Convergence and stability of a three-step iterative algorithm for a general quasi-variational inequality problem

ITERATIVE ALGORITHM FOR A GENERAL

QUASI-VARIATIONAL INEQUALITY PROBLEM

Received 11 February 2005; Revised 10 September 2005; Accepted 13 September 2005

We consider a general quasi-variational inequality problem involving nonlinear, non-convex and nondifferentiable term in uniformly smooth Banach space. Using retraction mapping and fixed point method, we study the existence of solution of general quasi-variational inequality problem and discuss the convergence analysis and stability of a three-step iterative algorithm for general quasi-variational inequality problem. The the-orems presented in this paper generalize, improve, and unify many previously known results in the literature.

Copyright © 2006 K. R. Kazmi and M. I. Bhat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Many problems arising in physics, mechanics, elasticity and engineering sciences can be formulated in variational inequalities involving nonlinear, nonconvex and nondiff eren-tiable term, see for example Baiocchi and Capelo [4], Duvaut and Lions [8] and Kikuchi and Oden [15]. The proximal (resolvent) method used to study the convergence analy-sis of iterative algorithms for variational inclusions, see [14,20], cannot be adopted for studying such classes of variational inequalities due to the presence of nondifferentiable term.

There are some methods, for example projection method and auxiliary principle method which can be used to study such classes of variational inequalities, see [7,17–

19] and the relevent references cited therein. It is remarked that most of the work, us-ing projection method and auxiliary principle method, has been done in the settus-ing of Hilbert space. Recently, Alber and Yao [3] and Chen et al. [6] studied some classes of co-variational inequality and co-complementarity problems in Banach spaces. Therefore, the study of other classes of variational inequalities using projection method and aux-iliary principle method in the setting of Banach spaceremains an interesting problem. Very recently, Chidume et al. [7] studied some classes of variational inequalities involving

nonlinear, convex and nondifferentiable term, using auxiliary principle method in the set-ting of reflexive Banach space.

In recent years, one step and two-step iteration algorithms (including Mann Iteration and Ishikawa iteration processes as the most important cases) have been extensively stud-ied by many authors to solve the nonlinear operator equations and variational inequality problems in Hilbert spaces and Banach spaces, see for example [3,6,7,12–14,16,18–

20,23–25,27,28] and the references therein. Noor [21,22] introduced and analyzed three-step iterative methods to study the approximate solutions of variational inequali-ties (inclusions) in Hilbert spaces by using the techniques of updating the solution and the auxiliary principle. Further Xu and Noor [26] and Liu et al. [17] used three step iterative algorithms to study nonlinear operator equations and variational inequality problems, respectively. A similar idea goes back to the so calledθ-schemes introduced by Glowinski and Le Tallec [9] to find a zero of sum of two (or more) maximal monotone operators by using the Lagrangian multiplier. Glowinski and Le Tallec [9] used three-step iterative algorithms to find the approximate solutions of the elastoviscoplasticity problem, liquid crystal theory, and eigenvalue computation, and they showed that three-step approxi-mations perform better numerically. Haubruge et al. [11] studied the convergence anal-ysis of three-step iterative algorithms of Glowinski and Le Tallec [9] and applied these algorithms to obtain new splitting-type algorithms for solving variational inequalities, separable convex programming, and minimization of a sum of convex functions. They also proved that three-step iterations lead to highly parallelized algorithms under certain conditions.

It has been shown in [11,21, 22] that three step iterative algorithms are a natural generalization of the splitting methods for solving partial differential equations (inclu-sions). For applications of splitting and decomposition methods, see [9,11,21,22] and the references therein. Thus one can conclude that three-step iterative algorithms play an important and significant part in solving various problems, which aries in pure and applied sciences. On the other hand there are no such three-step iterative algorithm for solving quasi-variational inequality problems in Banach spaces.

Motivated by these facts and the recent work going in this direction, we consider a general quasi-variational inequality problem (in short, GQVIP) involving nonlinear, nonconvex and nondifferentiable term, inuniformly smooth Banach space. Using sunny retraction mapping, we establish that GQVIP is equivalent to some relations. Further, using these relations, we suggest a three-step iterative algorithm for finding the approxi-mate solution of GQVIP. Furthermore, using fixed point method, we prove the existence of unique solution of GQVIP and discuss the convergence analysis and stability of the three-step iterative algorithm. The theorems presented in this paper generalize, improve and unify the results given in [5,12,13,18,24–27] and in the relevant references cited therein.

2. Preliminaries and formulation of problem

Throughout this paper, unless the contrary is stated, we assume thatE is a real uni-formly smooth Banach space equipped with norm · ;·,·is the dual pair between

J(x),x = J(x)2

E = x2E ∀x∈Eand CC(E) be the family of all nonempty, closed and convex subsets ofE. We note that ifE≡H, a Hilbert space, thenJbecomes identity mapping.

First we recall the following concepts and results which are needed in the sequel. Definition 2.1. A single-valued mappingg:E→Eis said to be

(i)k-strongly accretiveif there exists a constantk >0 such that

g(u)−g(v),J(u−v)≥ku−v2_{, ∀u,v∈E; (2.1)}

(ii)δ-Lipschitz continuousif there exists a constantδ >0 such that

_{g(u)−g(v)≤δu−v, ∀u,v∈E. (2.2)}

Definition 2.2. A mappingN(·,·,·) :E×E×E→Eis said to be

(i)α-strongly accretivein the first argument if there exists a constantα >0 such that

N(u,·,·)−N(v,·,·),J(u−v)≥αu−v2_{, ∀u,v∈E; (2.3)}

(ii)β-Lipschitz continuousin the first argument if there exists a constantβ >0 such that

_{N(u,·,·)−N(v,·,·)≤βu−v, ∀u,v∈E. (2.4)}

Definition 2.3[2,6,10]. LetK⊂Ebe a nonempty closed convex set. A mappingRK:E→

Kis said to be (i)retractionif

R2_K=RK; (2.5)

(ii)nonexpansive retractionif

_{RKu−RKv≤ u−v, ∀u,v∈E; (2.6)}

(iii)sunny retractionif

RKRKu−tu−RKu=RKu, ∀u∈E,t∈R. (2.7)

Lemma2.4 [6,10]. A retractionRKis sunny and nonexpansive if and only if

u−RK(u),J

RK(u)−v

Lemma2.5 [2,6,10]. For allu,v∈E, we have (i)u+v2_{≤ u}2_{+ 2v,J(u+v),} (ii)u−v,Ju−Jv ≤ 2d2_ρ

E(4u−v/d), where d=

(u2_+v2_{)/2 ρ}

E(t)= sup{((u+v)/2)−1 :u =1, v =t}is called the modulus of smoothness ofE.

Definition 2.6[23]. LetEbe a Banach space; letT:E→Ebe a mapping, and letu0∈E. Assume thatun+1= f(T,un) defines an iteration procedure which yields a sequence of points{un}∞

n=0⊆E. Suppose thatF(T)= {u∈H:T(u)=u} = ∅and that{un}∞n=0⊆E converges to somex∈F(T). Let{zn}∞

n=0⊆Eandn= zn+1−f(T,zn). If limn→∞n=0 implies limn→∞zn=x, then the iteration procedure defined byun+1= f(T,un) is said to beT-stableorstable with respect toT.

Lemma2.7 [16]. Let{an},{bn}, and{cn}be sequences of nonnegative real numbers satis-fying

an+1=

1−λn

an+bnλn+cn, ∀n≥0, (2.9)

where∞_n=0λn= ∞,{λn} ⊂[0, 1],limn→∞bn=0,∞n=0cn<∞. Thenlimn→∞an=0. We remark thatLemma 2.7is the particular case of Lemma 1 of Alber [1].

LetN:E×E×E→Eandg,h,A,B,C:E→Ebe single-valued mappings and letK:

E→CC(E) be a set-valued mapping. We consider the following general quasi-variational inequality problem (GQVIP): Findu∈Esuch thatg(u)∈K(u) and

hg(u),Jv−g(u)+ρb(u,v)−ρbu,g(u)

≥h(u),Jv−g(u)−ρNA(u),B(u),C(u)−f,Jv−g(u), (2.10)

∀v∈K(u), whereρ >0 is a constant; f ∈Eandb(·,·) :E×E→Ris a nonlinear, non-convex and nondifferentiable form satisfying the following conditions.

Condition 2.8. (i)b(·,·) is linear in the first argument; (ii) there exists a constantν>0 such that

b(u,v)≤νuv, ∀u,v∈E; (2.11)

(iii)b(u,v)−b(u,w)≤b(u,v−w),∀u,v∈E. Remark 2.9. (i)Condition 2.8(i)-(ii) implies that

−b(u,v)≤νuv, ∀u,v∈E. (2.12)

(ii) AlsoCondition 2.8(i)–(iii) imply that

_{b(u,v)−b(u,w) ≤νuv−w, ∀u,v,w∈E, (2.13)}

that is,b(u,v) is continuous with respect to the second argument.

2.1. Some special cases of GQVIP (2.10). (I) If f ≡Θ, where Θ is the zero element in E;N(u,v,w)≡u,∀u,v,w∈E, then GQVIP (2.10) reduces to the following quasi-variational inequality problem: Findu∈Esuch thatg(u)∈K(u) and

hg(u),Jv−g(u)+ρb(u,v)−ρbu,g(u)

≥h(u),Jv−g(u)−ρA(u),Jv−g(u), ∀v∈K(u), (2.14)

which appears to be new. Problem (2.14) has been studied by Zeng [27] in the setting of Hilbert space.

(II) If f ≡Θ; b≡0, a zero mapping, and N(u,v,w)≡u+v, ∀u,v,w∈E, then GQVIP (2.10) reduces to the following quasi-variational inequality problem: Findu∈E

such thatg(u)∈K(u) and

hg(u),Jv−g(u)

≥h(u),Jv−g(u)−ρ(A+B)(u),Jv−g(u), ∀v∈K(u), (2.15)

which appears to be new. Problem (2.15) has been studied by Verma [25] in the setting of Hilbert space.

(III) If f ≡Θ;b≡0, andN(u,v,w)≡u,∀u,v,w∈E, then GQVIP (2.10) reduces to the following quasi-variational inequality problem: Findu∈Esuch thatg(u)∈K(u) and

hg(u),Jv−g(u)

≥h(u),Jv−g(u)−ρA(u),Jv−g(u), ∀v∈K(u), (2.16)

which is also appears to be new. Problem (2.16) has been studied by Zeng [28] in the setting of Hilbert space.

We remark that for the appropriate and suitable choices of mappingsg,h,A,B,C,

3. A three-step iterative algorithm

First we prove the following important lemma.

Lemma3.1. Lett,ρ,λbe positive parameters witht≤1and letCondition 2.8be held. Then the following statements are equivalent:

(a)GQVIP (2.10) has a solutionu∈Ewithg(u)∈K(u); (b)there existsu∈Esuch thatg(u)∈K(u)and

u−Φ(u),Jv−g(u)≥0, ∀v∈K(u), (3.1)

where the mappingΦ:E→Eis defined by

Φ(u),J(v)=u,J(v)−hg(u),J(v)+h(u),J(v)

−ρNA(u),B(u),C(u)−f,J(v)−ρb(u,v), ∀u,v∈E; (3.2)

(c)there existsu∈Esuch thatg(u)∈K(u)and

g(u)=RK(u)

g(u)−λu+λΦ(u), (3.3)

where the mappingRK(u)is sunny retraction fromEontoK(u); (d)the mappingF:E→Edefined by

F(u)=(1−t)u+tu−g(u) +RK(u)g(u)−λu+λΦ(u), (3.4)

for allv∈Ehas a fixed point.

Proof. (a)⇒(b). Let (a) hold, that is,u∈Esuch thatg(u)∈K(u) and

hg(u),Jv−g(u)+ρb(u,v)−ρbu,g(u)

≥h(u),Jv−g(u)−ρNA(u),B(u),C(u)−f,Jv−g(u), (3.5)

which can be rewritten as

u,Jv−g(u)≥u,Jv−g(u)−hg(u),Jv−g(u)+h(u),Jv−g(u) −ρbu,v−g(u)−ρNA(u),B(u),C(u)−f,Jv−g(u).

(3.6)

By using (3.2), the preceding inequality becomes

u−Φ(u),Jv−g(u)≥0, ∀v∈E. (3.7)

(b)⇒(a). It is immediately followed by retracing the above steps and usingCondition 2.8.

Since, forλ >0,

λu−Φ(u),Jv−g(u)=g(u)−g(u)−λu+λΦ(u),Jv−g(u), ∀u,v∈E.

(3.8)

Therefore, from (3.8) andLemma 2.4, it follows the statements (b) and (c) are equiv-alent. Moreover, one can easily prove that fort∈(0, 1], (c) and (d) are equivalent. This

completes the proof.

Based on the above lemma, we suggest the following three-step iterative algorithm for finding the approximate solution of GQVIP (2.10).

3.1. Three-step iterative algorithm (TSIA) (3.1). Letg,h,A,B,C:E→E;K:E→CC(E). Givenu0∈E, compute the sequence{un}defined by the following iterative schemes:

un+1=

1−αn

un+αn

vn−g

vn

+RK(vn)

gvn

−λvn+λΦ

vn

+αnrn,

vn=

1−βn

uC+βn

wn−g

wn

+RK(wn)

gwn

−λwn+λΦ

wn

+βnqn; (3.9)

wn=

1−γn

un+γn

un−g

un

+RK(un)

gun

−λun+λΦ

un

+γnpn, (3.10) forn=0, 1, 2, 3,. . ., whereΦis given by

Φun

,Jvn

=un,J

vn

−hgun

,Jvn

+hun

,Jvn

−ρNAun

,Bun

,Cun

−f,Jvn

−ρbun,vn

, ∀vn∈K

un

; (3.11)

λ >0 is a parameter;{pn},{qn},{rn}are sequences of elements inEintroduced to take into account the possible inexact computations of the retraction points, and{αn},{βn}, {γn}are the sequences of real numbers satisfying the condition

∞

i=0

αn= ∞, 0≤αn,βn,γn≤1,∀n≥0. (3.12)

4. Existence of solution, convergence analysis, and stability

In this section, first we establish the existence of unique solution for GQVIP (2.10) and discuss the convergence analysis of TSIA (3.1).

Theorem4.1. LetEbe a uniformly smooth Banach space withρE(t)≤ct2for some constant

c >0. Letλ be a positive parameter; let the mappings g,h,A,B,C:E→E beq-Lipschitz continuous,m-Lipschitz continuous,r-Lipschitz continuous,s-Lipschitz continuous andξ -Lipschitz continuous, respectively; letgbe p-strongly accretive; let the mappingN:E×E×

E→Ebeβ-Lipschitz continuous,σ-Lipschitz continuous andτ-Lipschitz continuous in the first, second and third arguments, respectively, and beα-strongly accretive with respect toA

(i)

_{RK(u)(z)−RK(v)(z)≤μu−v, ∀u,v∈E; (4.1)}

(ii)b(·,·) :E×E→RsatisfyCondition 2.8(i)–(iii); (iii)

θ:=λk+iρ+1−2ρα+ρ2_d2_;

i:=ν+σs+τξ; d2_:=64cβ2_r2_, (4.2)

where

k:=λ−11−2p+ 64cq2_+μ+λ2_{−2λp+ 64cq}2

+m(q+ 1). (4.3)

Further assume thatCondition 4.2orCondition 4.3below hold.

Condition 4.2. Forρ >0,

ρi < λ−1_{−k≤1, (4.4)}

and one of the following conditions holds.

d > i, _α−

λ−1_{−ki >1−λ−}1_−k2_d2_−i2_,

ρ−α−

λ−1_−ki d2_−i2

<

α−λ−1_−ki2_−1−λ−1_−k2_d2_−i2

d2_−i2 ;

(4.5)

d=i,

α >λ−1_−ki,

ρ >1−λ−1_−k2_/2α−λ−1_−ki;

(4.6)

d < i,

ρ−

λ−1_−ki−α i2_−d2

>

i2_−d2_1−λ−1_−k2_+λ−1_−ki−α2

i2_−d2 .

(4.7)

Condition 4.3. Forρ >0,

and one of the following conditions holds:

d > i,

ρ−α−

λ−1_−ki d2_−i2

<

α−(λ−1_−k)i2_−1−λ−1_−k2_d2_−i2

d2_−i2 ;

(4.9)

d=i,

α <λ−1−ki,

ρ <λ−1_−k2_−1/2λ−1_−ki−α;

(4.10)

d < i,

_λ−1_{−ki−α >λ−}1_−k2_−1d2_−i2_,

ρ−

λ−1_−ki−α i2_−d2

>

i2_−d2_1−λ−1_−k2_+λ−1_−ki−α2

i2_−d2 .

(4.11)

Then GQVIP (2.10) has a unique solutionu∈E. Further, the sequence{un}generated by TSIA (3.1), converges strongly touprovided that

lim

n→∞βnγnpn=n→∞limβnqn=n→∞limrn=0. (4.12)

Proof. From (3.4), (4.1) andLemma 2.4, we estimateF(u)−F(v):

_{F(u)−F(v)=(1−t)u+t}

u−g(u) +RK(u)

g(u)−λu+λΦ(u)

+ (1−t)v+tv−g(v) +RK(v)

g(v)−λv+λΦ(v) ≤(1−t)u−v+tu−v−g(u)−g(v)

+tRK(u)

g(u)−λu+λΦ(u)−RK(u)

g(v)−λv+λΦ(v)

+tRK(u)

g(v)−λv+λΦ(v)−RK(v)

g(v)−λv+λΦ(v) ≤(1−t)u−v+tu−v−g(u)−g(v)

+tg(u)−g(v)−λ(u−v) +λΦ(u)−Φ(v)+tμu−v ≤(1−t)u−v+tu−v−g(u)−g(v)

+tg(u)−g(v)−λ(u−v)+tλΦ(u)−Φ(v)+tμu−v.

Now sincegisp-strongly accretive andq-Lipschitz continuous then by usingLemma 2.5, we have

_u−v−

g(u)−g(v)2

≤ u−v2_{−2g(u)−g(v),Ju−v−g(u)−g(v)}

= u−v2_{−2g(u)−g(v),J(u−v)}

+ 2g(u)−g(v),J(u−v)−Ju−v−g(u)−g(v) ≤1−2p+ 64cq2_u−v2_,

(4.14)

and similarly, we have

_{g(u)−g(v)−λ(u−v)≤λ}2_{−2λp+ 64cq}2_{u−v. (4.15)}

Now, using (2.14),Condition 2.8(i), andRemark 2.9(ii), we have

_{Φ(u)−Φ(v)}2

= Φ(u)−Φ(v),JΦ(u)−Φ(v)

= u−v,JΦ(u)−Φ(v)−ρbu,Φ(u)−Φ(v)+ρbv,Φ(u)−Φ(v)

−hg(u)−hg(v),JΦ(u)−Φ(v)+h(u)−h(v),JΦ(u)−Φ(v)

−ρNA(u),B(u),C(u)−NA(v),B(v),C(v),JΦ(u)−Φ(v)

≤ u−v−hg(u)−hg(v)+h(u)−h(v)

−ρNA(u),B(u),C(u)−NA(v),B(v),C(v),JΦ(u)−Φ(v)

+ρ bu−v,Φ(u)−Φ(v)

≤u−v−ρNA(u),B(u),C(u)−NA(v),B(v),C(v)

+hg(u)−hg(v)+h(u)−h(v)Φ(u)−Φ(v)

+ρ bu−v,Φ(u)−Φ(v)

≤u−v−ρNA(u),B(u),C(u)−NA(v),B(v),C(v)

+hg(u)−hg(v)+h(u)−h(v)+ρνu−vΦ(u)−Φ(v).

Now, sinceg andhareq-Lipschitz continuous andm-Lipschitz continuous, respec-tively, the preceding inequality reduces to

_{Φ(u)−Φ(v)≤u−v−ρ}

NA(u),B(u),C(u)−NA(v),B(v),C(v)

+m(q+ 1) +ρνu−v. (4.17)

Next, we have the following estimate: _u−v−ρ

NA(u),B(u),C(u)−NA(v),B(v),C(v) ≤u−v−ρNA(u),B(u),C(u)−NA(v),B(u),C(u)

+ρNA(v),B(u),C(u)−NA(v),B(v),C(u) +ρNA(v),B(v),C(u)−NA(v),B(v),C(v).

(4.18)

SinceN isα-strongly accretive with respect toAin the first argument, and N isβ -Lipschitz continuous,σ-Lipschitz continuous andτ-Lipschitz continuous with respect to the first, second and third arguments, respectively, we can easily obtain the following estimates:

_u−v−ρ

NA(u),B(u),C(u)−NA(v),B(u),C(u)

≤1−2ρα+ 64cρ2_β2_r2_u−v; (4.19)

_N

A(v),B(u),C(u)−NA(v),B(v),C(u)≤σsu−v; (4.20) _N

A(v),B(v),C(u)−NA(v),B(v),C(v)≤τξu−v. (4.21)

Combining (4.18)–(4.21), we have _u−v−ρ

NA(u),B(u),C(u)−NA(v),B(v),C(v)

≤1−2ρα+ 64cρ2_β2_r2_{+ρ(σs+τξ)u−v.} (4.22)

From (4.17) and (4.22), we have

_{Φ(u)−Φ(v)≤1−2ρα+ 64cρ}2_β2_r2_+ρ(ν_{+σs+τξ) +m(q+ 1)u−v. (4.23)}

From (4.13)–(4.15) and (4.23), we have

_{F(u)−F(v)≤1−t+t1−2p+ 64cq}2_+λ2_{−2λp+ 64cq}2_{+μ+λm(q+ 1)}

+λρ(ν+σs+τξ) +λ1−2ρα+ 64cρ2_β2_r2_u−v

=1−t(1−θ)u−v.

Now,

θ <1⇐⇒1−2ρα+ρ2_d2_<λ−1_−k−iρ

⇐⇒ρ2_d2_−i2_{−2ρα−λ}−1_−ki<λ−1_−k2_−1. (4.25)

From eitherCondition 4.2 orCondition 4.3, and (4.25), it follows thatθ <1. Since

t∈(0, 1], it follows from (4.24) thatFis a contraction mapping. Therefore, by Banach contraction principle,Fhas a unique fixed pointuinE. Thus it follows fromLemma 3.1

that GQVIP (2.10) has a unique solutionuinE. Further, we observe thatusatisfies

u=1−αn

u+αn

u−g(u) +RK(u)

g(u)−λu+λΦ(u);

u=1−βn

u+βn

u−g(u) +RK(u)

g(u)−λu+λΦ(u);

u=1−γnu+γnu−g(u) +RK(u)

g(u)−λu+λΦ(u),

(4.26)

for alln=0, 1, 2, 3,. . . .

Using (3.2), (4.26) and repeating the above arguments, we obtain _Φ

un

−Φ(u)≤1−2ρα+ρ2_d2_{+ρi+m(q+ 1)u}

n−u. (4.27)

It follows from (3.10), (4.26), (4.27), andLemma 2.4that _wn−u≤

1−γnun−u+γnun−u−

gun

−g(u)+γnpn +γnRK(un)

gun

−λun+λΦ

un

−RK(u)

g(u)−λu+λΦ(u) ≤1−γn+γn

1−2p+ 64cρ2_q2_+γ

nμ

un−u+γnpn +gun

−g(u)−λun−u+γnλΦ

un

−Φ(u) ≤1−(1−θ)γnun−u+γnpn

≤un−u+γnpn, since1−(1−θ)γn≤1.

(4.28)

By using similar arguments as above and (4.27), we have the following estimates: _vn−u≤

1−βnun−u+βnwn−u−gwn−g(u)+βnqn +βnRK(wn)

gwn−λwn+λΦwn−RK(u)g(u)−λu+λΦ(u)

≤1−βnun−u+θβnwn−u+βnqn ≤1−(1−θ)βnun−u+θβnγnpn+βnqn ≤un−u+θβnγnpn+βnqn, since

1−(1−θ)βn

≤1,

and by using (4.29), we have _un+1−u≤

1−αnun−u+αnvn−u−

gvn

−g(u)+αnrn +αnRK(vn)

gvn

−λvn+λΦ

vn

−RK(u)

g(u)−λu+λΦ(u) ≤1−αnun−u+θαnvn−u+αnrn

≤1−(1−θ)αnun−u+αn

θ2_β

nγnpn+θβnqn+rn.

(4.30)

Setting

an=un−u; λn=(1−θ)αn;

bn=(1−θ)−1θ2βnγnpn+θβnqn+rn;

cn=0, ∀n.

(4.31)

It follows fromLemma 2.7, (3.12), (4.12), and (4.30) thatan→0 as n→ ∞, that is,

un→uasn→ ∞. This completes the proof.

Finally, we discuss the stability of TSIA (3.1).

Corollary4.4. LetE,g,h,A,B,C,N, andKbe same as inTheorem 4.1. Let the assump-tions (4.1)–(4.3), (4.12), and eitherCondition 4.2orCondition 4.3ofTheorem 4.1hold. Let {zn}be any sequence inEand let{δn} ⊆[0,∞)be defined as

δn=zn+1−

1−αn

zn−αn

yn−g

yn

+RK(yn)

gyn

−λyn+λΦ

yn

−αnrn, (4.32) where

yn=

1−βn

zn+βn

xn−g

xn

+RK(xn)

gxn

−λxn+λΦ

xn

+βnqn;

xn=

1−γn

zn+γn

zn−g

zn

+RK(zn)

gzn

−λzn+λΦ

zn

+γnpn,

(4.33)

for alln=0, 1, 2, 3,. . ., andΦis defined by (3.11). If there existsω >0such that

αn≥ω, ∀n≥0, (4.34)

then

lim

n→∞zn=u, iffn→∞limδn=0. (4.35) Proof. Using the arguments used inTheorem 4.1for obtaining (4.28) and (4.29), we have

_xn−u≤

1−(1−θ)γnzn−u+γnpn≤zn−u+γnpn; _yn−u≤

1−(1−θ)βnzn−u+θβnγnpn+βnqn,

and

_1−αn

zn+αn

yn−g

yn

+RK(yn)

gyn

−λyn+λΦ

yn

+αnrn−u ≤1−αnzn−u+θαnyn−u+αnrn

≤1−(1−θ)αnzn−u+αn

θ2βnγnpn+θβnqn+rn.

(4.37)

Suppose that limn→∞δn=0.

Using (4.34) and (4.37), we estimate that _{zn+1−u≤1−αn}

zn+αnyn−gyn+RK(yn)

gyn−λyn+λΦyn+αnrn−u +zn+1−1−αnzn−αnyn−gyn+RK(yn)

gyn−λyn+λΦyn−αnrn ≤1−(1−θ)αnzn−u+αnθ2βnγnpn+θβnqn+rn+α−n1δn ≤1−(1−θ)αnzn−u+αnθ2βnγnpn+θβnqn+rn+ω−1δn.

(4.38)

Setting

an=zn−u; λn=(1−θ)αn;

bn=(1−θ)−1

θ2_β

nγnpn+θβnqn+rn+ω−1δn

;

cn=0, ∀n.

(4.39)

It follows fromLemma 2.7, (3.12), (4.12), and (4.38) thatan→0 as n→ ∞, that is,

zn→uasn→ ∞.

Conversely, suppose that limn→∞zn=u. Then (4.38), (4.12) and (4.34) ensure that

δn≤zn+1−u

+1−αn

zn+αn

yn−g

yn

+RK(yn)

gyn

−λyn+λΦ

yn

+αnrn−u ≤1−(1−θ)αnzn−u+αnβnγnpn+βnqn+rn

≤zn+1−u+

1−(1−θ)ωzn−u+θ2βnγnpn+θβnqn+rn −→0 asn−→ ∞.

(4.40)

Hence,δn→0 asn→ ∞. This completes the proof.

Acknowledgment

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K. R. Kazmi: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

E-mail address:[email protected]

M. I. Bhat: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Current address: Department of Applied Mathematics, Baba Ghulam Shah Badshah University, Rajouri, Jammu and Kashmir, India