QUASIVARIATIONAL INCLUSIONS
YA-PING FANG, NAN-JING HUANG, JUNG IM KANG, AND YEOL JE CHO
Received 10 August 2003
We introduce and study a new class of generalized nonlinear implicit quasivariational inclusions involving relaxed Lipschitzian mappings. We prove the existence of solution for the generalized nonlinear implicit quasivariational inclusions and construct some new stable perturbed iterative algorithms with errors. We also give an application to a class of generalized nonlinear implicit variational inequalities.
1. Introduction
Variational inequality theory and complementarity problem theory are very powerful tools of the current mathematical technology. In recent years, classical variational in-equality and complementarity problems have been extended and generalized to study a wide class of problems generated in mechanics, physics, optimization and control, non-linear programming, economics and transportation equilibrium, and engineering sci-ences, and so forth. A useful and important generalization of variational inequalities is a variational inclusion. Using the resolvent operator technique, many authors have studied various variational inequalities and inclusions with applications (see [1,2,5,6,8,9,10, 11,12,13,14,16,18,19,20,21] and the references therein).
In 1997, Verma [19] studied the solvability, based on an iterative algorithm, of a class of generalized nonlinear variational inequalities involving relaxed Lipschitz and relaxed monotone operators. Recently, Huang [9,10] introduced and studied the Mann- and Ish-ikawa-type perturbed iterative sequence with errors for the generalized nonlinear implicit quasivariational inequalities and inclusions. On the other hand, Huang et al. [12] and Shim et al. [16] proved some existence theorems of solutions for the generalized non-linear mixed quasivariational inequalities (inclusions) and convergence theorems of the iterative sequences generated by the perturbed algorithms with errors.
Inspired and motivated by the recent papers [1,9,10,11,12,16,19], in this paper, we introduce and study a new class of generalized nonlinear implicit quasivariational inclu-sions involving relaxed Lipschitz mappings and construct some new perturbed iterative algorithms with errors. We discuss the convergence and stability of perturbed iterative sequences with errors generated by the algorithms for solving the generalized nonlinear
Copyright©2005 Hindawi Publishing Corporation
implicit quasivariational inclusions. We also give an application to a class of generalized nonlinear implicit variational inequalities.
2. Preliminaries
LetHbe a real Hilbert space endowed with a norm · and an inner product·,·, re-spectively. For given mappings f,g,p:H→H, andN:H×H→H. LetM:H×H→2H be a set-valued mapping such that, for each fixedt∈H,M(·,t) :H→2H is a maximal monotone mapping and Range(p)Dom(M(·,t))= ∅. We consider the following prob-lem. Findu∈Hsuch that
p(u)∈DomM·,g(u),
0∈ f(u)−N(u,u) +Mp(u),g(u), (2.1)
which is called thegeneralized nonlinear implicit quasivariational inclusion. Some special cases of the problem (2.1) are as follows.
(1) IfM(x,t)=M(x) for allx,t∈H, then the problem (2.1) is equivalent to finding
u∈Hsuch that
p(u)∈Dom(M),
0∈ f(u)−N(u,u) +Mp(u), (2.2)
whereM:H→2H is a maximal monotone mapping. The problem (2.2) was considered by Adly [1] and Huang [10], respectively.
(2) IfM(·,t)=∂ϕ(·,t) for eacht∈H, then the problem (2.1) is equivalent to finding
u∈Hsuch that
p(u)∈Dom∂ϕ·,g(u),
f(u)−N(u,u),v−p(u)≥ϕp(u),g(u)−ϕv,g(u) (2.3)
for allv∈H, whereϕ:H×H→R{+∞}such that for eacht∈H,ϕ(·,t) :H→R{+∞} is a proper convex lower semicontinuous function with
Range(p)Dom∂ϕ(·,t)= ∅. (2.4)
Whengis the identity mapping, the problem (2.3) was considered by Ding [6].
(3) If f =0 andg is the identity mapping, then the problem (2.1) is equivalent to findingu∈Hsuch that
p(u)∈DomM(·,u),
0∈ −N(u,u) +Mp(u),u (2.5)
(4) If f =p andM(·,t)=∂ϕfor allt∈H, then the problem (2.1) is equivalent to findingu∈Hsuch that
p(u)∈Dom(∂ϕ),
p(u)−N(u,u),v−p(u)≥ϕp(u)−ϕ(v) (2.6)
for allv∈H, where∂ϕdenotes the subdifferential of a proper convex lower semicontin-uous functionϕ:H→R{+∞}.
(5) If f =p,N(x,y)=Sx−T yfor allx,y∈HandM(·,t)=δK(t)for allt∈H, then
the problem (2.1) is equivalent to findingu∈Hsuch that
p(u)∈Kg(u),
p(u)−(Su−Tu),v−p(u)≥0 (2.7)
for allv∈K(g(u)), whereS,T:H→H are two single-valued mappings,K:H→2His a set-valued mapping with nonempty closed convex values, andδK(t)denotes the indicator
function ofK(t) for each fixedt∈H.
Remark 2.1. For a suitable choice of f,p,g,N,M, and the spaceH, a number of classes of variational inequalities, complementarity problems, and variational inclusions can be obtained as special cases of the generalized nonlinear implicit quasivariational inclusion (2.1).
In the sequel, we give some concepts and lemmas. Definition 2.2. A mapping f :H→His said to be
(i)strongly monotoneif there exists a constantr >0 such that
f
(u)−f(v),u−v≥ru−v2 (2.8)
for allu,v∈H,
(ii)Lipschitzian continuousif there exists a constants >0 such that
f(u)−f(v)≤su−v (2.9)
for allu,v∈H.
Definition 2.3. A mappingN:H×H→His said to berelaxed Lipschitzianwith respect to the first argument if there exists a constantt >0 such that
N(u,·)−N(v,·),u−v≤ −tu−v2 (2.10)
Definition 2.4. A mappingN:H×H→H is said to be Lipschitzian continuous with respect to the first argument if there exists a constantα >0 such that
N(u,·)−N(v,·)≤αu−v (2.11)
for allu,v∈H.
In a similar way, we can define the Lipschitzian continuity of the mappingN(·,·) with respect to the second argument.
Definition 2.5. Let{Mn}andMbe maximal monotone mappings forn=0, 1, 2,.... The sequence{Mn}is said to begraph-convergencetoM(writeMn G→M) if, for every (x,y)∈ Graph(M), there exists a sequence (xn,yn)∈Graph(Mn) such thatxn→xandyn→yas
n→ ∞.
Lemma2.6 [4]. Let{Mn}andMbe maximal monotone mappings forn=0, 1, 2,.... Then
Mn G→Mif and only if
JMn
λ (x)−→JλM(x) (2.12)
for everyx∈Handλ >0, whereJλM=(I+λM)−1.
Lemma2.7. Let{an},{bn}, and{cn}be three sequences of nonnegative numbers satisfying the following conditions: there exists a positive integern0such that
an+1≤
1−tnan+bntn+cn (2.13)
forn≥n0, where
tn∈[0, 1], ∞
n=0
tn=+∞, nlim→∞bn=0, ∞
n=0
cn<+∞. (2.14)
Thenan→0asn→+∞.
Proof. Letσ=inf{an:n≥n0}. Thenσ≥0. Suppose thatσ >0. Thenan≥σ >0 for all
n≥n0. It follows from (2.7) that
an+1≤an−σtn+tnbn+cn=an− 1
2σ−bn
tn−1
2σtn+cn (2.15)
for alln≥n0. Sincebn→0 asn→ ∞, there existsn1≥n0such that
1
2σ≥bn (2.16)
for alln≥n1. Combining (2.13) and (2.15), we have
an+1≤an−1
for alln≥n1, which implies that
1 2σ
∞
n=n1
tn≤an1+ ∞
n=n1
cn<+∞. (2.18)
This is a contradiction. Therefore,σ=0 and so there exists a subsequence{anj} ⊂ {an}
such thatanj→0 asj→ ∞. It follows from (2.7) that
anj+1≤anj+bnjtnj+cnj, (2.19)
and soanj+1→0 asj→ ∞. A simple induction leads toanj+k→0 as j→ ∞for allk≥1 and this means thatan→0 asn→ ∞. This completes the proof. Lemma2.8 (see [11,12]). u∈His a solution of the problem (2.1) if and only if
p(u)=JM(·,g(u))
ρ p(u)−ρ f(u) +ρN(u,u), (2.20)
whereρ >0is a constant and
JM(·,g(u))
ρ =I+ρM·,g(u)−1. (2.21)
3. Existence and uniqueness theorems
In this section, we show the existence and uniqueness of solution for the generalized nonlinear implicit quasivariational inclusion problem (2.1) in terms ofLemma 2.8. Theorem3.1. LetN:H×H→H be Lipschitzian continuous with respect to the first and second arguments with constantsα,β, respectively, and let it be relaxed Lipschitzian with respect to the first argument with a constantt >0. Let f,p,g:H→H be Lipschitzian con-tinuous with constantsσ,s, andl, respectively, and letpbe strongly monotone with a constant
r >0. Suppose that there exist constantsλ >0andρ >0such that for eachx,y,z∈H,
JM(·,x)
ρ (z)−JρM(·,y)(z)≤λx−y, (3.1)
21−2r+s2+1−2ρt+ρ2α2+ρβ+ρσ+λl<1. (3.2)
Then the problem (2.1) has a unique solutionu∗∈H.
Proof. ByLemma 2.8, it is enough to show that the mappingF:H→H has a unique fixed pointu∗∈H, whereFis defined as follows:
for allu∈H. From (3.1) and (3.3), we have F(u)−F(v)
=u−p(u) +JρM(·,g(u))p(u)−ρ f(u) +ρN(u,u)
−v−p(v) +JρM(·,g(v))p(v)−ρ f(v) +ρN(v,v)
≤u−v−p(u)−p(v)
+JM(·,g(u))
ρ p(u)−ρ f(u) +ρN(u,u)−JM(·,g(v))
ρ p(v)−ρ f(v) +ρN(v,v)
≤u−v−p(u)−p(v)
+JρM(·,g(u))p(u)−ρ f(u) +ρN(u,u)−JρM(·,g(u))p(v)−ρ f(v) +ρN(v,v)
+JρM(·,g(u))p(v)−ρ f(v) +ρN(v,v)−JM(·,g(v))
ρ p(v)−ρ f(v) +ρN(v,v)
≤u−v−p(u)−p(v)+λg(u)−g(v)
+p(u)−ρ f(u) +ρN(u,u)−p(v)−ρ f(v) +ρN(v,v)
≤u−v+ρN(u,u)−N(v,v)
+ 2u−v−p(u)−p(v)+ρf(u)−f(v)+λlu−v ≤2u−v−p(u)−p(v)+u−v+ρN(u,u)−N(v,u)
+ρN(v,u)−N(v,v)+ρσ+λlu−v.
(3.4)
By the Lipschitzian continuity and strong monotonicity ofp, we have u−v−
p(u)−p(v)2
= u−v2−2u−v,p(u)−p(v)+p(u)−p(v)2
≤1−2r+s2u−v2.
(3.5)
SinceN is Lipschitzian continuous with respect to the first and second arguments and relaxed Lipschitzian with respect to the first argument, we obtain
u−v+ρ
N(u,u)−N(v,u)2
= u−v2+ 2ρu−v,N(u,u)−N(v,u)+ρ2N(u,u)−N(v,u)2
≤1−2ρt+ρ2α2u−v2,
N(v,u)−N(v,v)≤βu−v.
(3.6)
From (3.4), (3.5), and (3.6), we have
F(u)−F(v)≤hu−v, (3.7)
where
From (3.2), we know that 0< h <1. Therefore, there exists a uniqueu∗∈H such that
F(u∗)=u∗. This completes the proof.
Remark 3.2. If there is a constantρ >0 such that
ρ−t−α(12 −k)(β+σ) −(β+σ)2
<
t
+ (k−1)(β+σ)2−k(2−k)α2−(β+σ)2
α2−(β+σ)2 ,
t >(1−k)(β+σ) +
k(2−k)α2−(β+σ)2, α > β+σ,
ρ(β+σ)<1−k, k=21−2r+s2+λl, k <1,
(3.9)
then it is easy to check that condition (3.2) is satisfied. FromTheorem 3.1, we can obtain the following theorem.
Theorem3.3. LetN, p, and f be the same as inTheorem 3.1. Suppose that there exists a constantρ >0such that (3.2) holds fork=2√1−2r+s2. Then the problem (2.2) has a
unique solutionu∗∈H.
4. Perturbed algorithms and stability
In this section, we construct some new perturbed iterative algorithms with errors for solving the generalized nonlinear implicit quasivariational inclusion problem (2.1) and prove the convergence and stability of the iterative sequences generated by the perturbed iterative algorithms with errors.
Definition 4.1. Let T be a self-mapping ofH, x0∈H and xn+1= f(T,xn) define an
iteration procedure which yields a sequence of points {xn} in H. Suppose that {x∈
H:Tx=x} = ∅ and {xn} converges to a fixed pointx∗ of T. Let {yn} ⊂H and let n= yn+1−f(T,yn).
(i) If limn→∞n=0 implies that limn→∞yn=x∗, then the iteration procedure{xn} defined byxn+1=f(T,xn) is said to beT-stableorstablewith respect toT.
(ii) If∞n=0n<+∞implies that limn→∞yn=x∗, then the iteration procedure{xn}is
said to bealmostT-stable.
Some stability results of iteration algorithms have been established by several authors (see [3,7,12,15]). As was shown by Harder and Hicks [7], the study on the stability is both of theoretical and of numerical interest.
Remark 4.2. An iteration procedure{xn} which isT-stable is almostT-stable and an iteration procedure{xn}which is almostT-stable need not beT-stable [15].
Now, we give the perturbed iterative algorithms with errors for the generalized non-linear implicit quasivariational inclusion problem (2.1) as follows.
Algorithm 4.3. Let f,p,g:H→H andN:H×H→H be four single-valued mappings. Let {Mn} andM be set-valued mappings fromH×H into the power ofH such that for eacht∈H,Mn(·,t) andM(·,t) are maximal monotone mappings andMn(·,t)→G
as follows:
un+1=
1−αnun
+αn
vn−pvn+JρMn(·,g(vn))pvn−ρ fvn+ρNvn,vn+αnen+ln,
vn=1−βnun
+βn
un−pun+JM
n(·,g(un))
ρ pun−ρ fun+ρNun,un+fn
(4.1)
forn=0, 1, 2,..., where{αn}and{βn}are two sequences in [0, 1],{en},{ln}, and{fn}are three sequences inHsatisfying the following conditions:
lim
n→∞en=nlim→∞fn=0, ∞
n=0
αn=+∞, ∞
n=0
ln<+∞. (4.2)
From Algorithm 4.3, we obtain the following algorithm for the problem (2.2) as follows.
Algorithm 4.4. Let f,p:H→H andN:H×H→H be three single-valued mappings. Let{Mn}andMbe maximal monotone mappings fromHinto the power ofHsuch that
Mn G→M. For any givenu
0∈H, define the perturbed iterative sequence{un}with errors
as follows:
un+1=
1−αnun
+αn
vn−pvn+JρMnpvn−ρ fvn+ρNvn,vn+αnen+ln,
vn=1−βnun
+βn
un−pun+JρMnpun−ρ fun+ρNun,un+fn
(4.3)
forn=0, 1, 2,..., where{αn},{βn},{en},{ln}, and{fn}are the same as inAlgorithm 4.3. Remark 4.5. For a suitable choice off,p,g,N,Mn, andM,Algorithm 4.3includes several known algorithms in [1,5,6,9,10,12,16,18] as special cases.
Theorem4.6. Let f,p,g, andNbe the same as inTheorem 3.1. Suppose that{Mn}andM are set-valued mappings fromH×Hinto the power ofHsuch that for eacht∈H,Mn(·,t) andM(·,t)are maximal monotone mappings andMn(·,t)→G M(·,t). Assume that there exist constantsρ >0andλ >0such that for eachx,y,z∈Handn≥0,
JM(·,x)
ρ (z)−JρM(·,y)(z)≤λx−y,
JMn(·,x)
ρ (z)−JM
n(·,y)
ρ (z)≤λx−y,
and the condition (3.2) holds. Let{yn}be a sequence inHand define a sequence{n}of real numbers as follows:
n=yn+1−
1−αnyn+αn
xn−pxn+JρMn(·,g(xn))pxn−ρ fxn+ρNxn,xn
+αnen+ln,
xn=1−βnyn+βn
yn−pyn+JM
n(·,g(yn))
ρ p(yn)−ρ fyn+ρNyn,yn+fn, (4.5)
where{αn},{βn},{en},{ln}, and{fn}are the same as inAlgorithm 4.3. Then the following hold.
(1)The sequence{un}defined byAlgorithm 4.3converges strongly to the unique solution
u∗of the problem (2.1).
(2)Ifn=αn∆n+γnwith∞n=0γn<+∞andlimn→∞∆n=0, thenlimn→∞yn=u∗.
(3) limn→∞yn=u∗implies thatlimn→∞n=0.
Proof. Letu∗∈H be the unique solution of the problem (2.1). It is easy to see that the conclusion (1) follows from the conclusion (2). Now, we prove that (2) is true. It follows fromLemma 2.8that
u∗=1−αnu∗+αnu∗−pu∗+JM(·,g(u∗))
ρ pu∗−ρ fu∗+ρNu∗,u∗. (4.6)
From (4.1), (4.5), and (4.6), we have yn+1−u∗
=yn+1−1−αnu∗+αn
u∗−pu∗+JM(·,g(u∗))
ρ pu∗−ρ fu∗+ρNu∗,u∗
≤yn+1−1−αnyn+αn
xn−pxn+JM
n(·,g(xn))
ρ pxn−ρ fxn+ρNxn,xn
+αnen+ln
+1−αnyn+αn
xn−pxn+JM
n(·,g(xn))
ρ pxn−ρ fxn+ρNxn,xn
−1−αnu∗+αn
u∗−pu∗+JM(·,g(u∗))
ρ pu∗−ρ fu∗+ρNu∗,u∗ +αnen+ln
≤n+1−αnyn−u∗+αnxn−u∗−pxn−pu∗
+αnJM
n(·,g(xn))
ρ pxn−ρ fxn+ρNxn,xn
−JM(·,g(u∗))
ρ pu∗−ρ fu∗+ρNu∗,u∗
≤n+1−αnyn−u∗+αnxn−u∗−pxn−pu∗
+αnJM
n(·,g(xn))
ρ pxn−ρ fxn+ρNxn,xn
−JMn(·,g(xn))
ρ pu∗−ρ fu∗+ρNu∗,u∗
+αnJM
n(·,g(xn))
ρ pu∗−ρ fu∗+ρNu∗,u∗
−JMn(·,g(u∗))
ρ pu∗−ρ fu∗+ρNu∗,u∗
+αnJM
n(·,g(u∗))
ρ pu∗−ρ fu∗+ρNu∗,u∗
−JM(·,g(u∗))
ρ pu∗−ρ fu∗+ρNu∗,u∗+αnen+ln
≤n+1−αnyn−u∗+αnxn−u∗−pxn−pu∗
+αnpxn−pu∗−ρfxn−fu∗+ρNxn,xn−Nu∗,u∗
+αnλgxn−gu∗+αngn+αnen+ln
≤n+1−αnyn−u∗+ 2αnxn−u∗−pxn−pu∗
+αnxn−u∗+ρNxn,xn−Nu∗,xn+αnρNu∗,xn−Nu∗,u∗
+αnρfxn−fu∗+αnλlxn−u∗+αngn+αnen+ln
≤1−αnyn−u∗+ 2αnxn−u∗−pxn−pu∗
+αnxn−u∗+ρNxn,xn−Nu∗,xn+αnρNu∗,xn−Nu∗,u∗
+αnρσ+λlxn−u∗+αngn+en+∆n+ln+γn,
(4.7)
where
gn=JM
n(·,g(u∗))
ρ pu∗−ρ fu∗+ρNu∗,u∗
−JM(·,g(u∗))
ρ pu∗−ρ fu∗+ρNu∗,u∗−→0. (4.8)
It follows from (3.5) and (3.6) that
xn−u∗−pxn−pu∗2
≤1−2r+s2x
n−u∗2, xn−u∗+ρNxn,xn−Nu∗,xn2
≤1−2ρt+ρ2α2xn−u∗2, N
u∗,xn−Nu∗,u∗≤βxn−u∗.
(4.9)
Substituting (4.9) into (4.7), we have
yn+1−u∗≤1−αnyn−u∗+αnhxn−u∗
where
h=21−2r+s2+1−2ρt+ρ2α2+ρ(β+σ) +λl. (4.11)
From (3.2), we know that 0< h <1. Similarly, we have
xn−u∗≤1−βnyn−u∗+βnhyn−u∗+βngn+fn. (4.12)
Combining (4.10) and (4.12), we have
yn+1−u∗
≤1−αnyn−u∗+αnh1−βnyn−u∗+αnh2βnyn−u∗
+αnhβngn+αnhfn+αngn+en+∆n+ln+γn
≤1−1−hαnyn−u∗
+ (1−h)αn· 1 1−h
hβngn+hfn+gn+en+∆n+ln+γn.
(4.13)
Let
an=yn−u∗, cn= ln+γn, tn=(1−h)αn,
bn= 1
1−h
hβ
ngn+hfn+gn+en+∆n.
(4.14)
We can rewrite (4.13) as follows:
an+1≤
1−tnan+bntn+cn. (4.15)
From the assumptions, we know that{an},{bn},{cn}, and{tn}satisfy the conditions of Lemma 2.7. This implies thatan→0 and soyn→u∗.
Next, we prove the conclusion (3). Suppose that limn→∞yn=u∗. It follows from (4.2) and (4.12) thatxn→u∗. From (4.5), we have
n=yn+1−
1−αnyn+αn
xn−pxn+JρMn(·,g(xn))pxn−ρ fxn+ρNxn,xn
+αnen+ln
≤yn+1−u∗+αnen+ln
+1−αnyn+αn
xn−pxn+JM
n(·,g(xn))
As in the proof of (4.10), we have
1−αnyn+αn
xn−pxn+JM
n(·,g(xn))
ρ pxn−ρ fxn+ρNxn,xn−u∗
≤1−αnyn−u∗+αnhxn−u∗+αngn.
(4.17)
It follows from (4.16) and (4.17) that
n≤yn+1−u∗+αnen+ln+
1−αnxn−u∗+αnhxn−u∗+αngn. (4.18)
This implies that limn→∞n=0. This completes the proof. FromTheorem 4.6, we have the following theorem.
Theorem4.7. Let f,p, andNbe the same as inTheorem 4.6. Let{Mn}andMbe maximal monotone mappings fromHinto the power ofHsuch thatMn G→M. Assume that there exists a constantρ >0such that (3.2) holds fork=2√1−2r+s2. Let{yn}be a sequence inHand
define{n}as follows:
n=yn+1−
1−αnyn+αn
xn−pxn+JρMnpxn−ρ fxn+ρNxn,xn
+αnen+ln,
xn=1−βnyn+βn
yn−pyn+JρMnpyn−ρ fyn+ρNyn,yn+ fn,
(4.19)
where{αn},{βn},{en},{ln}, and{fn}are the same as inAlgorithm 4.4. Then
(1)the sequence{un}defined byAlgorithm 4.4converges strongly to the unique solution
u∗of the problem (2.2),
(2)ifn=αn∆n+γnwith∞n=0γn<+∞andlimn→∞∆n=0, thenlimn→∞yn=u∗,
(3) limn→∞yn=u∗implies thatlimn→∞n=0.
5. An application
In this section, we give an application to a class of generalized nonlinear implicit varia-tional inequalities.
Definition 5.1. LetAbe a single-valued mapping fromHtoH. The mappingAis said to behemicontinuousif the mapping from [0, 1] into (−∞, +∞) defined by
t−→A(1−t)u+tv,w (5.1)
is continuous for allu,v,w∈H.
Lemma 5.2 [17]. Letϕ:H →R{+∞} be a proper convex lower semicontinuous func-tion and letA:H→H be a single-valued monotone mapping such thatCL(Dom(ϕ))⊂
Theorem5.3. Let ϕ:H→R{+∞} be a proper convex lower semicontinuous function andA:H→His a single-valued monotone mapping such thatCL(Dom(ϕ))⊂Dom(A),S is hemicontinuous onCL(Dom(ϕ)), andDom(∂ϕ)is closed. LetN, f, andpbe the same as inTheorem 3.3. If there exists a constantρ >0such that (3.2) holds fork=2√1−2r+s2,
then there exists a uniqueu∗∈Hsuch that
p(u)∈Dom(∂ϕ),
Ap(u)+f(u)−N(u,u),v−p(u)≥ϕp(u)−ϕ(v) (5.2)
for allv∈H, which is called the generalized nonlinear implicit variational inequality. More-over, let{yn}be a sequence inH and define a sequence{n}of positive real numbers as follows:
n=yn+1−
1−αnyn+αn
xn−pxn+JρA+∂ϕpxn−ρ fxn+ρNxn,xn
+αnen+ln,
xn=1−βnyn+βn
yn−pyn+JρA+∂ϕpyn−ρ fyn+ρNyn,yn+fn, (5.3)
where{αn},{βn},{en},{ln}, and{fn}are the same as inAlgorithm 4.4. Then
(1)ifn=αn∆n+γnwith∞n=0γn<+∞andlimn→∞∆n=0, thenlimn→∞yn=u∗,
(2) limn→∞yn=u∗implies thatlimn→∞n=0.
Proof. The monotone mappingA+∂ϕ is maximal monotone byLemma 5.2. LetM= A+∂ϕ. FromTheorem 3.3, we know that there exists a uniqueu∗∈Hsuch that p(u)∈
Dom(A+∂ϕ) and
N(u,u)−f(u)−Ap(u)∈∂ϕp(u). (5.4)
Hencep(u)∈Dom(∂ϕ) and
Ap(u)+f(u)−N(u,u),v−p(u)≥ϕp(u)−ϕ(v) (5.5)
for allv∈H. The conclusions (1) and (2) follow fromTheorem 4.7. This completes the
proof.
Acknowledgments
References
[1] S. Adly,Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl.201(1996), no. 2, 609–630.
[2] R. P. Agarwal, Y. J. Cho, and N. J. Huang,Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett.13(2000), no. 6, 19–24.
[3] R. P. Agarwal, N. J. Huang, and Y. J. Cho,Stability of iterative processes with errors for nonlinear equations ofφ-strongly accretive type operators, Numer. Funct. Anal. Optim.22(2001), no. 5-6, 471–485.
[4] H. Attouch,Variational Convergence for Functions and Operators, Applicable Mathematics Se-ries, Pitman, Massachusetts, 1984.
[5] Y. J. Cho, J. H. Kim, N. J. Huang, and S. M. Kang,Ishikawa and mann iterative processes with errors for generalized strongly nonlinear implicit quasi-variational inequalities, Publ. Math. Debrecen58(2001), no. 4, 635–649.
[6] X. P. Ding,Perturbed proximal point algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl.210(1997), no. 1, 88–101.
[7] A. M. Harder and T. L. Hicks,Stability results for fixed point iteration procedures, Math. Japon. 33(1988), no. 5, 693–706.
[8] P. T. Harker and J.-S. Pang,Finite-dimensional variational inequality and nonlinear comple-mentarity problems: a survey of theory, algorithms and applications, Math. Programming48 (1990), no. 2, (Ser. B), 161–220.
[9] N. J. Huang,On the generalized implicit quasivariational inequalities, J. Math. Anal. Appl.216 (1997), no. 1, 197–210.
[10] ,Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions, Comput. Math. Appl.35(1998), no. 10, 1–7.
[11] ,Generalized nonlinear implicit quasivariational inclusion and an application to implicit variational inequalities, ZAMM Z. Angew. Math. Mech.79(1999), no. 8, 569–575. [12] N. J. Huang, M.-R. Bai, Y. J. Cho, and S. M. Kang,Generalized nonlinear mixed quasi-variational
inequalities, Comput. Math. Appl.40(2000), no. 2-3, 205–215.
[13] M. A. Noor,Some recent advances in variational inequalities. I. Basic concepts, New Zealand J. Math.26(1997), no. 1, 53–80.
[14] , Some recent advances in variational inequalities. II. Other concepts, New Zealand J. Math.26(1997), no. 2, 229–255.
[15] M. O. Osilike,Stability of the mann and Ishikawa iteration procedures forφ-strong pseudocon-tractions and nonlinear equations of theφ-strongly accretive type, J. Math. Anal. Appl.227 (1998), no. 2, 319–334.
[16] S. H. Shim, S. M. Kang, N. J. Huang, and J. Y. Cho,Perturbed iterative algorithms with errors for completely generalized strongly nonlinear implicit quasivariational inclusions, J. Inequal. Appl.5(2000), no. 4, 381–395.
[17] A. Taa,The maximality of the sum of monotone operators in banach space and an application to hemivariational inequalities, J. Math. Anal. Appl.204(1996), no. 3, 693–700.
[18] R. U. Verma,Iterative algorithms for variational inequalities and associated nonlinear equations involving relaxed Lipschitz operators, Appl. Math. Lett.9(1996), no. 4, 61–63.
[19] ,On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators, J. Math. Anal. Appl.213(1997), no. 1, 387–392.
[20] J. C. Yao,Applications of variational inequalities to nonlinear analysis, Appl. Math. Lett.4(1991), no. 4, 89–92.
Ya-Ping Fang: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China E-mail address:[email protected]
Nan-Jing Huang: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
E-mail address:[email protected]
Jung Im Kang: Department of Mathematics Education and the Research Institute of Natural Sci-ences, Gyeongsang National University, Chinju 660-701, Korea
E-mail address:[email protected]
Yeol Je Cho: Department of Mathematics Education and the Research Institute of Natural Sciences, Gyeongsang National University, Chinju 660-701, Korea
