ERRORS FOR A GENERALIZED IMPLICIT NONLINEAR
QUASIVARIATIONAL INCLUSIONS
M. K. AHMAD AND SALAHUDDIN
Received 2 January 2006; Revised 28 May 2006; Accepted 30 May 2006
We initiate and develop some new perturbed three-step approximation process with er-rors for solving generalized implicit nonlinear quasivariational inclusions. Also, the con-vergence and stability of the iterative sequences with errors generated by the algorithms are presented.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Variational inequality theory has become a rich source of inspiration in the pure and ap-plied mathematics. Variational inequalities not only have stimulated new results dealing with nonlinear partial differential equations, but also have been used in a large variety of problems arising in mechanics, physics, optimization and control nonlinear program-ming, economics and transportation equilibrium and engineering sciences, and so forth. In recent years variational inequalities have been generalized and applied in various di-rections. For details we refer to [2,5,6,20].
Recently Huang [10,11] constructed some new perturbed Ishikawa and Mann iterative algorithms to approximate the solution of some generalized implicit quasivariational in-clusions (inequalities), which includes many iterative algorithms for variational and qua-sivariational inequality problems as special cases.
On the other hand, Xu [19] revised the definition of Ishikawa and Mann iterative pro-cesses with errors and studied the convergence problem of Ishikawa and Mann iterative processes with errors for approximating the solutions of the generative strongly accretive operator equations.
Inspired and motivated by recent research works [4,7,12,14–16], in this paper we initiate and construct some perturbed three-step approximation processes with errors for solving a class of generalized implicit nonlinear quasivariational inclusions. We also discuss the convergence and stability of the iterative sequences generated by algorithms.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 43818, Pages1–15
2. Preliminaries
LetHbe a real Hilbert space endowed with a norm · and an inner product·,·, re-spectively. For a given maximal monotone mappingA(·,·) :H×H→2Hwith respect to the first argument, a nonlinear mappingN(·,·) :H×H→H, three single-valued map-pings,V,G,g:H→H, findu∈Hsuch thatg(u)∈domA(·,u) and
0∈g(u)−N(Vu,Gu) +Ag(u),u, (2.1)
where 2H denote the power subsets ofH. This variational inclusions is called the gener-alized implicit nonlinear quasivariational inclusions.
Remark 2.1. (1) We note thatA(g(u),u)≡A(g(u)) for allu∈H, then the problem (2.1) is equivalent to findingu∈Hsuch that
0∈g(u)−N(Vu,Gu) +Ag(u) (2.2)
is called general implicit nonlinear quasivariational inclusions.
(2) IfV andGare identity mappings, then (2.2) is equivalent to findingu∈H such that
0∈g(u)−N(u,u) +Ag(u). (2.3)
(3) We note thatN(u,u)≡0, zero mapping, then (2.3) is equivalent to the finding u∈Hsuch thatg(u)∈domA,
0∈g(u) +Ag(u) (2.4)
is called general variational inclusions considered by Huang et al. [11].
(4) We again note thatg≡I, an identity mapping, then (2.4) is equivalent to the clas-sical variational inclusions, for findingu∈Hsuch that
0∈u+A(u). (2.5)
(5) IfA(·,u)≡∂ϕ(·,u) is the subdifferential ofϕ(·,u), whereϕ(·,u) :H×H→R∪ {+∞}is a proper convex lower semicontinuous functional with respect to the first argu-ment, then the problem (2.1) is equivalent to findingu∈Hsuch thatg(u)∈dom∂ϕ(·,u) and
g(u)−N(Vu,Gu),v−g(u)≥ϕg(u),u−ϕ(v,u), ∀v∈H, (2.6)
(6) IfN(Vu,Gu)=V(u)−Gu, then (2.6) is equivalent to findingu∈H such that g(u)∈dom∂ϕ(·,u),
g(u)−(Vu−Gu),v−g(u)≥ϕg(u),u−ϕ(v,u), ∀v∈H, (2.7)
is called general strongly nonlinear quasivariational inclusions.
(7) If ∂ϕ(·,u)=∂ϕ(u), then (2.7) is equivalent to findingu∈H such that g(u)∈
domAand
g(u)−N(Vu−Gu),v−g(u)≥ϕg(u)−ϕ(v), ∀v∈H, (2.8)
is called a variant form of general strongly nonlinear quasivariational inclusions, which is the variant form of that of Khan et al. [13].
(8) IfV andGare identity mappings, then (2.6) is equivalent to findingu∈H such thatg(u)∈dom∂ϕ(·,u),
g(u)−N(u,u),v−g(u)≥ϕg(u),u−ϕ(v,u), ∀v∈H, (2.9)
is called generalized strongly nonlinear implicit quasivariational inclusions.
(9) Ifg(u)−N(u,u)≡T(u)−A(u) andϕ(·,u)=ϕ(u), then (2.9) reduces to the fol-lowing problem of findingu∈Hsuch thatg(u)∈dom∂ϕand
Tu−Au,v−g(u)≥ϕg(u)−ϕ(v), ∀v∈H, (2.10)
which is considered by Hassouni and Moudafi [9].
(10) IfKis a given closed convex subset ofHandϕ=IK is the indicator function of
K, defined by
IK(x)=
⎧ ⎨ ⎩
0, x∈K,
+∞, otherwise, (2.11)
then, problem (2.3) reduces to the following problem of findingu∈Hsuch that
g(u)−N(Vu,Gu),v−g(u)≥0, ∀v∈H, (2.12)
which is variant form of that of Verma [17].
(11) IfV andGare identity mappings, then (2.12) is equivalent to findingu∈Hsuch that
g
(u)−N(u,u),v−g(u)≥0, ∀v∈H. (2.13)
(12) Ifg(u)−N(u,u)=u−N(u,u), then (2.13) is equivalent to findingu∈H such that
u
−N(u,u),v−g(u)≥0, ∀v∈H, (2.14)
(13) Ifg≡Iidentity mapping,N(u,u)=T(u) for allu∈H, andT:H→His a single valued mapping, then (2.14) is equivalent to findingu∈Hsuch that
u−Tu,v−u ≥0, ∀v∈H, (2.15)
which is nonlinear variational inequality, considered by Verma [18].
(14) Ifu−N(u,u)=T(u), then (2.13) reduces to findingu∈Hsuch that
Tu,v−g(u)≥0, ∀v∈H, (2.16)
which is another classical variational inequality introduced by Hartman and Stampacchia [8].
(15) Ifg≡I, then (2.15) reduces to findingu∈Hsuch that
Tu,v−u ≥0, ∀v∈H, (2.17)
which is called classical variational inequality, considered by Hartman and Stampacchia [8].
Definition 2.2. A mappingg:H→His said to beα-strongly monotone if there exists a constantα >0 such that
g(u)−g(v),u−v≥αu−v2, ∀u,v∈H. (2.18)
Definition 2.3. A mappingg:H→His said to beβ-Lipschitz continuous if there exists a constantβ >0 such that
g(u)−g(v) ≤βu−v, ∀u,v∈H. (2.19)
Definition 2.4. LetV:H→H andN:H×H→Hbe the mappings,Nis said to be the following.
(i)σ-Lipschitz continuous in the first argument if there exists a constantσ >0 such that
N(u,w)−N(v,w) ≤σu−v, ∀u,v,w∈H. (2.20)
Similarly the Lipschitz continuity ofNcan be defined with respect to the second argument.
(ii)η-relaxed monotone with respect toV in the first argument if there exists a con-stantη >0 such that
N
(Vu,w)−N(Vv,w),u−v≥ −ηu−v2, ∀u,v,w∈H. (2.21)
Definition 2.5. LetG:H→H andN:H×H→H be the mappings. The mappingN is said to beκ-relaxed Lipschitz continuous with respect toGin the second argument if there exists a constantκ≥0 such that
Definition 2.6. LetH be a Hilbert space and letA:H→2H be a maximal monotone mapping. For any fixedλ >0, the mappingJλA:H→Hdefined by
JA
λ(u)=(I+λA)−1(u), ∀u∈H, (2.23)
is said to be the resolvent operator ofA, whereIis an identity mapping onH.
Lemma 2.7. LetA:H→2Hbe a maximal monotone mapping. Then the resolvent operator JA
λ :H→His nonexpansive, that is,
JA
λ(u)−JλA(v) ≤ u−v, ∀u,v∈H. (2.24)
Definition 2.8. Let {An} andAbe the maximal monotone mappings fromH into the power set ofHforn=0, 1, 2,.... The sequence{An}is said to be graph-convergence ofA (writeAn G−→A) if the following property holds.
For every (u,v)∈Graph(A), there exists a sequence (un,vn)∈Graph(An) such that
un→uandvn→vasn→ ∞.
Lemma 2.9 [2]. Let{An}andAbe the maximal monotone mappings fromHinto the power
set ofHforn=0, 1, 2,.... ThenAn G−→Aif and only if
JAn
λ (u)→JλA(u), (2.25)
for everyu∈Handλ >0.
Lemma 2.10 [15]. Let{an},{bn}, and{cn}be three sequences of nonnegative numbers satisfying the following conditions: there exists a positive integern0such that
an+1≤(1−tn)an+bntn+cn, n≥n0, (2.26)
where
tn∈[0, 1],
∞
n=0
tn=+∞, nlim
→∞bn=0, ∞
n=0
cn<+∞. (2.27)
Then limn→∞an=0.
Lemma 2.11. For a givenu∈H,uis a solution of the problem (2.1) if and only if
g(u)≡JA(·,u)
λ g(u)−λg(u)−N(Vu,Gu), (2.28)
Proof. Letu∈Hbe a solution of problem (2.1) if and only if for givenλ >0, a constant,
0∈g(u)−N(Vu,Gu) +Ag(u),u
⇐⇒0∈λg(u)−λN(Vu,Gu) +λAg(u),u
⇐⇒0∈ −g(u)−λg(u)−N(Vu,Gu)+I+λA(·,u)g(u)
⇐⇒g(u)−λg(u)−N(Vu,Gu)∈I+λA(·,u)g(u)
⇐⇒g(u)≡I+λA(·,u)−1[g(u)−λg(u)−N(Vu,Gu)
⇐⇒g(u)≡JA(·,u)
λ
g(u)−λg(u)−N(Vu,Gu),
(2.29)
which completes the proof.
3. The main results
Theorem 3.1. Letg:H→Hbe theα-strongly monotone andβ-Lipschitz continuous map-pings with constantsα >0 andβ >0, respectively. LetV,G:H→Hbe the twoξ-Lipschitz andμ-Lipschitz continuous mappings with constantsξ >0 andμ >0, respectively. Let bimap-pingN:H×H→Hbe theσ-Lipschitz continuous with respect to the first argument with constantσ >0 andδ-Lipschitz continuous with respect to the second argument with con-stantδ >0. LetN be theη-relaxed monotone with respect toV in the first argument with constantη >0 andκ-relaxed Lipschitz continuous with respect toGin the second argument with constantκ≥0. LetA:H×H→2Hbe the maximal monotone with respect to the first argument. It assume that for allu,v,w∈H,
JA(·,u)
λ (w)−JλA(·,v)(w) ≤ρu−v, (3.1)
whereρ >0 is a constant. If
λ−ησ−2ξp2(1−k) −p2
<
η−p(1−k)2−σ2ξ2−p2k(2−k) σ2ξ2−p2 , η > p(1−k) +σ2ξ2−p2k(2−k), k <1, p < σξ,η > p,
(3.2)
where
θ=k+1−2λη+λ2σ2ξ2+λp <1,
k=2Ω+ρ, p=Ω+ω, Ω=1−2α+β2, ω=1 + 2κ+δ2μ2.
(3.3)
Then the problem (2.1) has a unique solutionu∈H.
Proof. Define a mappingF:H→Has
By Lemmas2.11and 2.7, it is enough to show thatF is a contraction mapping. It follows from (3.4) that
F(u)−F(v) = u−v−
g(u)−g(v)
+ JλA(·,u)g(u)−λg(u)−N(Vu,Gu) −JA(·,u)
λ g(v)−λg(v)−N(Vv,Gv) + JλA(·,u)g(v)−λg(v)−N(Vv,Gv)
−JA(·,v)
λ
g(v)−λg(v)−N(Vv,Gv)
≤ u−v−g(u)−g(v)
+ g(u)−g(v)−λg(u)−N(Vu,Gu)+λg(v)−N(Vv,Gv) +ρu−v
≤2 u−v−g(u)−g(v)
+ u−v+λN(Vu,Gu)−N(Vv,Gu) +λ g(u)−g(v)−N(Vv,Gu)−N(Vv,Gv) +ρu−v
≤2 u−v−g(u)−g(v)
+ u−v+λN(Vu,Gu)−N(Vv,Gu) +λ u−v−g(u)−g(v)
+λ u−v−N(Vv,Gu)−N(Vv,Gv) +ρu−v.
(3.5)
Sincegisα-strongly monotone andβ-Lipschitz continuous, we have
u−v−
g(u)−g(v) 2≤1−2α+β2 u−v2. (3.6)
SinceNisσ-Lipschitz continuous with respect to first argument,Visξ-Lipschitz con-tinuous, andNisη-relaxed monotone with respect toV in the first argument with con-stantη >0, we have
u−v+λ
N(Vu,Gu)−N(Vv,Gu) 2
≤ u−v2+ 2λN(Vu,Gu)−N(Vv,Gu),u−v
+λ2 N(Vu,Gu)−N(Vv,Gu) 2
≤ u−v2−2ληu−v2+λ2σ2ξ2u−v2
≤1−2λη+λ2σ2ξ2u−v2.
(3.7)
argument with constantκ≥0, we get
u−v−N
(Vv,Gu)−N(Vv,Gu) 2
≤ u−v2−2N(Vv,Gu)−N(Vv,Gv),u−v
+ N(Vv,Gu)−N(Vv,Gv) 2
≤ u−v2+ 2κu−v2+δ2μ2u−v2
≤1 + 2κ+δ2μ2u−v2.
(3.8)
It follows from (3.5)–(3.8) that we have
F(u)−F(v)
≤21−2α+β2u−v+1−2λη+λ2σ2ξ2u−v
+λ1−2α+β2u−v+λ1 + 2κ+δ2μ2u−v+ρu−v
≤21−2α+β2+ρ+1−2λη+λ2σ2ξ2+λ1−2α+β2
+λ1 + 2κ+δ2μ2u−v
≤2Ω+ρ+1−2λη+λ2σ2ξ2+λΩ+λωu−v
≤k+1−2λη+λ2σ2ξ2+λ(Ω+ω)u−v
≤k+1−2λη+λ2σ2ξ2+λpu−v
≤θu−v,
(3.9)
where
θ=k+
1−2λη+λ2σ2ξ2+λp,
k=2Ω+ρ, p=Ω+ω, Ω=1−2α+β2, ω=1 + 2κ+δ2μ2.
(3.10)
It is easy to verify that (3.3) means 0< θ <1. HenceFis a contraction mapping and has a fixed pointu∈H. It follows fromLemma 2.11thatuis a unique solution of problem
(2.1). This completes the proof.
Algorithm 3.2. For any givenu0∈H, compute the approximate solution{un} by the perturbed iterative process with errors:
un+1=
1−αnun+αnvn−gvn+JλAn(·,vn)gvn−λgvn−NVvn,Gvn+αnen+ln,
vn=1−βnun+βnwn−gwn+JλAn(·,wn)gwn−λgwn−NVwn,Gwn+fn,
wn=1−γnun+γnun−gun+JλAn(·,un)gun−λgun−NVun,Gun+hn, (3.11)
for alln≥0, where{αn},{βn}, and{γn}are sequences in [0, 1]; and{en},{fn},{ln}, and {hn}are bounded sequences inH, satisfying suitable conditions:
n
n=0
αn=+∞, n
n=0
ln <+∞, lim
n→∞ en =nlim→∞ fn =nlim→∞ hn =0, (3.12)
andλ >0 is a constant.
If we remark thatγn=0 andhn=0, for n≥0, then Algorithm 3.2 reduces to the following.
Algorithm 3.3. For any givenu0∈H, compute the approximate solution{un} by the perturbed Ishikawa iterative process with errors:
un+1=
1−αnun+αnvn−gvn+JλAn(·,vn)gvn−λgvn−NVvn,Gvn+enαn+ln,
vn=1−βnun+βnun−gun+JλAn(·,un)gun−λgun−NVun,Gun+fn, (3.13)
where{αn},{βn},{en},{fn}, and{ln}are the same as inAlgorithm 3.2. Ifβn=0 and fn=0 forn≥0, thenAlgorithm 3.3reduces to the following.
Algorithm 3.4. For any givenu0∈H, compute the approximate solution{un} by the perturbed Mann iterative process with errors:
un+1=
1−αnun+αnun−gun+JλAn(·,un)gun−λgun−NVun,Gun
+enαn+ln,
(3.14)
where{αn},{en}, and{ln}are the same as inAlgorithm 3.2.
Algorithm 3.5. For any givenu0∈H, compute the approximate solution{un} by the iterative process:
un+1=1−αnun+αnvn−gvn+JλAn(·,vn)g(vn)−λgvn−NVvn,Gvn,
vn=1−βnun+βnwn−gwn+JλAn(·,wn)
gwn−λgwn−NVwn,Gwn,
wn=1−γnun+γnun−gun+JλAn(·,un)gun−λgun−NVun,Gun, (3.15) whereαn,βn, andγnare the same as inAlgorithm 3.2.
Now we discuss the convergence and stability of the iterative sequences with errors generated byAlgorithm 3.2; we first give some concepts.
LetT be a self map ofH,x0∈H, andxn+1= f(T,xn), define an iterative procedure which yields a sequence of points {xn} inH. Suppose that{x∈H:Tx=x} = ∅and {xn} converge to a fixed point x∈H. Let {yn} ⊂H and n= yn+1−f(T,yn). If limn→∞n=0 implies that limn→∞yn=x, then the iterative procedure{xn}defined by
xn+1=f(T,xn) is said to beT-stable or stable with respect toT. If∞n=0n<+∞implies that limn→∞yn=x, then the iterative procedure{xn}is said to almostT-stable.
Remark 3.6. An iterative procedure{xn} which isT-stable is almost T-stable, and an iterative procedure{xn}which is almostT-stable need not beT-stable, see [5].
Theorem 3.7. Letg,V,G, andNbe the same as inTheorem 3.1, and the conditions (3.1) and (3.3) hold. Let{An}andAbe maximal monotone mappings fromHinto power set ofH
such thatAn−→G A. Let{xn}be a sequence inHand define a sequence{n}of real numbers
as follows:
n= xn+1=
1−αnxn
+αnyn−gyn+JλAn(·,yn)gyn−λgyn−NV yn,Gyn +αnen+ln ,
yn=1−βnxn+βnzn−gzn+JλAn(·,zn)
gzn−λgzn−NVzn,Gzn+fn,
zn=1−γnxn+γnxn−gxn+JλAn(·,xn)gxn−λgxn−NVxn,Gxn+hn. (3.16) Then the following conditions hold.
(i) The sequence{un}generated byAlgorithm 3.2converges strongly to the unique so-lutionuof the problem (2.1).
(ii) Ifn=αnDn+qnwithn∞=0qn<+∞and limn→∞Dn=0, then lim
n→∞xn=u
. (3.17)
(iii) limn→∞xn=uimplies that lim
Proof. ByTheorem 3.1, we know that problem (2.1) has a unique solutionu∈H. It is easy to see that conclusion (i) follows from (ii). Now we prove (ii). It follows from the Lemma 2.11that
u=1−αnu+αnu−gu+JA(·,u)
λ gu−λgu−NVu,Gu =1−βnu+βnu−gu+JλA(·,u)gu−λgu−NVu,Gu
=1−γnu+γnu−gu+JλA(·,u)gu−λgu−NVu,Gu. (3.19)
From (3.16) and (3.19), we have
xn+1−u
≤ xn+1−
1−αnxn
+αnyn−gyn+JλAn(·,yn)gyn−λgyn−NV yn,Gyn +αnen+ln
+ 1−αnxn−u+αnyn−u−gyn−gu
+αnJλAn(·,yn)gyn−λgyn−NV yn,Gyn −JA(·,u)
λ gu−λgu−NVu,Gu +αn en + ln
≤1−αn xn−u +αn yn−u−gyn−gu
+αn JλAn(·,yn)gyn−λgyn−NV yn,Gyn −JAn(·,yn)
λ g(u)−λgu−NVu,Gu +αn JλAn(·,yn)guλgu−NVu,Gu
−JA(·,u)
λ gu−λgu−NVu,Gu +αn JλAn(·,u)gu−λgu−NVu,Gu
−JA(·,u)
λ gu−λgu−NVu,Gu +n+αn en + ln ≤1−αn xn−u + 2αn yn−u−gyn−gu
+αn yn−u+λNV yn,Gyn−NVu,Gyn +αnλ yn−u−gyn−gu
+αnλ yn−u−NVu,Gyn−NVu,Gu +αnρ yn−u +n+αnPn+αn en + ln ,
where
Pn= JAn(·,u ) λ
gu−λgu−NVu,Gu
−JA(·,u)
λ gu−λgu−NVu,Gu −→0.
(3.21)
As the proof of (3.6)–(3.8), we have
yn−u−gyn−gu ≤1−2α+β2 yn−u ,
yn−u+λNV yn,Gyn−NVu,Gyn ≤1−2λη+λ2σ2ξ2 yn−u ,
yn−u−NVu,Gyn−NVu,Gu ≤1 + 2κ+δ2μ2 yn−u .
(3.22)
It follows from (3.20)–(3.22) that
xn+1−u ≤1−αn xn−u +θαn yn−u +n+αnPn+αn en + ln , (3.23)
where
θ=k+1−2λη+λ2σ2ξ2+λp <1,
k=2Ω+ρ, p=Ω+ω, Ω=1−2α+β2, ω=1 + 2κ+δ2μ2.
(3.24)
Similarly, we have
yn−u ≤1−βn xn−u +βnθ zn−u +βnPn+ fn , (3.25)
zn−u ≤1−γn xn−u +γnθ xn−u +γnPn+ hn
≤1−γn1−θ xn−u +γnPn+ hn ≤ xn−u +γnPn+ hn ,
(3.26)
where
1−γn(1−θ)≤1. (3.27)
Substituting (3.26) into (3.25), we have
yn−u ≤1−βn xn−u +βnθ xn−u +βnγnPnθ+βnθ hn +βnPn+ fn
≤1−βn(1−θ) xn−u +βnγnPnθ+βnθ hn +βnPn+ fn ≤ xn−u + 2Pn+ hn + fn .
From (3.23) and (3.28),
xn+1−u ≤1−αn xn−u +αnθ xn−u +αnθ2Pn+ hn + fn
+αnΔn+qn+αnPn+αn en + ln ≤1−αn(1−θ) xn−u
+αn(1−θ) 1 1−θ
3Pn+ hn + fn +Δn+ en +qn+ ln . (3.29)
Setan=xn−u,bn=(1/(1−θ))(3Pn+hn+fn+Δn+en) andtn=αn(1−θ),
cn=qn+ln.
Then we can rewrite (3.29) as follows:
an+1≤
1−tnan+bntn+cn. (3.30)
From the assumption,we know that{an},{bn},{cn}, and{tn}satisfy the conditions of Lemma 2.10. It follows fromLemma 2.10thatan→0 and soxn→uasn→ ∞.
Next, we prove (iii). From (3.11) and (3.28), we know thatyn→uasn→ ∞. It follows from (3.16) that
n≤ xn−u
+ 1−αnxn+αnyn−gyn+JλAn(·,yn)gyn−λgyn−NV yn,Gyn−u
+αn en + ln .
(3.31)
As the proof of (3.23), we have
1−αn
xn+αnyn−gyn+JλAn(·,yn)gyn−λgyn−NV yn,Gyn−u
≤1−αn xn−u +αnθ yn−u +αnPn.
(3.32)
Substituting (3.32) and (3.31), we get
n≤ xn+1−u +
1−αn xn−u +αnθ yn−u +αnPn. (3.33)
Sincexn→u, yn→uandPn→0 asn→ ∞, it follows thatn→0. This completes
Acknowledgment
The authors are thankful to the anonymous referee for the valuable comments that re-sulted in the improvement of the manuscript.
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M. K. Ahmad: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
E-mail address:[email protected]
Salahuddin: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
