Báo cáo hóa học: " Research Article Hybrid Steepest Descent Method with Variable Parameters for General Variational Inequalities Yanrong Yu and Rudong Chen"

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 19270, 14 pages doi:10.1155/2007/19270

Research Article Hybrid Steepest Descent Method with Variable Parameters for General Variational Inequalities

Yanrong Yu and Rudong Chen

Received 16 April 2007; Accepted 2 August 2007

Recommended by Yeol Je Cho

We study the strong convergence of a hybrid steepest descent method with variable pa- rameters for the general variational inequality GVI(F,g,C). Consequently, as an applica- tion, we obtain some results concerning the constrained generalized pseudoinverse. Our results extend and improve the result of Yao and Noor (2007) and many others.

Copyright © 2007 Y. Yu and R. Chen. 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.

1. Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let F : H → H be an operator such that for some constants k,η > 0, F is k-Lipschitzian and η-strongly monotone on C; that is, F satisfies the following inequalities: (cid:3)Fx − F y(cid:3) ≤ k(cid:3)x − y(cid:3) and (cid:5)Fx − F y,x − y(cid:6) ≥ η(cid:3)x − y(cid:3)2 for all x, y ∈ C, respectively. Recall that T is nonexpansive if (cid:3)Tx − T y(cid:3) ≤ (cid:3)x − y(cid:3) for all x, y ∈ H.

We consider the following variational inequality problem: find a point u∗ ∈ C such

that

(cid:2) (cid:3)

(1.1)

F(u∗),v − v∗

≥ 0, ∀v ∈ C.

VI(F,C) :

Variational inequalities were introduced and studied by Stampacchia [1] in 1964. It is now well known that a wide class of problems arising in various branches of pure and applied sciences can be studied in the general and unified framework of variational inequalities. Several numerical methods including the projection and its variant forms, Wiener-Hofp equations, auxiliary principle, and descent type have been developed for solving the vari- ational inequalities and related optimization problems. The reader is referred to [1–18] and the references therein.

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Journal of Inequalities and Applications

It is well known that when F is strongly monotone on C, the VI(F,C) has a unique

solution and VI(F,C) is equivalent to the fixed point problem

(cid:5) ,

(1.2)

(cid:4) u∗ − μF(u∗)

u∗ = PC

where μ > 0 is an arbitrarily fixed constant and PC is the (nearest point) projection from H onto C. From (1.2), one can suggest a so-called projection method. Using the projection method, one establishes the equivalence between the variational inequalities and fixed- point problem. This alternative equivalence has been used to study the existence theory of the solution and to develop several iterative-type algorithms for solving variational inequalities. Under certain conditions, projection methods and their variant forms can be implemented for solving variational inequalities. However, there are some drawbacks of this method which rule out its problems in applications, for instance, the projection method involves the projection PC which may not be easily computed due to the com- plexity of the convex set C.

In order to reduce the complexity probably caused by the projection PC, Yamada [11]

introduced the following hybrid steepest descent method for solving the VI(F,C).

Algorithm 1.1. For a given u0 ∈ H, calculate the approximate solution un by the iterative scheme

(cid:5)

,

(1.3)

n ≥ 0,

un+1 = Tun − λn+1μF

(cid:4) Tun

where μ ∈ (0,2η/k2) and λn ∈ (0,1) satisfy the following conditions:

(cid:6)∞

= 0.

(1) limn→∞ λn = 0; n=1 λn = ∞; (2) (3) limn→∞(λn − λn+1)/λ2

n+1

Yamada [11] proved that the approximate solution {un}, obtained from Algorithm 1.1,

converges strongly to the unique solution of the VI(F,C).

Furthermore, Xu and Kim [12] and Zeng et al. [15] considered and studied the con- vergence of the hybrid steepest descent Algorithm 1.1 and its variant form. For details, please see [12, 15].

Let F : H → H be a nonlinear operator and let g : H → H be a continuous mapping. Now, we consider the following general variational inequality problem: find a point u∗ ∈ H such that g(u∗) ∈ C and

(cid:2) (cid:3)

(1.4)

F(u∗),g(v) − g(u∗)

≥ 0, ∀v ∈ H, g(v) ∈ C.

GVI(F,g,C) :

If g is the identity mapping of H, then the GVI(F,g,C) reduces to the VI(F,C).

Although iterative algorithm (1.3) has successfully been applied to finding the unique solution of the VI(F,C). It is clear that it can not be directly applied to computing solution of the GVI(F,g,C) due to the presence of g. Therefore, an important problem is how to apply hybrid steepest descent method to solving GVI(F,g,C). For this purpose, Zeng et al. [13] introduced a hybrid steepest descent method for solving the GVI(F,g,C) as follows.

Y. Yu and R. Chen 3

(cid:5)

(1.5)

(cid:5) ,

n ≥ 0,

Algorithm 1.2. Let {λn} ⊂ (0,1), {θn} ⊂ (0,1], and μ ∈ (0,2η/k2). For a given u0 ∈ H, calculate the approximate solution un by the iterative scheme (cid:5) Tun − θn+1g

(cid:4) 1 + θn+1 − λn+1μF (cid:4) Tun (cid:4) Tun

un+1 =

where F is η-strongly monotone and k-Lipschitzian and g is σ-Lipschitzian and δ-strongly monotone on C.

They also proved that the approximate solution {un} obtained from (1.5) converges strongly to the solution of the GVI(F,g,C) under some assumptions on parameters. Con- sequently, Yao and Noor [7] present a modified iterative algorithm for approximating solution of the GVI(F,g,C). But we note that all of the above work has imposed some additional assumptions on parameters or the iterative sequence {un}. There is a natural question that rises: could we relax it?

Our purpose in this paper is to suggest and analyze a hybrid steepest descent method with variable parameters for solving general variational inequalities. It is shown that the convergence of the proposed method can be proved under some mild conditions on pa- rameters. We also give an application of the proposed method for solving constrained generalized pseudoinverse problem.

2. Preliminaries

In the sequel, we will make use of the following results.

(cid:4)

(2.1)

n ≥ 0,

Lemma 2.1 [12]. Let {sn} be a sequence of nonnegative numbers satisfying the condition (cid:5) sn + αnβn,

sn+1 ≤

1 − αn

where {αn}, {βn} are sequences of real numbers such that (cid:6)∞

n=0 αn = ∞, (cid:6)∞

n=0 αnβn is convergent.

(i) {αn} ⊂ [0,1] and (ii) limsupn→∞ βn ≤ 0 or

Then, limn→∞ sn = 0.

Lemma 2.2 [19]. Let {xn} and {yn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0 < liminf n→∞ βn ≤ limsupn→∞ βn < 1. Suppose xn+1 = (1 − βn)yn + βnxn for all integers n ≥ 0 and limsupn→∞((cid:3)yn+1 − yn(cid:3) − (cid:3)xn+1 − xn(cid:3)) ≤ 0. Then, limn→∞ (cid:3)yn − xn(cid:3) = 0.

Lemma 2.3 [20] (demiclosedness principle). Assume that T is a nonexpansive self- mapping of a closed convex subset C of a Hilbert space H. If T has a fixed point, then I − T is demiclosed. That is, whenever {xn} is a sequence in C weakly converging to some x ∈ C and the sequence {(I − T)xn} strongly converges to some y, it follows that (I − T)x = y. Here, I is the identity operator of H.

The following lemma is an immediate consequence of an inner product.

Lemma 2.4. In a real Hilbert space H, there holds the inequality

(2.2)

(cid:3)x + y(cid:3)2 ≤ (cid:3)x(cid:3)2 + 2(cid:5)y,x + y(cid:6), ∀x, y ∈ H.

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Journal of Inequalities and Applications

3. Modified hybrid steepest descent method

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let F : H → H be k-Lipschitzian and η-strongly monotone mapping on C and let g : H → H be σ-Lipschitzian and δ-strongly monotone mapping on C for some constants σ > 0 and δ > 1. Assume also that the unique solution u∗ of the VI(F,C) is a fixed point of g.

Denote by PC the projection of H onto C. Namely, for each x ∈ H, PCx is the unique

element in C satisfying

(cid:8) (cid:9) (cid:7) (cid:7) = min

(3.1)

(cid:3)x − y(cid:3) : y ∈ C

.

(cid:7) (cid:7)x − PCx

It is known that the projection PC is characterized by inequality

(cid:3)

(3.2)

≤ 0, ∀y ∈ C. (cid:2) x − PCx, y − PCx

Thus, it follows that the GVI(F,g,C) is equivalent to the fixed point problem g(u∗) = PC(I − μF)g(u∗), where μ > 0 is an arbitrary constant.

(cid:10)

In this section, assume that Ti : H → H is a nonexpansive mapping for each 1 ≤ i ≤ N i=1 Fix(Ti) (cid:13)= ∅. Let δn1,δn2,...,δnN ∈ (0,1], n ≥ 1. We define, for each n ≥ 1,

N with mappings Un1,Un2,...,UnN by

Un1 = δn1T1 + Un2 = δn2T2Un1 +

(cid:4) (cid:5) 1 − δn1 I, (cid:5) (cid:4) 1 − δn2 I,

(3.3)

...

(cid:5) I,

Un,N −1 = δn,N −1TN −1Un,N −2 + Wn := UnN = δnN TN Un,N −1 +

(cid:4) 1 − δn,N −1 (cid:5) (cid:4) 1 − δnN I.

Such a mapping Wn is called the W-mapping generated by T1,...,TN and δn1,δn2,...,δnN . Nonexpansivity of Ti yields the nonexpansivity of Wn. Moreover, [21, Lemma 3.1] shows that

(cid:5)

(3.4)

= F. (cid:4) Fix Wn

Such property of Wn will be crucial in the proof on our result.

Now we suggest the following iterative algorithm for solving GVI(F,g,C).

(cid:5) (cid:4) (cid:5)

un − Wnun

+ αn+1

(3.5)

Algorithm 3.1. Let {αn} ⊂ [a,b] ⊂ (0,1), {λn} ⊂ (0,1), {θn} ⊂ (0,1], and {μn} ⊂ (0,2η/ k2). For a given u0 ∈ H, compute the approximate solution {un} by the iterative scheme (cid:4) Wnun (cid:5)(cid:5) ,

n ≥ 0.

un+1 = Wnun − λn+1μn+1F (cid:4) (cid:4) Wnun − g Wnun

+ θn+1

At this point, we state and prove our main result.

Theorem 3.2. Assume that 0 < a ≤ αn ≤ b < 1, 0 < μn < 2η/k2, and u∗ ∈ Fix(g). Let δn1, δn2,...,δnN be real numbers such that limn→∞(δn+1,i − δn,i) = 0 for all i = 1,2,...,N. Assume

Y. Yu and R. Chen 5

n=1 λn = ∞;

{λn} and {θn} satisfy the follwoing conditions: (cid:6)∞

(i) limn→∞ λn = 0, (ii) θn ∈ (0,2(1 − a)(δ − 1)/(σ 2 − 1)]; (iii) limn→∞ θn = 0,limn→∞ λn/θn = 0.

Then the sequence {un} generated by Algorithm 3.1 converges strongly to u∗ which is a solu- tion of the GVI(F,g,C).

Proof. Now we divide our proof into the following steps.

Step 1. First, we prove that {un} is bounded. From (3.5), we have

(cid:5) (cid:7) (cid:7) (cid:7) (cid:7) = (cid:7) (cid:7)un+1 − u∗ (cid:4) Wnun (cid:5) (cid:5) Wnun + αn+1un − θn+1g (cid:7) (cid:7) − u∗ (cid:5) (cid:5)(cid:4) (cid:5) (cid:5) (cid:7) (cid:7) = − u∗ (cid:4) Wnun (cid:5)

+ θn+1 (cid:5)

(3.6)

(cid:7) (cid:7) (cid:4) − θn+1 g (cid:4) Wnun − u∗ (cid:5)(cid:5) (cid:4) u∗ − F (cid:5) (cid:7) (cid:7) ≤ (cid:4) 1 − αn+1 + θn+1 (cid:4) − λn+1μn+1F Wnun (cid:4) Wnun − u∗ 1 − αn+1 (cid:5) (cid:4) un − u∗ (cid:4) (cid:4) Wnun F Wnun − u∗ (cid:5) (cid:5)(cid:7) (cid:7)

+

− u∗ (cid:5) − F(u∗) (cid:4) Wnun − u∗ − θn+1 − λn+1μn+1

(cid:4) g (cid:4) F (cid:7) (cid:7)F(u∗)

+ λn+1μn+1F(u∗) (cid:5)(cid:7) (cid:5) (cid:4) (cid:7) Wnun (cid:4) Wnun (cid:7) (cid:7).

+ αn+1 − λn+1μn+1 (cid:4) (cid:5)(cid:4) 1 − αn+1 (cid:7) (cid:7)θn+1 (cid:7) (cid:7)un − u∗ + αn+1

(cid:7) (cid:7) + λn+1μn+1

Observe that

(cid:5) (cid:5) (cid:5)(cid:4) (cid:7) (cid:7) (cid:5)(cid:7) (cid:7)2 − u∗ (cid:4) Wnun (cid:4) g (cid:7) (cid:7)2 = (cid:5) (cid:3) (cid:5) (cid:7) (cid:7)2

g

(cid:7) (cid:7)g − u∗ − g(u∗),Wnun − u∗ (cid:4) 1 − αn+1 (cid:4) 1 − αn+1 (cid:4) 1 − αn+1 (cid:4) Wnun − 2 (cid:11)(cid:4) ≤

1 − αn+1

n+1

(cid:11)(cid:4)

(cid:4) Wnun (cid:5) δθn+1 + σ 2θ2 (cid:5) δθn+1 + σ 2θ2 (cid:5)

1 − αn+1 (cid:5)

+ θ2 n+1 (cid:7) (cid:12)(cid:7) (cid:7)2 (cid:7)Wnun − u∗ (cid:7) (cid:12)(cid:7) (cid:7)2, (cid:7)un − u∗ (cid:5)(cid:7) (cid:7)2

[8pt]

F

(cid:7) (cid:7)θn+1 (cid:5) (cid:3) = θ2

F

n+1 − F(u∗) (cid:4) (cid:2) Wnun

n+1

− F(u∗),Wnun − u∗ (cid:5) (cid:7) (cid:7)2

Wnun − u∗ − θn+1 (cid:7) (cid:5)2 (cid:7)Wnun − u∗ (cid:2) (cid:5) θn+1 (cid:4) (cid:5)2 − 2 1 − αn+1 (cid:4) (cid:5)2 − 2 1 − αn+1 (cid:4) (cid:4) − λn+1μn+1 Wnun (cid:7) (cid:7)2 − 2θn+1λn+1μn+1 − F(u∗)

n+1μ2

n+1

n+1k2λn+1

≤ (cid:4) Wnun − u∗ (cid:7) (cid:7)Wnun − u∗ (cid:7) (cid:7)F (cid:4) Wnun (cid:7) (cid:7)2 ≤ − 2μn+1ηθn+1λn+1 + μ2

≤

+ λ2 (cid:4) θ2 n+1 (cid:4) θ2 n+1

− 2μn+1ηθn+1λn+1 + μ2 (cid:13)(cid:14) (cid:15)2 (cid:7) (cid:7)2

+

= θ2

.

μn+1k

n+1

(cid:5)(cid:7) (cid:7)Wnun − u∗ (cid:7) (cid:5)(cid:7) (cid:7)2 (cid:7)un − u∗ (cid:16) (cid:7) (cid:7)un − u∗

1 − λn+1 θn+1

n+1k2λn+1 2λn+1μn+1(k − η) θn+1

(3.7)

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Journal of Inequalities and Applications

From (3.7), we have

(cid:7) (cid:7)

(cid:7) (cid:7)un+1 − u∗ (cid:17)(cid:18)(cid:4) (cid:5) (cid:7) (cid:7) ≤ (cid:5)2 − 2 (cid:4) 1 − αn+1

δθn+1 + σ 2θ2

n+1 + αn+1

(cid:19)(cid:7) (cid:7)un − u∗

(cid:15)2

1 − αn+1 (cid:20) (cid:21) (cid:21) (cid:22)

(cid:7) (cid:7)

+

(cid:7) (cid:7)F(u∗) (cid:7) (cid:7)un − u∗ (cid:7) (cid:7) + λn+1μn+1

+ θn+1

(cid:14) 1 − λn+1μn+1k θn+1

2λn+1μn+1(k − η) θn+1

(cid:17)(cid:18)(cid:4) (cid:5) (cid:7) (cid:7) ≤ (cid:5)2 − 2

δθn+1 + σ 2θ2

(cid:19)(cid:7) (cid:7)un − u∗

1 − αn+1

n+1 + αn+1

(cid:15)(cid:24)(cid:14) (cid:14) (cid:15)2 (cid:4) 1 − αn+1 (cid:20) (cid:21) (cid:21) (cid:22) (cid:23) (cid:23) (cid:23) (cid:23) (cid:23) (cid:23) (cid:23)1 − λn+1μn+1k (cid:23)

1 +

+ θn+1

2λn+1μn+1(k − η) θn+1

1 − λn+1μn+1k θn+1

× (cid:7) (cid:7)F(u∗) (cid:7) (cid:7). (cid:7) (cid:7)un − u∗

θn+1 (cid:7) (cid:7) + λn+1μn+1

(3.8)

Now we can see that (iii) yields

(cid:14) (cid:15)(cid:24)(cid:14) (cid:15)

(3.9)

.

lim n→∞

− η k = − η k

λn+1μn+1k θn+1

1 − λn+1μn+1k θn+1

Hence, we infer that there exists an integer N0 ≥0 such that for all n ≥N0, (1/2)λn+1μn+1η < 1, and (λn+1μn+1k/θn+1 − η/k)/(1 − λn+1μn+1k/θn+1) < −η/2k. Thus we deduce that for all n ≥ N0,

(cid:14) (cid:15)(cid:24)(cid:14) (cid:15)2 (cid:20) (cid:21) (cid:21) (cid:22) (cid:23) (cid:23) (cid:23) (cid:23) (cid:23) (cid:23) (cid:23)1 − λn+1μn+1k (cid:23)

1 +

θn+1

1 − λn+1μn+1k θn+1

θn+1 (cid:14)

2λn+1μn+1(k − η) θn+1 (cid:14)

(cid:15)(cid:14) (cid:15)(cid:24)(cid:14) (cid:15)2(cid:15)

1 +

≤ θn+1

1 − λn+1μn+1k θn+1

λn+1μn+1(k − η) θn+1

1 − λn+1μn+1k θn+1

= θn+1 − λn+1μn+1k +

λn+1μn+1(k − η) 1 − λn+1μn+1k/θn+1

(3.10)

(cid:5)2 −λn+1μn+1k + (cid:4) λn+1μn+1k

/θn+1 + λn+1μn+1k − λn+1μn+1η

= θn+1 +

1 − λn+1μn+1k/θn+1

(cid:13)(cid:14) (cid:15)(cid:24)(cid:14) (cid:15)(cid:16)

= θn+1 + λn+1μn+1k − η k

λn+1μn+1k θn+1

1 − λn+1μn+1k θn+1

λn+1μn+1η.

≤ θn+1 − 1 2

Y. Yu and R. Chen 7

From (ii) and (iii), we can choose sufficient small θn+1 such that

(cid:5) (δ − 1) (cid:4) 1 − αn+1

≤ 2 =⇒ θn+1 (cid:5) (cid:5) (δ − 1) (cid:4) 1 − αn+1 =⇒ σ 2θn+1 − 2

n+1

− 2 =⇒ σ 2θ2

σ 2 − 1 (cid:4) (cid:5) 1 − αn+1 (cid:5) δ ≤ θn+1 − 2 (cid:5) δθn+1 (cid:5)

− 2θn+1

=⇒

(3.11)

n+1

≤ θ2 n+1 (cid:4) 1 − αn+1 (cid:5) δθn+1 + σ 2θ2 (cid:5) ≤

+ θ2

0 < θn+1 ≤ 2 (cid:4) σ 2 − 1 (cid:4) 1 − αn+1 (cid:4) 1 − αn+1 (cid:4) 1 − αn+1 (cid:4) 1 − αn+1 (cid:4) 1 − αn+1

n+1

(cid:4) 1 − αn+1 (cid:18)(cid:4) =⇒ (cid:5)2 − 2 (cid:5)2 − 2θn+1 (cid:5)2 − 2

1 − αn+1

n+1

(cid:4) 1 − αn+1 (cid:5) δθn+1 + σ 2θ2

≤ 1 − αn+1 − θn+1

(cid:18)(cid:4) =⇒ (cid:5)2 − 2

1 − αn+1

n+1

(cid:4) 1 − αn+1 (cid:5) δθn+1 + σ 2θ2

+ αn+1 + θn+1 ≤ 1.

Consequently it follows from (3.6) and (3.8)–(3.11), for all n ≥ N0, that

(cid:7) (cid:7) ≤

(3.12)

(cid:7) (cid:7)F(u∗) (cid:7) (cid:7). (cid:7) (cid:7)un+1 − u∗ (cid:15) (cid:7) (cid:7)un − u∗ (cid:7) (cid:7) + λn+1μn+1

λn+1μn+1η

(cid:14) 1 − 1 2

By induction, it easy to see that

(cid:26) (cid:25) (cid:7) (cid:7) (cid:7) (cid:7) ≤ max (cid:7) (cid:7),

,

(3.13)

(cid:7) (cid:7)F(u∗)

n ≥ 0.

(cid:7) (cid:7)un − u∗ (cid:7) (cid:7)ui − u∗

max 0≤i≤N0

2 η

Hence, {xn} is bounded, so are {Wnun}, {g(un)}, and {F(Wnun)}. We will use M to denote the possible different constants appearing in the following reasoning.

Define

(cid:5)

(3.14)

un+1 = αn+1un +

(cid:4) 1 − αn+1

yn.

8

Journal of Inequalities and Applications

(cid:5)

From the definition of yn, we obtain yn+1 − yn = un+2 − αn+2un+1 1 − αn+2 (cid:4) 1 − αn+2 + θn+2

− un+1 − αn+1un 1 − αn+1 (cid:5) Wn+1un+1 − θn+2g (cid:4) Wn+1un+1 =

1 − αn+2 (cid:5)

(cid:5) (cid:4) Wn+1un+1

λn+1μn+1F

(cid:4) Wnun − λn+2μn+2F

+

1 − αn+1 (cid:5) (cid:4) Wnun

1 − αn+2 (cid:4) 1 − αn+1 + θn+1

−

Wnun

Wn+1un+1 − θn+1 1 − αn+1

(cid:5) = Wn+1un+1 − Wnun + (cid:5)

(3.15)

(cid:5) Wnun − θn+1g 1 − αn+1 θn+2 1 − αn+2 − θn+2

+

g

g

(cid:4) Wn+1un+1 (cid:4) Wnun

(cid:5) (cid:5)

+

F

F

(cid:4) Wn+1un+1 (cid:4) Wnun

1 − αn+2 − λn+2μn+2 1 − αn+2

θn+1 1 − αn+1 λn+1μn+1 1 − αn+1

+

Wnun

(cid:5) − θn+2

+

g

g

= Wn+1un+1 − Wn+1un + Wn+1un − Wnun Wn+1un+1 − θn+1 1 − αn+1 (cid:5) (cid:4) Wnun (cid:4) Wn + 1un+1

(cid:5)

+

F

(cid:5) .

F

(cid:4) Wnun (cid:4) Wn+1un+1

1 − αn+2 − λn+2μn+2 1 − αn+2

θn+2 1 − αn+2 θn+1 1 − αn+1 λn+1μn+1 1 − αn+1

(cid:7) (cid:7)

(cid:7) (cid:7) ≤

It follows that (cid:7) (cid:7) (cid:7) (cid:7) − (cid:7)yn+1 − yn (cid:7)un+1 − un (cid:7) (cid:7)Wn+1un − Wnun

(3.16)

(cid:5)(cid:7) (cid:7) (cid:7) (cid:7) + (cid:7) (cid:7) +

+

(cid:7) (cid:7)Wn+1un+1 (cid:7) (cid:5)(cid:7) (cid:7)g (cid:7) + (cid:7) (cid:7)g (cid:4) Wnun (cid:4) Wn+1un+1

θn+2 1 − αn+2 θn+1 1 − αn+1 (cid:5)(cid:7) (cid:7) +

+

(cid:7) (cid:7)F

θn+2 1 − αn+2 (cid:5)(cid:7) (cid:7).

(cid:4) Wn+1un+1 (cid:7) (cid:7)Wnun (cid:7) (cid:4) (cid:7)F Wnun

λn+2μn+2 1 − αn+2

θn+1 1 − αn+1 λn+1μn+1 1 − αn+1

From (3.3), since Ti and Un,i for all i = 1,2,...,N are nonexpansive,

(cid:7) (cid:7) = (cid:4) 1 − δn+1,N (cid:5) un − δn,N TN Un,N −1un − (cid:5) un (cid:4) 1 − δn,N (cid:7) (cid:7) ≤

≤ (cid:4) TN Un+1,N −1un − TN Un,N −1un (cid:7) (cid:7)

(cid:7) (cid:7). (cid:7) (cid:7) (cid:7)Wn+1un − Wnun (cid:7) (cid:7) (cid:7)δn+1,N TN Un+1,N −1un + (cid:23) (cid:23) (cid:7) (cid:7) (cid:7) (cid:23)δn+1,N − δn,N (cid:23) (cid:7) + (cid:7)δn+1,N TN Un+1,N −1un − δn,N TN Un,N −1un (cid:7)un (cid:23) (cid:23) (cid:7) (cid:7) (cid:7) (cid:5)(cid:7) (cid:23)δn+1,N − δn,N (cid:23) (cid:7)δn+1,N (cid:7) + (cid:7)un (cid:7) (cid:23) (cid:7) (cid:23) (cid:7)TN Un,N −1un (cid:23)δn+1,N − δn,N (cid:23) + (cid:23) (cid:23) (cid:23) + δn+1,N (cid:23)δn+1,N − δn,N ≤ 2M (cid:7) (cid:7)Un+1,N −1un − Un,N −1un

(3.17)

Y. Yu and R. Chen 9

Again, from (3.3),

(cid:7) (cid:7) (cid:7) (cid:7)Un+1,N −1un − Un,N −1un = (cid:5) un (cid:7) (cid:7)

(cid:23) (cid:23) (cid:7) (cid:7) ≤ (cid:7) (cid:7)un (cid:7) (cid:7) (cid:7) (cid:7)δn+1,N −1TN −1Un+1,N −2un − δn,N −1TN −1Un,N −2un

(3.18)

(cid:23) (cid:23) (cid:7) (cid:7) ≤ (cid:7) (cid:7)un (cid:7) (cid:7) (cid:7) (cid:4) (cid:7)δn+1,N −1TN −1Un+1,N −2un + 1 − δn+1,N −1 (cid:4) (cid:5) − δn,N −1TN −1Un,N −2un − 1 − δn,N −1 un (cid:23) (cid:23)δn+1,N −1 − δn,N −1 + (cid:23) (cid:23)δn+1,N −1 − δn,N −1 (cid:7) (cid:7)TN −1Un+1,N −2un − TN −1Un,N −2un + δn+1,N −1

(cid:7) (cid:7) (cid:7) (cid:7)Un+1,N −2un − Un,N −2un

(cid:7) (cid:7).

+ ≤ 2M ≤ 2M

(cid:23) (cid:23)δn+1,N −1 − δn,N −1 (cid:23) (cid:23)δn+1,N −1 − δn,N −1 (cid:23) (cid:23)δn+1,N −1 − δn,N −1 (cid:23) (cid:23)M (cid:23) (cid:23) + δn+1,N −1 (cid:7) (cid:23) (cid:7)Un+1,N −2un − Un,N −2un (cid:23) +

Therefore, we have

(cid:7) (cid:7)

(cid:23) (cid:23) (cid:23) (cid:23)δn+1,N −2 − δn,N −2 (cid:23) (cid:23) + 2M (cid:7) (cid:7)

+

(cid:7) (cid:7)

(3.19)

(cid:23) (cid:23) +

(cid:7) (cid:7) = (cid:7) (cid:7)Un+1,N −1un − Un,N −1un (cid:23) (cid:23)δn+1,N −1 − δn,N −1 ≤ 2M (cid:7) (cid:7)Un+1,N −3un − Un,N −3un N −1(cid:27) (cid:7) (cid:7)Un+1,1un − Un,1un (cid:4) (cid:5) 1 − δn,1 un − δn,1T1un − (cid:5) un ≤ 2M i=2 (cid:7) (cid:7)δn+1,1T1un + N −1(cid:27) (cid:23) (cid:23)δn+1,i − δn,i (cid:4) 1 − δn+1,1 (cid:23) (cid:23),

+ 2M

i=2

(cid:23) (cid:23)δn+1,i − δn,i

then

(cid:23) (cid:23) (cid:7) (cid:7) ≤ (cid:7) (cid:7) + (cid:7) (cid:7) (cid:7)Un+1,N −1un − Un,N −1un (cid:7) (cid:7) (cid:7)un

(3.20)

+ 2M

i=2

i=1

(cid:23) (cid:23). (cid:23) (cid:23)δn+1,1 − δn,1 N −1(cid:27) (cid:23) (cid:23)δn+1,i − δn,i (cid:7) (cid:7)δn+1,1T1un − δn,1T1un N −1(cid:27) (cid:23) (cid:23) (cid:23)δn+1,i − δn,i (cid:23) ≤ 2M

Substituting (3.20) into (3.17), we have

N −1(cid:27)

i=1

(cid:23) (cid:23) (cid:7) (cid:7) ≤ 2M (cid:7) (cid:7)Wn+1un − Wnun (cid:23) (cid:23)δn+1,N − δn,N (cid:23) (cid:23) + 2δn+1,N M (cid:23) (cid:23)δn+1,i − δn,i

(3.21)

N(cid:27)

i=1

≤ 2M (cid:23) (cid:23). (cid:23) (cid:23)δn+1,i − δn,i

10

Journal of Inequalities and Applications

Since {un}, {F(Wnun)}, {g(Wnun)} are all bounded, it follows from (3.16), (3.21), (i),

and (iii) that

(cid:5) (cid:7) (cid:7) (cid:7) (cid:7) −

(3.22)

≤ 0. (cid:4)(cid:7) (cid:7)yn+1 − yn (cid:7) (cid:7)un+1 − un

limsup n→∞

Hence, by Lemma 2.2, we know

(3.23)

(cid:7) (cid:7) = 0. (cid:7) (cid:7)yn − un

lim n→∞

Consequently,

(3.24)

(cid:7) (cid:7) = 0. (cid:7) (cid:7)un+1 − un (cid:4) 1 − αn+1 (cid:5)(cid:7) (cid:7)yn − un (cid:7) (cid:7) = lim n→∞

lim n→∞

On the other hand,

(cid:7) (cid:7) ≤ (cid:7) (cid:7)un − Wnun (cid:7) (cid:7)un+1 − Wnun

≤ αn+1

(3.25)

(cid:5)(cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7)Wnun (cid:7) (cid:7)F (cid:7) (cid:7) (cid:4) Wnun

+

(cid:7) (cid:7) (cid:7) + (cid:7)un+1 − un (cid:7) (cid:7) (cid:7) + θn+1 (cid:7)un − Wnun (cid:7) (cid:5)(cid:7) (cid:4) (cid:7)g (cid:7) + λn+1μn+1 + θn+1 Wnun (cid:7) (cid:7) (cid:7), (cid:7)un+1 − un

this together with conditions (i), (iii), and (3.24) implies

(3.26)

(cid:7) (cid:7) = 0. (cid:7) (cid:7)un − Wnun

lim n→∞

We next show that

(cid:2) (cid:3)

(3.27)

≤ 0. − F(x∗),un − x∗

limsup n→∞

To prove this, we pick a subsequence {uni

} of {un} such that (cid:2) (cid:3) (cid:2) (cid:3)

(3.28)

− x∗

.

− F(x∗),un − x∗ − F(x∗),uni = lim i→∞

limsup n→∞

→ z weakly for some z ∈ H.

Without loss of generality, we may further assume that uni

By Lemma 2.3 and (3.26), we have

(cid:5) ,

(3.29)

(cid:4) z ∈ Fix Wn

this imply that

N(cid:28)

(3.30)

z ∈

i=1

(cid:5) . (cid:4) Fix Ti

Since x∗ solves VI(F,C). Then we obtain

(cid:2) (cid:3) (cid:2) (cid:3) =

(3.31)

− F(x∗),z − x∗ ≤ 0. − F(x∗),un − x∗

limsup n→∞

Y. Yu and R. Chen 11

Finally, we show that un → u∗ in norm. From (3.7)–(3.10) and Lemma 2.4, we have

(cid:5) (cid:5) (cid:5) (cid:5)(cid:4) (cid:7) (cid:4) (cid:7) (cid:7) (cid:7)2 = − u∗ (cid:7) (cid:7)un+1 − u∗ (cid:4) Wnun (cid:5)

(cid:4) − θn+1 g (cid:4) Wnun − u∗ (cid:5)

Wnun − u∗ (cid:5) + θn+1 (cid:5)

(cid:7) (cid:7) (cid:5) (cid:7) (cid:7) ≤ − u∗

+ λn+1μn+1F(u∗) (cid:5) (cid:4) Wnun (cid:5)

(3.32)

− F(u∗) (cid:4) (cid:5) − θn+1 g (cid:4) Wnun − u∗ (cid:5)(cid:7) (cid:7)2

(cid:7) (cid:7)2 ≤

λn+1μn+1η (cid:2)

(cid:3)

.

1 − αn+1 (cid:4) un − u∗ + αn+1 (cid:4) (cid:4) − λn+1μn+1 Wnun F (cid:4) (cid:5)(cid:4) Wnun − u∗ 1 − αn+1 (cid:5) (cid:4) un − u∗ + αn+1 + θn+1 (cid:5) (cid:4) (cid:4) − F(u∗) − λn+1μn+1 F Wnun (cid:3) (cid:2) − F(u∗),un+1 − u∗ (cid:15) (cid:7) (cid:7)un − u∗ − F(u∗),un+1 − u∗

+ 2λn+1μn+1 (cid:14) 1 − 1 2 + 2λn+1μn+1

An application of Lemma 2.1 combined with (3.31) yields that (cid:3)un − u∗(cid:3) → 0. This com- (cid:2) pletes the proof.

4. Application to constrained generalized pseudoinverse

Let K be a nonempty closed convex subset of a real Hilbert space H. Let A be a bounded linear operator on H. Given an element b ∈ H, consider the minimization problem

(4.1)

(cid:3)Ax − b(cid:3)2.

min x∈K

Let Sb denote the solution set. Then, Sb is closed and convex. It is known that Sb is nonempty if and only if PA(K)(b) ∈ A(K). In this case, Sb has a unique element with minimum norm; that is, there exists a unique point (cid:29)x ∈ Sb satisfying

(cid:8) (cid:9)

(4.2)

(cid:3)(cid:29)x(cid:3)2 = min

.

(cid:3)x(cid:3)2 : x ∈ Sb

Definition 4.1 [22]. The K-constrained pseudoinverse of A (symbol (cid:29)AK ) is defined as

(cid:5) (cid:8) (cid:9) (cid:4) =

,

(cid:5) ,

(4.3)

b ∈ D

D

(cid:4) (cid:29)AK (cid:29)AK (b) = (cid:29)x, (cid:29)Ak

b ∈ H : PA(K)(b) ∈ A(K)

where (cid:29)x ∈ Sb is the unique solution of (4.2).

Now we recall the K-constrained generalized pseudoinverse of A. Let θ : H → R be a differentiable convex function such that θ(cid:16) is a k-Lipschitzian and η-strongly monotone operator for some k > 0 and η > 0. Under these assumptions, there exists a unique point (cid:29)x0 ∈ Sb for b ∈ D( (cid:29)AK ) such that

(cid:5) (cid:8) (cid:9) = min

(4.4)

θ

.

(cid:4) (cid:29)x0

θ(x) : x ∈ Sb

Definition 4.2. The K-constrained generalized pseudoinverse of A associated with θ (symbol (cid:29)AK,θ) is defined as D( (cid:29)AK,θ) = D( (cid:29)AK ), (cid:29)AK,θ(b) = (cid:29)x0, and b ∈ D( (cid:29)AK,θ), where

12

Journal of Inequalities and Applications

(cid:29)x0 ∈ Sb is the unique solution to (4.4). Note that if θ(x) = (cid:3)x(cid:3)2/2, then the K-constrained generalized pseudoinverse (cid:29)AK,θ of A associated with θ reduces to the K-constrained pseu- doinverse (cid:29)AK of A in Definition 4.1.

We now apply the result in Section 3 to construct the K-constrained generalized pseudoinverse (cid:29)AK,θ of A. First observe that (cid:30)x ∈ K satisfies the minimization problem (4.1) if and only if there holds the following optimality condition: (cid:5)A∗(A(cid:30)x − b),x − (cid:30)x(cid:6) ≥ 0, x ∈ K, where A∗ is the adjoint of A. This for each λ > 0, is equivalent to,

(cid:2)(cid:11) (cid:3) ≥ 0, − (cid:30)x, (cid:30)x − x

(4.5)

(cid:12) λA∗b + (I − λA∗A)(cid:30)x x ∈ K, (cid:5) (cid:4) λA∗b + (I − λA∗A)(cid:30)x = (cid:30)x.

PK

Define a mapping T : H → H by

(cid:5)

,

(4.6)

(cid:4) A∗b + (I − λA∗A)x

x ∈ H.

Tx = PK

Lemma 4.3 [12]. If λ ∈ (0,2(cid:3)A(cid:3)−2) and if b ∈ D( (cid:29)AK ), then T is attracting nonexpansive and Fix(T) = Sb.

The proofs of the following Theorems 4.4 and 4.5 are obtained easily; we omit them.

Theorem 4.4. Assume that 0 < μn < 2η/k2. Assume {λn} and {θn} satisfy the following conditions:

n=1 λn = ∞;

(cid:6)∞

(i) limn→∞ λn = 0, (ii) θn ∈ (0,2(1 − a)(δ − 1)/(σ 2 − 1)]; (iii) limn→∞ θn = 0,limn→∞ λn/θn = 0.

Given an initial guess u0 ∈ H, let {un} be the sequence generated by the algorithm

(cid:16) (cid:4)

(cid:5) (cid:5) (cid:4) un − Tun

+ αn+1

un+1 = Tun − λn+1μn+1θ (cid:5)

(cid:4)

(4.7)

Tun (cid:5) ,

g

n ≥ 0,

− θn+1 (cid:4) Tun − Tun

where T is given in (4.6). Suppose that the unique solution (cid:29)u0 of (4.4) is also a fixed point of g. Then {un} strongly converges to (cid:29)AK,θ(b).

Theorem 4.5. Assume that 0 < μn < 2η/k2. Assume that the restrictions (ii) and (iii) hold for {θn} and also that the control condition (i) holds for {λn}. Given an initial guess u0 ∈ H, suppose that the unique solution (cid:31)u0 of (4.4) is also a fixed point of g. Then the sequence {un} generated by the algorithm

(cid:4) (cid:5) (cid:5) (cid:4) un − Wnun

+ αn+1

(cid:5)

(4.8)

Wnun (cid:5) ,

un+1 = Wnun − λn+1μn+1θ(cid:16) (cid:4) g

n ≥ 0,

(cid:4) Wnun − θn+1 − Wnun

converges to (cid:29)AK,θ(b).

Y. Yu and R. Chen 13

References

[1] G. Stampacchia, “Formes bilin´eaires coercitives sur les ensembles convexes,” Comptes Rendus de

l’Acad´emie des Sciences, vol. 258, pp. 4413–4416, 1964.

[2] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Ap- plications, vol. 88 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1980. [3] M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp.

119–122, 1988.

[4] M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and

Computation, vol. 152, no. 1, pp. 199–277, 2004.

[5] M. A. Noor and K. I. Noor, “Self-adaptive projection algorithms for general variational inequal-

ities,” Applied Mathematics and Computation, vol. 151, no. 3, pp. 659–670, 2004.

[6] M. A. Noor, “Wiener-Hopf equations and variational inequalities,” Journal of Optimization The-

ory and Applications, vol. 79, no. 1, pp. 197–206, 1993.

[7] Y. Yao and M. A. Noor, “On modified hybrid steepest-descent methods for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1276–1289, 2007.

[8] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Com-

putational Physics, Springer, New York, NY, USA, 1984.

[9] P. Jaillet, D. Lamberton, and B. Lapeyre, “Variational inequalities and the pricing of American

options,” Acta Applicandae Mathematicae, vol. 21, no. 3, pp. 263–289, 1990.

[10] E. Zeidler, Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Opti-

mization, Springer, New York, NY, USA, 1985.

[11] I. Yamada, “The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001. [12] H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational in- equalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003. [13] L. C. Zeng, N. C. Wong, and J. C. Yao, “Convergence of hybrid steepest-descent methods for generalized variational inequalities,” Acta Mathematica Sinica, vol. 22, no. 1, pp. 1–12, 2006. [14] M. A. Noor, “Wiener-Hopf equations and variational inequalities,” Journal of Optimization The-

ory and Applications, vol. 79, no. 1, pp. 197–206, 1993.

[15] L. C. Zeng, N. C. Wong, and J. C. Yao, “Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities,” Journal of Optimization Theory and Applications, vol. 132, no. 1, pp. 51–69, 2007.

[16] Y. Song and R. Chen, “An approximation method for continuous pseudocontractive mappings,”

Journal of Inequalities and Applications, vol. 2006, Article ID 28950, 9 pages, 2006.

[17] R. Chen and H. He, “Viscosity approximation of common fixed points of nonexpansive semi- groups in Banach space,” Applied Mathematics Letters, vol. 20, no. 7, pp. 751–757, 2007. [18] R. Chen and Z. Zhu, “Viscosity approximation fixed points for nonexpansive and m-accretive operators,” Fixed Point Theory and Applications, vol. 2006, Article ID 81325, 10 pages, 2006. [19] T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Ap- plications, vol. 305, no. 1, pp. 227–239, 2005.

[20] K. Geobel and W. A. Kirk, Topics on Metric Fixed Point Theory, Cambridge University Press,

Cambridge, UK, 1990.

14

Journal of Inequalities and Applications

[21] S. Atsushiba and W. Takahashi, “Strong convergence theorems for a finite family of nonexpan- sive mappings and applications,” Indian Journal of Mathematics, vol. 41, no. 3, pp. 435–453, 1999.

[22] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, vol. 13, Kluwer

Academic Publishers, Dordrecht, The Netherlands, 2000.

Yanrong Yu: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address: tjcrd@yahoo.com.cn

Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address: chenrd@tjpu.edu.cn