Abstract and Applied Analysis

Volume 2013 (2013), Article ID 852760, 21 pages

http://dx.doi.org/10.1155/2013/852760

Research Article

Modified Hybrid Steepest-Descent Methods for General Systems of Variational Inequalities with Solutions to Zeros of -Accretive Operators in Banach Spaces

Lu-Chuan Ceng1,2 and Ching-Feng Wen³

¹Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

²Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

³Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 1 July 2013; Accepted 28 July 2013

Academic Editor: Chi-Ming Chen

Copyright © 2013 Lu-Chuan Ceng and Ching-Feng Wen. 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.

Abstract.

The purpose of this paper is to introduce and analyze modified hybrid steepest-descent methods for a general system of variational inequalities (GSVI), with solutions being also zeros of an -accretive operator in the setting of real uniformly convex and 2-uniformly smooth Banach space . Here the modified hybrid steepest-descent methods are based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI and the set of zeros of in . Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

1. Introduction

Let be a real Banach space whose dual space is denoted by . The normalized duality mapping is defined by

where denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that is nonempty for each . Let be a nonempty closed convex subset of . A mapping is called nonexpansive if for every . The set of fixed points of is denoted by . We use the notation to indicate the weak convergence and to indicate the strong convergence. A mapping is said to be

(i) accretive if for each there exists such that

(ii) -strongly accretive if for each there exists such that

for some ;

(iii) -inverse strongly accretive if for each there exists such that

for some;

(iv) -strictly pseudocontractive [1] (see also [2]) if for each there exists such that

for some .

It is worth emphasizing that 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 example, [3–5]. Let denote the unite sphere of . A Banach space is said to be uniformly convex if for each there exists such that for all

It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space is said to be smooth if the limit

exists for all ; in this case, is also said to have a Gateaux differentiable norm. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for ; in this case, is also said to have a uniformly Frechet differentiable norm. The norm of is said to be the Frechet differential if for each this limit is attained uniformly for . In the meantime, we define a function called the modulus of smoothness of as follows:

It is known that is uniformly smooth if and only if . Let be a fixed real number with . Then a Banach space is said to be -uniformly smooth if there exists a constant such that for all . As pointed out in [6], no Banach space is -uniformly smooth for . In addition, it is also known that is single-valued if and only if is smooth, whereas if is uniformly smooth, then the mapping is norm-to-norm uniformly continuous on bounded subsets of .

Let be an operator with domain and range in is said to be accretive if for each and there exists such that

An accretive operator is said to be -accretive if for all . Denote by the resolvent of ; that is, for each ,

It is known that is a nonexpansive mapping from to which will be assumed convex. In 2008, Chen and Zhu [7] derived the following strong convergence theorems for viscosity approximation methods for accretive operators in a uniformly smooth Banach space .

Theorem  CZ1. Let be a uniformly smooth Banach space. Suppose that is an -accretive operator in such that is convex and is a fixed contractive map. For each , is defined by

Then as , converges strongly to a zero of .

Theorem  CZ2. Let be a uniformly smooth Banach space. Suppose that is an -accretive operator in such that is convex and is a fixed contractive map. The sequence is defined by

where and satisfy the following conditions:

(i), , and ,(ii) for all and .

Then as , converges strongly to a zero of .

In the meantime, Ceng et al. [8] derived some strong convergence theorems of composite iterative schemes for zeros of -accretive operators in uniformly smooth Banach spaces. Furthermore, motivated by strong convergence results for hybrid steepest-descent methods in [9, 10], Ceng et al. [11] established some strong convergence theorems for hybrid steepest-descent methods for nonexpansive and -accretive operators in a uniformly smooth Banach space . Subsequently, Ceng et al. [12] introduced hybrid viscosity approximation method for finding zeros of -accretive operators, which combine viscosity approximation method with hybrid steepest-descent method, and obtained the following strong convergence theorems.

Theorem  CASY1. Let be a uniformly smooth Banach space, let be an -accretive operator in with , and let be a contractive map. Assume that is -strongly accretive and -strictly pseudocontractive with . For each and each integer , let be defined by

where for some and with . Then as converges strongly to a zero of , which is a unique solution of the variational inequality problem (VIP)

Theorem  CASY2. Let be a uniformly smooth Banach space, let be an -accretive operator in with , and let be a contractive map. Assume that is -strongly accretive and -strictly pseudocontractive with . Given sequences in in , and in for some , suppose that there hold the following conditions:

(i) and ,(ii),(iii) for some ,(iv), , , and .

Then for any given point , the sequence generated by

converges strongly to a zero of , which is a unique solution of the VIP as above.

On the other hand, Cai and Bu [13] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space , which involves finding such that

where is a nonempty, closed and convex subset of , are two nonlinear mappings, and and are two positive constants. Here the set of solutions of GSVI (16) is denoted by GSVI . In particular, if , a real Hilbert space, then GSVI (16) reduces to the following GSVI of finding such that

where and are two positive constants. The set of solutions of problem (17) is still denoted by GSVI . In particular, if , then problem (17) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [14]. Further, if additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding such that

The solution set of the VIP (18) is denoted by VI . Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [15]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the condition that the involved operator must be strongly monotone and Lipschitz continuous.

Recently, Ceng et al. [16] transformed problem (17) into a fixed point problem in the following way.

Lemma 1 (see [16]). For given is a solution of problem (17) if and only if is a fixed point of the mapping defined by

where and is the the projection of onto .

In particular, if the mapping is -inverse strongly monotone for , then the mapping is nonexpansive provided for .

In 1976, Korpelevič [17] proposed an iterative algorithm for solving the VIP (18) in Euclidean space :

with being a given number, which is known as the extragradient method (see also [18]). The literature on the VIP is vast, and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [3, 13, 19–32] and references therein, to name but a few.

In particular, whenever is still a real smooth Banach space, , and , then GSVI (16) reduces to the variational inequality problem (VIP) of finding such that

which was considered by Aoyama et al. [33]. Note that VIP (21) is connected with the fixed point problem for nonlinear mapping (see, e.g., [34]), the problem of finding a zero point of a nonlinear operator (see, e.g., [35]), and so on. It is clear that VIP (21) extends VIP (18) from Hilbert spaces to Banach spaces.

In order to find a solution of VIP (21), Aoyama et al. [33] introduced the following iterative scheme for an accretive operator :

where is a sunny nonexpansive retraction from onto . Then they proved a weak convergence theorem.

Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (16) which contains VIP (21) as a special case. Very recently, Cai and Bu [13] constructed an iterative algorithm for solving GSVI (16) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach space .

Lemma 2 (see [36]). Let be a 2-uniformly smooth Banach space. Then

where is the 2-uniformly smooth constant of and is the normalized duality mapping from into .

Define the mapping as follows:

The fixed point set of is denoted by . Then their strong convergence theorem on the proposed method is stated as follows.

Theorem  CB (see [13, Theorem 3.1]). Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse strongly accretive with for . Let be a contraction of into itself with coefficient . Let be an infinite family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping defined by (24). For arbitrarily given , let be the sequence generated by

Suppose that and are two sequences in satisfying the following conditions:

(i) and ,(ii).

Assume that for any bounded subset of , let be a mapping of into defined by for all , and suppose that . Then converges strongly to , which solves the VIP:

It is easy to see that the iterative scheme in Theorem CB is essentially equivalent to the following two-step iterative scheme:

For the convenience of implementing the argument techniques in [16], the authors [13] have used the following inequality in a real smooth and uniform convex Banach space .

Proposition 3 (see [37]). Let be a real smooth and uniform convex Banach space and let . Then there exists a strictly increasing, continuous and convex function , such that

where .

Let be a nonempty closed convex subset of a real uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be a contraction with coefficient . Motivated and inspired by the research going on in this area, we introduce and analyze modified hybrid steepest-descent methods for the GSVI (16), with solutions being also zeros of an -accretive operator in . Here the modified hybrid steepest-descent methods are based on Korpelevich’s extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI (16) and the set of zeros of in . Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature [12, 13, 16, 32].

2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma 4 can be found in [38]. Lemma 5 is an immediate consequence of the subdifferential inequality of the function .

Lemma 4. Letbe a sequence of nonnegative real numbers such that

where and are sequences of real numbers satisfying the following conditions:

(i) and ,(ii)either or .

Then, .

Lemma 5. In a real smooth Banach space , there holds the inequality

where is the normalized duality mapping.

Let be a subset of and let be a mapping of into . Then is said to be sunny if

whenever for and . A mapping of into itself is called a retraction if . If a mapping ofinto itself is a retraction, then for every , where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto . The following lemma concerns the sunny nonexpansive retraction.

Lemma 6 (see [39]). Let be a nonempty closed convex subset of a real smooth Banach space . Let be a nonempty subset of . Let be a retraction of onto . Then the following are equivalent:

(i) is sunny and nonexpansive,(ii),(iii).

It is well known that if , a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto ; that is, . If is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space and if is a nonexpansive mapping with the fixed point set , then the set is a sunny nonexpansive retract of . The following lemma follows easily from Lemma 6.

Lemma 7. Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be nonlinear mappings. For given is a solution of GSVI (16) if and only if , where .

In terms of Lemma 7, we observe that

which implies that is a fixed point of the mapping . Throughout this paper, the set of fixed points of the mapping is denoted by .

Lemma 8 is the resolvent identity which can be found in [40], and Lemma 9 can be found in [41].

Lemma 8. For , there holds the identity

Lemma 9. Assume that . Then for all .

Lemma 10 (see [42]). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose that is nonempty. Let be a sequence of positive numbers with . Then a mapping on defined by for is defined well, nonexpansive, and holds.

Let be a mean if is a continuous linear functional on satisfying . Then we know that is a mean on if and only if

for every . According to time and circumstances, we use instead of . A mean on is called a Banach limit if and only if

for every . We know that if is a Banach limit, then

for every . So if , and (resp., ), as , we have

Further, it is well known that there holds the following result.

Lemma 11 (see [43]). Let be a nonempty closed convex subset of a uniformly smooth Banach space . Let be a bounded sequence of , let be a mean on , and let . Then

if and only if

where is the normalized duality mapping of .

Let be a nonempty closed convex subset of a Banach space and let be a nonexpansive mapping with . Let be the set of all contractive self-mappings on . For and , let be the unique fixed point of the contraction on ; that is,

Lemma 12 (see [34]). Let be a uniformly smooth Banach space. Let be a nonempty closed convex subset of , let be a nonexpansive mapping with , and let . Then the net defined by converges strongly to a point in . If one defines a mapping by for all , then solves the VIP:

The following proposition will be used frequently throughout this paper.

Proposition 13 (see [11]). Let be a real smooth Banach space and let be a mapping.

(i)If is -strictly pseudocontractive, then is Lipschitzian with constant .(ii)If is -strongly accretive and -strictly pseudocontractive with , then is contractive with constant .(iii)If is -strongly accretive and -strictly pseudocontractive with , then, for any fixed number , is contractive with constant .

3. Main Results

In this section, we introduce our modified hybrid steepest-descent schemes and show the strong convergence theorems. We will need the following useful lemmas in the sequel.

Lemma 14 (see [13, Lemma 2.8]). Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space . Let the mapping be -inverse strongly accretive. Then, one has

for , where . In particular, if , then is nonexpansive for .

Lemma 15 (see [13, Lemma 2.9]). Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse strongly accretive for . Let be the mapping defined by

If   for , then is nonexpansive.

Let be a real smooth Banach space and let be an -accretive operator in such that is convex. Let be a sunny nonexpansive retraction from onto . Let be -strongly accretive and -strictly pseudocontractive with . Let the mapping be -inverse strongly accretive for and let be a contractive map with coefficient . In this section, we will consider the problem of finding a point , which is a unique solution of the VIP:

where is the fixed point set of the mapping with for . For each and each integer , we choose a number arbitrarily and then consider the following mapping defined as

Then, is a contractive map. Indeed, utilizing Proposition 13(iii) and Lemma 15, we have for all

and hence is contractive due to . By Banach’s Contraction Mapping Principle, there exists a unique fixed point of   in ; that is,

Theorem 16. Let be a uniformly convex and 2-uniformly smooth Banach space and let be an -accretive operator in such that is convex. Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse strongly accretive for , let be a contractive map with coefficient , and let be -strongly accretive and -strictly pseudocontractive with . Assume that , where is the fixed point set of the mapping with for . For each and each integer , let be defined by (47), where for some and with . Then as converges strongly to , which is a unique solution of the VIP (44).

Proof. First let us show that for some is bounded. Indeed, since with , there exists some such that for all .

Take . Then utilizing Proposition 13, we have

and, hence, for all

Thus, this implies that is bounded and so are , , and .

Let us show that as .

Indeed, for simplicity we put , , , and . Then it is clear that and . Taking into account , we get

From Lemma 14 we have

From the last two inequalities, we obtain

which, together with (50), implies that

So it immediately follows that

Since for , we have

Utilizing Proposition 3 and Lemma 6, we have

which implies that

In the same way, we derive

which implies that

Substituting (57) for (59), we get

which, together with (50), implies that

So it immediately follows that

Hence, from (55) we conclude that

Utilizing the properties of and , we get

which leads to

That is,

Note that for all and that is bounded and so are , , and . Hence, we have

as . Also, observe that

This, together with (66) and (67), implies that

Utilizing the nonexpansivity of , we obtain from (67) and (69) that

Since for all , utilizing Lemma 9 we have

For any integer , for simplicity put for all . Now let be a sequence in that converges to as and define a function on by

where is a Banach limit. Define the set

and the mapping

where is a constant in . Then by Lemma 10, we know that . We observe that

So from (70) and (71) we obtain

Since is a uniformly smooth Banach space, is a nonempty bounded closed convex subset of ; for more details, see [43]. We claim that is also invariant under the nonexpansive mapping . Indeed, noticing (76), we have for

Since every nonempty closed bounded convex subset of uniformly smooth Banach space has the fixed point property for nonexpansive mappings and is a nonexpansive mapping of has a fixed point in , say . Utilizing Lemma 11, we get

Putting we have

Since , we get

It follows that

Since , from (79) and the boundedness of sequences , it follows that

Therefore, it is known that for any sequence in there exists a subsequence which is still denoted by that converges strongly to some fixed point of . To prove that the net converges strongly to as , suppose that there exists another subsequence such that as , and then we also have due to (76).

Since the sets and are bounded and the duality map is single-valued and norm-to-norm uniformly continuous on bounded sets of uniformly smooth Banach space , for any , from   () we obtain

as . Therefore,

Since ,

Utilizing Lemma 5, we have

Consequently, from the last two inequalities we deduce that

and hence

Noticing (84), from and the boundedness of sequences , we conclude that

Interchanging and leads to

Interchanging and leads to

This implies that

and hence

Taking into account , we obtain . Furthermore, by the careful analysis of the above proof, we can readily see that is also a unique solution of the VIP:

This completes the proof.

Remark 17. In the assertion of Theorem 16, “as , converges strongly to ;” this does not depend on . Indeed, it is known that there holds the condition that for some . Moreover, in the proof of Theorem 16 it can be readily seen that is first found out as a fixed point of the nonexpansive self-mapping of . This shows that depends on neither nor .

Theorem 18. Let be a uniformly convex and 2-uniformly smooth Banach space and let be an -accretive operator in such that is convex. Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse strongly accretive for , be a contractive map with coefficient , and let be -strongly accretive and -strictly pseudocontractive with . Assume that , where is the fixed point set of the mapping with for . Given sequences in , in , and in for some , suppose that there hold the following conditions:

(i) and ,(ii),(iii) for some ,(iv), , , and .

Then for any given point , the sequence generated by

converges strongly to , which is a unique solution of the VIP (44).

Proof. First, let us show that is bounded. Indeed, taking a fixed arbitrarily, we have

So for all . Thus, by Proposition 13(iii), we have

By induction,

Thus, is bounded and so is . Because and are nonexpansive for all , is contractive, and is Lipschitzian, , , , , , and are bounded. From conditions (i), (ii) we have

Now, we claim that

In order to prove (100), we estimate first. From (95) we have

Simple calculations show that

It follows that

On the other hand, if , using the resolvent identity in Lemma 8

we get

If , it is easy to see that

So combining the above cases we obtain

In the similar way we can derive

Therefore, we have

for all , where and for some .

Substituting (109) for (103), we obtain

where for some constant .

In the meantime, it follows from (95) that

Simple calculations show that

It follows from Proposition 13 (iii) and (109) that

where , for some .

Substituting (110) for (113), we get

Since it follows from conditions (i) and (iv) that and

Lemma 4 is applicable to (114) and we obtain

By condition (iii) and (95), we have

which implies that

This together with (99)-(100) implies that

So we obtain

which implies that

and hence

Next let us show that and .

Indeed, for simplicity, put , , and . Then , and for all . It is clear from (95) that

Utilizing Lemma 14, we have

Substituting (124) for (125), we obtain

which, together with (123), implies that

It immediately follows that

Since and are bounded and for , we deduce from (119) and condition (iii) that

Utilizing Proposition 3 and Lemma 6, we have

which implies that

In the same way, we derive

which implies that

Substituting (131) for (133), we get

which, together with (123), implies that

It immediately follows that

Since , , , and are bounded, we deduce from (119), (129), and condition (iii) that

Utilizing the properties of and , we get

which hence yields

That is,

Note that

So from (122) and (140) we have

which, together with (122), leads to

That is,

In addition, utilizing Lemma 9 we obtain from that

which, together with (142), implies that

Define a mapping

where is a constant in . Then by Lemma 10, we know that . We observe that

So from (144) and (146) we obtain

Now, we claim that

where , with being the fixed point of the contraction of into itself (due to Lemma 12). Then solves the fixed point equation . Thus we have

By Lemma 5 we conclude that

where

It follows from (152) that

Letting in (154) and noticing (153), we derive

where is a constant such that for all and . Taking in (155), we have

On the other hand, we have

It follows that

Taking into account that as , we have from (156)

Since has a uniformly Frechet differentiable norm, the duality mapping is norm-to-norm uniformly continuous on bounded subsets of . Consequently, the two limits are interchangeable and hence (150) holds. It is clear from (150) that

Finally, let us show that as . We observe that

and hence

Taking into account (160) and conditions (i), (ii), we obtain that and

Therefore, applying Lemma 4 to (162), we infer that

This completes the proof.

Remark 19. As pointed out in [12, Remark 3.2], the sequences , , and can be taken, which satisfy the conditions in Theorem 18. As a matter of fact, put , , and for all . Then there hold the following statements:

(i) and ,(ii),(iii), , and.

By the careful analysis of the proof of Theorem 18, we can obtain the following result. Because its proof is much simpler than that of Theorem 18, we omit its proof.

Theorem 20. Let be a uniformly convex and 2-uniformly smooth Banach space and let be an -accretive operator in such that is convex. Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse strongly accretive for , let be a contractive map with coefficient , and let be -strongly accretive and -strictly pseudocontractive with . Assume that , where is the fixed point set of the mapping with for . Given sequences in , in , and in for some , suppose that there hold the following conditions:

(i) and ,(ii) and ,(iii) for some ,(iv), , and .

Then for any given point , the sequence generated by

converges strongly to , which is a unique solution of the VIP (44).

Remark 21. Our Theorems 16–20 improve, extend, supplement, and develop Cai and Bu [13, Theorem 3.1] and Ceng et al. [12, Theorems 3.1–3.3] in the following aspects.

(i)The problem of finding a point in our Theorems 16–20 is very different from everyone of both the problem of finding a point in Cai and Bu [13, Theorem 3.1] and the problem of finding a point in Ceng et al. [12, Theorems 3.1–3.3]. There is no doubt that our problem of finding a point is more general than the problem of finding a point in [12, Theorems 3.1–3.3].(ii)Compared with the choice of iterative parameters in [12, Theorems 3.1–3.3], the choice of iterative parameters in our Theorems 16–20 is the same as that in [12, Theorems 3.1–3.3].(iii)The iterative schemes in [12, Theorems 3.1–3.3] are extended to develop the iterative schemes in our Theorems 16–20 by virtue of the iterative scheme of [13, Theorems 3.1]. The iterative schemes in our Theorems 16–20 are more advantageous and more flexible than the iterative schemes of [12, Theorems 3.1–3.3] because they involve solving two problems: the GSVI (16) and the problem of finding zeros of an -accretive operator.(iv)The iterative schemes in our Theorems 16–20 are very different from everyone in both [13, Theorem 3.1] and [12, Theorems 3.1–3.3] because the iterative scheme in our Theorem 16 is implicit and because the mapping in [13, Theorem 3.1] and the mapping in [12, Theorems 3.1–3.3] are replaced by the same composite mapping in the iterative schemes of our Theorems 16–20.(v)Cai and Bu’s proof in [13, Theorem 3.1] depends on the argument techniques in [16], the inequality in 2-uniformly smooth Banach spaces (see Lemma 2), and the inequality in smooth and uniform convex Banach spaces (see Proposition 3). Because the composite mapping appears in the iterative schemes in our Theorems 16–20, the proof of our Theorems 16–20 depends on the argument techniques in [16], the inequality in 2-uniformly smooth Banach spaces (see Lemma 2), the inequality in smooth and uniform convex Banach spaces (see Proposition 3), and the properties of the resolvent of an-accretive operator (see Lemmas 8 and 9), the Banach limit (see Lemma 11) and the strongly accretive and strictly pseudocontractive mapping (see Proposition 13).

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This research was partially supported by a Grant from NSC 102-2115-M-037-001.

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