be set-valued mappings. A mapping N : E × E × E × Ω → E issaid to be

(i) (L(N,1), L(N,2), L(N,3), lN )-mixed Lipschitz continuous, if there exist constantsL(N,1), L(N,2), L(N,3), lN > 0 such that

‖N(x1, y1, z1, λ1)−N(x2, y2, z2, λ2)‖ ≤ L(N,1)‖x1 − x2‖+ L(N,2)‖y1 − y2‖+

L(N,3)‖z1 − z2‖+ lN‖λ1 − λ2‖, ∀(x1, y1, z1, λ1), (x2, y2, z2, λ2) ∈ E × E × E × Ω;

(ii) ξ-strongly mixed P (g − m)-accretive with respect to A, B and C, if thereexists a constant ξ > 0 such that

〈N(u1, v1, w1, λ)−N(u2, v2, w2, λ), J(P(g−m)(x, λ)−P(g−m)(y, λ))〉 ≥ ξ‖x−y‖2,

∀(x, y, λ) ∈ E×E×Ω, u1 ∈ A(x, λ), u2 ∈ A(y, λ), v1 ∈ B(x, λ), v2 ∈ B(y, λ),

w1 ∈ C(x, λ), w2 ∈ C(y, λ);

(iii) σ-generalized mixed P (g −m)-pseudocontractive with respect to A, B andC, if there exists a constant σ > 0 such that

〈N(u1, v1, w1, λ)−N(u2, v2, w2, λ), J(P(g−m)(x, λ)−P(g−m)(y, λ))〉 ≤ σ‖x−y‖2,

∀(x, y, λ) ∈ E×E×Ω, u1 ∈ A(x, λ), u2 ∈ A(y, λ), v1 ∈ B(x, λ), v2 ∈ B(y, λ),

w1 ∈ C(x, λ), w2 ∈ C(y, λ);

(iv) ν-relaxed mixed P (g − m)-Lipschitz with respect to A, B and C, if thereexists a constant ν > 0 such that

〈N(u1, v1, w1, λ)−N(u2, v2, w2, λ), J(P(g−m)(x, λ)−P(g−m)(y, λ))〉 ≤ −ν‖x−y‖2,

∀(x, y, λ) ∈ E×E×Ω, u1 ∈ A(x, λ), u2 ∈ A(y, λ), v1 ∈ B(x, λ), v2 ∈ B(y, λ),

w1 ∈ C(x, λ), w2 ∈ C(y, λ), where P (g −m) denotes P composition (g −m).

Remark 4.1. The concepts given in Definition 4.3 generalize the concepts givenby many authors, see for details [5-15,18,19,22].

Now, we transfer PGSIQVLIP (3.1) into a parametric fixed point problem.

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GENERALIZED SET-VALUED IMPLICIT 77

Lemma 4.1. For each (f, λ) ∈ E × Ω, (x, u, v, w, r, s, t, z) with x = x(λ) ∈ E,u = u(x, λ) ∈ A(x, λ), v = v(x, λ) ∈ B(x, λ), w = w(x, λ) ∈ C(x, λ), r = r(x, λ) ∈R(x, λ), s = s(x, λ) ∈ S(x, λ), t = t(x, λ) ∈ T (x, λ) and z = z(x, λ) ∈ D(x, λ) suchthat (g − m)(x, λ) ∈ domain W (., z, λ) is a solution of PGSIQVLIP (3.1) if andonly if the set-valued mapping F : E × Ω→ 2E defined by

F (l, λ) =⋃

u∈A(l,λ), v∈B(l,λ), w∈C(l,λ), r∈R(l,λ), s∈S(l,λ), t∈T (l,λ), z∈D(l,λ)

[l−(g−m)(l, λ)

+JW (.,z,λ)ρ [P (g−m)(l, λ)−ρN(u, v, w, λ)+ρM(r, s, t, λ)+ρf ]

], l ∈ E,

(4.1)has a fixed point x = x(λ) ∈ E, where P : E → E and P (g − m) denotes P

composition (g −m); JW (.,z,λ)ρ = (P + ρW (., z, λ))−1, and ρ > 0 is a constant.

Proof. For each (f, λ) ∈ E×Ω, PGSIQVLIP (3.1) has a solution (x, u, v, w, r, s, t, z)with x = x(λ) ∈ E, u = u(x, λ) ∈ A(x, λ), v = v(x, λ) ∈ B(x, λ), w = w(x, λ) ∈C(x, λ), r = r(x, λ) ∈ R(x, λ), s = s(x, λ) ∈ S(x, λ), t = t(x, λ) ∈ T (x, λ) andz = z(x, λ) ∈ D(x, λ) such that (g −m)(x, λ) ∈ domain W (., z, λ) if and only if

f ∈ N(u, v, w, λ)−M(r, s, t, λ)+W ((g−m)(x, λ), z, λ)

⇔ P(g−m)(x, λ)−ρN(u, v, w, λ)+ρM(r, s, t, λ)+ρf ∈ (P+ρW (., z, λ))((g−m)(x, λ)).

Since for each (z, λ) ∈ E × Ω, W (., z, λ) is P -η-accretive, by definition of P -

η-proximal-point mapping JW (.,z,λ)ρ of W (., z, λ), preceding inclusion holds if and

only if

(g −m)(x, λ) = JW (.,z,λ)ρ [P (g −m)(x, λ)− ρN(u, v, w, λ) + ρM(r, s, t, λ) + ρf ],

that is, x ∈ F (x, λ). This completes the proof.

Now, we prove the following theorem which ensures that the solution set S(λ)of PGSIQVLIP (3.1) is nonempty and closed for each λ ∈ Ω.

Theorem 4.1. Let E be a real uniformly smooth Banach space with ρE(q) ≤ cq2

for some c > 0. Let the set-valued mappings A,B,C,R, S, T,D : E×Ω→ C(E) beH-Lipschitz continuous in the first argument with constants LA, LB , LC , LR, LS ,LT , LD, respectively; let the mapping η : E × E → E be τ -Lipschitz continuousand P : E → E be δ-strongly η-accretive. Let the mappings g,m : E×Ω→ E suchthat (g − m) is s-strongly accretive and L(g−m)-Lipschitz continuous in the firstargument, and let the mapping P (g −m) be LP(g−m)-Lipschitz continuous inthe first argument. Let the mapping N : E ×E ×E ×Ω→ E be ξ-strongly mixedP (g − m)-accretive with respect to A, B and C, and (L(N,1), L(N,2), L(N,3))-mixed Lipschitz continuous; let the mapping M : E × E × E × Ω → E be σ-generalized mixed P (g −m)-pseudocontractive with respect to R, S and T , and(L(M,1), L(M,2), L(M,3))-mixed Lipschitz continuous. Suppose that the set-valued

mapping W : E × E × Ω → 2E is such that for each (y, λ) ∈ E × Ω, W (., y, λ) :E → 2E is P -η-accretive with range (g −m)(E × λ) ∩ domain W (., y, λ) 6= ∅.Suppose that there exist constants k1, k2 > 0 such that

‖JW (.,x,λ)ρ (z)−JW (.,y,λ)

ρ (z)‖ ≤ k1‖x−y‖+k2‖λ−λ‖, ∀x, y, z ∈ E; λ, λ ∈ Ω, (4.2)

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78 F.A. KHAN, A.M. ALANAZI, J. ALI

and suppose that there is a constant ρ > 0 such that

θ := q+ε(ρ); where q := k1LD+√

1− 2s+ 64cL2(g−m) ;

ε(ρ) := r−1√L2P(g−m) − 2ρ(ξ − σ) + 132ρ2c(L2

N + L2M ) ; r :=

δ

τ;

LN := (LAL(N,1)+LBL(N,2)+LCL(N,3)); LM := (LRL(M,1)+LSL(M,2)+LTL(M,3));∣∣∣∣ρ− (ξ − σ)

132c(L2N + L2

M )

∣∣∣∣ <√

(ξ − σ)2 − 132c(L2N + L2

M )(L2P(g−m) − r2(1− q)2

)132c(L2

N + L2M )

,

ξ > σ +√

132c(L2N + L2

M )(L2P(g−m) − r2(1− q)2

); ξ > σ, q ∈ (0, 1). (4.3)

Then, for each fixed f ∈ E, the set-valued mapping F defined by (4.1) is a compact-valued uniform θ-H-contraction mapping with respect to λ ∈ Ω, where θ is givenby (4.3). Moreover, for each λ ∈ Ω, the solution set S(λ) of PGSIQVLIP (3.1) isnonempty and closed.

Proof. Let (x, λ) be an arbitrary element in E × Ω. Since A,B,C,R, S, T,D arecompact-valued, then for any sequences un ⊂ A(x, λ), vn ⊂ B(x, λ), wn ⊂C(x, λ), rn ⊂ R(x, λ), sn ⊂ S(x, λ), tn ⊂ T (x, λ), zn ⊂ D(x, λ), there ex-ist subsequences uni ⊂ un, vni ⊂ vn, wni ⊂ wn, rni ⊂ rn, sni ⊂sn, tni ⊂ tn, zni ⊂ zn and elements u ∈ A(x, λ), v ∈ B(x, λ), w ∈C(x, λ), r ∈ R(x, λ), s ∈ S(x, λ), t ∈ T (x, λ), z ∈ D(x, λ) such that uni

→u, vni

→ v, wni→ w, rni

→ r, sni→ s, tni

→ t, zni→ z as i → ∞. By using

Theorem 2.2 and (4.2) and the mixed Lipschitz continuity of N and M , we estimate

‖JW (.,zni,λ)

ρ [P(g−m)(x, λ)−ρN(uni , vni , wni , λ)+ρM(rni , sni , tni , λ)+ρf ]

−JW (.,z,λ)ρ [P (g−m)(x, λ)−ρN(u, v, w, λ)+ρM(r, s, t, λ)+ρf ]‖

≤ ‖JW (.,zni,λ)

ρ [P(g−m)(x, λ)−ρN(uni, vni

, wni, λ)+ρM(rni

, sni, tni

, λ)+ρf ]

−JW (.,z,λ)ρ [P(g−m)(x, λ)−ρN(uni

, vni, wni

, λ)+ρM(rni, sni

, tni, λ)+ρf ]‖

+‖JW (.,z,λ)ρ [P(g−m)(x, λ)−ρN(uni

, vni, wni

, λ)+ρM(rni, sni

, tni, λ)+ρf ]

−JW (.,z,λ)ρ [P (g−m)(x, λ)−ρN(u, v, , w, λ)+ρM(r, s, t, λ)+ρf ]‖

≤ k1‖zni−z‖+ρτ

δ

[‖N(uni

, vni, wni

, λ)−N(u, v, w, λ)‖+‖M(rni, sni

, tni, λ)−M(r, s, t, λ)‖

]≤ k1‖zni

−z‖+ρτδ

[L(N,1)‖uni

−u‖+L(N,2)‖vni−v‖+L(N,3)‖wni

−w‖

+L(M,1)‖rni−r‖+L(M,2)‖sni

−s‖+L(M,3)‖tni−t‖

]→ 0 as i→∞. (4.4)

Thus (4.1) and (4.4) yield that F (x, λ) ∈ C(E).

Now, for each fixed λ ∈ Ω, we prove that F (x, λ) is a uniform θ-H-contractionmapping. Let (x1, λ), (x2, λ) be arbitrary elements in E×Ω and any l1 ∈ F (x1, λ),there exist u1 = u1(x1, λ) ∈ A(x1, λ), v1 = v1(x1, λ) ∈ B(x1, λ), w1 = w1(x1, λ) ∈C(x1, λ), r1 = r1(x1, λ) ∈ R(x1, λ), s1 = s1(x1, λ) ∈ S(x1, λ), t1 = t1(x1, λ) ∈T (x1, λ) and z1 = z1(x1, λ) ∈ D(x1, λ) such that

l1 = x1−(g−m)(x1, λ)+JW (.,z1,λ)ρ [P(g−m)(x1, λ)−ρN(u1, v1, w1, λ)+ρM(r1, s1, t1, λ)+ρf ].

(4.5)

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GENERALIZED SET-VALUED IMPLICIT 79

It follows from the compactness of A(x2, λ), B(x2, λ), C(x2, λ), R(x2, λ), S(x2, λ),T (x2, λ) and D(x2, λ), and H-Lipschitz continuity of A,B,C,R, S, T,D that thereexist u2 = u2(x2, λ) ∈ A(x2, λ), v2 = v2(x2, λ) ∈ B(x2, λ), w2 = w2(x2, λ) ∈C(x2, λ), r2 = r2(x2, λ) ∈ R(x2, λ), s2 = s2(x2, λ) ∈ S(x2, λ), t2 = t2(x2, λ) ∈T (x2, λ) and z2 = z2(x2, λ) ∈ D(x2, λ) such that

‖u1 − u2‖ ≤ H(A(x1, λ), A(x2, λ)) ≤ LA‖x1 − x2‖,‖v1 − v2‖ ≤ H(B(x1, λ), B(x2, λ)) ≤ LB‖x1 − x2‖,‖w1 − w2‖ ≤ H(C(x1, λ), C(x2, λ)) ≤ LC‖x1 − x2‖,‖r1 − r2‖ ≤ H(R(x1, λ), R(x2, λ)) ≤ LR‖x1 − x2‖,‖s1 − s2‖ ≤ H(S(x1, λ), S(x2, λ)) ≤ LS‖x1 − x2‖,‖t1 − t2‖ ≤ H(T (x1, λ), T (x2, λ)) ≤ LT ‖x1 − x2‖,‖z1 − z2‖ ≤ H(D(x1, λ), D(x2, λ)) ≤ LD‖x1 − x2‖. (4.6)

Let

l2 = x2−(g−m)(x2, λ)+JW (.,z2,λ)ρ [P(g−m)(x2, λ)−ρN(u2, v2, w2, λ)+ρM(r2, s2, t2, λ)+ρf ],

(4.7)

then we have l2 ∈ F (x2, λ).

Next, using Theorem 2.2 and (4.2), we estimate

‖l1−l2‖ ≤ ‖x1−x2−((g−m)(x1, λ)−(g−m)(x2, λ))‖

+‖JW (.,z1,λ)ρ [P (g −m)(x1, λ)− ρN(u1, v1, w1, λ) + ρM(r1, s1, t1, λ) + ρf ]

−JW (.,z2,λ)ρ [P (g −m)(x1, λ)− ρN(u1, v1, w1, λ) + ρM(r1, s1, t1, λ) + ρf ]‖

+‖JW (.,z2,λ)ρ [P (g −m)(x1, λ)− ρN(u1, v1, w1, λ) + ρM(r1, s1, t1, λ) + ρf ]

−JW (.,z2,λ)ρ [P (g −m)(x2, λ)− ρN(u2, v2, w2, λ) + ρM(r2, s2, t2, λ) + ρf ]‖

≤ ‖x1 − x2 − ((g −m)(x1, λ)− (g −m)(x2, λ))‖+ k1‖z1 − z2‖

δ‖P (g −m)(x1, λ)− P (g −m)(x2, λ)

−ρ(N(u1, v1, w1, λ)−N(u2, v2, w2, λ)−M(r1, s1, t1, λ) +M(r2, s2, t2, λ)

)‖.

(4.8)Since (g −m) is s-strongly accretive and L(g−m)-Lipschitz continuous, we have

‖x1−x2−((g−m)(x1, λ)−(g−m)(x2, λ))‖2

≤ ‖x1−x2‖2−2〈(g−m)(x1, λ)−(g−m)(x2, λ), J(x1−x2−((g−m)(x1, λ)−(g−m)(x2, λ))

)〉

= ‖x1 − x2‖2 − 2〈(g −m)(x1, λ)− (g −m)(x2, λ), J(x1 − x2)〉−2〈(g−m)(x1, λ)−(g−m)(x2, λ), J

(x1−x2−((g−m)(x1, λ)−(g−m)(x2, λ))

)−J(x1−x2)〉

≤ ‖x1−x2‖2−2s‖x1−x2‖2+64cL2(g−m)‖x1−x2‖2

≤ (1− 2s+ 64cL2(g−m))‖x1 − x2‖2. (4.9)

Since N and M are mixed Lipschitz continuous and H-Lipschitz continuity ofset-valued mappings A,B,C,R, S, T , we have

‖N(u1, v1, w1, λ)−N(u2, v2, w2, λ)‖ ≤ L(N,1)‖u1−u2‖+L(N,2)‖v1−v2‖+L(N,3)‖w1−w2‖

≤ L(N,1)H(A(x1, λ), A(x2, λ))+L(N,2)H(B(x1, λ), B(x2, λ))+L(N,2)H(C(x1, λ), C(x2, λ))

≤ (LAL(N,1) + LBL(N,2) + LCL(N,3))‖x1 − x2‖, (4.10)

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80 F.A. KHAN, A.M. ALANAZI, J. ALI

and

‖M(r1, s1, t1, λ)−M(r2, s2, t2, λ)‖ ≤ L(M,1)‖r1−r2‖+L(M,2)‖s1−s2‖+L(M,3)‖t1−t2‖

≤ (LRL(M,1) + LSL(M,2) + LTL(M,3))‖x1 − x2‖.(4.11)

Further, since N is ξ-strongly mixed P (g − m)-accretive with respect to A,B and C; M is σ-generalized mixed P (g −m)-pseudocontractive with respect toR, S and T ; P (g −m) is LP(g−m)-Lipschitz continuous, then using ‖x+ y‖2 ≤2(‖x‖2 + ‖y‖2), we have

‖P(g−m)(x1, λ)−P(g−m)(x2, λ)

−ρ(N(u1, v1, w1, λ)−N(u2, v2, w2, λ)−M(r1, s1, t1, λ)+M(r2, s2, t2, λ))‖2

≤ ‖P(g−m)(x1, λ)−P(g−m)(x2, λ)‖2−2ρ〈N(u1, v1, w1, λ)−N(u2, v2, w2, λ)

−M(r1, s1, t1, λ)+M(r2, s2, t2, λ), J(P (g−m)(x1, λ)−P (g−m)(x2, λ)

−ρ(N(u1, v1, w1, λ)−N(u2, v2, w2, λ)−M(r1, s1, t1, λ)+M(r2, s2, t2, λ)))〉

≤ L2P(g−m)‖x1−x2‖2

−2ρ〈N(u1, v1, w1, λ)−N(u2, v2, w2, λ), J(P (g−m)(x1, λ)−P (g−m)(x2, λ))〉+2ρ〈M(r1, s1, t1, λ)−M(r2, s2, t2, λ), J(P (g −m)(x1, λ)− P (g −m)(x2, λ))〉−2ρ〈(N(u1, v1, w1, λ)−N(u2, v2, w2, λ))−(M(r1, s1, t1, λ)−M(r2, s2, t2, λ)),

J(P(g−m)(x1, λ)−P(g−m)(x2, λ)−ρ(N(u1, v1, w1, λ)−N(u2, v2, w2, λ)

−M(r1, s1, t1, λ) +M(r2, s2, t2, λ)))− J

(P (g −m)(x1, λ)− P (g −m)(x2, λ)

)〉

≤ L2P(g−m)‖x1−x2‖2−2ρξ‖x1−x2‖2+2ρσ‖x1−x2‖2

+64ρ2c‖(N(u1, v1, w1, λ)−N(u2, v2, w2, λ))− (M(r1, s1, t1, λ)−M(r2, s2, t2, λ))‖2

≤ L2P(g−m)‖x1−x2‖2−2ρ(ξ−σ)‖x1−x2‖2+132ρ2c

[(L(N,1)LA+L(N,2)LB+L(N,3)LC)2

+(L(M,1)LR+L(M,2)LS+L(M,3)LT )2]‖x1−x2‖2

=(L2P(g−m)−2ρ(ξ−σ)+132ρ2c

[(L(N,1)LA+L(N,2)LB+L(N,3)LC)2

+(L(M,1)LR + L(M,2)LS + L(M,3)LT )2])‖x1 − x2‖2. (4.12)

Now, from (4.8)-(4.12), it follows that

‖l1 − l2‖ ≤ θ ‖x1 − x2‖,(4.13)

where θ := q+ε(ρ); q := k1LD+√

(1− 2s+ 64cL2(g−m)) ;

ε(ρ) := r−1√L2P(g−m) − 2ρ(ξ − σ) + 132ρ2c(L2

N + L2M ) ; r :=

δ

τ;

LN := (L(N,1)LA+L(N,2)LB+L(N,3)LC); LM := (L(M,1)LR+L(M,2)LS+L(M,3)LT ).

Hence, we have

d(l1, F (x2, λ)

)= infl2∈F (x2,λ)

‖l1 − l2‖ ≤ θ‖x1 − x2‖. (4.14)

Since l1 ∈ F (x1, λ) is arbitrary, we obtain

supl1∈F (x1,λ)

d(l1, F (x2, λ)

)≤ θ‖x1 − x2‖. (4.15)

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GENERALIZED SET-VALUED IMPLICIT 81

By using same argument, we can prove

supl2∈F (x2,λ)

d(l2, F (x1, λ)

)≤ θ‖x1 − x2‖. (4.16)

By the definition of the Hausdorff metric H on C(E), we obtain that

H(F (x1, λ), F (x2, λ)

)≤ θ‖x1 − x2‖, ∀(x1, λ), (x2, λ) ∈ E × Ω, (4.17)

that is, F (x, λ) is a uniform θ-H-contraction mapping with respect to λ ∈ Ω.Also, it follows from condition (4.3) that θ < 1 and hence F (x, λ) is a set-valuedcontraction mapping which is uniform with respect to λ ∈ Ω. By Lemma 2.2, foreach λ ∈ Ω, F (x, λ) has a fixed point x = x(λ) ∈ E, that is, x = x(λ) ∈ F (x, λ),and hence Lemma 4.1 ensures that the solution set S(λ) 6= ∅. Further, for anysequence xn ⊂ S(λ) with lim

n→∞xn = x0, we have xn ∈ F (xn, λ) for all n ≥ 1.

Next, by virtue of (4.17), we have that

d(x0, F (x0, λ)

)≤ ‖x0 − xn‖+H

(F (xn, λ), F (x0, λ)

)≤ (1 + θ) ‖xn − x0‖ → 0 as n→∞,

that is, x0 ∈ F (x0, λ) and hence x0 ∈ S(λ). Thus S(λ) is closed in E.

Theorem 4.2. Let E be a real uniformly smooth Banach space with ρE(q) ≤ cq2

for some c > 0. Let the mappings A,B,C,R, S, T,D, P,W, g,m, η be same asin Theorem 4.1. Let the mapping N be (L(N,1), L(N,2), L(N,3))-mixed Lipschitzcontinuous and let the mapping M be ν-relaxed mixed P (g −m)-Lipschitz withrespect to R, S and T , and (L(M,1), L(M,2), L(M,3))-mixed Lipschitz continuous. Ifcondition (4.3) is satisfied, then there exists a constant ρ > 0 such that

θ1 := q+ε(ρ); where q := k1LD+√

1− 2s+ 64cL2(g−m) ;

ε(ρ) := r−1[ρLN+

√L2P(g−m) − 2ρν + 64ρ2c(L2

N + L2M )]

; r :=δ

τ, q ∈ (0, 1);

LN := (LAL(N,1) + LBL(N,2) + LCL(N,3)); LM := (LRL(M,1) + LSL(M,2) + LTL(M,3));∣∣∣∣ρ− ν − r(1− q)LN64cL2

M − L2N

∣∣∣∣ <√(

ν − r2(1− q)2LN)−(L2P(g−m) − r2(1− q2)

)(64cL2

M − L2N )

64cL2M − L2

N

,

ν − r2(1− q)2LN >(L2P(g−m) − r2(1− q)2)(64cL2

M − L2N ). (4.18)

Then, for given f ∈ E, the set-valued mapping F defined by (4.1) is a compact-valueduniform θ1-H-contraction mapping with respect to λ ∈ Ω, where θ1 is given by (4.18).Moreover, for each λ ∈ Ω, the solution set S(λ) of PGSIQVLIP (3.1) is nonempty andclosed.

Proof. As in the proof of Theorem 4.1, we see that F is compact-valued and (4.4)-(4.7) and (4.9)-(4.11) hold. Since N is (L(N,1), L(N,2), L(N,3))-mixed Lipschitz contin-uous. M is ν-relaxed mixed P (g − m)-Lipschitz with respect to R, S and T , and(L(M,1), L(M,2), L(M,3))-mixed Lipschitz continuous. It follows that

‖P(g−m)(x1, λ)−P(g−m)(x2, λ)+ρ(M(r1, s1, t1, λ)−M(r2, s2, t2, λ))‖2

≤ ‖P(g−m)(x1, λ)−P(g−m)(x2, λ)‖2

+2ρ〈M(r1, s1, t1, λ)−M(r2, s2, t2, λ), J(P (g −m)(x1, λ)− P (g −m)(x2, λ)

)〉

+2ρ〈M(r1, s1, t1, λ)−M(r2, s2, t2, λ), J(P (g −m)(x1, λ)− P (g −m)(x2, λ)

+ρ(M(r1, s1, t1, λ)−M(r2, s2, t2, λ)))− J

(P (g −m)(x1, λ)− P (g −m)(x2, λ)

)〉

≤ L2P(g−m)‖x1−x2‖2−2ρν‖x1−x2‖2+64ρ2c‖M(r1, s1, t1, λ)−M(r2, s2, t2, λ)‖2

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82 F.A. KHAN, A.M. ALANAZI, J. ALI

≤(L2P(g−m) − 2ρν + 64ρ2c[LRL(M,1) + LSL(M,2) + LTL(M,3)]

2)‖x1 − x2‖2. (4.19)

It follows from (4.4)-(4.11) and (4.19), we have

‖l1−l2‖ ≤ ‖x1−x2−((g−m)(x1, λ)−(g−m)(x2, λ))‖+k1‖z1−z2‖

δ

[‖P(g−m)(x1, λ)−P(g−m)(x2, λ)+ρ

(M(r1, s1, t1, λ)−M(r2, s2, t2, λ)

)‖

+ρ‖N(u1, v1, w1, λ)−N(u2, v2, w2, λ)‖]

≤ (1− 2s+ 64cL2(g−m))

12 ‖x1 − x2‖+ k1LD‖x1 − x2‖+

τ

δ‖x1 − x2‖

×[(L2P(g−m) − 2ρν + 64ρ2c[LRL(M,1) + LSL(M,2) + LTL(M,3)]

2) 1

2

+ρ[LAL(N,1)+LBL(N,1)+LCL(N,3)]]

≤(k1LD+

√1− 2s+ 64cL2

(g−m)+τ

δ

[ρ[LAL(N,1)+LBL(N,1)+LCL(N,3)]

+√L2P(g−m) − 2ρν + 64ρ2c[LRL(M,1) + LSL(M,2) + LTL(M,3)]2

])‖x1−x2‖

≤ θ1 ‖x1−x2‖.(4.20)

The rest of the proof follows precisely as in the proof of Theorem 4.1.

Next, we prove that the solution set S(λ) of PGSIQVLIP (3.1) is Lipschitz continuouswith respect to the parameter λ.

Theorem 4.3. Let E be a real uniformly smooth Banach space with ρE(q) ≤ cq2 for somec > 0. Let the set-valued mappings A,B,C,R, S, T,D be H-mixed Lipschitz continuouswith pairs of constants (LA, lA), (LB , lB), (LC , lC), (LR, lR), (LS , lS), (LT , lT ), (LD, lD),respectively. Let η : E ×E → E be a τ -Lipschitz continuous mapping and let P : E → Ebe a δ-strongly η-accretive mapping. Let the mappings (g − m), P (g − m) be mixedLipschitz continuous with pairs of constants (L(g−m), l(g−m)) and (LP(g−m), lP(g−m)),respectively. Let the mapping N be (L(N,1), L(N,2), L(N,3), lN )-mixed Lipschitz continuousand let the mapping M be ν-relaxed mixed P (g −m)-Lipschitz with respect to R, Sand T , and (L(M,1), L(M,2), L(M,3), lM )-mixed Lipschitz continuous. Suppose that the set-valued mapping W is same as in Theorem 4.1 and conditions (4.2), (4.3) and (4.18) hold,then for each λ ∈ Ω, the solution set S(λ) of PGSIQVLIP (3.1) is a H-Lipschitz continuousmapping from Ω into E.

Proof. For each λ, λ ∈ Ω, it follows from Theorem 4.2, S(λ) and S(λ) are both nonemptyand closed subsets of E. By Theorem 4.2, F (x, λ) and F (x, λ) are both set-valued θ1-H-contraction mappings with same contractive constant θ1 ∈ (0, 1). By Lemma 2.3, weobtain

H(S(λ), S(λ)) ≤( 1

1− θ1

)supx∈E

H(F (x, λ), F (x, λ)), (4.21)

where θ1 is given by (4.18).

Now, for any i1 ∈ F (x, λ), there exist u = u(x, λ) ∈ A(x, λ), v = v(x, λ) ∈ B(x, λ), w =w(x, λ) ∈ C(x, λ), r = r(x, λ) ∈ R(x, λ), s = s(x, λ) ∈ S(x, λ), t = t(x, λ) ∈ T (x, λ) andz = z(x, λ) ∈ D(x, λ) satisfying

i1 = x− (g−m)(x, λ) +JW (.,z,λ)ρ [P (g−m)(x, λ)−ρN(u, v, w, λ) +ρM(r, s, t, λ) +ρf ].

(4.22)

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GENERALIZED SET-VALUED IMPLICIT 83

It is easy to see that there exist u = u(x, λ) ∈ A(x, λ), v = v(x, λ) ∈ B(x, λ), w =w(x, λ) ∈ C(x, λ), r = r(x, λ) ∈ R(x, λ), s = s(x, λ) ∈ S(x, λ), t = t(x, λ) ∈ T (x, λ) andz = z(x, λ) ∈ D(x, λ) such that

‖u− u‖ ≤ H(A(x, λ), A(x, λ)) ≤ lA‖λ− λ‖,

‖v − v‖ ≤ H(B(x, λ), B(x, λ)) ≤ lB‖λ− λ‖,‖w − w‖ ≤ H(C(x, λ), C(x, λ)) ≤ lC‖λ− λ‖,‖r − r‖ ≤ H(R(x, λ), R(x, λ)) ≤ lR‖λ− λ‖,‖s− s‖ ≤ H(S(x, λ), S(x, λ)) ≤ lS‖λ− λ‖,‖t− t‖ ≤ H(T (x, λ), T (x, λ)) ≤ lT ‖λ− λ‖,‖z − z‖ ≤ H(D(x, λ), D(x, λ)) ≤ lD‖λ− λ‖. (4.23)

Let

i2 = x− (g−m)(x, λ) +JW (.,z,λ)ρ [P (g−m)(x, λ)−ρN(u, v, w, λ) +ρM(r, s, t, λ) +ρf ].

(4.24)Clearly, i2 ∈ F (x, λ).

Since N and M are mixed Lipschitz continuous and in view of (4.2) and (4.21)-(4.24)and with a = P (g −m)(x, λ)− ρN(u, v, w, λ) + ρM(r, s, t, λ) + ρf, we have

‖i1−i2‖ ≤ ‖(g−m)(x, λ)−(g−m)(x, λ)‖

+‖JW (.,z,λ)ρ [P (g −m)(x, λ)− ρN(u, v, w, λ) + ρM(r, s, t, λ) + ρf ]− JW (.,z,λ)

ρ (a)‖

+‖JW (.,z,λ)ρ (a)−JW (.,z,λ)

ρ (a)‖+‖JW (.,z,λ)ρ (a)−JW (.,z,λ)

ρ (a)‖

≤ ‖(g−m)(x, λ)−(g−m)(x, λ)‖+τ

δ‖P (g−m)(x, λ)−P (g−m)(x, λ)‖

δρ[‖N(u, v, w, λ)−N(u, v, w, λ)‖+‖M(r, s, t, λ)−M(r, s, t, λ)‖

]+k1‖z−z‖+k2‖λ−λ‖

≤ l(g−m)‖λ−λ‖+τ

δlP(g−m)‖λ−λ‖+

τ

δρ(lAL(N,1)+lBL(N,2)+lCL(N,3)+lN

+lRL(M,1) + lSL(M,2) + lTL(M,3) + lM)‖λ− λ‖+ k1lD‖λ− λ‖+ k2‖λ− λ‖

≤ θ2 ‖λ−λ‖,(4.25)

where

θ2 := l(g−m) + k2 + k1lD +τ

δ

[lP(g−m) + ρ

(lAL(N,1) + lBL(N,2) + lCL(N,3) + lN

+lRL(M,1) + lSL(M,2) + lTL(M,3) + lM)].

Hence, we obtain

supi1∈F (x,λ)

d(i1, F (x, λ)

)≤ θ2‖λ− λ‖.

By using similar argument, we have

supi2∈F (x,λ)

d(F (x, λ), i2

)≤ θ2‖λ− λ‖.

Hence, it follows that

H(F (x, λ), F (x, λ)

)≤ θ2‖λ− λ‖, ∀ (x, λ), (x, λ) ∈ E × Ω.

By Lemma 2.3, we obtain

H(S(λ), S(λ)

)≤( θ2

1− θ1

)‖λ− λ‖. (4.26)

This implies that S(λ) is H-Lipschitz continuous with respect to λ ∈ Ω.

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84 F.A. KHAN, A.M. ALANAZI, J. ALI

Remark 4.2. For k1, k2, ρ > 0, it is clear that ξ > σ; q ∈ (0, 1); δ = rτ ; ξ −σ >

√132c(L2

N + L2M )(L2P(g−m) − r2(1− q)2

), where LN := (LAL(N,1) + LBL(N,2) +

LCL(N,3)) and LM := (LRL(M,1) + LSL(M,2) + LTL(M,3)). Further, θ ∈ (0, 1) and condi-tion (4.3) of Theorem 4.1 holds for some suitable values of constants.

Remark 4.3. Since the PGSIQVLIP (3.1) includes many known classes of parametricgeneralized variational inclusions as special cases, Theorems 4.1-4.3 improve and generalizethe known results given in [2,3,5,6,8-13,15,19,21,22].

Conflict of Interests. The authors declare no conflict of interests.

Acknowledgments. The authors would like to thank the anonymous referees for theircomments that helped us to improve this paper.

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FAIZAN A. KHANDepartment of Mathematics, University of Tabuk, Tabuk-71491, KSA

E-mail address: [email protected]

ABDULAZIZ M. ALANAZI

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

E-mail address: [email protected]

Javid Ali

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaE-mail address: [email protected]