Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 580158, 18 pages
doi:10.1155/2012/580158
Research Article
Algorithms for a System of General Variational
Inequalities in Banach Spaces
Jin-Hua Zhu,1 Shih-Sen Chang,2 and Min Liu1
1
2
Department of Mathematics, Yibin University, Sichuan, Yibin 644007, China
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan,
Kunming 650221, China
Correspondence should be addressed to Shih-Sen Chang, changss@yahoo.cn
Received 21 December 2011; Accepted 6 February 2012
Academic Editor: Zhenyu Huang
Copyright q 2012 Jin-Hua Zhu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The purpose of this paper is using Korpelevich’s extragradient method to study the existence
problem of solutions and approximation solvability problem for a class of systems of finite
family of general nonlinear variational inequality in Banach spaces, which includes many kinds
of variational inequality problems as special cases. Under suitable conditions, some existence
theorems and approximation solvability theorems are proved. The results presented in the paper
improve and extend some recent results.
1. Introduction
Throughout this paper, we denote by N and R the sets of positive integers and real numbers,
respectively. We also assume that E is a real Banach space, E∗ is the dual space of E, C is a
nonempty closed convex subset of E, and ·, · is the pairing between E and E∗ .
In this paper, we are concerned a finite family of a general system of nonlinear
variational inequalities in Banach spaces, which involves finding x1∗ , x2∗ , . . . , xn∗ ∈ C × C ×
· · · × C such that
λ1 A1 x2∗ x1∗ − x2∗ , j x − x1∗ ≥ 0, ∀x ∈ C,
λ2 A2 x3∗ x2∗ − x3∗ , j x − x2∗ ≥ 0, ∀x ∈ C,
λ3 A3 x4∗ x3∗ − x4∗ , j x − x3∗ ≥ 0, ∀x ∈ C,
1.1
..
.
∗
∗
∗
∗
λN−1 AN−1 xN
xN−1
− xN
, j x − xN−1
≥ 0, ∀x ∈ C,
∗
∗
− x1∗ , j x − xN
≥ 0, ∀x ∈ C,
λN AN x1∗ xN
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where {Ai : C → E, i 1, 2, . . . , N} is a finite family of nonlinear mappings and λi i 1,
2, . . . , N are positive real numbers.
As special cases of the problem 1.1, we have the following.
I If E is a real Hilbert space and N 2, then 1.1 reduces to
λ1 A1 x2∗ x1∗ − x2∗ , x − x1∗ ≥ 0,
λ2 A2 x1∗ x2∗ − x1∗ , x − x2∗ ≥ 0,
∀x ∈ C,
∀x ∈ C,
1.2
which was considered by Ceng et al. 1. In particular, if A1 A2 A, then the problem 1.2
reduces to finding x1∗ , x2∗ ∈ C × C such that
λ1 Ax2∗ x1∗ − x2∗ , x − x1∗ ≥ 0,
λ2 Ax1∗ x2∗ − x1∗ , x − x2∗ ≥ 0,
∀x ∈ C,
∀x ∈ C,
1.3
which is defined by Verma 2. Furthermore, if x1∗ x2∗ , then 1.3 reduces to the following
variational inequality VI of finding x∗ ∈ C such that
Ax∗ , x − x∗ ≥ 0,
∀x ∈ C.
1.4
This problem is a fundamental problem in variational analysis and, in particular, in
optimization theory. Many algorithms for solving this problem are projection algorithms
that employ projections onto the feasible set C of the VI or onto some related set, in order
to iteratively reach a solution. In particular, Korpelevich’s extragradient method which was
introduced by Korpelevich 3 in 1976 generates a sequence {xn } via the recursion
xn1
yn PC xn − λAxn ,
PC xn − λAyn , n ≥ 0,
1.5
where PC is the metric projection from Rn onto C, A : C → H is a monotone operator, and λ
is a constant. Korpelevich 3 proved that the sequence {xn } converges strongly to a solution
of V IC, A. Note that the setting of the space is Euclid space Rn .
The literature on the VI is vast, and Korpelevich’s extragradient method has received
great attention by many authors, who improved it in various ways. See, for example, 4–16
and references therein.
II If E is still a real Banach space and N 1, then the problem 1.1 reduces to finding
x∗ ∈ C such that
∗
Ax , jx − x∗ ≥ 0,
∀x ∈ C,
1.6
which was considered by Aoyama et al. 17. Note that this problem is connected with the
fixed point problem for nonlinear mapping, the problem of finding a zero point of a nonlinear
operator, and so on. It is clear that problem 1.6 extends problem 1.4 from Hilbert spaces
to Banach spaces.
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3
In order to find a solution for problem 1.6, Aoyama et al. 17 introduced the
following iterative scheme for an accretive operator A in a Banach space E:
xn1 αn xn 1 − αn ΠC xn − λn Axn ,
n ≥ 1,
1.7
where ΠC is a sunny nonexpansive retraction from E to C. Then they proved a weak convergence theorem in a Banach space. For related works, please see 18 and the references therein.
It is an interesting problem of constructing some algorithms with strong convergence
for solving problem 1.1 which contains problem 1.6 as a special case.
Our aim in this paper is to construct two algorithms for solving problem 1.1. For this
purpose, we first prove that the system of variational inequalities 1.1 is equivalent to a fixed
point problem of some nonexpansive mapping. Finally, we prove the strong convergence of
the proposed methods which solve problem 1.1.
2. Preliminaries
In the sequel, we denote the strong convergence and weak convergence of the sequence {xn }
by xn → x and xn x, respectively.
∗
For q > 1, the generalized duality mapping Jq : E → 2E is defined by
Jq x f ∈ E∗ : x, f xq , f xq−1
2.1
for all x ∈ E. In particular, J J2 is called the normalized duality mapping. It is known that
Jq x ||x||q−2 for all x ∈ E. If E is a Hilbert space, then J I, the identity mapping. Let
U {x ∈ E : ||x|| 1}. A Banach space E is said to be uniformly convex if, for any ε ∈ 0, 2,
there exists δ > 0 such that, for any x, y ∈ U,
x − y ≥ ε implies x y ≤ 1 − δ.
2 2.2
It is known that a uniformly convex Banach space is reflexive and strictly convex. A
Banach space E is said to be smooth if the limit
x ty − x
lim
n→0
t
2.3
exists for all x, y ∈ U. It is also said to be uniformly smooth if the previous limit is attained
uniformly for x, y ∈ U. The norm of E is said to be Fréchet differentiable if, for each x ∈ U,
the previous limit is attained uniformly for all y ∈ U. The modulus of smoothness of E is
defined by
ρτ sup
1 x y x − y − 1 : x, y ∈ E, x 1, y τ ,
2
2.4
where ρ : 0, ∞ → 0, ∞ is function. It is known that E is uniformly smooth if and only if
limτ → 0 ρτ/τ 0. Let q be a fixed real number with 1 < q ≤ 2. Then a Banach space E is
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Journal of Applied Mathematics
said to be q-uniformly smooth if there exists a constant c > 0 such that ρτ ≤ cτ q for all τ > 0.
Note the following.
1 E is a uniformly smooth Banach space if and only if J is single valued and uniformly
continuous on any bounded subset of E.
p
2 All Hilbert spaces, Lp or lp spaces p ≥ 2 and the Sobolev spaces Wm p ≥ 2 are
P
spaces 1 < p ≤ 2 are p-uniformly
2-uniformly smooth, while Lp or lp and Wm
smooth.
3 Typical examples of both uniformly convex and uniformly smooth Banach spaces
are Lp , where p > 1. More precisely, Lp is min{p, 2}-uniformly smooth for every
p > 1.
In our paper, we focus on a 2-uniformly smooth Banach space with the smooth constant K.
Let E be a real Banach space, C a nonempty closed convex subset of E, T : C → C a
mapping, and FT the set of fixed points of T .
Recall that a mapping T : C → C is called nonexpansive if
T x − T y ≤ x − y ,
∀x, y ∈ C.
2.5
A bounded linear operator F : C ∈ E is called strongly positive if there exists a constant γ > 0
with the property
Fx, jx ≥ γx2 ,
∀x ∈ C.
2.6
A mapping A : C → E is said to be accretive if there exists jx − y ∈ Jx − y such that
Ax − Ay, j x − y ≥ 0,
2.7
for all x, y ∈ C, where J is the duality mapping.
A mapping A of C into E is said to be α-strongly accretive if, for α > 0,
2
Ax − Ay, j x − y ≥ αx − y ,
2.8
for all x, y ∈ C.
A mapping A of C into E is said to be α-inverse-strongly accretive if, for α > 0,
2
Ax − Ay, j x − y ≥ αAx − Ay ,
2.9
for all x, y ∈ C.
Remark 2.1. Evidently, the definition of the inverse strongly accretive mapping is based on
that of the inverse strongly monotone mapping, which was studied by so many authors; see,
for instance, 6, 19, 20.
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5
Let D be a subset of C, and let Π be a mapping of C into D. Then Π is said to be
sunny if
ΠΠx tx − Πx Πx
2.10
whenever Πx tx − Πx ∈ C for x ∈ C and t ≥ 0. A mapping Π of C into itself is called
a retraction if Π2 Π. If a mapping Π of C into itself is a retraction, then Πz z for every
z ∈ RΠ, where RΠ is the range of Π. A subset D of C is called a sunny nonexpansive
retract of C if there exists a sunny nonexpansive retraction from C onto D. Then following
lemma concerns the sunny nonexpansive retraction.
Lemma 2.2 see 21. Let C be a closed convex subset of a smooth Banach space E, let D be a
nonempty subset of C, and let Π be a retraction from C onto D. Then Π is sunny and nonexpansive if
and only if
u − Πu, j y − Πu ≤ 0,
2.11
for all u ∈ C and y ∈ D.
Remark 2.3. 1 It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction ΠC is coincident with the metric projection from E onto C.
2 Let C be a nonempty closed convex subset of a uniformly convex and uniformly
smooth Banach space E, and let T be a nonexpansive mapping of C into itself with the set
FT / ∅. Then the set FT is a sunny nonexpansive retract of C.
In what follows, we need the following lemmas for proof of our main results.
Lemma 2.4 see 22. Assume that {αn } is a sequence of nonnegative real numbers such that
αn1 ≤ 1 − γn αn δn ,
2.12
where {γn } is a sequence in 0, 1 and {δn } is a sequence such that
a Σ∞
n1 γn ∞,
b lim supn → ∞ δn /γn ≤ 0 or Σ∞
n1 |δn | < ∞.
Then limn → ∞ αn 0.
Lemma 2.5 see 23. Let X be a Banach space, {xn }, {yn } be two bounded sequences in X and
{βn } be a sequence in 0, 1 satisfying
0 < lim inf βn ≤ lim sup βn < 1.
n→∞
n→∞
2.13
Suppose that xn1 βn xn 1 − βn yn , for all n ≥ 1 and
lim sup yn1 − yn − xn1 − xn ≤ 0,
n→∞
then limn → ∞ yn − xn 0.
2.14
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Journal of Applied Mathematics
Lemma 2.6 see 24. Let E be a real 2-uniformly smooth Banach space with the best smooth
constant K. Then the following inequality holds:
x y2 ≤ x2 2 y, Jx 2Ky2 ,
∀x, y ∈ E.
2.15
Lemma 2.7 see 25. Let C be a nonempty bounded closed convex subset of a uniformly convex
Banach space E, and let G be a nonexpansive mapping of C into itself. If {xn } is a sequence of C such
that xn x and xn − Gxn → 0, then x is a fixed point of G.
Lemma 2.8 see 26. Let C be a nonempty closed convex subset of a real Banach space E. Assume
that the mapping F : C → E is accretive and weakly continuous along segments (i.e., Fx ty Fx as t → 0). Then the variational inequality
x∗ ∈ C,
∗
Fx , jx − x∗ ≥ 0,
x ∈ C,
2.16
x ∈ C.
2.17
is equivalent to the dual variational inequality
x∗ ∈ C,
Fx, jx − x∗ ≥ 0,
Lemma 2.9. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E.
Let ΠC be a sunny nonexpansive retraction from E onto C. Let {Ai : C → E, i 1, 2, . . . , N} be
a finite family of γi -inverse-strongly accretive. For given x1∗ , x2∗ , . . . , xn∗ ∈ C × C × · · · × C, where
∗
∗
, i ∈ {i, 2, . . . , N −1}, xN
ΠC I −λN AN x1∗ , then x1∗ , x2∗ , . . . , xn∗ x∗ x1∗ , xi∗ ΠC I −λi Ai xi1
∗
is a solution of the problem 1.1 if and only if x is a fixed point of the mapping Q defined by
Qx ΠC I − λ1 A1 ΠC I − λ2 A2 · · · ΠC I − λN AN x,
2.18
where λi i 1, 2, . . . , N are real numbers.
Proof. We can rewrite 1.1 as
x1∗ − x2∗ − λ1 A1 x2∗ , j x − x1∗ ≥ 0,
x2∗ − x3∗ − λ2 A2 x3∗ , j x − x2∗ ≥ 0,
∗ ∗
x3 − x4 − λ3 A3 x4∗ , j x − x3∗ ≥ 0,
∀x ∈ C,
∀x ∈ C,
∀x ∈ C,
..
.
∗
∗
∗
∗
xN−1 − xN − λN−1 AN−1 xN
, j x − xN−1
≥ 0, ∀x ∈ C,
∗ ∗
∗
≥ 0, ∀x ∈ C.
xN − x1 − λN AN x1∗ , j x − xN
2.19
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By Lemma 2.2, we can check 2.19 is equivalent to
x1∗ ΠC I − λ1 A1 x2∗ ,
x2∗ ΠC I − λ2 A2 x3∗ ,
..
.
∗
∗
ΠC I − λN−1 AN−1 xN
,
xN−1
2.20
∗
ΠC I − λN AN x1∗ .
xN
⇐⇒
Qx∗ ΠC I − λ1 A1 ΠC I − λ2 A2 · · · ΠC I − λN AN x∗ x∗ .
This completes the proof.
Throughout this paper, the set of fixed points of the mapping Q is denoted by Ω.
Lemma 2.10. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space
E. Let ΠC be a sunny nonexpansive retraction from E onto C. Let {Ai : C → E, i 1, 2, . . . , N} be
a finite family of γi -inverse-strongly accretive. Let Q be defined as Lemma 2.9. If 0 ≤ λi ≤ γi /K 2 , then
Q : C → C is nonexpansive.
Proof. First, we show that for all i ∈ {i, 2, . . . , N}, the mapping ΠC I − λi Ai is nonexpansive.
Indeed, for all x, y ∈ C, from the condition λi ∈ 0, γi /K 2 and Lemma 2.6, we have
ΠC I − λi Ai x − ΠC I − λi Ai y2 ≤ I − λi Ai x − I − λi Ai y2
2
x − y − λi Ai x − Ai y 2
≤ x − y − 2λi Ai x − Ai y, j x − y
2
2K 2 λ2i Ai x − Ai y
2
2
2
≤ x − y − 2λi γi Ai x − Ai y 2K 2 λ2i Ai x − Ai y
2
2
x − y 2λi K 2 λi − γi Ai x − Ai y
2
≤ x − y ,
2.21
which implies for all i ∈ {1, 2, . . . , N}, the mapping ΠC I − λi Ai is nonexpansive, so is the
mapping Q.
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Journal of Applied Mathematics
3. Main Results
In this section, we introduce our algorithms and show the strong convergence theorems.
Algorithm 3.1. Let C be a nonempty closed convex subset of a uniformly convex and 2uniformly smooth Banach space E. Let ΠC be a sunny nonexpansive retraction from E to
C. Let {Ai : C → E, i 1, 2, . . . , N} be a finite family of γi -inverse-strongly accretive.
Let B : C → E be a strongly positive bounded linear operator with coefficient α > 0 and
F : C → E be a strongly positive bounded linear operator with coefficient ρ ∈ 0, α. For any
t ∈ 0, 1, define a net {xt } as follows:
xt ΠC tF I − tByt ,
3.1
yt ΠC I − λ1 A1 ΠC I − λ2 A2 · · · ΠC I − λN AN xt ,
where, for any i, λi ∈ 0, γi /K 2 is a real number.
Remark 3.2. We notice that the net {xt } defined by 3.1 is well defined. In fact, we can define
a self-mapping Wt : C → C as follows:
Wt x : ΠC tF I − tBΠC I − λ1 A1 ΠC I − λ2 A2 · · · ΠC I − λN AN x,
∀x ∈ C.
3.2
From Lemma 2.10, we know that if, for any i, λi ∈ 0, γi /K 2 , the mapping ΠC I −
λ1 A1 ΠC I − λ2 A2 · · · ΠC I − λN AN Q is nonexpansive and ||I − tB|| ≤ 1 − tα. Then, for any
x, y ∈ C, we have
Wt x − Wt y ΠC tF I − tBQx − ΠC tF I − tBQ y ≤ tF I − tBQx − tF I − tBQy
t Fx − Fy I − tB Qx − Qy ≤ tρx − y I − tBQx − Qy
≤ tρx − y 1 − tαx − y
1 − α − ρ t x − y.
3.3
This shows that the mapping Wt is contraction. By Banach contractive mapping principle, we
immediately deduce that the net 3.1 is well defined.
Theorem 3.3. Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly
smooth Banach space E. Let ΠC be a sunny nonexpansive retraction from E to C. Let {Ai : C →
E, i 1, 2, . . . , N} be a finite family of γi -inverse-strongly accretive. Let B : C → E be a strongly
positive bounded linear operator with coefficient α > 0, and let F : C → E be a strongly positive
bounded linear operator with coefficient ρ ∈ 0, α. Assume that Ω /
∅ and λi ∈ 0, γi /K 2 . Then the
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net {xt } generated by the implicit method 3.1 converges in norm, as t → 0 to the unique solution
x of VI
x ∈ Ω,
jz − x
≥ 0,
B − Fx,
z ∈ Ω.
3.4
Proof. We divide the proof of Theorem 3.3 into four steps.
I Next we prove that the net {xt } is bounded.
Take that x∗ ∈ Ω, we have
xt − x∗ ΠC tF I − tByt − x∗ ΠC tF I − tByt − ΠC x∗ ≤ tF I − tByt − x∗ t F yt − Fx∗ I − tB yt − x∗ tFx∗ − tBx∗ ≤ tF yt − Fx∗ I − tByt − x∗ tFx∗ − Bx∗ ≤ tρyt − x∗ 1 − tαyt − x∗ tFx∗ − Bx∗ 1 − α − ρ t yt − x∗ tFx∗ − Bx∗ 1 − α − ρ t Qxt − Qx∗ tFx∗ − Bx∗ ≤ 1 − α − ρ t xt − x∗ tFx∗ − Bx∗ .
3.5
It follows that
xt − x∗ ≤
Fx∗ − Bx∗ .
α−ρ
3.6
Therefore, {xt } is bounded. Hence, {yt }, {Byt }, {Ai xt }, and {Fyt } are also bounded. We
observe that
xt − yt ΠC tF I − tByt − ΠC yt ≤ tF I − tByt − yt tF yt − B yt 3.7
−→ 0.
From Lemma 2.10, we know that Q : C → C is nonexpansive. Thus, we have
yt − Q yt Qxt − Q yt ≤ xt − yt −→ 0.
3.8
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Therefore,
limxt − Qxt 0.
3.9
t→0
II {xt } is relatively norm-compact as t → 0 .
Let {tn } ⊂ 0, 1 be any subsequence such that tn → 0 as n → ∞. Then, there exists
a positive integer n0 such that 0 < tn < 1/2, for all n ≥ n0 . Let xn : xtn . It follows from 3.9
that
xn − Qxn −→ 0.
3.10
xt ΠC tF I − tByt − tF I − tByt tF I − tByt .
3.11
We can rewrite 3.1 as
For any x∗ ∈ Ω ⊂ C, by Lemma 2.2, we have
tF I − tByt − xt , jx∗ − xt tF I − tByt − ΠC tF I − tByt , j x∗ − ΠC tF I − tByt ≤ 0.
3.12
With this fact, we derive that
xt − x∗ 2 x∗ − xt , jx∗ − xt x∗ − tF I − tByt , jx∗ − xt tF I − tByt − ΠC tF I − tByt , jx∗ − xt ≤ tF I − tB x∗ − yt , jx∗ − xt t Bx∗ − Fx∗ , jx∗ − xt ≤ 1 − t α − ρ x∗ − yt x∗ − xt t Bx∗ − Fx∗ , jx∗ − xt 1 − t α − ρ Qx∗ − Qxt x∗ − xt t Bx∗ − Fx∗ , jx∗ − xt ≤ 1 − t α − ρ x∗ − xt x∗ − xt t Bx∗ − Fx∗ , jx∗ − xt ≤ 1 − t α − ρ x∗ − xt 2 t Bx∗ − Fx∗ , jx∗ − xt .
3.13
It turns out that
xt − x∗ 2 ≤
1 Bx∗ − Fx∗ , jx∗ − xt ,
α−ρ
x∗ ∈ Ω.
3.14
1 Bx∗ − Fx∗ , jx∗ − xn ,
α−ρ
x∗ ∈ Ω.
∗∗
In particular,
xn − x∗ 2 ≤
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Since {xn } is bounded, without loss of generality, xn x ∈ C can be assumed.
Noticing 3.10, we can use Lemma 2.7 to get x ∈ Ω FQ. Therefore, we can substitute
x for x∗ in ∗∗ to get
2≤
xn − x
1 Bx
− Fx,
jx − xn .
α−ρ
3.15
Consequently, the weak convergence of {xn } to x actually implies that xn → x strongly. This
has proved the relative norm compactness of the net {xt } as t → 0 .
III Now, we prove that x solves the variational inequality 3.4. From 3.1, we have
xt ΠC tF I − tByt − tF I − tByt tF I − tByt
⇒ xt ΠC tF I − tByt − tF I − tByt − tF I − tB xt − yt
tFxt I − tBxt ⇒ Fxt − Bxt 1
tF I − tByt − ΠC tF I − tByt − tF I − tB yt − xt .
t
3.16
For any z ∈ Ω, we obtain
1
Fxt − Bxt , jz − xt tF I − tByt − ΠC tF I − tByt , jz − xt t
−
1
tF I − tB yt − xt , jz − xt t
≤−
1
tF I − tB yt − xt , jz − xt t
−
1
yt − xt , jz − xt B − F yt − xt , jz − xt .
t
3.17
Now we prove that yt − xt , jz − xt ≥ 0. In fact, we can write yt Qxt . At the same
time, we note that z Qz, so
yt − xt , jz − xt z − Qz − xt − Qxt , jz − xt .
3.18
Since I − Q is accretive this is due to the nonexpansivity of Q, we can deduce immediately
that
yt − xt , jz − xt z − Qz − xt − Qxt , jz − xt ≥ 0.
3.19
Fxt − Bxt , jz − xt ≤ B − F yt − xt , jz − xt .
3.20
Therefore,
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Journal of Applied Mathematics
Since B, F is strongly positive, we have
0 ≤ α − ρ z − xt 2 ≤ B − Fz − xt , jz − xt Fxt − Bxt − Fz − Bz, jz − xt .
3.21
Fz − Bz, jz − xt ≤ Fxt − Bxt , jz − xt .
3.22
It follows that
Combining 3.20 and 3.22, we get
Fz − Bz, jz − xt ≤ B − F yt − xt , jz − xt .
3.23
Now replacing t in 3.23 with tn and letting n → ∞, noticing that xtn − ytn → 0, we obtain
Fz − Bz, jz − x
≤ 0,
z ∈ Ω,
3.24
which is equivalent to its dual variational inequality see Lemma 2.8
jz − x
≥ 0,
B − Fx,
z ∈ Ω,
3.25
that is, x ∈ Ω is a solution of 3.4.
IV Now we show that the solution set of 3.4 is singleton.
As a matter of fact, we assume that x∗ ∈ Ω is also a solution of 3.4 Then, we have
B − Fx∗ , jx − x∗ ≥ 0.
3.26
≥ 0.
jx∗ − x
B − Fx,
3.27
≥0
jx∗ − x
B − Fx∗ , jx − x∗ B − Fx,
≥0
⇒ B − Fx − x∗ , jx∗ − x
jx∗ − x
≤0
⇒ B − Fx∗ − x,
⇒ α − ρ x∗ − x
2 ≤ 0.
3.28
From 3.25, we have
So,
Therefore, x∗ x.
In summary, we have shown that each cluster point of {xt } as t → 0
equals x.
Therefore, xt → x as t → 0. This completes the proof.
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13
Next, we introduce our explicit method which is the discretization of the implicit
method 3.1.
Algorithm 3.4. Let C be a nonempty closed convex subset of a uniformly convex and 2uniformly smooth Banach space E. Let ΠC be a sunny nonexpansive retraction from E to
C. Let {Ai : C → E, i 1, 2, . . . , N} be a finite family of γi -inverse-strongly accretive. Let
B : C → E be a strongly positive bounded linear operator with coefficient α > 0, and let
F : C → E be a strongly positive bounded linear operator with coefficient ρ ∈ 0, α. For
arbitrarily given x0 ∈ C, let the sequence {xn } be generated iteratively by
xn1 βn xn 1 − βn ΠC αn F I − αn BΠC I − λ1 A1 ΠC I − λ2 A2 · · · ΠC I − λN AN xn ,
n ≥ 0,
3.29
where {αn } and {βn } are two sequences in 0, 1 and, for any i, λi ∈ 0, γi /K 2 is a real number.
Theorem 3.5. Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly
smooth Banach space E, and let ΠC be a sunny nonexpansive retraction from E to C. Let {Ai : C →
E, i 1, 2, . . . , N} be a finite family of γi -inverse-strongly accretive.Let B : C → E be a strongly
positive bounded linear operator with coefficient α > 0, and let F : C → E be a strongly positive
bounded linear operator with coefficient ρ ∈ 0, α. Assume that Ω /
∅. For given x0 ∈ C, let {xn } be
generated iteratively by 3.29. Suppose the sequences {αn } and {βn } satisfy the following conditions:
1 limn → ∞ αn 0 and Σ∞
n1 αn ∞,
2 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn ≤ 1.
Then {xn } converges strongly to x ∈ Ω which solves the variational inequality 3.4.
Proof. Set yn ΠC I − λ1 A1 ΠC I − λ2 A2 · · · ΠC I − λN AN xn for all n ≥ 0. Then xn1 βn xn 1 − βn ΠC αn F I − αn Byn for all n ≥ 0. Pick up x∗ ∈ Ω.
From Lemma 2.10, we have
yn − x∗ Qxn − Qx∗ ≤ xn − x∗ .
Hence, it follows that
xn1 − x∗ βn xn 1 − βn ΠC αn F I − αn Byn − x∗ βn xn − x∗ 1 − βn ΠC αn F I − αn Byn − x∗ 3.30
14
Journal of Applied Mathematics
≤ βn xn − x∗ 1 − βn ΠC αn F I − αn Byn − ΠC x∗ ≤ βn xn − x∗ 1 − βn αn F I − αn Byn − x∗ βn xn − x∗ 1 − βn αn F I − αn B yn − x∗ αn Fx∗ − Bx∗ ≤ βn xn − x∗ 1 − βn αn ρ 1 − αn α yn − x∗ 1 − βn αn Fx∗ − Bx∗ Fx∗ − Bx∗ .
≤ 1 − αn 1 − βn α − ρ xn − x∗ αn 1 − βn α − ρ
α−ρ
3.31
By induction, we deduce that
Fx∗ − Bx∗ .
xn1 − x∗ ≤ max x0 − x∗ ,
α−ρ
3.32
Therefore, {xn } is bounded. Hence, {Ai xi } i 1, 2, . . . , N, {yn }, {Byn }, and {Fyn } are also
bounded. We observe that
yn1 − yn Qxn1 − Qxn ≤ xn1 − xn .
3.33
Set xn1 βn xn 1 − βn zn for all n ≥ 0. Then zn ΠC αn F I − αn Byn . It follows that
zn1 − zn ΠC αn1 F I − αn1 Byn1 − ΠC αn F I − αn Byn ≤ αn1 F I − αn1 Byn1 − αn F I − αn Byn yn1 − yn αn1 F yn1 − B yn1 − αn F yn − B yn ≤ yn1 − yn αn1 F yn1 − B yn1 − αn F yn − B yn ≤ xn1 − xn αn1 F yn1 − B yn1 − αn F yn − B yn .
3.34
This implies that
lim supzn1 − zn − xn1 − xn ≤ 0.
n→∞
3.35
Hence, by Lemma 2.5, we obtain limn → ∞ zn − xn 0. Consequently,
lim xn1 − xn lim 1 − βn zn − xn 0.
n→∞
n→∞
3.36
Journal of Applied Mathematics
15
At the same time, we note that
zn − yn ΠC αn F I − αn Byn − yn ΠC αn F I − αn Byn − ΠC yn ≤ αn F I − αn Byn − yn αn F yn − B yn 3.37
−→ 0.
It follows that
lim xn − yn 0.
n→∞
3.38
From Lemma 2.10, we know that Q : C → C is nonexpansive. Thus, we have
yn − Q yn Qxn − Q yn ≤ xn − yn −→ 0.
3.39
Thus, limn → ∞ xn − Qxn 0. We note that
zn − Qzn ≤ zn − xn xn − Qxn Qxn − Qzn ≤ 2zn − xn xn − Qxn 2ΠC αn F I − αn Byn − ΠC xn xn − Qxn ≤ 2 yn − xn αn F yn − B yn xn − Qxn 3.40
−→ 0.
Next, we show that
lim sup Fx
≤ 0,
− Bx,
jzn − x
n→∞
3.41
where x ∈ Ω is the unique solution of VI3.4.
To see this, we take a subsequence {znj } of {zn } such that
lim Fx
− Bx,
j znj − x .
lim Fx
− Bx,
jzn − x
n→∞
nj → ∞
3.42
We may also assume that znj z. Note that z ∈ Ω in virtue of Lemma 2.7 and 3.40. It
follows from the variational inequality 3.4 that
− Bx,
j znj − x
lim Fx
lim Fx
− Bx,
jzn − x
n→∞
nj → ∞
Fx
− Bx,
jz − x
≤ 0.
3.43
16
Journal of Applied Mathematics
Since zn ΠC αn F I − αn Byn , according to Lemma 2.2, we have
αn F I − αn Byn − ΠC αn F I − αn Byn , jx − zn ≤ 0.
3.44
From 3.44, we have
2 ΠC αn F I − αn Byn − x,
jzn − x
zn − x
ΠC αn F I − αn Byn − αn F I − αn Byn , jzn − x
αn F I − αn Byn − x,
jzn − x
≤ αn F I − αn Byn − x,
jzn − x
αn F I − αn B yn − x , jzn − x
αn Fx
− Bx,
jzn − x
≤ 1 − αn α − ρ yn − xzn − x
αn Fx
− Bx,
jzn − x
3.45
≤
2
1 − αn α − ρ
yn − x2 1 zn − x
2 αn Fx
.
− Bx,
jzn − x
2
2
It follows that
2
2 ≤ 1 − αn α − ρ yn − x 2αn Fx
,
− Bx,
jzn − x
zn − x
≤ 1 − αn α − ρ xn − x
2 2αn Fx
.
− Bx,
jzn − x
3.46
From xn1 βn xn 1 − βn zn and 3.46, we have
Finally, we prove xn → x.
2 ≤ βn xn − x
2 1 − βn zn − x
2
xn1 − x
≤ βn xn − x
2 1 − βn
2 2αn Fx
1 − αn α − ρ xn − x
− Bx,
jzn − x
1 − αn 1 − βn α − ρ xn − x
2 αn 1 − βn α − ρ
2 ×
.
Fx
− Bx,
jzn − x
α−ρ
3.47
This completes the
We can apply Lemma 2.4 to the relation 3.47 and conclude that xn → x.
proof.
Acknowledgments
The authors would like to express their thanks to the referees and the editor for their
helpful suggestion and comments. This work was supported by the Scientific Reserch Fund
of Sichuan Provincial Education Department 09ZB102,11ZB146 and Yunnan University of
Finance and Economics.
Journal of Applied Mathematics
17
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