Convergence and stability of perturbed three-step iterative algorithm for completely generalized nonlinear quasivariational inequalities

Applied Mathematics and Computation 149 (2004) 245–258

www.elsevier.com/locate/amc

Convergence and stability of perturbedthree-step iterative algorithm forcompletely generalized nonlinear

quasivariational inequalities

Zeqing Liu a, Shin Min Kang b,*

a Department of Mathematics, Liaoning Normal University, P.O. Box 200,

Dalian, Liaoning 116029, PR Chinab Department of Mathematics and Research Institute of Natural Science, Gyeongsang

National University, Chinju 660-701, South Korea

Abstract

In this paper, we introduce a new class of completely generalized nonlinear quasi-

variational inequalities and obtain its equivalence with a class of fixed point problems

by using the resolvent operator technique. Using this equivalence, we develop perturbed

three-step iterative algorithm for this class of completely generalized quasivariational

inequalities. We establish a few existence theorems of unique solution for the class of

completely generalized quasivariational inequalities involving relaxed monotone, gen-

eralized pseudocontractive and strongly monotone mappings and prove some conver-

gence and stability results of iterative sequence generated by perturbed three-step

iterative algorithm.

� 2003 Elsevier Inc. All rights reserved.

Keywords: Completely generalized nonlinear quasivariational inequality; Perturbed three-step

iterative algorithm; Relaxed monotone mapping; Generalized pseudocontractive mapping;

Stability

* Corresponding author.

E-mail addresses: [email protected] (Z. Liu), [email protected] (S.M. Kang).

0096-3003/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0096-3003(03)00137-1

mail to: [email protected]

246 Z. Liu, S.M. Kang / Appl. Math. Comput. 149 (2004) 245–258

1. Introduction

Variational inequality theory has become a rich source of inspiration in pure

and applied mathematics. Variational inequalities have been used in a large

variety of problems arising in elasticity, structural analysis, economics, opti-

mization, oceanography, and regional, physical and engineering sciences, etc.

For details, we refer to [1–3,5,6,8,9] and the references therein.

In 2000, Huang et al. [6] introduced and studied a perturbed iterative al-

gorithm for the generalized nonlinear mixed quasivariational inequalities and

proved the convergence and stability of the iterative sequence generated by theperturbed iterative algorithm. In 2001, Cho et al. [3] obtained a few existence

theorems of solutions and convergence theorems of the Mann and Ishikawa

iterative sequences with errors for a class of generalized strongly nonlinear

implicit quasivariational inequalities involving Lipschitzian generalized

pseudocontractive mappings.

Inspired and motivated by the recent research works [3,6], in this paper, we

introduce and study a new class of completely generalized nonlinear quasi-

variational inequalities, which includes as special cases, the classes of varia-tional and quasivariational inequalities studied in [1,3,5,6,8,9]. Making use of

the resolvent operator technique, we establish the equivalence between the

completely generalized nonlinear quasivariational inequality and the fixed-

point problem. We also suggest perturbed three-step iterative algorithm for

solving the class of completely generalized nonlinear quasivariational in-

equalities and obtain a few existence theorems of unique solution for the

completely generalized nonlinear quasivariational inequality involving relaxed

monotone, generalized pseudocontractive and strongly monotone mappingsand some convergence and stability results of iterative sequences generated by

three-step perturbed iterative algorithm. The results presented in this paper

extend, improve and unify the recent results due to Adly [1], Cho et al.

[3], Hassouni and Moudafi [5], Huang et al. [6], Siddiqi and Ansari [8] and

Verma [9].

2. Preliminaries

Let H be a real Hilbert space endowed with a norm k � k and an inner

product h�; �i, 2H , CBðHÞ and CCðHÞ denote the familities of all nonempty

subsets, all nonempty closed bounded subsets and all nonempty closed convex

subsets of H respectively. Let Hð�; �Þ stand for the Hausdorff metric on CBðHÞand I be the identity mapping on H . Assume that g; a; b; c; d : H ! H ;N ;M : N � N ! N , are singlevalued mappings, W : H � H ! 2H is a multi-

valued mapping such that, for each fixed y 2 H , W ð�; yÞ : H ! 2H is maximal

Z. Liu, S.M. Kang / Appl. Math. Comput. 149 (2004) 245–258 247

monotone mapping and gðHÞ \ domðW ð�; yÞÞ 6¼ ;. Given f 2 H , we consider

the following problem:Find u 2 H such that gðuÞ 2 domðW ð�; uÞÞ 6¼ ; and

f 2 NðaðuÞ; bðuÞÞ �MðcðuÞ; dðuÞÞ þ W ðgðuÞ; ðuÞÞ; ð2:1Þ

which is know as the completely generalized nonlinear quasivariational in-

equality.

Some special cases of problem (2.1) are as follows:

(i) If c ¼ d ¼ I and Mðu; uÞ ¼ �f for all u 2 H , then problem (2.1) is equiv-

alent to finding u 2 H such that gðuÞ 2 domðW ð�; uÞÞ and
0 2 NðaðuÞ; bðuÞÞ þ W ðgðuÞ; uÞ; ð2:2Þ
which is called the generalized nonlinear mixed quasivariational inequality

introduced and studied by Huang et al. [6].

(ii) If f ¼ 0, b ¼ d ¼ I , Nðu; vÞ ¼ Mðu; vÞ ¼ u and W ðu; vÞ ¼ W ðuÞ for all

u; v 2 H , where W : H ! 2H is a maximal monotone mapping withgðHÞ \ domðW Þ 6¼ ;, then problem (2.1) collapses to finding u 2 H with

gðuÞ 2 domðW Þ such that

0 2 aðuÞ � cðuÞ þ W ðgðuÞÞ: ð2:3Þ

Inequalities like (2.3) have been studied in [1].

(iii) If c ¼ d ¼ I , f ¼ Mðu; vÞ ¼ 0, Nðu; vÞ ¼ u� v and W ðu; vÞ ¼ ouðuÞ for allu; v 2 H , where ou stands for the subdifferential of a proper, convex and

lower semicontinuous function u : H ! R [ fþ1g, then problem (2.1)

is equivalent to the following problem studied in [5]:

Find u 2 H such that gðuÞ 2 domðouÞ 6¼ ; and

haðuÞ � bðuÞ; v� gðuÞiPuðgðuÞÞ � uðvÞ; v 2 H : ð2:4Þ

(iv) If K : H ! CCðHÞ, f ¼ 0, a ¼ b ¼ c ¼ d ¼ I , Nðu; uÞ ¼ u,W ðu; vÞ ¼ ouðu; vÞ and uðu; vÞ ¼ IkðvÞðuÞ for all u; v 2 H , where IKðvÞð�Þ is

the indicator function of KðvÞ defined by

IKðvÞðuÞ ¼0; if u 2 KðvÞ;þ1; if u 62 KðvÞ;

�

then problem (2.1) is equivalent to finding u 2 H such that gðuÞ 2 KðuÞ andhu�Mðu; uÞ; v� gðuÞiP 0; v 2 KðuÞ; ð2:5Þ

which is called the generalized strongly nonlinear implicit quasivariational

inequality and studied by Cho et al. [3].

(v) If f ¼ 0, a ¼ b ¼ c ¼ g ¼ I , Nðu; uÞ ¼ Mðu; vÞ ¼ u, W ðu; vÞ ¼ ouðuÞ and

uðuÞ ¼ IKðuÞ for all u; v 2 H , where K 2 CCðHÞ and IK is the indicator

function of K, then problem (2.1) collapses to finding u 2 K such that


hu� cðuÞ; v� uiP 0; v 2 K; ð2:6Þ

which is called the nonlinear variational inequality and introduced in [9].

It is clear that the completely generalized nonlinear quasivariational in-

equality (2.1) includes many kinds of variational inequalities and quasivari-

ational inequalities of [1,3,5,6,8,9] as special cases.Let W : H ! 2H be a maximal monotone mapping. For any fixed q > 0, the

mapping JWq : H � H defined by

JWq ðxÞ ¼ ðI þ qW Þ�1ðxÞ; x 2 H

is known as the resolvent operator of W . It is known that the resolvent operator

JWq is singlevalued and nonexpansive.

Definition 2.1. Let W ;Wn : H ! 2H be maximal monotone mapings for nP 0.

The sequence fWngnP 0 is said to be graph-convergence to W (write Wn!GW ) if

for each ðx; yÞ 2 GraphðW Þ, there exists a sequence ðxn; ynÞ 2 GraphðWnÞ suchthat xn ! x and yn ! y in H .

Definition 2.2. Let N : H � H ! H and g : H ! H be mappings.

(a) g is said to be r-strongly monotone and s-Lipschitz continuous if there exist

constants r > 0; s > 0 such that, respectively,

hgðuÞ � gv; u� viP rku� vk2 and kgðuÞ � gvk6 sku� vk; u; v 2 H ;

(b) g is said to be r-strongly monotone with respect to the first argument of N if

there exist a constant r > 0 such that

hNðgðuÞ; xÞ � Nðgv; xÞ; u� viP rku� vk2; u; v; x 2 H ;

(c) g is said to be r-relaxed monotone with respect to the second argument of N if

there exists a constant r > 0 such that

hNðx; gðuÞÞ � Nðy; gvÞ; u� viP � rku� vk2; u; v; x 2 H ;

(d) g is said to be r-generalized pseudocontractive with respect to the first argu-ment of N if there exists a constant r > 0 such that

hNðgðuÞ; xÞ � Nðgv; xÞ; u� vi6 rku� vk2; u; v; x 2 H ;

(e) N is called r-Lipschitz continuous in the first argument if there exists a con-

stant r > 0 such that

kNðu; xÞ � Nðv; xÞk6 rku� vk; u; v; x 2 H :


In a similar way, we can define Lipschitz continuity of N in the second ar-

gument.

Definition 2.3. Let T : H ! H be a mapping and x0 2 H . Assume that

xnþ1 ¼ f ðT ; xnÞ define an iteration procedure which yields a sequence of points

fxngnP 0 in H . Suppose that F ðT Þ ¼ fx 2 H : x ¼ Txg 6¼ ; and fxngnP 0 con-

verges to some u 2 F ðT Þ. Let fzngnP 0 � H and en ¼ kznþ1 � f ðT ; znÞk. If

limn!1 en ¼ 0 implies that limn!1 zn ¼ u, then the iteration procedure defined

by xnþ1 ¼ f ðxn; T Þ is said to be T -stable or stable with respect to T .

Harder and Hicks [4] proved how such a sequence fzngnP 0 could arise in

practice and demonstrated the importance of investigating the stability of

various iterative schemes for various classes of nonlinear mappings.

Lemma 2.1 (Liu, 1995 [7]). Let fangnP 0, fbngnP 0 and fcngnP 0 be nonnegativesequences satisfying

anþ1 6 ð1� dnÞan þ dnbn þ cn; nP 0;

where fdngnP 0 � ½0; 1�,P1

n¼0 dn ¼ 1, limn!1 bn ¼ 0 andP1

n¼0 cn < 1. Thenlimn!1 an ¼ 0.

Lemma 2.2 (Attouch, 1974 [2]). Let Wn and W be maximal monotone mappingsfor nP 0. Then Wn!

GW if and only if JWn

q ðxÞ ! JWq ðxÞ for every x 2 H and q > 0.

3. Perturbed three-step iterative algorithm

The following result is very useful in approximation and numerical analysis

of completely generalized nonlinear quasivariational inequalities.

Lemma 3.1. Let t and q be positive parameter. Then the following statements areequivalent:

(a) the completely generalized nonlinear quasivariational inequality (2.1) has asolution u 2 H with gðuÞ 2 domðW ð�; uÞÞ;

(b) there exists u 2 H satisfying

gðuÞ ¼ JW ð�;uÞq ðgðuÞ � qNðaðuÞ; bðuÞÞ þ qMðcðuÞ; dðuÞÞ þ qf Þ; ð3:1Þ

where JW ð�;uÞq is the resolvent operator;

(c) the mapping F : H ! H defined by

F ðxÞ ¼ ð1� tÞxþ tðx� gðxÞ þ JW ð�;xÞq ðgðxÞ � qNðaðxÞ; bðxÞÞ

þ qMðcðxÞ; dðxÞÞ þ qf Þ; x 2 H ð3:2Þ

has a fixed point u 2 H .


Proof. Notice that t > 0. It is easy to see that u ¼ Fu if and only if (3.1) holds.

On the other hand, (a) holds if and only if

gðuÞ � qNðaðuÞ; bðuÞÞ þ qMðcðuÞ; dðuÞÞ þ qf 2 ðI þ qW ð�; uÞÞðgðuÞÞ:ð3:3Þ

From the definition of the resolvent operator JW ð�;uÞq it follows that (3.3) and

(3.1) are equivalent. This completes the proof. �

Remark 3.1. Lemma 3.1 extends Lemma 3.1 in [1,8], Lemma 2.1 in [5] andTheorem 2.1 in [9].

Based on Lemma 3.1, we suggest the following perturbed three-step iterative

algorithm for the completely generalized nonlinear quasivariational inequality

(2.1).

Algorithm 3.1. Let g; a; b; c; d : H ! H , N ;M : H � H ! H be mappings and

f 2 H . Let Ex ¼ gx� qNðax; bxÞ þ qMðcx; dxÞ þ qf for all x 2 H . Given

u0 2 H , the iterative sequence fungnP 0 is defined by

unþ1 ¼ ð1� an � bnÞun þ anðun � gðvnÞ þ JWnð�;vnÞq ðEðvnÞÞÞ þ bnpn;

vn ¼ ð1� a0n � b0nÞun þ a0nðwn � gðwnÞ þ JWnð�;wnÞq ðEðwnÞÞÞ þ b0nqn;

wn ¼ ð1� a00n � b00nÞun þ a00nðun � gðunÞ þ JWnð�;unÞq ðEðunÞÞÞ þ b00nrn

ð3:4Þ

for all nP 0, where each Wn : H � H ! 2H is a multivalued mapping such that,

for each y 2 H , Wnð�; yÞ : H ! 2H is a maximal monotone mapping, q is a

constant, fpngnP 0 and fqngnP 0, frngnP 0 are bounded sequences in H intro-

duced to take into account possible in exact computation and the sequencesfangnP 0, fbngnP 0, fa0ngnP 0, fb0ngnP 0, fa00ngnP 0 and fb00ngnP 0 are in ½0; 1� andsatisfy

maxfan þ bn; a0n þ b0n; a00n þ b00ng6 1; nP 0; ð3:5Þ

X1n¼0

an ¼ 1; limn!1

a0nb00n ¼ lim

n!1b0n ¼ 0 ð3:6Þ

and one of the following conditions:

X1n¼0

bn < 1; ð3:7Þ

there exists a nonnegative sequence

fhngnP 0 with limn!1

hn ¼ 0 and bn ¼ anhn for all nP 0: ð3:8Þ


Remark 3.2. If a00n ¼ b00n ¼ 0 for all nP 0, then perturbed three-step iterative

algorithm reduces to the Ishikawa type perturbed iterative algorithm. Fur-thermore, if a0n ¼ b0n ¼ 0 for all nP 0, then the Ishikawa type perturbed iter-

ative algorithm reduces to the Mann type perturbed iterative algorithm.

Remark 3.3. In case a00n ¼ b00n ¼ 0 and Wnð�; yÞ ¼ ou;uð�Þ ¼ IKðuÞð�Þ for all nP 0

and y 2 H , where K : H ! CCðHÞ is a multivalued mapping, then perturbed

three-step iterative algorithm yields the Ishikawa iteration process with error

introduced in [3].

4. Existence, convergence and stability

In this section, we discuss those conditions under which the approximatesolutions un obtained from perturbed three-step iterative algorithm converge

strongly to the exact solution u 2 H of the completely generalized nonlinear

quasivariational inequality (2.1) and the convergence are stable.

Theorem 4.1. Let a; b; c; d : H ! H be p-Lipschitz continuous, q-Lipschitzcontinuous, r-Lipschitz continuous, s-Lipschitz continuous, respectively,g : H ! H satisfy that I � g is l-Lipschitz continuous. Let N : H � H ! H bea-Lipschitz continuous in the first argument, b-Lipschitz continuous in the secondargument and a be n-strongly monotone with respect to the first argument of N .Let M : H � H ! H be c-Lipschitz continuous in the first argument, d-Lipschitzcontinuous in the second argument and c is g-generalized pseudocontractive withrespect to the first argument of M . Suppose that Wn, W : H � H ! 2H are suchthat, for each y 2 H and nP 0, Wnð�; yÞ;W ð�; yÞ : H ! 2H are maximal mono-tone, Wnð�; yÞ!

GW ð�; yÞ and

supfkJW ð�;xÞq ðzÞ � JW ð�;yÞ

q ðzÞk; kJWnð�;xÞq ðzÞ � JWnð�;yÞ

q ðzÞk : nP 0g6 lkx� yk;x; y; z 2 H ; ð4:1Þ

where l > 0 is a constant. Let fxngnP 0 be any sequence in H and definefengnP 0 � ½0;1Þ by

en ¼ kxnþ1 � ½ð1� an � bnÞxn þ anðyn � gðynÞ þ JWnð�;ynÞq ðEðynÞÞÞ þ bnpn�k;

yn ¼ ð1� a0n � b0nÞxn þ a0nðzn � gðznÞ þ JWnð�;znÞq ðEðznÞÞÞ þ b0nqn;

zn ¼ ð1� a00n � b00nÞxn þ a00nðxn � gðxnÞ þ JWnð�;xnÞq ðEðxnÞÞÞ þ b00nrn

ð4:2Þ

for all nP 0. Let k ¼ 2lþ l and j ¼ bqþ ds. If there exists a constant q > 0

satisfying


k þ qj < 1 ð4:3Þ

and one of the following conditions:

apþ cr > j; jn� g� ð1� kÞjj >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikð2� kÞððapþ crÞ2 � j2Þ

q;

q

�� n� g� ð1� kÞjðapþ crÞ2 � j2

�� <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn� g� ð1� kÞjÞ2 � kð2� kÞððapþ crÞ2 � j2Þ

qðapþ crÞ2 � j2

;

ð4:4Þ

ap þ cr ¼ j; n� g > ð1� kÞj; q >kð2� kÞ

2ðn� g� ð1� kÞjÞ ; ð4:5Þ

apþ cr < j;

q

�� ð1� kÞj� nþ g

j2 � ðapþ crÞ2

�� >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikð2� kÞðj2 � ðapþ crÞ2Þ þ ðð1� kÞj� nþ gÞ2

qj2 � ðapþ crÞ2

;

ð4:6Þ

then the completely generalized nonlinear quasivariational inequality (2.1) has aunique solution u 2 H and the sequence fungnP 0 defined by Algorithm 3.1 con-verges strongly to u. Moreover, if there exists a constant A > 0 satisfying

an PA; nP 0; ð4:7Þ

then limn!1 xn ¼ u if and only if limn!1 �n ¼ 0.

Proof. First we show that the completely generalized nonlinear quasivari-

ational inequality (2.1) has a unique solution u 2 H . According to Lemma 3.1,

it is enough to prove that the mapping F : H ! H defined by (3.2) has a

unique fixed point u 2 H , where t 2 ð0; 1� is a parameter. Let x; y be arbitrary

elements in H . Since a and c are p-Lipschitz continuous and r-Lipschitz con-

tinuous, respectively, N and M are a-Lipschitz continuous and c-Lipschitzcontinuous, respectively, a is n-strongly monotone with respect to the first

argument of N and c is g-generalized pseudocontractive with respect to the first

argument of M , it follows that

kx� y � qðNðaðxÞ; bðyÞÞ � NðaðyÞ; bðxÞÞ �MðcðxÞ; dðxÞÞ þMðcðyÞ; dðxÞÞÞk2

¼ kx� yk2 � 2qhNðaðxÞ; bðxÞÞ � NðaðyÞ; bðxÞÞ; x� yiþ 2qhMðcðxÞ; dðxÞÞ �MðcðyÞ; dðxÞÞ; x� yi þ q2kNðaðxÞ; bðxÞÞ� NðaðyÞ; bðxÞÞ �MðcðxÞ; dðxÞÞ þMðcðyÞ; dðxÞÞk2

6 ð1� 2qðn� gÞ þ q2ðap þ crÞ2Þkx� yk2: ð4:8Þ


Using (4.1), (4.8), the nonexpansivity of JW ð�;xÞq and Lipschitz continuity of

I � g, b, d and N and M in the second arguments, we infer that

kF ðxÞ � F ðyÞk6 ð1� tÞkx� yk þ tkx� y � ðgðxÞ � gðyÞÞk

þ tkJW ð�;xÞq ðEðxÞÞ � JW ð�;xÞ

q ðEðyÞÞk þ tkJW ð�;xÞq ðEðyÞÞ � JW ð�;yÞ

q ðEðyÞÞk6 ð1� t þ tlÞkx� yk þ tkEðxÞ � EðyÞk þ tlkx� yk6 ð1� t þ tðlþ lÞÞkx� yk þ tkgðxÞ � gðyÞ � ðx� yÞk

þ tkx� y � qðNðaðxÞ; bðxÞÞ � NðaðyÞ; bðxÞÞ �MðcðxÞ; dðxÞÞþMðcðyÞ; dðxÞÞk þ tqkNðaðyÞ; bðxÞÞ � NðaðyÞ; bðyÞÞkþ tqkNðcðyÞ; dðxÞÞ � NðcðyÞ; dðyÞÞk

6 ð1� t þ tð2lþ lÞÞkx� yk

þ tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2qðn� gÞ þ q2ðap þ crÞ2

qkx� yk þ tqðbqþ dsÞkx� yk

¼ ð1� ð1� hÞtÞkx� yk; ð4:9Þ

where

h ¼ k þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2qðn� gÞ þ q2ðap þ crÞ2

qþ qj: ð4:10Þ

In view of (4.3) and (4.10), we have

h < 1 ()ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2qðn� gÞ þ q2ðap þ crÞ2

q< 1� k � qj

() ½ðap þ crÞ2 � j2�q2 � 2qðn� g� ð1� kÞjÞ < k2 � 2k:ð4:11Þ

It follows from (4.11) and one of (4.4)–(4.6) that h < 1. Since t 2 ð0; 1�, by(4.9), we infer that F is a contraction mapping. Hence it has a unique fixed

point u 2 H , which is a unique solution of the completely generalized nonlinear

quasivariational inequality (2.1).Now we show that limn!1 un ¼ u. Notice that

u ¼ ð1� an � bnÞuþ anðu� gðuÞ þ JW ð�;uÞq ðEðuÞÞÞ þ bnu

¼ ð1� a0n � b0nÞuþ a0nðu� gðuÞ þ JW ð�;uÞq ðEðuÞÞÞ þ b0nu

¼ ð1� a00n � b00nÞuþ b00nðu� gðuÞ þ JW ð�;uÞq ðEðuÞÞÞ þ b00nu: ð4:12Þ

Put dn ¼ kJWnð�;uÞq ðEðuÞÞ � JW ð�;uÞ

q ðEðuÞÞk and L ¼ supfkpn � uk; kqn � uk;krn � uk : nP 0g. Lemma 3.2 ensures that

limn!1

dn ¼ 0: ð4:13Þ


Using (3.4), (3.5), (4.8) and (4.12), we know that

kunþ1 � uk6 ð1� an � bnÞkun � uk þ ankvn � u� ðgðvnÞ � gðuÞÞk

þ ankJWnð�;vnÞq ðEðvnÞÞ � JWnð�;vnÞ

q ðEðuÞÞkþ ankJWnð�;vnÞ

q ðEðuÞÞ � JWnð�;uÞq ðEðuÞÞk

þ ankJWnð�;uÞq ðEðuÞÞ � JW ð�;uÞ

q ðEðuÞÞk þ bnkpn � uk6 ð1� an � bnÞkun � uk þ anlkvn � uk þ ankEðvnÞ � EðuÞk

þ anlkvn � uk þ andn þ bnL

6 ð1� an � bnÞkun � uk þ ankkvn � uk� ankvn � u� qðNðaðvnÞ; bðvnÞÞ � NðaðuÞ; bðvnÞÞ

�MðcðvnÞ; dðvnÞÞ þMðcðuÞ; dðvnÞÞkþ anqkNðaðuÞ; bðvnÞÞ � NðaðuÞ; bðuÞÞkþ anqkMðcðuÞ; dðvnÞÞ �MðcðuÞ; dðuÞÞk þ andn þ bnL

6 ð1� an � bnÞkun � uk þ ankkvn � uk

þ an

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2qðn� gÞ þ q2ðap þ crÞ2

qkvn � uk

þ anqðbqþ dsÞkvn � uk þ andn þ bnL

¼ ð1� an � bnÞkun � uk þ anhkvn � uk þ andn þ bnL: ð4:14Þ

Similarly, we have

kvn � uk6 ð1� a0n � b0nÞkun � uk þ a0nhkwn � uk þ a0ndn þ b0nL;

kwn � uk6 ð1� a00n � b00nÞkun � uk þ a00nhkun � uk þ a00ndn þ b00nL:ð4:15Þ

Substituting (4.15) into (4.14), we get that

kunþ1 � uk6 ½1� an � bn þ anhð1� a0n � b0n þ a0nhð1� a00n � b00nþ a00nhÞÞ�kun � uk þ an½ha0nðha00ndn þ hLb00n þ dnÞ þ hLb0n þ dn� þ bnL

6 ð1� ð1� hÞanÞ þ an½a0nð2dn þ Lb00nÞ þ Lb0n þ dn� þ bnL:

ð4:16Þ

Suppose that (3.7) holds. Set an ¼ kun � uk, dn ¼ ð1� hÞan; bn ¼ ð1� hÞ�1 �½a0nð2dn þ Lb00n þ Lb0n þ dnÞ� and cn ¼ bnL for all nP 0. It follows from Lemma

2.1, (3.6), (3.7), (4.13) and (4.16) that limn!1 an ¼ 0. That is, limn!1 un ¼ u.Suppose that (3.8) holds. Set an ¼ kun � uk, dn ¼ ð1� hÞan, bn ¼

ð1� hÞ�1½a0nð2dn þ Lb00nÞ þ Lb0n þ dn þ Lbn� and cn ¼ 0 for all nP 0. Using

Lemma 2.1, (3.6), (3.8), (4.13) and (4.16), we obtain that limn!1 un ¼ u.Now we assume that (4.7) holds. Observe that either (3.7) or (3.8) yields that

limn!1

bn ¼ 0: ð4:17Þ


As in the proof of (4.14), by (4.7), we conclude that

kð1� an � bnÞxn þ anðyn � gðynÞ þ JWnð�;ynÞq ðEðynÞÞÞ þ bnpn � uk

6 ð1� ð1� hÞanÞkxn � uk þ an½a0nð2dn þ Lb00nÞ þ Lb0n þ dn� þ bnL

6 ð1� ð1� hÞAÞkxn � uk þ a0nð2dn þ Lb00nÞ þ Lb0n þ dn� þ bnL:

ð4:18Þ

Suppose that limn!1 xn ¼ u. By virtue of (3.6), (4.2), (4.13), (4.17), (4.18)

and one of (3.7) and (3.8), we obtain that

e6 kxnþ1 � uk þ kð1� an � bnÞxnþ anðyn � gðynÞ þ JWnð�;ynÞ

q ðEðynÞÞÞ þ bnpn � uk6 kxnþ1 � uk þ ð1� ð1� hÞAÞkxn � uk

þ a0nð2dn þ Lb00nÞ þ Lb0n þ dn þ bnL ! 0

as n ! 1. That is, limn!1 en ¼ 0.Conversely, suppose that limn!1 en ¼ 0. In light of (4.2) and (4.18), we know

that

kxnþ1 � uk6 kð1� an � bnÞxn þ anðyn � gðynÞ þ JWnð�;ynÞ

q ðEðynÞÞÞ þ bnpn � uk þ en

6 ð1� ð1� hÞAÞkxn � uk þ a0nð2dn þ Lb00nÞ þ Lb0n þ dn þ bnLþ en:

ð4:19Þ

Let an ¼ kxn � uk, dn ¼ ð1� hÞA, bn ¼ ð1� hÞ�1A�1½a0nð2dn þ Lb00nÞ þ Lb0n þdn þ bnLþ en� and cn ¼ 0 for all nP 0. It follows from Lemma 2.1, (3.6), (4.13),

(4.17) and (4.19) that limn!1 xn ¼ u. This completes the proof. �

Theorem 4.2. Let k, a, b, c, d, g, N , M , W , fWngnP 0, fxngnP 0 and fengnP 0 be asin Theorem 4.1. Suppose that d is f-realxed monotone with respect to the secondargument of M . Let j ¼ bqþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2gþ c2r2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2fþ d2s2

p. If there exists a

constant q > 0 satisfying (4.3) and one of the following conditions:

ap > j;

jn� ð1� kÞjj >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikð2� kÞða2p2 � j2Þ

p;

q

�� n� ð1� kÞja2p2 � j2

�� <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn� ð1� kÞjÞ2 � kð2� kÞða2p2 � j2Þ

qa2p2 � j2

; ð4:20Þ

ap ¼ j; n > ð1� kÞj; q >kð2� kÞ

2ðn� ð1� kÞjÞ ; ð4:21Þ


ap < j; q

�� ð1� kÞj� nj2 � a2p2

�� >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikð2� kÞðj2 � a2p2Þ þ ðð1� kÞj� nÞ2

qj2 � a2p2

;

ð4:22Þ
then the completely generalized nonlinear quasivariational inequality (2.1) has aunique solution u 2 H and the sequence fungnP 0 defined by Algorithm 3.1 con-verges strongly to u. Moreover, if there exists a constant A > 0 satisfying (4.7),then limn!1 xn ¼ u if and only if limn!1 en ¼ 0.
Proof. Since d is s-Lipschitz continuous, f-relaxed monotone with respect to

the second argument of M and M is d-Lipschitz continuous in the second ar-

gument, it follows that

kMðcðyÞ; dðxÞÞ �MðcðyÞ; dðyÞÞ � ðx� yÞk2

¼ kx� yk2 � 2hMðcðyÞ; dðxÞÞ �MðcðyÞ; dðyÞÞ; x� yiþ kMðcðyÞ; dðxÞÞ �MðcðyÞ; dðyÞÞk2

6 ð1þ 2fþ d2s2Þkx� yk2: ð4:23Þ
As in the proof of Theorem 4.1, by (4.23), we know that
kF ðxÞ � F ðyÞk6 ð1� t þ tlÞkx� yk þ tkEðxÞ � EðyÞk þ tlkx� yk6 ð1� t þ tkÞkx� yk þ tkx� y � qðNðaðxÞ; bðxÞÞ � NðaðyÞ; bðxÞÞÞk

þ tqkNðaðyÞ; bðxÞÞ � NðaðyÞ; bðyÞÞk þ tqkMðcðxÞ; dðxÞÞ �MðcðyÞ; dðxÞÞþ x� yk þ tqkMðcðyÞ; dðxÞÞ �MðcðyÞ; dðyÞÞ � ðx� yÞk

6 ð1� ð1� hÞtÞkx� yk;
where h ¼ k þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2qnþ q2a2p2

pþ qj. Thus, (4.3) and one of (4.20)–(4.22)

ensure that h < 1. That is, F has a unique fixed point u 2 H , which is a unique

solution of the completely generalized nonlinear quasivariational inequality

(2.1).

Similarly, we can show that

kunþ1 � uk6 ð1� ð1� hÞanÞkun � uk þ an½a0nð2dn þ Lb00nÞ þ Lb0n þ dn� þ bnL:

The rest of the argument follows as in the proof of Theorem 4.1 and is

therefore omitted. This completes the proof. �

Remark 4.1. Theorems 4.1 and 4.2 extend Theorems 3.1–3.6 in [3] and The-

orem 2.2 in [9] in the following ways:

(a) The Mann iterative scheme in [9] and the Ishikawa iterative scheme with

errors in [3] are replaced by the more general perturbed three-step iterative

algorithm.


(b) The generalized strongly nonlinear implicit quasivariational inequalities in

[3] and the nonlinear quasivariational inequalities in [9] are replaced by themore general completely generalized nonlinear quasivariational inequali-

ties.

(c) Conditions (4.4)–(4.6) are weaker than conditions (3.3)–(3.5) in [3].

(d) Conditions (3.6)–(3.8) are weaker than conditions (1)–(3) in [3]. On

the other hand, the authors in [3] used condition limn!1 bn ¼ 0, so they

could not establish the stability of the Ishikawa iterative scheme with er-

rors for the generalized strongly nonlinear implicit quasivariational in-

equalities.

Replacing the Lipschitz continuity of I � g by the Lipschitz continuity and

the strong monotonicity of g in Theorems 4.1 and 4.2, we have the following

results.

Theorem 4.3. Let j, a, b, c, d, N , M , W , fWngnP 0, fxngnP 0 and fengnP 0 be as inTheorem 4.1. Let g : H ! H be l-Lipschitz continuous and m-strongly monotoneand k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2mþ l2

pþ l. If there exists a constant q > 0 satisfying (4.3) and

one of (4.4)–(4.6), then the completely generalized nonlinear quasivariationalinequality (2.1) has a unique solution u 2 H and the sequence fungnP 0 defined byAlgorithm 3.1 converges strongly to u. Moreover, if there exists a contant A > 0

satisfying (4.7), then limn!1 xn ¼ u if and only if limn!1 en ¼ 0.

Proof. Since g is l-Lipschitz continuous and m-strongly monotone, it follows

that

kx� y � ðgðxÞ � gðyÞÞk2 ¼ kx� yk2 � 2hgðxÞ � gðyÞ; x� yi þ kgðxÞ � gðyÞk2

6 ð1� 2mþ l2Þkx� yk2:

By a similar argument used in the proof of Theorem 3.1, the result follows.

This completes the proof. �

A proof similar to that of Theorem 3.1 gives the following result and is thus

omitted.

Theorem 4.4. Let j, a, b, c, d, N , M , W , fWngnP 0, fxngnP 0 and fengnP 0 be as inTheorem 4.2. Let k; g be as in Theorem 4.3. If there exists a constant q > 0

satisfying (4.3) and one of (4.20)–(4.22), then the completely generalized non-linear quasivariational inequality (2.1) has a unique solution u 2 H and the se-quence fungnP 0 defined by Algorithm 2.1 converges strongly to u. Moreover, ifthere exists a constant A > 0 satisfying (4.7), then limn!1 xn ¼ u if and only iflimn!1 en ¼ 0.


Remark 4.2. Theorems 4.3 and 4.4 include Theorem 3.4 in [1], Theorem 2.1 in

[5], Theorem 5.1 in [6], Theorem 3.1 in [8] and Theorem 2.2 in [9] as specialcases.

Acknowledgement

This work was supported by Korea Research Foundation Grant (KRF-2001-005-D00002).

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Documents

Convergence and stability of perturbed three-step iterative algorithm for completely generalized nonlinear quasivariational inequalities