Volume 2007, Article ID 29863,12pages doi:10.1155/2007/29863
Research Article
Perturbed Iterative Algorithms for Generalized Nonlinear
Set-Valued Quasivariational Inclusions Involving Generalized
m
-Accretive Mappings
Mao-Ming Jin
Received 24 August 2006; Revised 10 January 2007; Accepted 14 January 2007
Recommended by H. Bevan Thompson
A new class of generalized nonlinear set-valued quasivariational inclusions involving gen-eralizedm-accretive mappings in Banach spaces are studied, which included many varia-tional inclusions studied by others in recent years. By using the properties of the resolvent operator associated with generalizedm-accretive mappings, we established the equiva-lence between the generalized nonlinear set-valued quasi-variational inclusions and the fixed point problems, and some new perturbed iterative algorithms, proved that its prox-imate solution converges strongly to its exact solution in real Banach spaces.
Copyright © 2007 Mao-Ming Jin. This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In 1994, Hassouni and Moudafi [1] introduced and studied a class of variational inclu-sions and developed a perturbed algorithm for finding approximate solutions of the vari-ational inclusions. Since then, Adly [2], Ding [3], Ding and Luo [4], Huang [5,6], Huang et al. [7], Ahmad and Ansari [8] have obtained some important extensions of the results in various different assumptions. For more details, we refer to [1–29] and the references therein.
In 2001, Huang and Fang [16] were the first to introduce the generalizedm-accretive mapping and give the definition of the resolvent operator for the generalizedm-accretive mappings in Banach spaces. They also showed some properties of the resolvent operator for the generalizedm-accretive mappings in Banach spaces. For further works, we refer to Huang [15], Huang et al. [19] and Huang et al. [20].
η-monotone mapping. They also introduced and studied a new class of nonlinear varia-tional inclusions involving maximalη-monotone mapping in Hilbert spaces.
Motivated and inspired by the research work going on in this field, we introduce and study a new class of generalized nonlinear set-valued quasivariational inclusions involv-ing generalizedm-accretive mappings in Banach spaces, which include many variational inclusions studied by others in recent years. By using the properties of the resolvent op-erator associated with generalized m-accretive mappings, we establish the equivalence between the generalized nonlinear set-valued quasivariational inclusions and the fixed point problems, and some new perturbed iterative algorithms, prove that its proximate solution converges to its exact solution in real Banach spaces. The results presented in this paper extend and improve the corresponding results in the literature.
2. Preliminaries
Throughout this paper, we assume thatX is a real Banach space equipped with norm
· ,X∗is the topological dual space ofX,CB(X) is the family of all nonempty closed and bounded subset ofX, 2Xis a power set ofX,D(·,·) is the Hausdorffmetric onCB(X)
defined by
D(A,B)=max
sup
u∈A
d(u,B), sup
v∈B
d(A,v)
∀A,B∈CB(X), (2.1)
whered(u,B)=infv∈Bd(u,v) andd(A,v)=infu∈Ad(u,v).
Suppose that·,·is the dual pair betweenXandX∗,J:X→2X∗
is the normalized duality mapping defined by
J(x)=f ∈X∗:x,f = x2,x = f, x∈X, (2.2)
and jis a selection of normalized duality mappingJ.
Definition 2.1. A single-valued mapping g:X→X is said to bek-strongly accretive if there existsk >0 such that for anyx,y∈X, there existsj(x−y)∈J(x−y) such that
g(x)−g(y),j(x−y)≥kx−y2. (2.3)
Definition 2.2. A single-valued mappingN:X×X→Xis said to beγ-Lipschitz contin-uous with respect to the first argument if there exists a constantγ >0 such that
N(x,·)−N(y,·)≤γx−y ∀x,y∈X. (2.4)
In a similar way, we can define Lipschitz continuity ofN(·,·) with respect to the second argument.
Definition 2.3. A set-valued mappingS:X→2Xis said to beξ-D-Lipschitz continuous if
there existsξ >0 such that
Definition 2.4. A mappingη:X×X→X∗is said to be (i) accretive if for anyx,y∈X,
x−y,η(x,y)≥0; (2.6)
(ii) strictly accretive if for anyx,y∈X,
x−y,η(x,y)≥0, (2.7)
and equality holds if and only ifx=y;
(iii)α-strongly accretive if there exists a constantα >0 such that
x−y,η(x,y)≥αx−y2 ∀x,y∈X; (2.8)
(iv)β-Lipschitz continuous if there exists a constantβ >0 such that
η(x,y)≤βx−y ∀x,y∈X. (2.9)
Definition 2.5[16]. Letη:X×X→X∗be a single-valued mapping. A set-valued map-pingM:X→2Xis said to be
(i)η-accretive if for anyx,y∈X,
u−v,η(x,y)≥0, u∈M(x), v∈M(y); (2.10)
(ii) strictlyη-accretive if for anyx,y∈X,
u−v,η(x,y)≥0, u∈M(x), v∈M(y), (2.11)
and equality holds if and only ifx=y;
(iii)μ-stronglyη-accretive if there exists a constantμ >0 such that
u−v,η(x,y)≥μx−y2 ∀x,y∈X,u∈M(x),v∈M(y); (2.12)
(iv) generalizedm-accretive ifM isη-accretive and (I+ρM)(X)=X for anyρ >0, whereIis the identity mapping.
Remark 2.6. IfX is a smooth Banach space, η(x,y)=J(x−y) for all x,y in X, then
Definition 2.5reduces to the usual definitions of accretiveness of the set-valued mapping Min smooth Banach spaces.
Lemma2.7 [30]. LetXbe a real Banach space and letJ:X→2X∗
be the normalized duality mapping. Then for anyx,y∈X,
Lemma2.8 [16]. Letη:X×X→Xbe a strictly accretive mapping and letM:X→2Xbe a generalizedm-accretive mapping. Then the following conclusions hold:
(1)x−y,η(u,v)≥0∀(y,v)∈graph(M)implies(x,u)∈graph(M), wheregraph(M)= {(x,u)∈X×X:x∈M(u)};
(2)the mapping(I+ρM)−1is single-valued for anyρ >0.
Based onLemma 2.8, we can define the resolvent operator for a generalized m-accre-tive mappingMas follows:
JM
ρ (z)=(I+ρM)−1(z) ∀z∈X, (2.14)
whereρ >0 is a constant andη:X×X→X∗is a strictly accretive mapping.
Lemma2.9 [16]. Letη:X×X→X∗be aδ-strongly accretive andτ-Lipschitz continuous mapping. LetM:X→2Xbe a generalizedm-accretive mapping. Then the resolvent operator
JM
ρ forMisτ/δ-Lipschitz continuous, that is, JM
ρ (u)−JρM(v)≤
τ
δu−v ∀u,v∈X. (2.15)
3. Variational inclusions
In this section, by using the resolvent operator for the generalizedm-accretive mapping and the results obtained inSection 2, we introduce and study a new class of general-ized nonlinear set-valued quasivariational inclusion problem involving generalgeneral-ized m-accretive mappings, and prove that its proximate solution converges strongly to its exact solution in real Banach spaces.
LetS,T,G:X→CB(X) andM(·,·) :X×X→2Xbe set-valued mappings such that for
any givent∈X,M(t,·) :X→2Xis a generalizedm-accretive mapping. Letg:X→Xand
N(·,·) :X×X→Xbe nonlinear mappings. For any f ∈X, we consider the following problem.
Findx∈X,w∈S(x),y∈T(x),z∈G(x) such that
f ∈N(w,y) +M z,g(x), (3.1)
which is called the generalized nonlinear set-valued quasivariational inclusion problem involving generalizedm-accretive mappings.
Some special cases of problem (3.1) are as follows.
(I) IfS,T,G:X→Xis a single-valued mapping, then problem (3.1) reduced to find-ingx∈Xsuch that
f ∈N S(x),T(x)+M G(x),g(x), (3.2)
which is called the nonlinear quasivariational inclusion problem.
(II) IfX=His a Hilbert space andη(u,v)=u−v, then problem (3.1) becomes the usual nonlinear quasivariational inclusion with a maximal monotone mappingM.
(inclusion) can be obtained as special cases of generalized nonlinear set-valued quasivari-ational inclusion (3.1).
Lemma 3.2. Problem (3.1) has a solution (x,w,y,z), where x∈X,w∈S(x), y∈T(x),
z∈G(x)if and only if(p,x,w,y,z), wherep∈X, is a solution of implicit resolvent equation
p=g(x)−ρ N(w,y)−f, g(x)=JM(z,·)
ρ (p), (3.3)
whereJρM(z,·)=(I+ρM(z,·))−1andρ >0is a constant.
Proof. This directly follows from the definition ofJρM(z,·).
NowLemma 3.2and Nadler’s theorem [31] allow us to suggest the following iterative algorithm.
Algorithm3.3. Assume thatS,T,G:X→CB(X), andM(·,·) :X×X→2Xare set-valued mappings such that for any givent∈X,M(t,·) :X→2Xis a generalizedm-accretive map-ping andg:X→Xis a strongly accretive and Lipschitz continuous mapping. LetN(·,·) : X×X→X be a nonlinear mapping. For any f ∈X and for given p0∈X,x0∈X and
w0∈S(x0), y0∈T(x0),z0∈G(x0), compute the sequences{pn},{xn},{wn},{yn}, and {zn}defined by the iterative schemes
g xn
=JM(zn,·)
ρ pn,
wn∈S xn
, wn−wn+1≤ 1 + (1 +n)−1
D S xn
,S xn+1
,
yn∈T xn
, yn−yn+1≤ 1 + (1 +n)−1
D T xn
,T xn+1
,
zn∈G xn
, zn−zn+1≤ 1 + (1 +n)−1
D G xn
,G xn+1
,
pn+1=(1−λ)pn+λ g xn
−ρN wn,yn
+ρ f+λen,
n=0, 1, 2,. . ., (3.4)
where0< λ≤1is a constant anden∈Xis the errors while considering the approximation in computation.
IfS,T,G:X→Xare single-valued mappings, thenAlgorithm 3.3can be degenerated to the following algorithm for problem (3.2).
Algorithm3.4. For any f ∈X and for given p0∈X,x0∈X, we can obtain sequences
{pn},{xn}satisfying
g xn
=JM(G(xn),·)
ρ pn,
pn+1=(1−λ)pn+λ g xn
−ρN S xn
,T xn
+ρ f+λen,
n=0, 1, 2,. . ., (3.5)
where0< λ≤1is a constant anden∈Xis the errors while considering the approximation in computation.
Remark 3.5. If we choose suitableS,T,G,N,M,g, and the spaceX, thenAlgorithm 3.3
Theorem 3.6. Let X be a real Banach space. Let η:X×X→X∗ be δ-strongly accre-tive andτ-Lipschitz continuous, letS,T,G:X→CB(X)be α,β andγ-D-Lipschitz con-tinuous, respectively, letg:X→X bek-strongly accretive andξ-Lipschitz continuous. Let
N(·,·) :X×X→X ber,t-Lipschitz continuous with respect to the first and second argu-ments, respectively. LetM:X×X→2Xbe such that for each fixedt∈X,M(t,·)is a gener-alizedm-accretive mapping. Suppose that there exist constantsρ >0andμ >0such that for eachx,y,z∈X,
JM(x,·)
ρ (z)−J M(y,·)
ρ (z)≤μx−y, (3.6)
ρ <
k+ 1.5−μ2γ2−bξ
b(rα+tβ) , bξ <
k+ 1.5−μ2γ2, b=τ
δ, (3.7)
lim
n→∞en=0, ∞
n=0
en+1−en<∞. (3.8)
Then there existp,x∈X,w∈S(x),y∈T(x),z∈G(x)satisfy the implicit resolvent equa-tion (3.3) and the iterative sequences {pn}, {xn}, {wn}, {yn}, and {zn} generated by Algorithm 3.3converge strongly top,x,w,y, andzinX, respectively.
Proof. From condition (3.6),Lemma 2.9, andγ-Lipschitz continuity ofG, we have
JM(zn+1,·)
ρ pn+1−JρM(zn,·)pn ≤JM(zn+1,·)
ρ pn+1−JρM(zn,·)pn+1+JρM(zn,·)pn+1−JρM(zn,·)pn ≤μzn+1−zn+τδpn+1−pn
≤μγ
1 +1 n
xn+1−xn+τ
δpn+1−pn.
(3.9)
Sinceg isk-strongly accretive mapping, fromAlgorithm 3.3,Lemma 2.7, and (3.9), for anyj(xn+1−xn)∈J(xn+1−xn), we have
xn+1−xn2
=xn+1−xn+ g xn+1
−g xn− JρM(zn+1,·)pn+1−JρM(zn,·)pn2 ≤JM(zn+1,·)
ρ pn+1−JρM(zn,·)pn2−2g xn+1−g xn+xn+1−xn,j xn+1−xn
≤
μγ
1 +1 n
xn+1−xn+τ
δpn+1−pn
2
−2g xn+1
−g xn
,j xn+1−xn
−2xn+1−xn,j xn+1−xn
≤
2μ2γ2
1 +1 n
2
−2k−2xn+1−xn2+ 2τ
2
δ2pn+1−pn 2
,
which implies
xn+1−xn≤ b
k+ 1.5−μ2γ2 1 + (1/n)2
pn+1−pn, (3.11)
whereb=τ/δ.
SinceNisr,t-Lipschitz continuous with respect to the first, second arguments, respec-tively,S,Tareα,β-Lipschitz continuous, respectively, andgisξ-Lipschitz continuous, by (3.4), we obtain
pn+2−pn+1=(1−λ)pn+1+λ
g xn+1
−ρN wn+1,yn+1
+ρ f
+λen+1−(1−λ)pn+λg xn−ρN wn,yn+ρ f+λen ≤(1−λ)pn+1−pn+λg xn+1−g xn
+λρ N wn+1,yn+1
−N wn,yn+1+N wn,yn+1
−N wn,yn
+λen+1−en≤(1−λ)pn+1−pn
+λ
ξ+ρ
1 +1 n
(rα+tβ)xn+1−xn+λen+1−en.
(3.12)
It follows from (3.11) and (3.12) that
pn+2−pn+1≤1−λ+λbξ+ρ 1 + (1/n) rα+tβ
k+ 1.5−μ2γ2 1 + (1/n)2
pn+1−pn+λen+1−en
=1−λ 1−hnpn+1−pn+λen+1−en
=θnpn+1−pn+λen+1−en,
(3.13)
where
θn=1−λ 1−hn
, hn=
bξ+ρ 1 + (1/n)(rα+tβ)
k+ 1.5−μ2γ2 1 + (1/n)2
. (3.14)
Letting
θ=1−λ(1−h), h=b
ξ+ρ(rα+tβ)
k+ 1.5−μ2γ2 , (3.15)
we know thathn→handθn→θasn→ ∞. It follows from (3.7) and 0< λ≤1 that 0<
alln≥N. Therefore, for alln≥N, by (3.13), we now know that
pn+2−pn+1≤θ∗pn+1−pn+λen+1−en
≤θ∗ θ∗pn−pn−1+λen−en−1+λen+1−en
=θ2
∗pn−pn−1+λθ∗en−en−1+λen+1−en
≤ ··· ≤θn∗+1−NpN+1−pN+ n+1−N
i=1
θ∗i−1λen+1−(i−1)−en+1−i,
(3.16)
which implies that for anym > n > N, we have
pm−pn≤m−1
j=n
pj+1−pj
≤ m−1
j=n
θ∗j+1−NpN+1−pN+ m−1
j=n j+1−N
i=1
θi−1
∗ λen+1−(i−1)−en+1−i.
(3.17)
Since 0< λ≤1 and θ∗∈(0, 1), it follows from (3.8) and (3.17) that limm,n→∞pm−
pn =0, and hence{pn}is a Cauchy sequence inX. Letpn→pasn→ ∞. From (3.11),
we know that sequence{xn}is also a Cauchy sequence inX. Letxn→xasn→ ∞.
On the other hand, from the Lipschitzian continuity ofS,T,G, andAlgorithm 3.3, we have
wn−wn+1≤1 + 1 n+ 1
D S xn
,S xn+1
≤
1 + 1 n+ 1
αxn−xn+1,
yn−yn+1≤1 + 1 n+ 1
D T xn,T xn+1
≤
1 + 1 n+ 1
βxn−xn+1,
zn−zn+1≤1 + 1 n+ 1
D G xn
,G xn+1
≤1 + 1 n+ 1
γxn−xn+1.
(3.18)
Since{xn}is a Cauchy sequence, from (3.18), we know that{wn},{yn}, and{zn}are also
Cauchy sequences. Letwn→w,yn→y, andzn→zasn→ ∞. FromAlgorithm 3.3,
pn+1=(1−λ)pn+λ g xn−ρN wn,yn+ρ f+λen. (3.19)
By the assumptions and limn→∞en =0, we have
p=g(x)−ρ N(w,y)−f,
g xn
=JM(zn,·)
ρ pn=⇒g(x)=JρM(z,·)p.
(3.20)
Now we will prove thatw∈S(x),y∈T(x), andz∈G(x). In fact, sincewn∈S(xn) and
d wn,S(x)
≤max
d wn,S(x)
, sup
v∈S(x)
d S xn
,v
≤max
sup
u∈S(xn)
u,S(x), sup
v∈S(x)
d S xn
,v
=D S xn
,S(x),
(3.21)
we have
d w,S(x)≤w−wn+d wn,S(x)≤w−wn+D S xn,S(x)
≤w−wn+γxn−x−→0.
(3.22)
This implies thatw∈S(x). Similarly, we know that y∈T(x) andz∈G(x). This
com-pletes the proof.
IfS,T,G:X→Xare single-valued mappings, thenTheorem 3.6can be degenerated to the following theorem.
Theorem3.7. LetX,g,η,N(·,·),M(·,·)be the same as inTheorem 3.6, and letS,T,G: X→X be α,β,γ-Lipschitz continuous single-valued mappings, respectively. If conditions (3.6)–(3.8) hold, then the sequences{xn}generated byAlgorithm 3.4converges strongly to the unique solutionxof problem (3.2).
Proof. ByTheorem 3.6, problem (3.2) has a solutionx∈X andxn→xasn→ ∞. Now
we prove thatxis a unique solution of problem (3.2). Letx∗∈Xbe another solution of problem (3.2). Then
g x∗=JρM(G(x∗),·)m x∗
, m x∗=g x∗−ρ N S x∗,T x∗−f. (3.23)
We have
x−x∗2
=x−x∗+ g(x)−g x∗− JM(G(x),·)
ρ m(x)−JM(G(x ∗),·)
ρ m x∗
2
≤JρM(G(x),·)m(x)−JM(G(x ∗),·)
ρ m x∗
2
−2g(x)−g x∗+x−x∗,j x−x∗
≤ JρM(G(x),·)m(x)−JM(G(x ∗),·)
ρ m(x)+JM(G(x ∗),·)
ρ m(x)−JM(G(x ∗),·)
ρ m x∗
2
−2g(x)−g x∗,j x−x∗−2x−x∗,j x−x∗
≤
μG(x)−G x∗+τ
δm(x)−m x
∗2−2(k+ 1)x−x∗2
≤2 μ2γ2−k−1x−x∗2
+ 2τ
2
δ2m(x)−m x∗ 2
.
This implies that
x−x∗≤ b k+ 1.5−μ2γ2
m(x)−m x∗, (3.25)
whereb=τ/δ. Furthermore,
m(x)−m x∗=g(x)−g x∗−ρ N S(x),T(x)−N S x∗),T x∗
≤g(x)−g x∗+ρN S(x),T(x)−N S x∗,T(x)
+N S x∗,T(x)−N S x∗,T x∗
≤ξ+ρ(rα+tβ)x−x∗.
(3.26)
Combining (3.25) and (3.26), we have
x−x∗≤b
ξ+ρ(rα+tβ)
k+ 1.5−μ2γ2
x−x∗=hx−x∗, (3.27)
where
h=b
ξ+ρ(rα+tβ)
k+ 1.5−μ2γ2 . (3.28)
It follows from (3.7) that 0< h <1 and sox=x∗. This completes the proof.
Acknowledgments
The author would like to thank the referees for their valuable comments and suggestions leading to the improvements of this paper. This work was supported by the National Nat-ural Science Foundation of China (10471151) and the Educational Science Foundation of Chongqing, Chongqing, China.
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Mao-Ming Jin: Department of Mathematics, Yangtze Normal University, Chongqing 408003, Fuling, China